A Practical Introduction to Data Structures and Algorithm Analysis

Sec. 9.1 Searching Unsorted and Sorted Arrays 319Let p i be the probability that K is in position i of L. When K is not in L,sequential search will require n comparisons. Let p 0 be the probability that K isnot in L. Then the average cost T(n) will beT(n) = np 0 +n∑ip i .What happens to the equation if we assume all the p i ’s are equal (except p 0 )?i=1T(n) = p 0 n += p 0 n + pn∑ipi=1n∑ii=1n(n + 1)= p 0 n + p2= p 0 n + 1 − p 0nn(n + 1)2= n + 1 + p 0(n − 1)2Depending on the value of p 0 , n+12≤ T(n) ≤ n.For large collections of records that are searched repeatedly, sequential searchis unacceptably slow. One way to reduce search time is to preprocess the recordsby sorting them. Given a sorted array, an obvious improvement over simple linearsearch is to test if the current element in L is greater than K. If it is, then we knowthat K cannot appear later in the array, and we can quit the search early. But thisstill does not improve the worst-case cost of the algorithm.We can also observe that if we look first at position 1 in sorted array L and findthat K is bigger, then we rule out positions 0 as well as position 1. Because more isoften better, what if we look at position 2 in L and find that K is bigger yet? Thisrules out positions 0, 1, and 2 with one comparison. What if we carry this to theextreme and look first at the last position in L and find that K is bigger? Then weknow in one comparison that K is not in L. This is very useful to know, but what iswrong with this approach? While we learn a lot sometimes (in one comparison wemight learn that K is not in the list), usually we learn only a little bit (that the lastelement is not K).The question then becomes: What is the right amount to jump? This leads usto an algorithm known as Jump Search. For some value j, we check every j’th