Given an integer n and an array a of length n, your task is to apply the following mutation to a:     Array a mutates into a new array b of length n.     For each i from 0 to n - 1, b[i] = a[i - 1] + a[i] + a[i + 1].     If some element in the sum a[i - 1] + a[i] + a[i + 1] does not exist, it should be set to 0. For example, b[0] should be equal to 0 + a[0] + a[1].

1. Asymptotics. Given an array A of n integers, you'd like to output a two-dimensional n x n array B in which B[i, j] = max {A[i], A[i + 1],..., A[j]} for each i < j. For i j the value of B[i, j] can be left as is. for i = 1,2, η for j = i + 1, n " Compute the maximum of the entries A[i], A[i + 1], , A[j]. Store the maximum value in B[i, j]. (a) Find a function of such that the running time of the algorithm is O(f(n)), and clearly explain why. (b) For the same function f argue that the running time of the algorithm is also (f(n)). (This establishes an asymptotically tight bound (f(n)).) (c) Design and analyze a faster algorithm for this problem. You should give an algorithm with running O(g(n)), where lim→∞ g(n)/f(n) = 0.

Consider sorting n numbers stored in array A[1:n] by first finding the smallestelement of A[1:n] and exchanging it with the element in A[1]. Then find thesmallest element of A[2:n], and exchange it with A[2]. Then find the smallestelement of A[3:n], and exchange it with A[3]. Continue in this manner for thefirst n-1 elements of A. Write pseudocode for this algorithm, which is knownas selection sort. What loop invariant does this algorithm maintain? Why does itneed to run for only the first n-1 elements, rather than for all n elements? Give theworst-case running time of selection sort in big theta notation. Is the best-case runningtime any better?

it's not a programming question.