Problem 1. (13 marks) On sorting lower bound.

a. (5 marks) Consider the Searching problem which takes as input a sorted array A with n elements and a key k, and we return the largest element in A that is less than or equal to k. From Exercise I of Assignment #1, we know that it can be solved by binary-search in O(log n) time. Show that any algorithm for solving this Searching problem that accesses the array only via comparisons (that is, only by checking whether must take time.

b. (8 marks) Given an array A of n integers, propose an algorithm that can sort array A in time, where x is the minimum integer in A and y is the maximum integer in A. Prove the running time of your algorithm and brief explain why the lower bound doesn't apply here.

A[i]