Introduction to Algorithms(3rd Edition)

Implement a program (recursive) that finds all possible permutations of a string (Java) Using Figure 2.2 as a model, illustrate the operation ofINSERTION-SORT on the array A- (31,41, 59, 26, 41,58) Consider the searchingproblem Input: A sequence ofn numbers A = (a1, a2, , an) and a value v Output: An index i such that v=A[i] or the special value NIL if v does not appear in A. Write pseudocode for linear search, which scans through the sequence, lookingforv. Using a loop invariant, prove thatyour algorithm is correct. Make sure that your loop invariant fulfills the three necessary properties 2.2-2 Consider sorting n numbers stored in array A by first finding the smallest element of A and exchangingit with the element in A[1]. Then find the second smallest element of A, and exchange it with A[2]. Continue in this manner for the first n-1 elements of A. Write pseudocode for this algorithm, which is known as selection sort. What loop invariant does this algorithm maintain? Why does it need to run for only the first n- 1 elements, rather than for all n elements? Give the best-case and worst-case running times of selection sort in . notation