Following is pseudocode for an algorithm to find the maximum element in an array of integers  a) Determine a loop invariant for this algorithm  b) Use your loop invariant to prove the correctness of this algorithm    MAX(A) n= A.length  largest = - infinity  for i = 1 to n    If A[i] > largest       largest = A[i]    return largest

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.

Given an integer array A[1..n] you are to list the k smallest elements. What method will you use when (i)k=0(1) Explain why you chose the algorithm you chose for parts (i)

Let A be an array of n integers: (a) Write a pseudo code for a divide-and-conquer algorithm that computes the MAX and MIN values in array A . (b) Write the recurrence for this algorithm. (c) Solve the recurrence using the substitution method.