The American Museum of Natural History in New York City contains more than 32 million specimens and artifacts in its various collections, including the world’s largest collection of dinosaur fossils. Many of these are in storage away from public view, but all must be carefully inventoried. a. Suppose the inventory is unordered (!) and a sequential search is done to locate a specific artifact. Given that the search is executed on a computer that can do 12,000 comparisons per second, about how much time on the average would the search require? b. Assuming the inventory is sorted, about how much time would a binary search require?

a. Use Gauss's approach to find the following sum: $2 + 4 + 6 + \ldots + 100$ b. Use Gauss's approach to find a formula for the sum of the even numbers from 2 to 2n: $2+4+6+\ldots+2 n$ Your formula will be an expression involving n.

Refer to the pattern-matching algorithm. a. What is the output of the algorithm as it currently stands if our text is Text: We must band together and handle adversity and we search for the pattern “and”? b. How could we modify the algorithm so that it finds only the complete word and rather than the occurrence of the character sequence a, n, and d that is contained within another word, such as band? Get values for n and m, the size of the text and the pattern, respectively. Get values for both the text $T_{1} T_{2} \dots T_{n}$ and the pattern $P_{1} P_{2} \ldots P_{m}.$ Set k, the starting location for the attempted match, to 1. While $(k \leq(n-m+1))$ do. Set the value of i to 1. Set the value of Mismatch to NO. While both $(i \leq m)$ and (Mismatch = NO) do. If $P_{i} \neq T_{k+(j-1)}$ then. Set Mismatch to yes. Else. Increment i by 1 (to move to the next character). End of the loop. If Mismatch = NO then. Print the message 'There is a match at position'. Print the value of k. Increment k by 1. End of the loop. Stop, we are finished.

Here is a list of seven names: Sherman, Jane, Ted, Elise, Raul, Maki, John Search this list for each name in turn, using sequential search and counting the number of comparisons for each name. Now take the seven comparison counts and find their average. Did you get a number that you expected? Why?

The selection sort algorithm could be modified to stop when the unsorted section of the list contains only one number, because that one number must be in the correct position. Show that this modification would have no effect on the number of comparisons required to sort an n-element list.

A tennis tournament has 342 players. A single match involves 2 players. The winner of a match will play the winner of a match in the next round, whereas losers are eliminated from the tournament. The 2 players who have won all previous rounds play in the final game, and the winner wins the tournament. What is the total number of matches needed to determine the winner? a. Here is one algorithm to answer this question. Compute 342 / 2 = 171 to get the number of pairs (matches) in the first round, which results in 171 winners to go on to the second round. Compute 171 / 2 = 85 with 1 left over, which results in 85 matches in the second round and 85 winners, plus the 1 left over, to go on to the third round. For the third round compute 86 / 2 = 43, so the third round has 43 matches, and so on. The total number of matches is 171 + 85 + 43 + . . . . Finish this process to find the total number of matches. b. Here is another algorithm to solve this problem. Each match results in exactly one loser, so there must be the same number of matches as losers in the tournament. Compute the total number of losers in the entire tournament. (Hint: This isn’t really a computation; it is a one-sentence argument.) c. What are your opinions on the relative clarity, elegance, and efficiency of the two algorithms?