A Practical Introduction to Data Structures and Algorithm Analysis

320 Chap. 9 Searchingelement in L, that is, we check elements L[j], L[2j], and so on. So long as K isgreater than the values we are checking, we continue on. But when we reach avalue in L greater than K, we do a linear search on the piece of length j − 1 thatwe know brackets K if it is in the list.If mj ≤ n < (m+1)j, then the total cost of this algorithm is at most m+j −13-way comparisons. Therefore, the cost to run the algorithm on n items with a jumpof size j is⌊ ⌋ nT(n, j) = m + j − 1 = + j − 1.jWhat is the best value that we can pick for j? We want to minimize the cost:{⌊ ⌋ }nmin + j − 11≤j≤n jTake the derivative and solve for f ′ (j) = 0 to find the minimum, which isj = √ n. In this case, the worst case cost will be roughly 2 √ n.This example teaches us some lessons about algorithm design. We want tobalance the work done while selecting a sublist with the work done while searchinga sublist. In general, it is a good strategy to make subproblems of equal effort. Thisis an example of a divide and conquer algorithm.What if we extend this idea to three levels? We would first make jumps ofsome size j to find a sublist of size j − 1 whose end values bracket value K. Wwwould then work through this sublist by making jumps of some smaller size, sayj 1 . Finally, once we find a bracketed sublist of size j 1 − 1, we would do sequentialsearch to complete the process.This probably sounds convoluted to do two levels of jumping to be followed bya sequential search. While it might make sense to do a two-level algorithm (that is,jump search jumps to find a sublist and then does sequential search on the sublist),it almost never seems to make sense to do a three-level algorithm. Instead, whenwe go beyond two levels, we nearly always generalize by using recursion. Thisleads us to the most commonly used search algorithm for sorted arrays, the binarysearch described in Section 3.5.If we know nothing about the distribution of key values, then binary searchis the best algorithm available for searching a sorted array (see Exercise 9.22).However, sometimes we do know something about the expected key distribution.Consider the typical behavior of a person looking up a word in a large dictionary.Most people certainly do not use sequential search! Typically, people use a modifiedform of binary search, at least until they get close to the word that they are