Consider the following basic problem. You’re given an array A consisting of n integers A[1...

Consider the following basic problem. You’re given an array A consisting of n integers A[1], A[2],… A[n]. You’d like to output a two-dimensional n-by-n array B in which B[i, j] (for i < j) contains the sum of array entries A[i] through A[j]—that is, the sum A[i]+ A[i + 1]+ …+ A[j]. (The value of array entry B[i, j] is left unspecified whenever i ≥ j, so it  doesn’t matter what is output for these values.)

Here’s a simple algorithm to solve this problem.

For i = 1, 2,…, n

   For j = i + 1, i + 2,…, n

       Add up array entries A[i] through A[j]

       Store the result in B[i, j]

   Endfor

Endfor

(a) For some function f that you should choose, give a bound of the form O(f (n)) on the running time of this algorithm on an input of size n (i.e., a bound on the number of operations performed by the algorithm).

(b) For this same function f , show that the running time of the algorithm on an input of size n is also Ω(f (n)). (This shows an asymptotically tight bound of Θ(f (n)) on the running time.)

(c) Although the algorithm you analyzed in parts (a) and (b) is the most natural way to solve the problem—after all, it just iterates through the relevant entries of the array B, filling in a value for each—it contains some highly unnecessary sources of inefficiency. Give a different algorithm to solve this problem, with an asymptotically better running time. In other words, you should design an algorithm with running time O(g(n)), where limn→∞ g(n)/f (n)= 0.