Question

7. (10) Given an array of integers A[1..n], such that, for all i, 1 <i< n, we have |Ali]- Ali+1]| < 1. Let A[1] = and Alny such that r < y. Using the divide-and-conquer technique, describe in English algorithm to find j such that Alj] z for a given value z, xz < y. Show that your algorithm's running time is o(n) and that it is correct o(n) search an 2 8. (10) Solve the recurrence in asymptotically tight big Oh function where k > 1: k I(lа,п), t(n) —D п +

Answer 1

As HOMEWORKLIB Guidelines, I'll answer one question (first question) although two questions are given.

Divide the array of n elements into two parts: one of length 1 and another one of length (n-1). If the element in the part of size 1 is equal to z, then it's done and return the index of that element, otherwise recurs in the (n-1) element part. The algorithm can be written as follows:

FindZ (A, 1);

FindZ (Array A, j)

if (A[j] = z)

return j;

if (j = N)

Can't find Z and return;

FindZ (A,j+1);

The running time of the algorithm can be represented bu using a recurrence equation as follows:

T(n) = T(n-1) + 1 when n > 1

= 1 when n = 1

The recurrence equation can be solved as follows:

T(n) = T(n-1) + 1

= [T(n-2) + 1] + 1

= n = O(n)

The above explanation shows that the algorithm has running time of O(n).