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Data Structures

Terms in this set (223)

tree data structure. a specialized tree-based data structure that satisfies the heap property: If A is a parent node of B then the key (the value) of node A is ordered with respect to the key of node B with the same ordering applying across the heap. A heap can be classified further as either a "max heap" or a "min heap". In a max heap, the keys of parent nodes are always greater than or equal to those of the children and the highest key is in the root node. In a min heap, the keys of parent nodes are less than or equal to those of the children and the lowest key is in the root node. Heaps are crucial in several efficient graph algorithms such as Dijkstra's algorithm, and in the sorting algorithm heapsort. A common implementation of a heap is the binary heap, in which the tree is a complete binary tree (see figure).
-In a heap, the highest (or lowest) priority element is always stored at the root. A heap is not a sorted structure and can be regarded as partially ordered. As visible from the heap-diagram, there is no particular relationship among nodes on any given level, even among the siblings. When a heap is a complete binary tree, it has a smallest possible height—a heap with N nodes always has log N height. A heap is a useful data structure when you need to remove the object with the highest (or lowest) priority.
-Note that, as shown in the graphic, there is no implied ordering between siblings or cousins and no implied sequence for an in-order traversal (as there would be in, e.g., a binary search tree). The heap relation mentioned above applies only between nodes and their parents, grandparents, etc. The maximum number of children each node can have depends on the type of heap, but in many types it is at most two, which is known as a binary heap.
-The heap is one maximally efficient IMPLEMENTATION of an abstract data type called a priority queue, and in fact priority queues are often referred to as "heaps", regardless of how they may be implemented.
-A heap data structure should not be confused with the heap which is a common name for the pool of memory from which dynamically allocated memory is allocated. The term was originally used only for the data structure.
-Types
Heap
Binary heap
Weak heap
Binomial heap
Fibonacci heap
AF-heap
Leonardo Heap
2-3 heap
Soft heap
Pairing heap
Leftist heap
Treap
Beap
Skew heap
Ternary heap
D-ary heap
Brodal queue
primitive data type. a data type, having two values (usually denoted true and false), intended to represent the truth values of logic and Boolean algebra. It is named after George Boole, who first defined an algebraic system of logic in the mid 19th century. The Boolean data type is primarily associated with conditional statements, which allow different actions and change control flow depending on whether a programmer-specified Boolean condition evaluates to true or false. It is a special case of a more general logical data type; logic does not always have to be Boolean. In programming languages that have a built-in Boolean data type, such as Pascal and Java, the comparison operators such as > and ≠ are usually defined to return a Boolean value. Conditional and iterative commands may be defined to test Boolean-valued expressions.

Languages without an explicit Boolean data type, like C90 and Lisp, may still represent truth values by some other data type. Common Lisp uses an empty list for false, and any other value for true. C uses an integer type, where relational expressions like i > j and logical expressions connected by && and || are defined to have value 1 if true and 0 if false, whereas the test parts of if, while, for, etc., treat any non-zero value as true.[1][2] Indeed, a Boolean variable may be regarded (and be implemented) as a numerical variable with a single binary digit (bit), which can store only two values. It is worth noting that the implementation of Booleans in computers are most likely represented as a full word, rather than a bit; this is usually due to the ways computers transfer blocks of information.

Most programming languages, even those that do not have an explicit Boolean type, have support for Boolean algebraic operations such as conjunction (AND, &, *), disjunction (OR, |, +), equivalence (EQV, =, ==), exclusive or/non-equivalence (XOR, NEQV, ^, !=), and negation (NOT, ~, !).

In some languages, like Ruby, Smalltalk, and Alice the "true" and "false" values belong to separate classes—i.e. True and False, resp.—so there is no single Boolean "type."

In SQL, which uses a three-valued logic for explicit comparisons because of its special treatment of Nulls, the Boolean data type (introduced in SQL:1999) is also defined to include more than two truth values, so that SQL "Booleans" can store all logical values resulting from the evaluation of predicates in SQL. A column of Boolean type can also be restricted to just TRUE and FALSE though.
composite data type. a value that may have any of several representations or formats; or it is a data structure that consists of a variable that may hold such a value. Some programming languages support special data types, called union types, to describe such values and variables. In other words, a union type definition will specify which of a number of permitted primitive types may be stored in its instances, e.g., "float or long integer". Contrast with a record (or structure), which could be defined to contain a float and an integer; in a union, there is only one value at any given time.
-A union can be pictured as a chunk of memory that is used to store variables of different data types. Once a new value is assigned to a field, the existing data is overwritten with the new data. The memory area storing the value has no intrinsic type (other than just bytes or words of memory), but the value can be treated as one of several abstract data types, having the type of the value that was last written to the memory area.
-Because of the limitations of their use, untagged unions are generally only provided in untyped languages or in a type-unsafe way (as in C). They have the advantage over simple tagged unions of not requiring space to store a data type tag.
-The name "union" stems from the type's formal definition. If a type is considered as the set of all values that that type can take on, a union type is simply the mathematical union of its constituting types, since it can take on any value any of its fields can. Also, because a mathematical union discards duplicates, if more than one field of the union can take on a single common value, it is impossible to tell from the value alone which field was last written.
-However, one useful programming function of unions is to map smaller data elements to larger ones for easier manipulation. A data structure consisting, for example, of 4 bytes and a 32-bit integer, can form a union with an unsigned 64-bit integer, and thus be more readily accessed for purposes of comparison etc.
-In C and C++, untagged unions are expressed nearly exactly like structures (structs), except that each data member begins at the same location in memory. The data members, as in structures, need not be primitive values, and in fact may be structures or even other unions.
-The primary use of a union is allowing access to a common location by different data types, for example hardware input/output access, perhaps bitfield and word sharing. Unions also provide crude polymorphism. However, there is no checking of types, so it is up to the programmer to be sure that the proper fields are accessed in different contexts. The relevant field of a union variable is typically determined by the state of other variables, possibly in an enclosing struct.
abstract data type. an abstract data type that generalizes a queue, for which elements can be added to or removed from either the front (head) or back (tail). It is also often called a head-tail linked list, though properly this refers to a specific data structure implementation.
-This differs from the queue abstract data type or First-In-First-Out List (FIFO), where elements can only be added to one end and removed from the other. This general data class has some possible sub-types:
1) An input-restricted deque is one where deletion can be made from both ends, but insertion can be made at one end only.
2) An output-restricted deque is one where insertion can be made at both ends, but deletion can be made from one end only.
-Both the basic and most common list types in computing, queues and stacks can be considered specializations of deques, and can be implemented using deques.
-There are at least two common ways to efficiently implement a deque: with a modified dynamic array or with a doubly linked list.
The dynamic array approach uses a variant of a dynamic array that can grow from both ends, sometimes called array deques. These array deques have all the properties of a dynamic array, such as constant-time random access, good locality of reference, and inefficient insertion/removal in the middle, with the addition of amortized constant-time insertion/removal at both ends, instead of just one end. Three common implementations include:
1) Storing deque contents in a circular buffer, and only resizing when the buffer becomes full. This decreases the frequency of resizings.
2) Allocating deque contents from the center of the underlying array, and resizing the underlying array when either end is reached. This approach may require more frequent resizings and waste more space, particularly when elements are only inserted at one end.
3) Storing contents in multiple smaller arrays, allocating additional arrays at the beginning or end as needed. Indexing is implemented by keeping a dynamic array containing pointers to each of the smaller arrays.
abstract data type. an abstract data type that is meant to implement the undirected graph and directed graph concepts from mathematics.
-A graph data structure consists of a finite (and possibly mutable) set of vertices or nodes or points, together with a set of unordered pairs of these vertices for an undirected graph or a set of ordered pairs for a directed graph. These pairs are known as edges, arcs, or lines for an undirected graph and as arrows, directed edges, directed arcs, or directed lines for a directed graph. The vertices may be part of the graph structure, or may be external entities represented by integer indices or references.
-A graph data structure may also associate to each edge some edge value, such as a symbolic label or a numeric attribute (cost, capacity, length, etc.).
-Different data structures for the representation of graphs are used in practice:
1) Adjacency list
Vertices are stored as records or objects, and every vertex stores a list of adjacent vertices. This data structure allows the storage of additional data on the vertices. Additional data can be stored if edges are also stored as objects, in which case each vertex stores its incident edges and each edge stores its incident vertices.
2) Adjacency matrix
A two-dimensional matrix, in which the rows represent source vertices and columns represent destination vertices. Data on edges and vertices must be stored externally. Only the cost for one edge can be stored between each pair of vertices.
3) Incidence matrix
A two-dimensional Boolean matrix, in which the rows represent the vertices and columns represent the edges. The entries indicate whether the vertex at a row is incident to the edge at a column.
-look at table for time complexity cost
-Adjacency lists are generally preferred because they efficiently represent sparse graphs. An adjacency matrix is preferred if the graph is dense, that is the number of edges |E | is close to the number of vertices squared, |V |2, or if one must be able to quickly look up if there is an edge connecting two vertices.
-some properties of abstract data types: order, unique, associative
array data structure. A bitboard, often used for boardgames such as chess, checkers, othello and word games, is a specialization of the bit array data structure, where each bit represents a game position or state, designed for optimization of speed and/or memory or disk use in mass calculations. Bits in the same bitboard relate to each other in the rules of the game, often forming a game position when taken together. Other bitboards are commonly used as masks to transform or answer queries about positions. The "game" may be any game-like system where information is tightly packed in a structured form with "rules" affecting how the individual units or pieces relate.
-Bitboards are used in many of the world's highest-rated chess playing programs such as Houdini, Stockfish, and Critter. They help the programs analyze chess positions with few CPU instructions and hold a massive number of positions in memory efficiently.
-Bitboards allow the computer to answer some questions about game state with one logical operation. For example, if a chess program wants to know if the white player has any pawns in the center of the board (center four squares) it can just compare a bitboard for the player's pawns with one for the center of the board using a logical AND operation. If there are no center pawns then the result will be zero.
-Query results can also be represented using bitboards. For example, the query "What are the squares between X and Y?" can be represented as a bitboard. These query results are generally pre-calculated, so that a program can simply retrieve a query result with one memory load.
However, as a result of the massive compression and encoding, bitboard programs are not easy for software developers to either write or debug.
array data structure. a mapping from some domain (for example, a range of integers) to bits, that is, values which are zero or one. It is also called a bit array or bitmap index. In computer graphics, when the domain is a rectangle (indexed by two coordinates) a bitmap gives a way to store a binary image, that is, an image in which each pixel is either black or white (or any two colors).
-The more general term pixmap refers to a map of pixels, where each one may store more than two colors, thus using more than one bit per pixel. Often bitmap is used for this as well. In some contexts, the term bitmap implies one bit per pixel, while pixmap is used for images with multiple bits per pixel
-A bitmap is a type of memory organization or image file format used to store digital images. The term bitmap comes from the computer programming terminology, meaning just a map of bits, a spatially mapped array of bits. Now, along with pixmap, it commonly refers to the similar concept of a spatially mapped array of pixels. Raster images in general may be referred to as bitmaps or pixmaps, whether synthetic or photographic, in files or memory.
-Many graphical user interfaces use bitmaps in their built-in graphics subsystems
-Similarly, most other image file formats, such as JPEG, TIFF, PNG, and GIF, also store bitmap images (as opposed to vector graphics), but they are not usually referred to as bitmaps, since they use compressed formats internally.
-In typical uncompressed bitmaps, image pixels are generally stored with a color depth of 1, 4, 8, 16, 24, 32, 48, or 64 bits per pixel. Pixels of 8 bits and fewer can represent either grayscale or indexed color. An alpha channel (for transparency) may be stored in a separate bitmap, where it is similar to a grayscale bitmap, or in a fourth channel that, for example, converts 24-bit images to 32 bits per pixel.
-The bits representing the bitmap pixels may be packed or unpacked (spaced out to byte or word boundaries), depending on the format or device requirements. Depending on the color depth, a pixel in the picture will occupy at least n/8 bytes, where n is the bit depth.
array data structure. a data structure that uses a single, fixed-size buffer as if it were connected end-to-end. This structure lends itself easily to buffering data streams.
-a data buffer (or just buffer) is a region of a physical memory storage used to temporarily store data while it is being moved from one place to another.
-the useful property of a circular buffer is that it does not need to have its elements shuffled around when one is consumed. (If a non-circular buffer were used then it would be necessary to shift all elements when one is consumed.) In other words, the circular buffer is well-suited as a FIFO buffer while a standard, non-circular buffer is well suited as a LIFO buffer.
-Circular buffering makes a good implementation strategy for a queue that has fixed maximum size. Should a maximum size be adopted for a queue, then a circular buffer is a completely ideal implementation; all queue operations are constant time. However, expanding a circular buffer requires shifting memory, which is comparatively costly. For arbitrarily expanding queues, a linked list approach may be preferred instead.
-In some situations, overwriting circular buffer can be used, e.g. in multimedia. If the buffer is used as the bounded buffer in the producer-consumer problem then it is probably desired for the producer (e.g., an audio generator) to overwrite old data if the consumer (e.g., the sound card) is unable to momentarily keep up. Also, the LZ77 family of lossless data compression algorithms operates on the assumption that strings seen more recently in a data stream are more likely to occur soon in the stream. Implementations store the most recent data in a circular buffer.
-A circular buffer can be implemented using four pointers, or two pointers and two integers:
buffer start in memory
buffer end in memory, or buffer capacity
start of valid data (index or pointer)
end of valid data (index or pointer), or amount of data currently in the buffer (integer)
-A circular-buffer implementation may be optimized by mapping the underlying buffer to two contiguous regions of virtual memory. (Naturally, the underlying buffer's length must then equal some multiple of the system's page size.) Reading from and writing to the circular buffer may then be carried out with greater efficiency by means of direct memory access; those accesses which fall beyond the end of the first virtual-memory region will automatically wrap around to the beginning of the underlying buffer. When the read offset is advanced into the second virtual-memory region, both offsets—read and write—are decremented by the length of the underlying buffer.
array data structure. a random access, variable-size list data structure that allows elements to be added or removed. It is supplied with standard libraries in many modern mainstream programming languages.
-A dynamic array is not the same thing as a dynamically allocated array, which is an array whose size is fixed when the array is allocated, although a dynamic array may use such a fixed-size array as a back end.
-The simplest dynamic array is constructed by allocating a fixed-size array and then dividing it into two parts: the first stores the elements of the dynamic array and the second is reserved, or unused. We can then add or remove elements at the end of the dynamic array in constant time by using the reserved space, until this space is completely consumed. The number of elements used by the dynamic array contents is its logical size or size, while the size of the underlying array is called the dynamic array's capacity or physical size, which is the maximum possible size without relocating data.
-In applications where the logical size is bounded, the fixed-size data structure suffices. This may be short-sighted, as more space may be needed later. A philosophical programmer may prefer to write the code to make every array capable of resizing from the outset, then return to using fixed-size arrays during program optimization. Resizing the underlying array is an expensive task, typically involving copying the entire contents of the array.
-To avoid incurring the cost of resizing many times, dynamic arrays resize by a large amount, such as doubling in size, and use the reserved space for future expansion.
-The dynamic array has performance similar to an array, with the addition of new operations to add and remove elements:
1) Getting or setting the value at a particular index (constant time)
2) Iterating over the elements in order (linear time, good cache performance)
3) Inserting or deleting an element in the middle of the array (linear time)
4) Inserting or deleting an element at the end of the array (constant amortized time)
-Dynamic arrays benefit from many of the advantages of arrays, including good locality of reference and data cache utilization, compactness (low memory use), and random access. They usually have only a small fixed additional overhead for storing information about the size and capacity. This makes dynamic arrays an attractive tool for building cache-friendly data structures. However, in languages like Python or Java that enforce reference semantics, the dynamic array generally will not store the actual data, but rather it will store references to the data that resides in other areas of memory. In this case, accessing items in the array sequentially will actually involve accessing multiple non-contiguous areas of memory, so the many advantages of the cache-friendliness of this data structure are lost.
-Compared to linked lists, dynamic arrays have faster indexing (constant time versus linear time) and typically faster iteration due to improved locality of reference; however, dynamic arrays require linear time to insert or delete at an arbitrary location, since all following elements must be moved, while linked lists can do this in constant time. This disadvantage is mitigated by the gap buffer and tiered vector variants discussed under Variants below. Also, in a highly fragmented memory region, it may be expensive or impossible to find contiguous space for a large dynamic array, whereas linked lists do not require the whole data structure to be stored contiguously.
-A balanced tree can store a list while providing all operations of both dynamic arrays and linked lists reasonably efficiently, but both insertion at the end and iteration over the list are slower than for a dynamic array, in theory and in practice, due to non-contiguous storage and tree traversal/manipulation overhead.
array data structure. a dynamic array data-structure published by Edward Sitarski in 1996,[1] maintaining an array of separate memory fragments (or "leaves") to store the data elements, unlike simple dynamic arrays which maintain their data in one contiguous memory area. Its primary objective is to reduce the amount of element copying due to automatic array resizing operations, and to improve memory usage patterns.
-Whereas simple dynamic arrays based on geometric expansion waste linear (Ω(n)) space, where n is the number of elements in the array, hashed array trees waste only order O(√n) storage space. An optimization of the algorithm allows to eliminate data copying completely, at a cost of increasing the wasted space.
-It can perform access in constant (O(1)) time, though slightly slower than simple dynamic arrays. The algorithm has O(1) amortized performance when appending a series of objects to the end of a hashed array tree. Contrary to its name, it does not use hash functions.
-As defined by Sitarski, a hashed array tree has a top-level directory containing a power of two number of leaf arrays. All leaf arrays are the same size as the top-level directory. This structure superficially resembles a hash table with array-based collision chains, which is the basis for the name hashed array tree. A full hashed array tree can hold m2 elements, where m is the size of the top-level directory. The use of powers of two enables faster physical addressing through bit operations instead of arithmetic operations of quotient and remainder[1] and ensures the O(1) amortized performance of append operation in the presence of occasional global array copy while expanding.
-Brodnik et al. presented a dynamic array algorithm with a similar space wastage profile to hashed array trees. Brodnik's implementation retains previously allocated leaf arrays, with a more complicated address calculation function as compared to hashed array trees.
array data structure. a raster image used to store values, such as surface elevation data, for display in 3D computer graphics. A heightmap can be used in bump mapping to calculate where this 3D data would create shadow in a material, in displacement mapping to displace the actual geometric position of points over the textured surface, or for terrain where the heightmap is converted into a 3D mesh.
-A heightmap contains one channel interpreted as a distance of displacement or "height" from the "floor" of a surface and sometimes visualized as luma of a grayscale image, with black representing minimum height and white representing maximum height. When the map is rendered, the designer can specify the amount of displacement for each unit of the height channel, which corresponds to the "contrast" of the image. Heightmaps can be stored by themselves in existing grayscale image formats, with or without specialized metadata, or in specialized file formats such as Daylon Leveller, GenesisIV and Terragen documents.
-One may also exploit the use of individual color channels to increase detail. For example, a standard RGB 8-bit image can only show 256 values of grey and hence only 256 heights. By using colors, a greater number of heights can be stored (for an 24-bit image, 2563 = 16,777,216 heights can be represented (2564 = 4,294,967,296 if the alpha channel is also used)). This technique is especially useful where height varies slightly over a large area. Using only grey values, because the heights must be mapped to only 256 values, the rendered terrain appears flat, with "steps" in certain places.
-Heightmap of Earth's surface (including water and ice) in equirectangular projection, normalized as 8-bit grayscale
Heightmaps are commonly used in geographic information systems, where they are called digital elevation models.
array data structure. an array that replaces runtime computation with a simpler array indexing operation. The savings in terms of processing time can be significant, since retrieving a value from memory is often faster than undergoing an "expensive" computation or input/output operation. The tables may be precalculated and stored in static program storage, calculated (or "pre-fetched") as part of a program's initialization phase (memorization), or even stored in hardware in application-specific platforms. Lookup tables are also used extensively to validate input values by matching against a list of valid (or invalid) items in an array and, in some programming languages, may include pointer functions (or offsets to labels) to process the matching input.
-Examples
1) Simple lookup in an array, an associative array or a linked list (unsorted list)
This is known as a linear search or brute-force search, each element being checked for equality in turn and the associated value, if any, used as a result of the search. This is often the slowest search method unless frequently occurring values occur early in the list. For a one-dimensional array or linked list, the lookup is usually to determine whether or not there is a match with an 'input' data value.
2) Binary search in an array or an associative array (sorted list)
An example of a "divide and conquer algorithm", binary search involves each element being found by determining which half of the table a match may be found in and repeating until either success or failure. This is only possible if the list is sorted but gives good performance even if the list is lengthy.
3) Trivial hash function
For a trivial hash function lookup, the unsigned raw data value is used directly as an index to a one-dimensional table to extract a result. For small ranges, this can be amongst the fastest lookup, even exceeding binary search speed with zero branches and executing in constant time.
array data structure. a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. The individual items in a matrix are called its elements or entries.
-Applications of matrices are found in most scientific fields. In every branch of physics, including classical mechanics, optics, electromagnetism, quantum mechanics, and quantum electrodynamics, they are used to study physical phenomena, such as the motion of rigid bodies. In computer graphics, they are used to project a 3D model onto a 2 dimensional screen. In probability theory and statistics, stochastic matrices are used to describe sets of probabilities; for instance, they are used within the PageRank algorithm that ranks the pages in a Google search.[5] Matrix calculus generalizes classical analytical notions such as derivatives and exponentials to higher dimensions.
-A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old and is today an expanding area of research. Matrix decomposition methods simplify computations, both theoretically and practically. Algorithms that are tailored to particular matrix structures, such as sparse matrices and near-diagonal matrices, expedite computations in finite element method and other computations. Infinite matrices occur in planetary theory and in atomic theory. A simple example of an infinite matrix is the matrix representing the derivative operator, which acts on the Taylor series of a function.
array data structure. an array data structure in which each element is sorted in numerical, alphabetical, or some other order, and placed at equally spaced addresses in computer memory. It is typically used in computer science to implement static lookup tables to hold multiple values which have the same data type. Sorting an array is useful in organising data in ordered form and recovering them rapidly.
-There are many well-known methods by which an array can be sorted, which include, but are not limited to: selection sort, bubble sort, insertion sort, merge sort, quicksort, heapsort, and counting sort.
-Sorted arrays are the most space-efficient data structure with the best locality of reference for sequentially stored data.[citation needed]
-Elements within a sorted array are found using a binary search, in O(log n); thus sorted arrays are suited for cases when one needs to be able to look up elements quickly, e.g. as a set or multiset data structure. This complexity for lookups is the same as for self-balancing binary search trees.
-In some data structures, an array of structures is used. In such cases, the same sorting methods can be used to sort the structures according to some key as a structure element; for example, sorting records of students according to roll numbers or names or grades.
-If one is using a sorted dynamic array, then it is possible to insert and delete elements. The insertion and deletion of elements in a sorted array executes at O(n), due to the need to shift all the elements following the element to be inserted or deleted; in comparison a self-balancing binary search tree inserts and deletes at O(log n). In the case where elements are deleted or inserted at the end, a sorted dynamic array can do this in amortized O(1) time while a self-balancing binary search tree always operates at O(log n).
-Elements in a sorted array can be looked up by their index (random access) at O(1) time, an operation taking O(log n) or O(n) time for more complex data structures.
array data structure. an array in which most of the elements have the default value (usually 0 or null). The occurrence of zero-value elements in a large array is inefficient for both computation and storage. An array in which there is a large number of zero elements is referred to as being sparse.
-In the case of sparse arrays, one can ask for a value from an "empty" array position. If one does this, then for an array of numbers, a value of zero should be returned, and for an array of objects, a value of null should be returned.
-A naive implementation of an array may allocate space for the entire array, but in the case where there are few non-default values, this implementation is inefficient. Typically the algorithm used instead of an ordinary array is determined by other known features (or statistical features) of the array. For instance, if the sparsity is known in advance or if the elements are arranged according to some function (e.g., the elements occur in blocks).
A heap memory allocator in a program might choose to store regions of blank space in a linked list rather than storing all of the allocated regions in, say a bit array.
-An obvious question that might be asked is why we need a linked list to represent a sparse array if we can represent it easily using a normal array. The answer to this question lies in the fact that while representing a sparse array as a normal array, a lot of space is allocated for zero or null elements. For example, consider the following array declaration:
double arr[1000][1000]
-When we define this array, enough space for 1,000,000 doubles is allocated. If each double requires 8 bytes of memory, this array will require 8 million bytes of memory. Because this is a sparse array, most of its elements will have a value of zero (or null). Hence, defining this array will soak up all this space and waste memory (compared to an array in which memory has been allocated only for the nonzero elements). An effective way to overcome this problem is to represent the array using a linked list which requires less memory as only elements having non-zero value are stored. This involves a time-space trade-off: though less memory is used, average access and insertion time becomes linear in the number of elements stored because the previous elements in the list must be traversed to find the desired element. A normal array has constant access and insertion time.
-A sparse array as a linked list contains nodes linked to each other. In a one-dimensional sparse array, each node includes the non-zero element's "index" (position), the element's "value", and a node pointer "next" (for linking to the next node). Nodes are linked in order as per the index. In the case of a two-dimensional sparse array, each node contains a row index, a column index (which together give its position), a value at that position and a pointer to the next node.
array data structure. a matrix in which most of the elements are zero. By contrast, if most of the elements are nonzero, then the matrix is considered dense. The number of zero-valued elements divided by the total number of elements (e.g., m × n for an m × n matrix) is called the sparsity of the matrix (which is equal to 1 minus the density of the matrix).
-Conceptually, sparsity corresponds to systems which are loosely coupled. Consider a line of balls connected by springs from one to the next: this is a sparse system as only adjacent balls are coupled. By contrast, if the same line of balls had springs connecting each ball to all other balls, the system would correspond to a dense matrix. The concept of sparsity is useful in combinatorics and application areas such as network theory, which have a low density of significant data or connections.
-Large sparse matrices often appear in scientific or engineering applications when solving partial differential equations.
-When storing and manipulating sparse matrices on a computer, it is beneficial and often necessary to use specialized algorithms and data structures that take advantage of the sparse structure of the matrix. Operations using standard dense-matrix structures and algorithms are slow and inefficient when applied to large sparse matrices as processing and memory are wasted on the zeroes. Sparse data is by nature more easily compressed and thus require significantly less storage. Some very large sparse matrices are infeasible to manipulate using standard dense-matrix algorithms.
array data structure. a data structure used to implement multi-dimensional arrays. An Iliffe vector for an n-dimensional array (where n ≥ 2) consists of a vector (or 1-dimensional array) of pointers to an (n − 1)-dimensional array. They are often used to avoid the need for expensive multiplication operations when performing address calculation on an array element. They can also be used to implement jagged arrays, such as triangular arrays, triangular matrices and other kinds of irregularly shaped arrays. The data structure is named after John K. Iliffe.

Their disadvantages include the need for multiple chained pointer indirections to access an element, and the extra work required to determine the next row in an n-dimensional array to allow an optimising compiler to prefetch it. Both of these are a source of delays on systems where the CPU is significantly faster than main memory.

The Iliffe vector for a 2-dimensional array is simply a vector of pointers to vectors of data, i.e., the Iliffe vector represents the columns of an array where each column element is a pointer to a row vector.

Multidimensional arrays in languages such as Java, Python (multidimensional lists), Ruby, Visual Basic .NET, Perl, PHP, JavaScript, Objective-C (when using NSArray, not a row-major C-style array), Swift, and Atlas Autocode are implemented as Iliffe vectors.

Iliffe vectors are contrasted with dope vectors in languages such as Fortran, which contain the stride factors and offset values for the subscripts in each dimension.
List data structure. a linear collection of data elements, called nodes, pointing to the next node by means of a pointer. It is a data structure consisting of a group of nodes which together represent a sequence. Under the simplest form, each node is composed of data and a reference (in other words, a link) to the next node in the sequence. This structure allows for efficient insertion or removal of elements from any position in the sequence during iteration. More complex variants add additional links, allowing efficient insertion or removal from arbitrary element references.
-Linked lists are among the simplest and most common data structures. They can be used to implement several other common abstract data types, including lists (the abstract data type), stacks, queues, associative arrays, and S-expressions, though it is not uncommon to implement the other data structures directly without using a list as the basis of implementation.
-The principal benefit of a linked list over a conventional array is that the list elements can easily be inserted or removed without reallocation or reorganization of the entire structure because the data items need not be stored contiguously in memory or on disk, while an array has to be declared in the source code, before compiling and running the program. Linked lists allow insertion and removal of nodes at any point in the list, and can do so with a constant number of operations if the link previous to the link being added or removed is maintained during list traversal.
-On the other hand, simple linked lists by themselves do not allow random access to the data, or any form of efficient indexing. Thus, many basic operations — such as obtaining the last node of the list (assuming that the last node is not maintained as separate node reference in the list structure), or finding a node that contains a given datum, or locating the place where a new node should be inserted — may require sequential scanning of most or all of the list elements. The advantages and disadvantages of using linked lists are given below.
List data structure. a data structure that allows fast search within an ordered sequence of elements. Fast search is made possible by maintaining a linked hierarchy of subsequences, each skipping over fewer elements. Searching starts in the sparsest subsequence until two consecutive elements have been found, one smaller and one larger than or equal to the element searched for. Via the linked hierarchy, these two elements link to elements of the next sparsest subsequence, where searching is continued until finally we are searching in the full sequence. The elements that are skipped over may be chosen probabilistically [2] or deterministically,[3] with the former being more common.
-A schematic picture of the skip list data structure. Each box with an arrow represents a pointer and a row is a linked list giving a sparse subsequence; the numbered boxes at the bottom represent the ordered data sequence. Searching proceeds downwards from the sparsest subsequence at the top until consecutive elements bracketing the search element are found.
-A skip list is built in layers. The bottom layer is an ordinary ordered linked list. Each higher layer acts as an "express lane" for the lists below, where an element in layer i appears in layer i+1 with some fixed probability p (two commonly used values for p are 1/2 or 1/4). On average, each element appears in 1/(1-p) lists, and the tallest element (usually a special head element at the front of the skip list) in all the lists, {\displaystyle \log _{1/p}n\,} \log _{1/p}n\, of them.

A search for a target element begins at the head element in the top list, and proceeds horizontally until the current element is greater than or equal to the target. If the current element is equal to the target, it has been found. If the current element is greater than the target, or the search reaches the end of the linked list, the procedure is repeated after returning to the previous element and dropping down vertically to the next lower list. The expected number of steps in each linked list is at most 1/p, which can be seen by tracing the search path backwards from the target until reaching an element that appears in the next higher list or reaching the beginning of the current list. Therefore, the total expected cost of a search is (log _{1/p}n)/p, which is O(log n) when p is a constant. By choosing different values of p, it is possible to trade search costs against storage costs.
List data structure. a persistent data structure designed by Phil Bagwell in 2002 that combines the fast indexing of arrays with the easy extension of cons-based (or singly linked) linked lists.
-Like arrays, VLists have constant-time lookup on average and are highly compact, requiring only O(log n) storage for pointers, allowing them to take advantage of locality of reference. Like singly linked or cons-based lists, they are persistent, and elements can be added to or removed from the front in constant time. Length can also be found in O(log n) time.
-The primary operations of a VList are:
1) Locate the kth element (O(1) average, O(log n) worst-case)
2) Add an element to the front of the VList (O(1) average, with an occasional allocation)
3) Obtain a new array beginning at the second element of an old array (O(1))
4) Compute the length of the list (O(log n))
-The primary advantage VLists have over arrays is that different updated versions of the VList automatically share structure. Because VLists are immutable, they are most useful in functional programming languages, where their efficiency allows a purely functional implementation of data structures traditionally thought to require mutable arrays, such as hash tables.
-However, VLists also have a number of disadvantages over their competitors:
...While immutability is a benefit, it is also a drawback, making it inefficient to modify elements in the middle of the array.
...Access near the end of the list can be as expensive as O(log n); it is only constant on average over all elements. This is still, however, much better than performing the same operation on cons-based lists.
...Wasted space in the first block is proportional to n. This is similar to linked lists, but there are data structures with less overhead. When used as a fully persistent data structure, the overhead may be considerably higher and this data structure may not be appropriate.
-VList may be modified to support the implementation of a growable array. In the application of a growable array, immutability is no longer required. Instead of growing at the beginning of the list, the ordering interpretation is reversed to allow growing at the end of the array.
List data structure. a data structure that takes advantage of the bitwise XOR operation to decrease storage requirements for doubly linked lists. A bitwise XOR takes two bit patterns of equal length and performs the logical exclusive OR operation on each pair of corresponding bits. The result in each position is 1 if only the first bit is 1 or only the second bit is 1, but will be 0 if both are 0 or both are 1. In this we perform the comparison of two bits, being 1 if the two bits are different, and 0 if they are the same.
0101 (decimal 5)
XOR 0011 (decimal 3)
= 0110 (decimal 6)
The bitwise XOR may be used to invert selected bits in a register (also called toggle or flip). Any bit may be toggled by XORing it with 1. For example, given the bit pattern 0010 (decimal 2) the second and fourth bits may be toggled by a bitwise XOR with a bit pattern containing 1 in the second and fourth positions:
0010 (decimal 2)
XOR 1010 (decimal 10)
= 1000 (decimal 8)
This technique may be used to manipulate bit patterns representing sets of Boolean states.
-Assembly language programmers sometimes use XOR as a short-cut to setting the value of a register to zero. Performing XOR on a value against itself always yields zero, and on many architectures this operation requires fewer clock cycles and memory than loading a zero value and saving it to the register.
-An ordinary doubly linked list stores addresses of the previous and next list items in each list node, requiring two address fields:
... A B C D E ...
-> next -> next -> next ->
<- prev <- prev <- prev <-
An XOR linked list compresses the same information into one address field by storing the bitwise XOR (here denoted by ⊕) of the address for previous and the address for next in one field:
... A B C D E ...
<-> A⊕C <-> B⊕D <-> C⊕E <->
When you traverse the list from left to right: supposing you are at C, you can take the address of the previous item, B, and XOR it with the value in the link field (B⊕D). You will then have the address for D and you can continue traversing the list. The same pattern applies in the other direction.
-To start traversing the list in either direction from some point, you need the address of two consecutive items, not just one. If the addresses of the two consecutive items are reversed, you will end up traversing the list in the opposite direction.
-Drawbacks:
...General-purpose debugging tools cannot follow the XOR chain, making debugging more difficult; [1]
...The price for the decrease in memory usage is an increase in code complexity, making maintenance more expensive;
...Most garbage collection schemes do not work with data structures that do not contain literal pointers;
...XOR of pointers is not defined in some contexts (e.g., the C language), although many languages provide some kind of type conversion between pointers and integers;
...The pointers will be unreadable if one isn't traversing the list — for example, if the pointer to a list item was contained in another data structure;
...While traversing the list you need to remember the address of the previously accessed node in order to calculate the next node's address.
...XOR linked lists do not provide some of the important advantages of doubly linked lists, such as the ability to delete a node from the list knowing only its address or the ability to insert a new node before or after an existing node when knowing only the address of the existing node.
-Computer systems have increasingly cheap and plentiful memory, and storage overhead is not generally an overriding issue outside specialized embedded systems. Where it is still desirable to reduce the overhead of a linked list, unrolling provides a more practical approach (as well as other advantages, such as increasing cache performance and speeding random access).
List data structure. a technique of representing an aggregate data structure so that it is convenient for writing programs that traverse the structure arbitrarily and update its contents, especially in purely functional programming languages. The zipper was described by Gérard Huet in 1997.[1] It includes and generalizes the gap buffer technique sometimes used with arrays.
-The zipper technique is general in the sense that it can be adapted to lists, trees, and other recursively defined data structures. Such modified data structures are usually referred to as "a tree with zipper" or "a list with zipper" to emphasize that the structure is conceptually a tree or list, while the zipper is a detail of the implementation.
-A layman's explanation for a tree with zipper would be an ordinary computer filesystem with operations to go to parent (often cd ..), and the possibility to go downwards (cd subdirectory). The zipper is the pointer to the current path. Behind the scenes the zippers are efficient when making (functional) changes to a data structure, where a new, slightly changed, data structure is returned from an edit operation (instead of making a change in the current data structure).
-Uses:
-The zipper is often used where there is some concept of focus or of moving around in some set of data, since its semantics reflect that of moving around but in a functional non-destructive manner.
-The zipper has been used in
...Xmonad, to manage focus and placement of windows
...Huet's papers cover a structural editor based on zippers and a theorem prover
...A filesystem (ZipperFS) written in Haskell offering "...transactional semantics; undo of any file and directory operation; snapshots; statically guaranteed the strongest, repeatable read, isolation mode for clients; pervasive copy-on-write for files and directories; built-in traversal facility; and just the right behavior for cyclic directory references."
...Clojure has extensive support for zippers.
List data structure. a data structure to represent an embedding of a planar graph in the plane, and polytopes in 3D
-This data structure provides efficient manipulation of the topological information associated with the objects in question (vertices, edges, faces). It is used in many algorithms of computational geometry to handle polygonal subdivisions of the plane, commonly called planar straight-line graphs (PSLG).[1] For example, a Voronoi diagram is commonly represented by a DCEL inside a bounding box.
-This data structure was originally suggested by Muller and Preparata[2] for representations of 3D convex polyhedra.
-Later a somewhat different data structuring was suggested, but the name "DCEL" was retained.
-For simplicity, only connected graphs are considered, however the DCEL structure may be extended to handle disconnected graphs as well.
-DCEL is more than just a doubly linked list of edges. In the general case, a DCEL contains a record for each edge, vertex and face of the subdivision. Each record may contain additional information, for example, a face may contain the name of the area. Each edge usually bounds two faces and it is therefore convenient to regard each edge as two half-edges. Each half-edge bounds a single face and thus has a pointer to that face. A half-edge has a pointer to the next half-edge and previous half-edge of the same face. To reach the other face, we can go to the twin of the half-edge and then traverse the other face. Each half-edge also has a pointer to its origin vertex (the destination vertex can be obtained by querying the origin of its twin, or of the next half-edge).
-Each vertex contains the coordinates of the vertex and also stores a pointer to an arbitrary edge that has the vertex as its origin. Each face stores a pointer to some half-edge of its outer boundary (if the face is unbounded then pointer is null). It also has a list of half-edges, one for each hole that may be incident within the face. If the vertices or faces do not hold any interesting information, there is no need to store them, thus saving space and reducing the data structure's complexity.
binary tree data structure. a form of balanced tree used for storing and retrieving ordered data efficiently. AA trees are named for Arne Andersson, their inventor.
-AA trees are a variation of the red-black tree, a form of binary search tree which supports efficient addition and deletion of entries. Unlike red-black trees, red nodes on an AA tree can only be added as a right subchild. In other words, no red node can be a left sub-child. This results in the simulation of a 2-3 tree instead of a 2-3-4 tree, which greatly simplifies the maintenance operations. The maintenance algorithms for a red-black tree need to consider seven different shapes to properly balance the tree (see image in wiki).
An AA tree on the other hand only needs to consider two shapes due to the strict requirement that only right links can be red (see image in wiki).
-Balancing rotations:
-Whereas red-black trees require one bit of balancing metadata per node (the color), AA trees require O(log(N)) bits of metadata per node, in the form of an integer "level". The following invariants hold for AA trees:
1. The level of every leaf node is one.
2. The level of every left child is exactly one less than that of its parent.
3. The level of every right child is equal to or one less than that of its parent.
4. The level of every right grandchild is strictly less than that of its grandparent.
5. Every node of level greater than one has two children.
A link where the child's level is equal to that of its parent is called a horizontal link, and is analogous to a red link in the red-black tree. Individual right horizontal links are allowed, but consecutive ones are forbidden; all left horizontal links are forbidden. These are more restrictive constraints than the analogous ones on red-black trees, with the result that re-balancing an AA tree is procedurally much simpler than re-balancing a red-black tree.

Insertions and deletions may transiently cause an AA tree to become unbalanced (that is, to violate the AA tree invariants). Only two distinct operations are needed for restoring balance: "skew" and "split". Skew is a right rotation to replace a subtree containing a left horizontal link with one containing a right horizontal link instead. Split is a left rotation and level increase to replace a subtree containing two or more consecutive right horizontal links with one containing two fewer consecutive right horizontal links. Implementation of balance-preserving insertion and deletion is simplified by relying on the skew and split operations to modify the tree only if needed, instead of making their callers decide whether to skew or split.
binary tree data structure. a self-balancing binary search tree. It was the first such data structure to be invented.[2] In an AVL tree, the heights of the two child subtrees of any node differ by at most one; if at any time they differ by more than one, rebalancing is done to restore this property. Lookup, insertion, and deletion all take O(log n) time in both the average and worst cases, where n is the number of nodes in the tree prior to the operation. Insertions and deletions may require the tree to be rebalanced by one or more tree rotations.
-The AVL tree is named after its two Soviet inventors, Georgy Adelson-Velsky and Evgenii Landis, who published it in their 1962 paper "An algorithm for the organization of information".
-AVL trees are often compared with red-black trees because both support the same set of operations and take O(log n) time for the basic operations. For lookup-intensive applications, AVL trees are faster than red-black trees because they are more rigidly balanced. Similar to red-black trees, AVL trees are height-balanced. Both are in general not weight-balanced nor μ-balanced for any μ≤1⁄2;[5] that is, sibling nodes can have hugely differing numbers of descendants.
-Both AVL trees and red-black trees are self-balancing binary search trees and they are very similar mathematically.[9] The operations to balance the trees are different, but both occur on the average in O(1) with maximum in O(log n). The real difference between the two is the limiting height.
-AVL trees are more rigidly balanced than red-black trees, leading to faster retrieval but slower insertion and deletion.
variation in binary search. a technique to speed up a sequence of binary searches for the same value in a sequence of related data structures. The first binary search in the sequence takes a logarithmic amount of time, as is standard for binary searches, but successive searches in the sequence are faster. The original version of fractional cascading, introduced in two papers by Chazelle and Guibas in 1986 (Chazelle & Guibas 1986a; Chazelle & Guibas 1986b), combined the idea of cascading, originating in range searching data structures of Lueker (1978) and Willard (1978), with the idea of fractional sampling, which originated in Chazelle (1983). Later authors introduced more complex forms of fractional cascading that allow the data structure to be maintained as the data changes by a sequence of discrete insertion and deletion events.
-In general, fractional cascading begins with a catalog graph, a directed graph in which each vertex is labeled with an ordered list. A query in this data structure consists of a path in the graph and a query value q; the data structure must determine the position of q in each of the ordered lists associated with the vertices of the path. For the simple example above, the catalog graph is itself a path, with just four nodes. It is possible for later vertices in the path to be determined dynamically as part of a query, in response to the results found by the searches in earlier parts of the path.
-To handle queries of this type, for a graph in which each vertex has at most d incoming and at most d outgoing edges for some constant d, the lists associated with each vertex are augmented by a fraction of the items from each outgoing neighbor of the vertex; the fraction must be chosen to be smaller than 1/d, so that the total amount by which all lists are augmented remains linear in the input size. Each item in each augmented list stores with it the position of that item in the unaugmented list stored at the same vertex, and in each of the outgoing neighboring lists. In the simple example above, d = 1, and we augmented each list with a 1/2 fraction of the neighboring items.
-A query in this data structure consists of a standard binary search in the augmented list associated with the first vertex of the query path, together with simpler searches at each successive vertex of the path. If a 1/r fraction of items are used to augment the lists from each neighboring item, then each successive query result may be found within at most r steps of the position stored at the query result from the previous path vertex, and therefore may be found in constant time without having to perform a full binary search.
binary tree data structure. a tree data structure in which each node has at most two children, which are referred to as the left child and the right child. A recursive definition using just set theory notions is that a (non-empty) binary tree is a triple (L, S, R), where L and R are binary trees or the empty set and S is a singleton set.[1] Some authors allow the binary tree to be the empty set as well.[2]
-From a graph theory perspective, binary (and K-ary) trees as defined here are actually arborescences.[3] A binary tree may thus be also called a bifurcating arborescence[3]—a term which actually appears in some very old programming books,[4] before the modern computer science terminology prevailed. It is also possible to interpret a binary tree as an undirected, rather than a directed graph, in which case a binary tree is an ordered, rooted tree.[5] Some authors use rooted binary tree instead of binary tree to emphasize the fact that the tree is rooted, but as defined above, a binary tree is always rooted.[6] A binary tree is a special case of an ordered K-ary tree, where k is 2.
-In computing, binary trees are seldom used solely for their structure. Much more typical is to define a labeling function on the nodes, which associates some value to each node.[7] Binary trees labelled this way are used to implement binary search trees and binary heaps, and are used for efficient searching and sorting. The designation of non-root nodes as left or right child even when there is only one child present matters in some of these applications, in particular it is significant in binary search trees.[8] In mathematics, what is termed binary tree can vary significantly from author to author. Some use the definition commonly used in computer science,[9] but others define it as every non-leaf having exactly two children and don't necessarily order (as left/right) the children either
binary tree data structure. introduced by Martínez and Roura subsequently to the work of Aragon and Seidel on treaps, stores the same nodes with the same random distribution of tree shape, but maintains different information within the nodes of the tree in order to maintain its randomized structure.
Rather than storing random priorities on each node, the randomized binary search tree stores a small integer at each node, the number of its descendants (counting itself as one); these numbers may be maintained during tree rotation operations at only a constant additional amount of time per rotation. When a key x is to be inserted into a tree that already has n nodes, the insertion algorithm chooses with probability 1/(n + 1) to place x as the new root of the tree, and otherwise it calls the insertion procedure recursively to insert x within the left or right subtree (depending on whether its key is less than or greater than the root). The numbers of descendants are used by the algorithm to calculate the necessary probabilities for the random choices at each step. Placing x at the root of a subtree may be performed either as in the treap by inserting it at a leaf and then rotating it upwards, or by an alternative algorithm described by Martínez and Roura that splits the subtree into two pieces to be used as the left and right children of the new node.
-The deletion procedure for a randomized binary search tree uses the same information per node as the insertion procedure, and like the insertion procedure it makes a sequence of O(log n) random decisions in order to join the two subtrees descending from the left and right children of the deleted node into a single tree. If the left or right subtree of the node to be deleted is empty, the join operation is trivial; otherwise, the left or right child of the deleted node is selected as the new subtree root with probability proportional to its number of descendants, and the join proceeds recursively.
binary tree data structure. a type of binary tree data structure that is used by main-memory databases, such as Datablitz, EXtremeDB, MySQL Cluster, Oracle TimesTen and MobileLite.
-A T-tree is a balanced index tree data structure optimized for cases where both the index and the actual data are fully kept in memory, just as a B-tree is an index structure optimized for storage on block oriented secondary storage devices like hard disks. T-trees seek to gain the performance benefits of in-memory tree structures such as AVL trees while avoiding the large storage space overhead which is common to them.
-T-trees do not keep copies of the indexed data fields within the index tree nodes themselves. Instead, they take advantage of the fact that the actual data is always in main memory together with the index so that they just contain pointers to the actual data fields.
-The 'T' in T-tree refers to the shape of the node data structures in the original paper that first described this type of index.
-Although T-trees seem to be widely used for main-memory databases, recent research indicates that they actually do not perform better than B-trees on modern hardware.
-The main reason seems to be that the traditional assumption of memory references having uniform cost is no longer valid given the current speed gap between cache access and main memory access.
-A T-tree node usually consists of pointers to the parent node, the left and right child node, an ordered array of data pointers and some extra control data. Nodes with two subtrees are called internal nodes, nodes without subtrees are called leaf nodes and nodes with only one subtree are named half-leaf nodes. A node is called the bounding node for a value if the value is between the node's current minimum and maximum value, inclusively.
-For each internal node, leaf or half leaf nodes exist that contain the predecessor of its smallest data value (called the greatest lower bound) and one that contains the successor of its largest data value (called the least upper bound). Leaf and half-leaf nodes can contain any number of data elements from one to the maximum size of the data array. Internal nodes keep their occupancy between predefined minimum and maximum numbers of elements
binary tree data structure. a data structure based on a binary tree for unrooted dynamic trees that is used mainly for various path-related operations. It allows simple divide-and-conquer algorithms. It has since been augmented to maintain dynamically various properties of a tree such as diameter, center and median.
-A top tree R is defined for an underlying tree T and a set delta T (?) of at most two vertices called as External Boundary Vertices
-The treap was first described by Cecilia R. Aragon and Raimund Seidel in 1989;[1][2] its name is a portmanteau of tree and heap. It is a Cartesian tree in which each key is given a (randomly chosen) numeric priority. As with any binary search tree, the inorder traversal order of the nodes is the same as the sorted order of the keys. The structure of the tree is determined by the requirement that it be heap-ordered: that is, the priority number for any non-leaf node must be greater than or equal to the priority of its children. Thus, as with Cartesian trees more generally, the root node is the maximum-priority node, and its left and right subtrees are formed in the same manner from the subsequences of the sorted order to the left and right of that node.

An equivalent way of describing the treap is that it could be formed by inserting the nodes highest-priority-first into a binary search tree without doing any rebalancing. Therefore, if the priorities are independent random numbers (from a distribution over a large enough space of possible priorities to ensure that two nodes are very unlikely to have the same priority) then the shape of a treap has the same probability distribution as the shape of a random binary search tree, a search tree formed by inserting the nodes without rebalancing in a randomly chosen insertion order. Because random binary search trees are known to have logarithmic height with high probability, the same is true for treaps.

Aragon and Seidel also suggest assigning higher priorities to frequently accessed nodes, for instance by a process that, on each access, chooses a random number and replaces the priority of the node with that number if it is higher than the previous priority. This modification would cause the tree to lose its random shape; instead, frequently accessed nodes would be more likely to be near the root of the tree, causing searches for them to be faster.

Naor and Nissim[3] describe an application in maintaining authorization certificates in public-key cryptosystems.
binary tree data structure. heap data structure. The treap was first described by Cecilia R. Aragon and Raimund Seidel in 1989;[1][2] its name is a portmanteau of tree and heap. It is a Cartesian tree in which each key is given a (randomly chosen) numeric priority. As with any binary search tree, the inorder traversal order of the nodes is the same as the sorted order of the keys. The structure of the tree is determined by the requirement that it be heap-ordered: that is, the priority number for any non-leaf node must be greater than or equal to the priority of its children. Thus, as with Cartesian trees more generally, the root node is the maximum-priority node, and its left and right subtrees are formed in the same manner from the subsequences of the sorted order to the left and right of that node.
-An equivalent way of describing the treap is that it could be formed by inserting the nodes highest-priority-first into a binary search tree without doing any rebalancing. Therefore, if the priorities are independent random numbers (from a distribution over a large enough space of possible priorities to ensure that two nodes are very unlikely to have the same priority) then the shape of a treap has the same probability distribution as the shape of a random binary search tree, a search tree formed by inserting the nodes without rebalancing in a randomly chosen insertion order. Because random binary search trees are known to have logarithmic height with high probability, the same is true for treaps.
-Aragon and Seidel also suggest assigning higher priorities to frequently accessed nodes, for instance by a process that, on each access, chooses a random number and replaces the priority of the node with that number if it is higher than the previous priority. This modification would cause the tree to lose its random shape; instead, frequently accessed nodes would be more likely to be near the root of the tree, causing searches for them to be faster.
-Naor and Nissim[3] describe an application in maintaining authorization certificates in public-key cryptosystems.
binary tree data structure. a self-balancing binary search tree. WAVL trees are named after AVL trees, another type of balanced search tree, and are closely related both to AVL trees and red-black trees, which all fall into a common framework of rank balanced trees. Like other balanced binary search trees, WAVL trees can handle insertion, deletion, and search operations in time O(log n) per operation.[1][2]
-WAVL trees are designed to combine some of the best properties of both AVL trees and red-black trees. One advantage of AVL trees over red-black trees is that they are more balanced: they have height at most log (phi?) n= approx 1.44 log _{2}n (for a tree with n data items, where phi is the golden ratio), while red-black trees have larger maximum height, 2log _{2}n. If a WAVL tree is created using only insertions, without deletions, then it has the same small height bound that an AVL tree has. On the other hand, red-black trees have the advantage over AVL trees that they perform less restructuring of their trees. In AVL trees, each deletion may require a logarithmic number of tree rotation operations, while red-black trees have simpler deletion operations that use only a constant number of tree rotations. WAVL trees, like red-black trees, use only a constant number of tree rotations, and the constant is even better than for red-black trees.[1][2]
-WAVL trees were introduced by Haeupler, Sen & Tarjan (2015). The same authors also provided a common view of AVL trees, WAVL trees, and red-black trees as all being a type of rank-balanced tree.
binary tree data structure. a type of self-balancing binary search trees that can be used to implement dynamic sets, dictionaries (maps) and sequences.[1] These trees were introduced by Nievergelt and Reingold in the 1970s as trees of bounded balance, or BB[α] trees.[2][3] Their more common name is due to Knuth.[4]

Like other self-balancing trees, WBTs store bookkeeping information pertaining to balance in their nodes and perform rotations to restore balance when it is disturbed by insertion or deletion operations. Specifically, each node stores the size of the subtree rooted at the node, and the sizes of left and right subtrees are kept within some factor of each other. Unlike the balance information in AVL trees (which store the height of subtrees) and red-black trees (which store a fictional "color" bit), the bookkeeping information in a WBT is an actually useful property for applications: the number of elements in a tree is equal to the size of its root, and the size information is exactly the information needed to implement the operations of an order statistic tree, viz., getting the n'th largest element in a set or determining an element's index in sorted order.[5]
-Weight-balanced trees are popular in the functional programming community and are used to implement sets and maps in MIT Scheme, SLIB and implementations of Haskell.
-A weight-balanced tree is a binary search tree that stores the sizes of subtrees in the nodes. That is, a node has fields
...key, of any ordered type
...value (optional, only for mappings)
...left, right, pointer to node
...size, of type integer.
By definition, the size of a leaf (typically represented by a nil pointer) is zero. The size of an internal node is the sum of sizes of its two children, plus one (size[n] = size[n.left] + size[n.right] + 1). Based on the size, one defines the weight as weight[n] = size[n] + 1
b-tree data structure. a tree data structure similar to B+ trees. It was invented by Hans Reiser, for use by the Reiser4 file system. As opposed to self-balancing binary search trees that attempt to keep their nodes balanced at all times, dancing trees only balance their nodes when flushing data to a disk (either because of memory constraints or because a transaction has completed).[1]

The idea behind this is to speed up file system operations by delaying optimization of the tree and only writing to disk when necessary, as writing to disk is thousands of times slower than writing to memory. Also, because this optimization is done less often than with other tree data structures, the optimization can be more extensive.

In some sense, this can be considered to be a self-balancing binary search tree that is optimized for storage on a slow medium, in that the on-disc form will always be balanced but will get no mid-transaction writes; doing so eases the difficulty (at the time) of adding and removing nodes, and instead performs these (slow) rebalancing operations at the same time as the (much slower) write to the storage medium.

However, a (negative) side effect of this behavior is witnessed in cases of unexpected shutdown, incomplete data writes, and other occurrences that may prevent the final (balanced) transaction from completing. In general, dancing trees will pose a greater difficulty for data recovery from incomplete transactions than a normal tree; though this can be addressed by either adding extra transaction logs or developing an algorithm to locate data on disk not previously present, then going through with the optimizations once more before continuing with any other pending operations/transactions.
b-tree data structure. multiway tree data structure. a type of tree data structure that implements an associative array on w-bit integers. When operating on a collection of n key-value pairs, it uses O(n) space and performs searches in O(logw n) time, which is asymptotically faster than a traditional self-balancing binary search tree, and also better than the van Emde Boas tree for large values of w. It achieves this speed by exploiting certain constant-time operations that can be done on a machine word. Fusion trees were invented in 1990 by Michael Fredman and Dan Willard.[1]
-Several advances have been made since Fredman and Willard's original 1990 paper. In 1999[2] it was shown how to implement fusion trees under a model of computation in which all of the underlying operations of the algorithm belong to AC0, a model of circuit complexity that allows addition and bitwise Boolean operations but disallows the multiplication operations used in the original fusion tree algorithm. A dynamic version of fusion trees using hash tables was proposed in 1996[3] which matched the original structure's O(logw n) runtime in expectation. Another dynamic version using exponential tree was proposed in 2007[4] which yields worst-case runtimes of O(logw n + log log u) per operation, where u is the size of the largest key. It remains open whether dynamic fusion trees can achieve O(logw n) per operation with high probability.
-A fusion tree is essentially a B-tree with branching factor of w1/5 (any small exponent is also possible), which gives it a height of O(logw n). To achieve the desired runtimes for updates and queries, the fusion tree must be able to search a node containing up to w1/5 keys in constant time. This is done by compressing ("sketching") the keys so that all can fit into one machine word, which in turn allows comparisons to be done in parallel.
heap data structure. a data structure for priority queue operations, consisting of a collection of heap-ordered trees. It has a better amortized running time than many other priority queue data structures including the binary heap and binomial heap. Michael L. Fredman and Robert E. Tarjan developed Fibonacci heaps in 1984 and published them in a scientific journal in 1987. They named Fibonacci heaps after the Fibonacci numbers, which are used in their running time analysis.
-For the Fibonacci heap, the find-minimum operation takes constant (O(1)) amortized time.[1] The insert and decrease key operations also work in constant amortized time.[2] Deleting an element (most often used in the special case of deleting the minimum element) works in O(log n) amortized time, where n is the size of the heap.[2] This means that starting from an empty data structure, any sequence of a insert and decrease key operations and b delete operations would take O(a + b log n) worst case time, where n is the maximum heap size. In a binary or binomial heap such a sequence of operations would take O((a + b) log n) time. A Fibonacci heap is thus better than a binary or binomial heap when b is smaller than a by a non-constant factor. It is also possible to merge two Fibonacci heaps in constant amortized time, improving on the logarithmic merge time of a binomial heap, and improving on binary heaps which cannot handle merges efficiently.
-Using Fibonacci heaps for priority queues improves the asymptotic running time of important algorithms, such as Dijkstra's algorithm for computing the shortest path between two nodes in a graph, compared to the same algorithm using other slower priority queue data structures.
|Structure|
-A Fibonacci heap is a collection of trees satisfying the minimum-heap property, that is, the key of a child is always greater than or equal to the key of the parent. This implies that the minimum key is always at the root of one of the trees. Compared with binomial heaps, the structure of a Fibonacci heap is more flexible. The trees do not have a prescribed shape and in the extreme case the heap can have every element in a separate tree. This flexibility allows some operations to be executed in a lazy manner, postponing the work for later operations. For example, merging heaps is done simply by concatenating the two lists of trees, and operation decrease key sometimes cuts a node from its parent and forms a new tree.
-However at some point some order needs to be introduced to the heap to achieve the desired running time. In particular, degrees of nodes (here degree means the number of children) are kept quite low: every node has degree at most O(log n) and the size of a subtree rooted in a node of degree k is at least Fk+2, where Fk is the kth Fibonacci number. This is achieved by the rule that we can cut at most one child of each non-root node. When a second child is cut, the node itself needs to be cut from its parent and becomes the root of a new tree (see Proof of degree bounds, below). The number of trees is decreased in the operation delete minimum, where trees are linked together.
-As a result of a relaxed structure, some operations can take a long time while others are done very quickly. For the amortized running time analysis we use the potential method, in that we pretend that very fast operations take a little bit longer than they actually do. This additional time is then later combined and subtracted from the actual running time of slow operations. The amount of time saved for later use is measured at any given moment by a potential function. The potential of a Fibonacci heap is given by
Potential = t + 2m
where t is the number of trees in the Fibonacci heap, and m is the number of marked nodes. A node is marked if at least one of its children was cut since this node was made a child of another node (all roots are unmarked). The amortized time for an operation is given by the sum of the actual time and c times the difference in potential, where c is a constant (chosen to match the constant factors in the O notation for the actual time).
-Thus, the root of each tree in a heap has one unit of time stored. This unit of time can be used later to link this tree with another tree at amortized time 0. Also, each marked node has two units of time stored. One can be used to cut the node from its parent. If this happens, the node becomes a root and the second unit of time will remain stored in it as in any other root.
an algorithm for finding the shortest paths between nodes in a graph, which may represent, for example, road networks. It was conceived by computer scientist Edsger W. Dijkstra in 1956 and published three years later.[1][2]

The algorithm exists in many variants; Dijkstra's original variant found the shortest path between two nodes,[2] but a more common variant fixes a single node as the "source" node and finds shortest paths from the source to all other nodes in the graph, producing a shortest-path tree.

For a given source node in the graph, the algorithm finds the shortest path between that node and every other.[3]:196-206 It can also be used for finding the shortest paths from a single node to a single destination node by stopping the algorithm once the shortest path to the destination node has been determined. For example, if the nodes of the graph represent cities and edge path costs represent driving distances between pairs of cities connected by a direct road, Dijkstra's algorithm can be used to find the shortest route between one city and all other cities. As a result, the shortest path algorithm is widely used in network routing protocols, most notably IS-IS and Open Shortest Path First (OSPF). It is also employed as a subroutine in other algorithms such as Johnson's.

Dijkstra's original algorithm does not use a min-priority queue and runs in time {\displaystyle O(|V|^{2})} O(|V|^{2}) (where {\displaystyle |V|} |V| is the number of nodes). The idea of this algorithm is also given in (Leyzorek et al. 1957). The implementation based on a min-priority queue implemented by a Fibonacci heap and running in {\displaystyle O(|E|+|V|\log |V|)} O(|E|+|V|\log |V|) (where {\displaystyle |E|} |E| is the number of edges) is due to (Fredman & Tarjan 1984). This is asymptotically the fastest known single-source shortest-path algorithm for arbitrary directed graphs with unbounded non-negative weights. However, specialized cases (such as bounded/integer weights, directed acyclic graphs etc.) can indeed be improved further as detailed in § Specialized variants.

In some fields, artificial intelligence in particular, Dijkstra's algorithm or a variant of it is known as uniform-cost search and formulated as an instance of the more general idea of best-first search
heap data structure. a variant on the simple heap data structure that has constant amortized time for 5 types of operations. This is achieved by carefully "corrupting" (increasing) the keys of at most a certain number of values in the heap. The constant time operations are:
1. create(S): Create a new soft heap
2. insert(S, x): Insert an element into a soft heap
3. meld(S, S' ): Combine the contents of two soft heaps into one, destroying both
4. delete(S, x): Delete an element from a soft heap
5. findmin(S): Get the element with minimum key in the soft heap
Other heaps such as Fibonacci heaps achieve most of these bounds without any corruption, but cannot provide a constant-time bound on the critical delete operation.
-The amount of corruption can be controlled by the choice of a parameter ε, but the lower this is set, the more time insertions require (O(log 1/ε) for an error rate of ε).
-More precisely, the guarantee offered by the soft heap is the following: for a fixed value ε between 0 and 1/2, at any point in time there will be at most ε*n corrupted keys in the heap, where n is the number of elements inserted so far. Note that this does not guarantee that only a fixed percentage of the keys currently in the heap are corrupted: in an unlucky sequence of insertions and deletions, it can happen that all elements in the heap will have corrupted keys. Similarly, we have no guarantee that in a sequence of elements extracted from the heap with findmin and delete, only a fixed percentage will have corrupted keys: in an unlucky scenario only corrupted elements are extracted from the heap.
-The soft heap was designed by Bernard Chazelle in 2000. The term "corruption" in the structure is the result of what Chazelle called "carpooling" in a soft heap. Each node in the soft heap contains a linked-list of keys and one common key. The common key is an upper bound on the values of the keys in the linked-list. Once a key is added to the linked-list, it is considered corrupted because its value is never again relevant in any of the soft heap operations: only the common keys are compared. This is what makes soft heaps "soft"; you can't be sure whether or not any particular value you put into it will be corrupted. The purpose of these corruptions is effectively to lower the information entropy of the data, enabling the data structure to break through information-theoretic barriers regarding heaps.
heap data structure. a type of heap data structure with relatively simple implementation and excellent practical amortized performance, introduced by Michael Fredman, Robert Sedgewick, Daniel Sleator, and Robert Tarjan in 1986.[1] Pairing heaps are heap-ordered multiway tree structures, and can be considered simplified Fibonacci heaps. They are considered a "robust choice" for implementing such algorithms as Prim's MST algorithm,[2] and support the following operations (assuming a min-heap):
1. find-min: simply return the top element of the heap.
2. merge: compare the two root elements, the smaller remains the root of the result, the larger element and its subtree is appended as a child of this root.
3. insert: create a new heap for the inserted element and merge into the original heap.
4. decrease-key (optional): remove the subtree rooted at the key to be decreased, replace the key with a smaller key, then merge the result back into the heap.
5. delete-min: remove the root and merge its subtrees. Various strategies are employed.
The analysis of pairing heaps' time complexity was initially inspired by that of splay trees.[1] The amortized time per delete-min is O(log n), and the operations find-min, merge, and insert run in O(1) amortized time.[3]
-Determining the precise asymptotic running time of pairing heaps when a decrease-key operation is needed has turned out to be difficult. Initially, the time complexity of this operation was conjectured on empirical grounds to be O(1),[4] but Fredman proved that the amortized time per decrease-key is at least Omega (\log \log n) for some sequences of operations.[5] Using a different amortization argument, Pettie then proved that insert, meld, and decrease-key all run in O(2^{{2{\sqrt {\log \log n}}}}) amortized time, which is o(\log n).[6] Elmasry later introduced a variant of pairing heaps for which decrease-key runs in O(\log \log n) amortized time and with all other operations matching Fibonacci heaps,[7] but no tight Theta (\log \log n) bound is known for the original data structure.[6][3] Moreover, it is an open question whether a o(\log n) amortized time bound for decrease-key and a O(1) amortized time bound for insert can be achieved simultaneously.[8]
-Although this is worse than other priority queue algorithms such as Fibonacci heaps, which perform decrease-key in O(1) amortized time, the performance in practice is excellent. Stasko and Vitter,[4] Moret and Shapiro,[9] and Larkin, Sen, and Tarjan[8] conducted experiments on pairing heaps and other heap data structures. They concluded that pairing heaps are often faster in practice than array-based binary heaps and d-ary heaps, and almost always faster in practice than other pointer-based heaps, including data structures like Fibonacci heaps that are theoretically more efficient.
tree data structure where each tree node compares a bit slice of key values. a kind of search tree -- an ordered tree data structure that is used to store a dynamic set or associative array where the keys are usually strings. Unlike a binary search tree, no node in the tree stores the key associated with that node; instead, its position in the tree defines the key with which it is associated. All the descendants of a node have a common prefix of the string associated with that node, and the root is associated with the empty string. Values are not necessarily associated with every node. Rather, values tend only to be associated with leaves, and with some inner nodes that correspond to keys of interest. For the space-optimized presentation of prefix tree, see compact prefix tree.
-In the example shown, keys are listed in the nodes and values below them. Each complete English word has an arbitrary integer value associated with it. A trie can be seen as a tree-shaped deterministic finite automaton. Each finite language is generated by a trie automaton, and each trie can be compressed into a deterministic acyclic finite state automaton.
-Though tries are usually keyed by character strings, they need not be. The same algorithms can be adapted to serve similar functions of ordered lists of any construct, e.g. permutations on a list of digits or shapes. In particular, a bitwise trie is keyed on the individual bits making up any fixed-length binary datum, such as an integer or memory address.
-As discussed below, a trie has a number of advantages over binary search trees.[6] A trie can also be used to replace a hash table, over which it has the following advantages:
1. Looking up data in a trie is faster in the worst case, O(m) time (where m is the length of a search string), compared to an imperfect hash table. An imperfect hash table can have key collisions. A key collision is the hash function mapping of different keys to the same position in a hash table. The worst-case lookup speed in an imperfect hash table is O(N) time, but far more typically is O(1), with O(m) time spent evaluating the hash.
2. There are no collisions of different keys in a trie.
3. Buckets in a trie, which are analogous to hash table buckets that store key collisions, are necessary only if a single key is associated with more than one value.
4. There is no need to provide a hash function or to change hash functions as more keys are added to a trie.
5. A trie can provide an alphabetical ordering of the entries by key.
-Tries do have some drawbacks as well:
1. Tries can be slower in some cases than hash tables for looking up data, especially if the data is directly accessed on a hard disk drive or some other secondary storage device where the random-access time is high compared to main memory.[7]
2. Some keys, such as floating point numbers, can lead to long chains and prefixes that are not particularly meaningful. Nevertheless, a bitwise trie can handle standard IEEE single and double format floating point numbers.
3. Some tries can require more space than a hash table, as memory may be allocated for each character in the search string, rather than a single chunk of memory for the whole entry, as in most hash tables.
Dictionary Representation
-A common application of a trie is storing a predictive text or autocomplete dictionary, such as found on a mobile telephone. Such applications take advantage of a trie's ability to quickly search for, insert, and delete entries; however, if storing dictionary words is all that is required (i.e., storage of information auxiliary to each word is not required), a minimal deterministic acyclic finite state automaton (DAFSA) would use less space than a trie. This is because a DAFSA can compress identical branches from the trie which correspond to the same suffixes (or parts) of different words being stored.
-Tries are also well suited for implementing approximate matching algorithms,[8] including those used in spell checking and hyphenation[4] software.
Term Indexing
-A discrimination tree term index stores its information in a trie data structure.
tree data structure where each tree node compares a bit slice of key values. a data structure that represents a space-optimized trie in which each node that is the only child is merged with its parent. The result is that the number of children of every internal node is at least the radix r of the radix tree, where r is a positive integer and a power x of 2, having x ≥ 1. Unlike in regular tries, edges can be labeled with sequences of elements as well as single elements. This makes radix trees much more efficient for small sets (especially if the strings are long) and for sets of strings that share long prefixes.
-Unlike regular trees (where whole keys are compared en masse from their beginning up to the point of inequality), the key at each node is compared chunk-of-bits by chunk-of-bits, where the quantity of bits in that chunk at that node is the radix r of the radix trie. When the r is 2, the radix trie is binary (i.e., compare that node's 1-bit portion of the key), which minimizes sparseness at the expense of maximizing trie depth—i.e., maximizing up to conflation of nondiverging bit-strings in the key. When r is an integer power of 2 greater or equal to 4, then the radix trie is an r-ary trie, which lessens the depth of the radix trie at the expense of potential sparseness.
-As an optimization, edge labels can be stored in constant size by using two pointers to a string (for the first and last elements).[1]
-Note that although the examples in this article show strings as sequences of characters, the type of the string elements can be chosen arbitrarily; for example, as a bit or byte of the string representation when using multibyte character encodings or Unicode.
tree data structure where each tree node compares a bit slice of key values. a compressed data structure for pattern matching. Compressed suffix arrays are a general class of data structure that improve on the suffix array.[1][2] These data structures enable quick search for an arbitrary string with a comparatively small index.
-Given a text T of n characters from an alphabet Σ, a compressed suffix array supports searching for arbitrary patterns in T. For an input pattern P of m characters, the search time is typically O(m) or O(m + log(n)). The space used is typically O(nH_{k}(T))+o(n), where H_{k}(T) is the k-th order empirical entropy of the text T. The time and space to construct a compressed suffix array are normally O(n).
-The original instantiation of the compressed suffix array[1] solved a long-standing open problem by showing that fast pattern matching was possible using only a linear-space data structure, namely, one proportional to the size of the text T, which takes O(n log|Sigma|}) bits. The conventional suffix array and suffix tree use Omega (n log n) bits, which is substantially larger. The basis for the data structure is a recursive decomposition using the "neighbor function," which allows a suffix array to be represented by one of half its length. The construction is repeated multiple times until the resulting suffix array uses a linear number of bits. Following work showed that the actual storage space was related to the zeroth-order entropy and that the index supports self-indexing.[4] The space bound was further improved achieving the ultimate goal of higher-order entropy; the compression is obtained by partitioning the neighbor function by high-order contexts, and compressing each partition with a wavelet tree.[3] The space usage is extremely competitive in practice with other state-of-the-art compressors,[5] and it also supports fast pattern matching.
-The memory accesses made by compressed suffix arrays and other compressed data structures for pattern matching are typically not localized, and thus these data structures have been notoriously hard to design efficiently for use in external memory. Recent progress using geometric duality takes advantage of the block access provided by disks to speed up the I/O time significantly[6] In addition, potentially practical search performance for a compressed suffix array in external-memory has been demonstrated.
tree data structure where each tree node compares a bit slice of key values. hash data structure. a concurrent thread-safe lock-free implementation of a hash array mapped trie. It is used to implement the concurrent map abstraction. It has particularly scalable concurrent insert and remove operations and is memory-efficient.[3] It is the first known concurrent data-structure that supports O(1), atomic, lock-free snapshots.
|Advantages|
Ctries have been shown to be comparable in performance with concurrent skip lists,[2][4] concurrent hash tables and similar data structures in terms of the lookup operation, being slightly slower than hash tables and faster than skip lists due to the lower level of indirections. However, they are far more scalable than most concurrent hash tables where the insertions are concerned.[1] Most concurrent hash tables are bad at conserving memory - when the keys are removed from the hash table, the underlying array is not shrunk. Ctries have the property that the allocated memory is always a function of only the current number of keys in the data-structure.[1]
-Ctries have logarithmic complexity bounds of the basic operations, albeit with a low constant factor due to the high branching level (usually 32).
-Ctries support a lock-free, linearizable, constant-time snapshot operation,[2] based on the insight obtained from persistent data structures. This is a breakthrough in concurrent data-structure design, since existing concurrent data-structures do not support snapshots. The snapshot operation allows implementing lock-free, linearizable iterator, size and clear operations - existing concurrent data-structures have implementations which either use global locks or are correct only given that there are no concurrent modifications to the data-structure. In particular, Ctries have an O(1) iterator creation operation, O(1) clear operation, O(1) duplicate operation and an amortized O(logn) size retrieval operation.
|Problems|
Most concurrent data structures require dynamic memory allocation, and lock-free concurrent data structures rely on garbage collection on most platforms. The current implementation[4] of the Ctrie is written for the JVM, where garbage collection is provided by the platform itself. While it's possible to keep a concurrent memory pool for the nodes shared by all instances of Ctries in an application or use reference counting to properly deallocate nodes, the only implementation so far to deal with manual memory management of nodes used in Ctries is the common-lisp implementation cl-ctrie, which implements several stop-and-copy and mark-and-sweep garbage collection techniques for persistent, memory-mapped storage. Hazard pointers are another possible solution for a correct manual management of removed nodes. Such a technique may be viable for managed environments as well, since it could lower the pressure on the GC. A Ctrie implementation in Rust makes use of hazard pointers for this purpose
multiway tree data structure. data structure for representing a forest, a set of rooted trees, and offers the following operations:
-Add a tree consisting of a single node to the forest.
Given a node in one of the trees, disconnect it (and its subtree) from the tree of which it is part.
Attach a node to another node as its child.
Given a node, find the root of the tree to which it belongs. By doing this operation on two distinct nodes, one can check whether they belong to the same tree.
The represented forest may consist of very deep trees, so if we represent the forest as a plain collection of parent pointer trees, it might take us a long time to find the root of a given node. However, if we represent each tree in the forest as a link/cut tree, we can find which tree an element belongs to in O(log(n)) amortized time. Moreover, we can quickly adjust the collection of link/cut trees to changes in the represented forest. In particular, we can adjust it to merge (link) and split (cut) in O(log(n)) amortized time.
-Link/cut trees divide each tree in the represented forest into vertex-disjoint paths, where each path is represented by an auxiliary tree (often splay trees, though the original paper predates splay trees and thus uses biased binary search trees). The nodes in the auxiliary trees are keyed by their depth in the corresponding represented tree. In one variation, Naive Partitioning, the paths are determined by the most recently accessed paths and nodes, similar to Tango Trees. In Partitioning by Size paths are determined by the heaviest child (child with the most children) of the given node. This gives a more complicated structure, but reduces the cost of the operations from amortized O(log n) to worst case O(log n). It has uses in solving a variety of network flow problems and to jive data sets.
-In the original publication, Sleator and Tarjan referred to link/cut trees as "dynamic trees", or "dynamic dyno trees".
multiway tree data structure. a tree data structure used in computer science, and more specifically graph algorithms, to represent the triconnected components of a graph. The SPQR tree of a graph may be constructed in linear time[1] and has several applications in dynamic graph algorithms and graph drawing.
-The basic structures underlying the SPQR tree, the triconnected components of a graph, and the connection between this decomposition and the planar embeddings of a planar graph, were first investigated by Saunders Mac Lane (1937); these structures were used in efficient algorithms by several other researchers[2] prior to their formalization as the SPQR tree by Di Battista and Tamassia (1989, 1990, 1996).
|Structure|
An SPQR tree takes the form of an unrooted tree in which for each node x there is associated an undirected graph or multigraph Gx. The node, and the graph associated with it, may have one of four types, given the initials SPQR:
1.In an S node, the associated graph is a cycle graph with three or more vertices and edges. This case is analogous to series composition in series-parallel graphs; the S stands for "series".[3]
2.In a P node, the associated graph is a dipole graph, a multigraph with two vertices and three or more edges, the planar dual to a cycle graph. This case is analogous to parallel composition in series-parallel graphs; the P stands for "parallel".[3]
3.In a Q node, the associated graph has a single real edge. This trivial case is necessary to handle the graph that has only one edge. In some works on SPQR trees, this type of node does not appear in the SPQR trees of graphs with more than one edge; in other works, all non-virtual edges are required to be represented by Q nodes with one real and one virtual edge, and the edges in the other node types must all be virtual.
4.In an R node, the associated graph is a 3-connected graph that is not a cycle or dipole. The R stands for "rigid": in the application of SPQR trees in planar graph embedding, the associated graph of an R node has a unique planar embedding.[3]
-Each edge xy between two nodes of the SPQR tree is associated with two directed virtual edges, one of which is an edge in Gx and the other of which is an edge in Gy. Each edge in a graph Gx may be a virtual edge for at most one SPQR tree edge.
-An SPQR tree T represents a 2-connected graph GT, formed as follows. Whenever SPQR tree edge xy associates the virtual edge ab of Gx with the virtual edge cd of Gy, form a single larger graph by merging a and c into a single supervertex, merging b and d into another single supervertex, and deleting the two virtual edges. That is, the larger graph is the 2-clique-sum of Gx and Gy. Performing this gluing step on each edge of the SPQR tree produces the graph GT; the order of performing the gluing steps does not affect the result. Each vertex in one of the graphs Gx may be associated in this way with a unique vertex in GT, the supervertex into which it was merged.
-Typically, it is not allowed within an SPQR tree for two S nodes to be adjacent, nor for two P nodes to be adjacent, because if such an adjacency occurred the two nodes could be merged into a single larger node. With this assumption, the SPQR tree is uniquely determined from its graph. When a graph G is represented by an SPQR tree with no adjacent P nodes and no adjacent S nodes, then the graphs Gx associated with the nodes of the SPQR tree are known as the triconnected components of G.
multiway tree data structure. a class of tree data structures used in Project Xanadu "Green" designs of the 1970s and 1980s. Enfilades allow quick editing, versioning, retrieval and inter-comparison operations in a large, cross-linked hypertext database. The Xanadu "Gold" design starting in the 1990s used a related data structure called the Ent.
-Although the principles of enfilades can be applied to any tree data structure, the particular structure used in the Xanadu system was much like a B-Tree. What distinguishes enfilades is the use of dsps and wids in the indexing information within tree nodes.
-Dsps are displacements, offsets or relative keys. A dsp is the difference in key between a containing node and that of a subtree or leaf. For instance, the leaf for a grid square in a map might have a certain longitude and latitude offset relative to the larger grid represented by the subtree the leaf is part of. The key of any leaf of an enfilade is found by combining all the dsps on the path down the tree to that leaf. Dsps can also be used for other context information that is imposed top-down on entire subtrees or ranges of content at once.
-Wids are widths, ranges, or bounding boxes. A wid is relative to the key of a subtree or leaf, but specifies a range of addresses covering all items within the subtree. Wids identify the interesting parts of sparsely populated address spaces. In some enfilades, the wids of subtrees under a given node can overlap, and in any case, a search for data within a range of addresses must visit any subtrees whose wids intersect the search range. Wids are combined from the leaves of the tree, upward through all layers to the root (although they are maintained incrementally). Wids can also contain other summaries such as totals or maxima of data.
-The relative nature of wids and dsps allows subtrees to be rearranged within an enfilade. By changing the dsp at the top of a subtree, the keys of all the data underneath are implicitly changed. Edit operations in enfilades are performed by "cutting," or splitting the tree down relevant access paths, inserting, deleting or rearranging subtrees, and splicing the pieces back together. The cost of cutting and splicing operations is generally log-like in 1-D trees and between log-like and square-root-like in 2-D trees.
-Subtrees can also be shared between trees, or be linked from multiple places within a tree. This makes the enfilade a fully persistent data structure with virtual copying and versioning of content. Each use of a subtree inherits a different context from the chain of dsps down to it. Changes to a copy create new nodes only along the cut paths, and leave the entire original in place. The overhead for a version is very small, a new version's tree is balanced and fast, and its storage cost is related only to changes from the original.
-One-dimensional enfilades are intermediate between arrays' direct addressability and linked lists' ease of insertion, deletion and rearrangement. Multidimensional enfilades resemble loose, rearrangeable, versionable Quad trees, Oct trees or k-d trees.
space partitioning or binary space partitioning data structure. a k-d tree that associates a minimum and maximum value with each of its nodes
-a k-d tree with two scalar values - a minimum and a maximum - assigned to its nodes. The minimum/maximum of an inner node is equal to the minimum/maximum of its children's minima/maxima.
|Construction|
Min/max kd-trees may be constructed recursively. Starting with the root node, the splitting plane orientation and position is evaluated. Then the children's splitting planes and min/max values are evaluated recursively. The min/max value of the current node is simply the minimum/maximum of its children's minima/maxima.
|Properties|
The min/max kdtree has - besides the properties of an kd-tree - the special property that an inner node's min/max values coincide each with a min/max value of either one child. This allows to discard the storage of min/max values at the leaf nodes by storing two bits at inner nodes, assigning min/max values to the children: Each inner node's min/max values will be known in advance, where the root node's min/max values are stored separately. Each inner node has besides two min/max values also two bits given, defining to which child those min/max values are assigned (0: to the left child 1: to the right child). The non-assigned min/max values of the children are the from the current node already known min/max values. The two bits may also be stored in the least significant bits of the min/max values which have therefore to be approximated by fractioning them down/up.
-The resulting memory reduction is not minor, as the leaf nodes of full binary kd-trees are one half of the tree's nodes.
space partitioning or binary space partitioning data structure. tree data structures used for spatial access methods, i.e., for indexing multi-dimensional information such as geographical coordinates, rectangles or polygons. The R-tree was proposed by Antonin Guttman in 1984[1] and has found significant use in both theoretical and applied contexts.[2] A common real-world usage for an R-tree might be to store spatial objects such as restaurant locations or the polygons that typical maps are made of: streets, buildings, outlines of lakes, coastlines, etc. and then find answers quickly to queries such as "Find all museums within 2 km of my current location", "retrieve all road segments within 2 km of my location" (to display them in a navigation system) or "find the nearest gas station" (although not taking roads into account). The R-tree can also accelerate nearest neighbor search[3] for various distance metrics, including great-circle distance.
-The key idea of the data structure is to group nearby objects and represent them with their minimum bounding rectangle in the next higher level of the tree; the "R" in R-tree is for rectangle. Since all objects lie within this bounding rectangle, a query that does not intersect the bounding rectangle also cannot intersect any of the contained objects. At the leaf level, each rectangle describes a single object; at higher levels the aggregation of an increasing number of objects. This can also be seen as an increasingly coarse approximation of the data set.
-Similar to the B-tree, the R-tree is also a balanced search tree (so all leaf nodes are at the same height), organizes the data in pages, and is designed for storage on disk (as used in databases). Each page can contain a maximum number of entries, often denoted as {\displaystyle M} M. It also guarantees a minimum fill (except for the root node), however best performance has been experienced with a minimum fill of 30%-40% of the maximum number of entries (B-trees guarantee 50% page fill, and B*-trees even 66%). The reason for this is the more complex balancing required for spatial data as opposed to linear data stored in B-trees.
-As with most trees, the searching algorithms (e.g., intersection, containment, nearest neighbor search) are rather simple. The key idea is to use the bounding boxes to decide whether or not to search inside a subtree. In this way, most of the nodes in the tree are never read during a search. Like B-trees, this makes R-trees suitable for large data sets and databases, where nodes can be paged to memory when needed, and the whole tree cannot be kept in main memory.
-The key difficulty of R-trees is to build an efficient tree that on one hand is balanced (so the leaf nodes are at the same height) on the other hand the rectangles do not cover too much empty space and do not overlap too much (so that during search, fewer subtrees need to be processed). For example, the original idea for inserting elements to obtain an efficient tree is to always insert into the subtree that requires least enlargement of its bounding box. Once that page is full, the data is split into two sets that should cover the minimal area each. Most of the research and improvements for R-trees aims at improving the way the tree is built and can be grouped into two objectives: building an efficient tree from scratch (known as bulk-loading) and performing changes on an existing tree (insertion and deletion).
-R-trees do not guarantee good worst-case performance, but generally perform well with real-world data.[5] While more of theoretical interest, the (bulk-loaded) Priority R-tree variant of the R-tree is worst-case optimal,[6] but due to the increased complexity, has not received much attention in practical applications so far.
-When data is organized in an R-tree, the k nearest neighbors (for any Lp-Norm) of all points can efficiently be computed using a spatial join.[7] This is beneficial for many algorithms based on the k nearest neighbors, for example the Local Outlier Factor. DeLi-Clu,[8] Density-Link-Clustering is a cluster analysis algorithm that uses the R-tree structure for a similar kind of spatial join to efficiently compute an OPTICS clustering.
space partitioning or binary space partitioning data structure. a method for looking up data using a location, often (x, y) coordinates, and often for locations on the surface of the earth. Searching on one number is a solved problem; searching on two or more, and asking for locations that are nearby in both x and y directions, requires craftier algorithms.
-Fundamentally, an R+ tree is a tree data structure, a variant of the R tree, used for indexing spatial information.
|Difference between R+ trees and R trees|
-R+ trees are a compromise between R-trees and kd-trees: they avoid overlapping of internal nodes by inserting an object into multiple leaves if necessary. Coverage is the entire area to cover all related rectangles. Overlap is the entire area which is contained in two or more nodes.[1] Minimal coverage reduces the amount of "dead space" (empty area) which is covered by the nodes of the R-tree. Minimal overlap reduces the set of search paths to the leaves (even more critical for the access time than minimal coverage). Efficient search requires minimal coverage and overlap.
R+ trees differ from R trees in that: nodes are not guaranteed to be at least half filled, the entries of any internal node do not overlap, and an object ID may be stored in more than one leaf node.
|Advantages|
Because nodes are not overlapped with each other, point query performance benefits since all spatial regions are covered by at most one node. A single path is followed and fewer nodes are visited than with the R-tree
|Disadvantages|
Since rectangles are duplicated, an R+ tree can be larger than an R tree built on same data set. Construction and maintenance of R+ trees is more complex than the construction and maintenance of R trees and other variants of the R tree.
space partitioning or binary space partitioning data structure. a variant of R-trees used for indexing spatial information. R*-trees have slightly higher construction cost than standard R-trees, as the data may need to be reinserted; but the resulting tree will usually have a better query performance. Like the standard R-tree, it can store both point and spatial data. It was proposed by Norbert Beckmann, Hans-Peter Kriegel, Ralf Schneider, and Bernhard Seeger in 1990
|Difference between R* Trees and R Trees|
Minimization of both coverage and overlap is crucial to the performance of R-trees. Overlap means that, on data query or insertion, more than one branch of the tree needs to be expanded (due to the way data is being split in regions which may overlap). A minimized coverage improves pruning performance, allowing to exclude whole pages from search more often, in particular for negative range queries. The R*-tree attempts to reduce both, using a combination of a revised node split algorithm and the concept of forced reinsertion at node overflow. This is based on the observation that R-tree structures are highly susceptible to the order in which their entries are inserted, so an insertion-built (rather than bulk-loaded) structure is likely to be sub-optimal. Deletion and reinsertion of entries allows them to "find" a place in the tree that may be more appropriate than their original location.
-When a node overflows, a portion of its entries are removed from the node and reinserted into the tree. (In order to avoid an indefinite cascade of reinsertions caused by subsequent node overflow, the reinsertion routine may be called only once in each level of the tree when inserting any one new entry.) This has the effect of producing more well-clustered groups of entries in nodes, reducing node coverage. Furthermore, actual node splits are often postponed, causing average node occupancy to rise. Re-insertion can be seen as a method of incremental tree optimization triggered on node overflow.
|Performance|
-Improved split heuristic produces pages that are more rectangular and thus better for many applications.
-Reinsertion method optimizes the existing tree, but increases complexity.
-Efficiently supports point and spatial data at the same time.
space partitioning or binary space partitioning data structure. an R-tree variant, is an index for multidimensional objects like lines, regions, 3-D objects, or high-dimensional feature-based parametric objects. It can be thought of as an extension to B+-tree for multidimensional objects.
-The performance of R-trees depends on the quality of the algorithm that clusters the data rectangles on a node. Hilbert R-trees use space-filling curves, and specifically the Hilbert curve, to impose a linear ordering on the data rectangles.
|Hilbert curve or Hilbert space-filling curve|
a continuous fractal space-filling curve first described by the German mathematician David Hilbert in 1891,[1] as a variant of the space-filling Peano curves discovered by Giuseppe Peano in 1890.[2]
-Because it is space-filling, its Hausdorff dimension is 2 (precisely, its image is the unit square, whose dimension is 2 in any definition of dimension; its graph is a compact set homeomorphic to the closed unit interval, with Hausdorff dimension 2).
-H_{n} is the nth approximation to the limiting curve. The Euclidean length of H_{n} is 2^{n} - (1/(2^{n})), i.e., it grows exponentially with n, while at the same time always being bounded by a square with a finite area.
|Back to Hilbert R-Trees|
-There are two types of Hilbert R-trees, one for static databases, and one for dynamic databases. In both cases Hilbert space-filling curves are used to achieve better ordering of multidimensional objects in the node. This ordering has to be 'good', in the sense that it should group 'similar' data rectangles together, to minimize the area and perimeter of the resulting minimum bounding rectangles (MBRs). Packed Hilbert R-trees are suitable for static databases in which updates are very rare or in which there are no updates at all.
-The dynamic Hilbert R-tree is suitable for dynamic databases where insertions, deletions, or updates may occur in real time. Moreover, dynamic Hilbert R-trees employ flexible deferred splitting mechanism to increase the space utilization. Every node has a well defined set of sibling nodes. By adjusting the split policy the Hilbert R-tree can achieve a degree of space utilization as high as is desired. This is done by proposing an ordering on the R-tree nodes. The Hilbert R-tree sorts rectangles according to the Hilbert value of the center of the rectangles (i.e., MBR). (The Hilbert value of a point is the length of the Hilbert curve from the origin to the point.) Given the ordering, every node has a well-defined set of sibling nodes; thus, deferred splitting can be used. By adjusting the split policy, the Hilbert R-tree can achieve as high utilization as desired. To the contrary, other R-tree variants have no control over the space utilization.
space partitioning or binary space partitioning data structure. a BSP tree which segregates data in a metric space by choosing a position in the space (the "vantage point") and dividing the data points into two partitions: those that are nearer to the vantage point than a threshold, and those that are not. By repeatedly applying this procedure to partition the data into smaller and smaller sets, a tree data structure is created where neighbors in the tree are likely to be neighbors in the space.[1]
-One of its declination is called the multi-vantage point tree, or MVP tree: an abstract data structure for indexing objects from large metric spaces for similarity search queries. It uses more than one point to partition each level
-The way a VP tree stores data can be represented by a circle.[5] First, understand that each node of this tree contains an input point and a radius. All the left children of a given node are the points inside the circle and all the right children of a given node are outside of the circle. The tree itself does not need to know any other information about what is being stored. All it needs is the distance function that satisfies the properties of the metric space.[5] Just imagine a circle with a radius. The left children are all located inside the circle and the right children are located outside the circle.
|Advantages|
1.Instead of inferring multidimensional points for domain before the index being built, we build the index directly based on the distance.[5] Doing this, avoids pre-processing steps.
2.Updating a VP tree is relatively easy compared to the fast-map approach. For fast maps, after inserting or deleting data, there will come a time when fast-map will have to rescan itself. That takes up too much time and it is unclear to know when the rescanning will start.
3.Distance based methods are flexible. It is "able to index objects that are represented as feature vectors of a fixed number of dimensions.
space partitioning or binary space partitioning data structure. a partitioning data structure similar to that of bounding volume hierarchies or kd-trees. Bounding interval hierarchies can be used in high performance (or real-time) ray tracing and may be especially useful for dynamic scenes.
-The BIH was first presented under the name of SKD-Trees,[1] presented by Ooi et al., and BoxTrees,[2] independently invented by Zachmann.
Bounding interval hierarchies (BIH) exhibit many of the properties of both bounding volume hierarchies (BVH) and kd-trees. Whereas the construction and storage of BIH is comparable to that of BVH, the traversal of BIH resemble that of kd-trees. Furthermore, BIH are also binary trees just like kd-trees (and in fact their superset, BSP trees). Finally, BIH are axis-aligned as are its ancestors. Although a more general non-axis-aligned implementation of the BIH should be possible (similar to the BSP-tree, which uses unaligned planes), it would almost certainly be less desirable due to decreased numerical stability and an increase in the complexity of ray traversal.
-The key feature of the BIH is the storage of 2 planes per node (as opposed to 1 for the kd tree and 6 for an axis aligned bounding box hierarchy), which allows for overlapping children (just like a BVH), but at the same time featuring an order on the children along one dimension/axis (as it is the case for kd trees).
-It is also possible to just use the BIH data structure for the construction phase but traverse the tree in a way a traditional axis aligned bounding box hierarchy does. This enables some simple speed up optimizations for large ray bundles [3] while keeping memory/cache usage low.
-Some general attributes of bounding interval hierarchies (and techniques related to BIH) as described by [4] are:
Very fast construction times
Low memory footprint
Simple and fast traversal
Very simple construction and traversal algorithms
High numerical precision during construction and traversal
Flatter tree structure (decreased tree depth) compared to kd-trees
space partitioning or binary space partitioning data structure. BSP is a method for recursively subdividing a space into convex sets (the region such that, for every pair of points within the region, every point on the straight line segment that joins the pair of points is also within the region) by hyperplanes (a subspace of one dimension less than its ambient space). This subdivision gives rise to a representation of objects within the space by means of a tree data structure known as a BSP tree.
-Binary space partitioning was developed in the context of 3D computer graphics,[1][2] where the structure of a BSP tree allows spatial information about the objects in a scene that is useful in rendering, such as their ordering from front-to-back with respect to a viewer at a given location, to be accessed rapidly. Other applications include performing geometrical operations with shapes (constructive solid geometry) in CAD,[3] collision detection in robotics and 3-D video games, ray tracing and other computer applications that involve handling of complex spatial scenes.
-Binary space partitioning is a generic process of recursively dividing a scene into two until the partitioning satisfies one or more requirements. It can be seen as a generalisation of other spatial tree structures such as k-d trees and quadtrees, one where hyperplanes that partition the space may have any orientation, rather than being aligned with the coordinate axes as they are in k-d trees or quadtrees. When used in computer graphics to render scenes composed of planar polygons, the partitioning planes are frequently (but not always) chosen to coincide with the planes defined by polygons in the scene.
-The specific choice of partitioning plane and criterion for terminating the partitioning process varies depending on the purpose of the BSP tree. For example, in computer graphics rendering, the scene is divided until each node of the BSP tree contains only polygons that can render in arbitrary order. When back-face culling is used, each node therefore contains a convex set of polygons, whereas when rendering double-sided polygons, each node of the BSP tree contains only polygons in a single plane. In collision detection or ray tracing, a scene may be divided up into primitives on which collision or ray intersection tests are straightforward.
-Binary space partitioning arose from the computer graphics need to rapidly draw three-dimensional scenes composed of polygons. A simple way to draw such scenes is the painter's algorithm, which produces polygons in order of distance from the viewer, back to front, painting over the background and previous polygons with each closer object. This approach has two disadvantages: time required to sort polygons in back to front order, and the possibility of errors in overlapping polygons. Fuchs and co-authors[2] showed that constructing a BSP tree solved both of these problems by providing a rapid method of sorting polygons with respect to a given viewpoint (linear in the number of polygons in the scene) and by subdividing overlapping polygons to avoid errors that can occur with the painter's algorithm. A disadvantage of binary space partitioning is that generating a BSP tree can be time-consuming. Typically, it is therefore performed once on static geometry, as a pre-calculation step, prior to rendering or other realtime operations on a scene. The expense of constructing a BSP tree makes it difficult and inefficient to directly implement moving objects into a tree.
-BSP trees are often used by 3D video games, particularly first-person shooters and those with indoor environments. Game engines utilising BSP trees include the Doom engine (probably the earliest game to use a BSP data structure was Doom), the Quake engine and its descendants. In video games, BSP trees containing the static geometry of a scene are often used together with a Z-buffer, to correctly merge movable objects such as doors and characters onto the background scene. While binary space partitioning provides a convenient way to store and retrieve spatial information about polygons in a scene, it does not solve the problem of visible surface determination.
-The canonical use of a BSP tree is for rendering polygons (that are double-sided, that is, without back-face culling) with the painter's algorithm. Each polygon is designated with a front side and a back side which could be chosen arbitrarily and only affects the structure of the tree but not the required result
space partitioning or binary space partitioning data structure. an algorithm designed to efficiently search nonconvex, high-dimensional spaces by randomly building a space-filling tree. The tree is constructed incrementally from samples drawn randomly from the search space and is inherently biased to grow towards large unsearched areas of the problem. RRTs were developed by Steven M. LaValle and James J. Kuffner Jr. [1] .[2] They easily handle problems with obstacles and differential constraints (nonholonomic and kinodynamic) and have been widely used in autonomous robotic path planning.
-RRTs can be viewed as a technique to generate open-loop trajectories for nonlinear systems with state constraints. An RRT can also be considered as a Monte-Carlo method to bias search into the largest Voronoi regions of a graph in a configuration space. Some variations can even be considered stochastic fractals.
-An RRT grows a tree rooted at the starting configuration by using random samples from the search space. As each sample is drawn, a connection is attempted between it and the nearest state in the tree. If the connection is feasible (passes entirely through free space and obeys any constraints), this results in the addition of the new state to the tree. With uniform sampling of the search space, the probability of expanding an existing state is proportional to the size of its Voronoi region. As the largest Voronoi regions belong to the states on the frontier of the search, this means that the tree preferentially expands towards large unsearched areas.
-The length of the connection between the tree and a new state is frequently limited by a growth factor. If the random sample is further from its nearest state in the tree than this limit allows, a new state at the maximum distance from the tree along the line to the random sample is used instead of the random sample itself. The random samples can then be viewed as controlling the direction of the tree growth while the growth factor determines its rate. This maintains the space-filling bias of the RRT while limiting the size of the incremental growth.
-RRT growth can be biased by increasing the probability of sampling states from a specific area. Most practical implementations of RRTs make use of this to guide the search towards the planning problem goals. This is accomplished by introducing a small probability of sampling the goal to the state sampling procedure. The higher this probability, the more greedily the tree grows towards the goal.
Application-specific tree data structure. a specialized variation of a minimax game tree for use in artificial intelligence systems that play two-player zero-sum games such as backgammon, in which the outcome depends on a combination of the player's skill and chance elements such as dice rolls. In addition to "min" and "max" nodes of the traditional minimax tree, this variant has "chance" ("move by nature") nodes, which take the expected value of a random event occurring.[1] In game theory terms, an expectiminimax tree is the game tree of an extensive-form game of perfect, but incomplete information.
-In the traditional minimax method, the levels of the tree alternate from max to min until the depth limit of the tree has been reached. In an expectiminimax tree, the "chance" nodes are interleaved with the max and min nodes. Instead of taking the max or min of the utility values of their children, chance nodes take a weighted average, with the weight being the probability that that child is reached.[1]
-The interleaving depends on the game. Each "turn" of the game is evaluated as a "max" node (representing the AI player's turn), a "min" node (representing a potentially-optimal opponent's turn), or a "chance" node (representing a random effect or player).[1]
-For example, consider a game in which each round consists of a single dice throw, and then decisions made by first the AI player, and then another intelligent opponent. The order of nodes in this game would alternate between "chance", "max" and then "min"
Application-specific tree data structure. a purely functional data structure used in efficiently implementing other functional data structures. A finger tree gives amortized constant time access to the "fingers" (leaves) of the tree, where data is stored, and also stores in each internal node the result of applying some associative operation to its descendants. This "summary" data stored in the internal nodes can be used to provide the functionality of data structures other than trees. For example, a priority queue can be implemented by labeling the internal nodes by the minimum priority of its children in the tree, or an indexed list/array can be implemented with a labeling of nodes by the count of the leaves in their children.
-Finger trees can provide amortized O(1) pushing, reversing, popping, O(log n) append and split; and can be adapted to be indexed or ordered sequences. And like all functional data structures, it is inherently persistent; that is, older versions of the tree are always preserved.
-They have since been used in the Haskell core libraries (in the implementation of Data.Sequence), and an implementation in OCaml exists[1] which was derived from a proven-correct Coq specification;[2] and a C# implementation of finger trees was published in 2008; the Yi text editor specializes finger trees to finger strings for efficient storage of buffer text. Finger trees can be implemented with or without lazy evaluation,[3] but laziness allows for simpler implementations.
-They were first published in 1977 by Leonidas J. Guibas,[4] and periodically refined since (e.g. a version using AVL trees,[5] non-lazy finger trees, simpler 2-3 finger trees,[6] B-Trees and so on)
Application-specific tree data structure. a data structure with performance characteristics that make it attractive for providing indexed access to files with high insert volume, such as transactional log data. LSM trees, like other search trees, maintain key-value pairs. LSM trees maintain data in two or more separate structures, each of which is optimized for its respective underlying storage medium; data is synchronized between the two structures efficiently, in batches.
-One simple version of the LSM tree is a two-level LSM tree.[1] As described by Patrick O'Neil, a two-level LSM tree comprises two tree-like structures, called C0 and C1. C0 is smaller and entirely resident in memory, whereas C1 is resident on disk. New records are inserted into the memory-resident C0 component. If the insertion causes the C0 component to exceed a certain size threshold, a contiguous segment of entries is removed from C0 and merged into C1 on disk. The performance characteristics of LSM trees stem from the fact that each component is tuned to the characteristics of its underlying storage medium, and that data is efficiently migrated across media in rolling batches, using an algorithm reminiscent of merge sort.
-Most LSM trees used in practice employ multiple levels. Level 0 is kept in main memory, and might be represented using a tree. The on-disk data is organized into sorted runs of data. Each run contains data sorted by the index key. A run can be represented on disk as a single file, or alternatively as a collection of files with non-overlapping key ranges. To perform a query on a particular key to get its associated value, one must search in the Level 0 tree, as well as each run.
-A particular key may appear in several runs, and what happens depends on the application. Some applications simply want the newest key-value pair with a given key. Some applications must combine the values in some way to get the proper aggregate value to return. For example, in Apache Cassandra, each value represents a row in a database, and different versions of the row may have different sets of columns.[2]
-In order to keep down the cost of queries, the system must avoid a situation where there are too many runs.
-Extensions to the 'levelled' method to incorporate B+ structures have been suggested, for example bLSM[3] and Diff-Index.[4]
-LSM trees are used in database management systems such as BigTable, HBase, LevelDB, MongoDB, SQLite4, RocksDB, WiredTiger,[5] Apache Cassandra, and InfluxDB
Hash data structure. a space-efficient probabilistic data structure, conceived by Burton Howard Bloom in 1970, that is used to test whether an element is a member of a set. False positive matches are possible, but false negatives are not, thus a Bloom filter has a 100% recall rate. In other words, a query returns either "possibly in set" or "definitely not in set". Elements can be added to the set, but not removed (though this can be addressed with a "counting" filter). The more elements that are added to the set, the larger the probability of false positives.

Bloom proposed the technique for applications where the amount of source data would require an impractically large amount of memory if "conventional" error-free hashing techniques were applied. He gave the example of a hyphenation algorithm for a dictionary of 500,000 words, out of which 90% follow simple hyphenation rules, but the remaining 10% require expensive disk accesses to retrieve specific hyphenation patterns. With sufficient core memory, an error-free hash could be used to eliminate all unnecessary disk accesses; on the other hand, with limited core memory, Bloom's technique uses a smaller hash area but still eliminates most unnecessary accesses. For example, a hash area only 15% of the size needed by an ideal error-free hash still eliminates 85% of the disk accesses, an 85-15 form of the Pareto principle.[1]

More generally, fewer than 10 bits per element are required for a 1% false positive probability, independent of the size or number of elements in the set.
Hash data structure. a computer programming technique used in hash tables to resolve hash collisions (a situation that occurs when two distinct pieces of data have the same hash value, checksum, fingerprint, or cryptographic digest. Collisions are unavoidable whenever members of a very large set (such as all possible person names, or all possible computer files) are mapped to a relatively short bit string. This is merely an instance of the pigeonhole principle, which states that if n items are put into m containers, with n > m, then at least one container must contain more than one item), in cases when two different values to be searched for produce the same hash key. It is a popular collision-resolution technique in open-addressed hash tables. Double hashing is implemented in many popular libraries.
-Like linear probing, it uses one hash value as a starting point and then repeatedly steps forward an interval until the desired value is located, an empty location is reached, or the entire table has been searched; but this interval is decided using a second, independent hash function (hence the name double hashing). Unlike linear probing and quadratic probing, the interval depends on the data, so that even values mapping to the same location have different bucket sequences; this minimizes repeated collisions and the effects of clustering.
-Given two randomly, uniformly, and independently selected hash functions h_{1} and h_{2}, the ith location in the bucket sequence for value k in a hash table T is: h(i,k)=(h_1(k) + i * h_2(k)) mod |T|. Generally, h_{1} and h_{2} are selected from a set of universal hash functions.
Hash data structure. typically a list of hashes of the data blocks in a file or set of files. Lists of hashes are used for many different purposes, such as fast table lookup (hash tables) and distributed databases (distributed hash tables). This article covers hash lists that are used to guarantee data integrity.
-A hash list is an extension of the old concept of hashing an item (for instance, a file). A hash list is usually sufficient for most needs, but a more advanced form of the concept is a hash tree.
-Hash lists can be used to protect any kind of data stored, handled and transferred in and between computers. An important use of hash lists is to make sure that data blocks received from other peers in a peer-to-peer network are received undamaged and unaltered, and to check that the other peers do not "lie" and send fake blocks.
-Usually a cryptographic hash function such as SHA-256 is used for the hashing. If the hash list only needs to protect against unintentional damage less secure checksums such as CRCs can be used.
-Hash lists are better than a simple hash of the entire file since, in the case of a data block being damaged, this is noticed, and only the damaged block needs to be redownloaded. With only a hash of the file, many undamaged blocks would have to be redownloaded, and the file reconstructed and tested until the correct hash of the entire file is obtained. Hash lists also protect against nodes that try to sabotage by sending fake blocks, since in such a case the damaged block can be acquired from some other source.
Hash data structure. introduced by Bender et al. in 2011. the quotient filter is a space-efficient probabilistic data structure used to test whether an element is a member of a set (an approximate member query filter, AMQ). A query will elicit a reply specifying either that the element is definitely not in the set or that the element is probably in the set. The former result is definitive; i.e., the test does not generate false negatives. But with the latter result there is some probability, ε, of the test returning "element is in the set" when in fact the element is not present in the set (i.e., a false positive). There is a tradeoff between ε, the false positive rate, and storage size; increasing the filter's storage size reduces ε. Other AMQ operations include "insert" and "optionally delete". The more elements are added to the set, the larger the probability of false positives.
-An approximate member query (AMQ) filter used to speed up answers in a key-value storage system. Key-value pairs are stored on a disk which has slow access times. AMQ filter decisions are much faster. However some unnecessary disk accesses are made when the filter reports a positive (in order to weed out the false positives). Overall answer speed is better with the AMQ filter than without it. Use of an AMQ filter for this purpose, however, does increase memory usage.
A typical application for quotient filters, and other AMQ filters, is to serve as a proxy for the keys in a database on disk. As keys are added to or removed from the database, the filter is updated to reflect this. Any lookup will first consult the fast quotient filter, then look in the (presumably much slower) database only if the quotient filter reported the presence of the key. If the filter returns absence, the key is known not to be in the database without any disk accesses having been performed.
-A quotient filter has the usual AMQ operations of insert and query. In addition it can also be merged and re-sized without having to re-hash the original keys (thereby avoiding the need to access those keys from secondary storage). This property benefits certain kinds of log-structured merge-trees.
Graph data structure. a collection of unordered lists used to represent a finite graph. Each list describes the set of neighbors of a vertex in the graph. This is one of several commonly used representations of graphs for use in computer programs.
|Data Structures|
For use as a data structure, the main alternative to the adjacency list is the adjacency matrix. Because each entry in the adjacency matrix requires only one bit, it can be represented in a very compact way, occupying only |V|^2/8 bytes of contiguous space, where | V| is the number of vertices of the graph. Besides avoiding wasted space, this compactness encourages locality of reference.
-However, for a sparse graph, adjacency lists require less space, because they do not waste any space to represent edges that are not present. Using a naïve array implementation on a 32-bit computer, an adjacency list for an undirected graph requires about 2(32/8)| E| = 8| E| bytes of space, where | E| is the number of edges of the graph.
-Noting that an undirected simple graph can have at most | V|^2/2 edges, allowing loops, we can let d = |E|/| V|^2 denote the density of the graph. Then, 8| E | > | V |^2/8 when | E|/| V|^2 > 1/64, that is the adjacency list representation occupies more space than the adjacency matrix representation when d > 1/64. Thus a graph must be sparse enough to justify an adjacency list representation.
-Besides the space trade-off, the different data structures also facilitate different operations. Finding all vertices adjacent to a given vertex in an adjacency list is as simple as reading the list. With an adjacency matrix, an entire row must instead be scanned, which takes O(| V |) time. Whether there is an edge between two given vertices can be determined at once with an adjacency matrix, while requiring time proportional to the minimum degree of the two vertices with the adjacency list.
Graph data structure. a directed acyclic graph (DAG) where each directed path represents a stack.
-The graph-structured stack is an essential part of Tomita's algorithm, where it replaces the usual stack of a pushdown automaton. This allows the algorithm to encode the nondeterministic choices in parsing an ambiguous grammar, sometimes with greater efficiency.
|Tomita's algorithm|
-A GLR parser (GLR standing for "generalized LR", where L stands for "left-to-right" and R stands for "rightmost (derivation)") is an extension of an LR parser algorithm to handle nondeterministic and ambiguous grammars. The theoretical foundation was provided in a 1974 paper[1] by Bernard Lang (along with other general Context-Free parsers such as GLL). It describes a systematic way to produce such algorithms, and provides uniform results regarding correctness proofs, complexity with respect to grammar classes, and optimization techniques. The first actual implementation of GLR was described in a 1984 paper by Masaru Tomita, it has also been referred to as a "parallel parser". Tomita presented five stages in his original work,[2] though in practice it is the second stage that is recognized as the GLR parser.
-Though the algorithm has evolved since its original forms, the principles have remained intact. As shown by an earlier publication,[3] Lang was primarily interested in more easily used and more flexible parsers for extensible programming languages. Tomita's goal was to parse natural language text thoroughly and efficiently. Standard LR parsers cannot accommodate the nondeterministic and ambiguous nature of natural language, and the GLR algorithm can.
|Advantages of GLR|
Recognition using the GLR algorithm has the same worst-case time complexity as the CYK algorithm and Earley algorithm: O(n3).[citation needed] However, GLR carries two additional advantages:
1.The time required to run the algorithm is proportional to the degree of nondeterminism in the grammar: on deterministic grammars the GLR algorithm runs in O(n) time (this is not true of the Earley[citation needed] and CYK algorithms, but the original Earley algorithms can be modified to ensure it)
2.The GLR algorithm is "online" - that is, it consumes the input tokens in a specific order and performs as much work as possible after consuming each token.
-In practice, the grammars of most programming languages are deterministic or "nearly deterministic", meaning that any nondeterminism is usually resolved within a small (though possibly unbounded) number of tokens. Compared to other algorithms capable of handling the full class of context-free grammars (such as Earley or CYK), the GLR algorithm gives better performance on these "nearly deterministic" grammars, because only a single stack will be active during the majority of the parsing process.
-GLR can be combined with the LALR(1) algorithm, in a hybrid parser, allowing still higher performance
Graph data structure. a general data structure commonly used by vector-based graphics editing applications and modern computer games, which arranges the logical and often (but not necessarily) spatial representation of a graphical scene. Examples of such programs include Acrobat 3D, Adobe Illustrator, AutoCAD, CorelDRAW, OpenSceneGraph, OpenSG, VRML97, X3D, Hoops and Open Inventor.
-A scene graph is a collection of nodes in a graph or tree structure. A tree node (in the overall tree structure of the scene graph) may have many children but often only a single parent, with the effect of a parent applied to all its child nodes; an operation performed on a group automatically propagates its effect to all of its members. In many programs, associating a geometrical transformation matrix (see also transformation and matrix) at each group level and concatenating such matrices together is an efficient and natural way to process such operations. A common feature, for instance, is the ability to group related shapes/objects into a compound object that can then be moved, transformed, selected, etc. as easily as a single object.
-It also happens that in some scene graphs, a node can have a relation to any node including itself, or at least an extension that refers to another node (for instance Pixar's PhotoRealistic RenderMan because of its usage of Reyes rendering algorithm, or Adobe Systems's Acrobat 3D for advanced interactive manipulation).
-The term scene graph is sometimes confused with Canvas (GUI), since some canvas implementations include scene graph functionality.
Graph data structure. a data structure that is used to represent a Boolean function. On a more abstract level, BDDs can be considered as a compressed representation of sets or relations. Unlike other compressed representations, operations are performed directly on the compressed representation, i.e. without decompression. Other data structures used to represent a Boolean function include negation normal form (NNF), and propositional directed acyclic graph (PDAG).
|Definition|
A Boolean function can be represented as a rooted, directed, acyclic graph, which consists of several decision nodes and terminal nodes. There are two types of terminal nodes called 0-terminal and 1-terminal. Each decision node N is labeled by Boolean variable V_N and has two child nodes called low child and high child. The edge from node V_N to a low (or high) child represents an assignment of V_N to 0 (resp. 1). Such a BDD is called 'ordered' if different variables appear in the same order on all paths from the root. A BDD is said to be 'reduced' if the following two rules have been applied to its graph:
1.Merge any isomorphic subgraphs.
2.Eliminate any node whose two children are isomorphic.
-In popular usage, the term BDD almost always refers to Reduced Ordered Binary Decision Diagram (ROBDD in the literature, used when the ordering and reduction aspects need to be emphasized). The advantage of an ROBDD is that it is canonical (unique) for a particular function and variable order.[1] This property makes it useful in functional equivalence checking and other operations like functional technology mapping.
-A path from the root node to the 1-terminal represents a (possibly partial) variable assignment for which the represented Boolean function is true. As the path descends to a low (or high) child from a node, then that node's variable is assigned to 0 (resp. 1).
Graph data structure. a generalization of a graph in which an edge can connect any number of vertices. Formally, a hypergraph {\displaystyle H} H is a pair {\displaystyle H=(X,E)} H = (X,E) where {\displaystyle X} X is a set of elements called nodes or vertices, and {\displaystyle E} E is a set of non-empty subsets of {\displaystyle X} X called hyperedges or edges. Therefore, {\displaystyle E} E is a subset of {\displaystyle {\mathcal {P}}(X)\setminus \{\emptyset \}} \mathcal{P}(X) \setminus\{\emptyset\}, where {\displaystyle {\mathcal {P}}(X)} {\mathcal {P}}(X) is the power set of {\displaystyle X} X.
-While graph edges are pairs of nodes, hyperedges are arbitrary sets of nodes, and can therefore contain an arbitrary number of nodes. However, it is often desirable to study hypergraphs where all hyperedges have the same cardinality; a k-uniform hypergraph is a hypergraph such that all its hyperedges have size k. (In other words, one such hypergraph is a collection of sets, each such set a hyperedge connecting k nodes.) So a 2-uniform hypergraph is a graph, a 3-uniform hypergraph is a collection of unordered triples, and so on.
-A hypergraph is also called a set system or a family of sets drawn from the universal set X. The difference between a set system and a hypergraph is in the questions being asked. Hypergraph theory tends to concern questions similar to those of graph theory, such as connectivity and colorability, while the theory of set systems tends to ask non-graph-theoretical questions, such as those of Sperner theory.
-There are variant definitions; sometimes edges must not be empty, and sometimes multiple edges, with the same set of nodes, are allowed.
-Hypergraphs can be viewed as incidence structures. In particular, there is a bipartite "incidence graph" or "Levi graph" corresponding to every hypergraph, and conversely, most, but not all, bipartite graphs can be regarded as incidence graphs of hypergraphs.
-Hypergraphs have many other names. In computational geometry, a hypergraph may sometimes be called a range space and then the hyperedges are called ranges.[1] In cooperative game theory, hypergraphs are called simple games (voting games); this notion is applied to solve problems in social choice theory. In some literature edges are referred to as hyperlinks or connectors.[2]
-Special kinds of hypergraphs include, besides k-uniform ones, clutters, where no edge appears as a subset of another edge; and abstract simplicial complexes, which contain all subsets of every edge.
-The collection of hypergraphs is a category with hypergraph homomorphisms as morphisms.
Other data structure. a computer representation of the topology of a two-dimensional or three-dimensional map, that is, a graph drawn on a (closed) surface.
The quad-edge data structure:
1.represents simultaneously both the map, its dual and mirror image.
2.can represent the most general form of a map, admitting vertices and faces of degree 1 and 2.
3.is a variant of the earlier winged edge data structure.
-The fundamental idea behind the quad-edge structure is the recognition that a single edge, in a closed polygonal mesh topology, sits between exactly two faces and exactly two vertices. Thus, it can represent a dual of the graph simply by reversing the convention on what is a vertex and what is a face.
|Uses|
Much like Winged Edge, quad-edge structures are used in programs to store the topology of a 2D or 3D polygonal mesh. The mesh itself does not need to be closed in order to form a valid quad-edge structure.
-Using a quad-edge structure, iterating through the topology is quite easy. Often, the interface to quad-edge topologies is through directed edges. This allows the two vertices to have explicit names (start and end), and this gives faces explicit names as well (left and right, relative to a person standing on start and looking in the direction of end). The four edges are also given names, based on the vertices and faces: start-left, start-right, end-left, and end-right. A directed edge can be reversed to generate the edge in the opposite direction.
-Iterating around a particular face only requires having a single directed edge to which that face is on the left (by convention) and then walking through all of the start-left edges until the original edge is reached.
Other data structure. a data structure used by a language translator such as a compiler or interpreter, where each identifier in a program's source code is associated with information relating to its declaration or appearance in the source.
|Implementation|
A common implementation technique is to use a hash table. A compiler may use one large symbol table for all symbols or use separated, hierarchical symbol tables for different scopes. There are also trees, linear lists and self-organizing lists which can be used to implement a symbol table. It also simplifies the classification of literals in tabular format. The symbol table is accessed by most phases of a compiler, beginning with the lexical analysis to optimization.
|Uses|
An object file will contain a symbol table of the identifiers it contains that are externally visible. During the linking of different object files, a linker will use these symbol tables to resolve any unresolved references.
-A symbol table may only exist during the translation process, or it may be embedded in the output of that process for later exploitation, for example, during an interactive debugging session, or as a resource for formatting a diagnostic report during or after execution of a program.
-While reverse engineering an executable, many tools refer to the symbol table to check what addresses have been assigned to global variables and known functions. If the symbol table has been stripped or cleaned out before being converted into an executable, tools will find it harder to determine addresses or understand anything about the program.
-At that time of accessing variables and allocating memory dynamically, a compiler should perform many works and as such the extended stack model requires the symbol table.