Given two sorted sequences with m, n elements, respectively, design and analyze an efficient divide-and-conquer algorithmto find the kth element in the merge of the two sequences. The best algorithm runs in time O(log(max(m, n))).

For any input array of n 3-digit numbers, prove that Radix Sort is guaranteed to correctly sort these n numbers, with the algorithm running in O(n) time.

Given a string S[1, . . . , n] and a dictionary of strings S, return true if S[1, . . . , n] can be segmented into a space-separated sequence of one or more dictionary words. Note that the same word in the dictionary may be reused multiple times in the segmentation. Precisely define the subproblem.Provide the recurrence equation.Describe the algorithm in pseudocode to compute the optimal value.Describe the algorithm in pseudocode to print out an optimal solution.

Consider a randomized algorithm SCRAMBLE-SEARCH that works by first randomly permuting the input array and then running the deterministic linear search given above on the resulting permuted array. Letting k be the number of indices i such that A[i] =x, give the worst-case and expected running times of SCRAMBLE-SEARCH for the cases 1. Generalize your solution to handle the case in which k > in which k O and k = 1.