Monge arraysAn m × n array A of real numbers is a Monge array if for all i, j, k, and l su...

Monge arrays

An m × n array A of real numbers is a Monge array if for all i, j, k, and l such that 1 ≤ i<k ≤ m and 1 ≤ j<l<n, we have

A[i,j] + A[k,l] ≤ A[i,l] + A[k,j].

In other words, whenever we pick two rows and two columns of a Monge array and consider the four elements at the intersections of the rows and the columns, the sum of the upper-left and lower-right elements is less or equal to the sum of the lower-left and upper-right elements. For example, the following array is Monge:

a. Prove that an array is Monge if and only if for all i = 1,2,…, m - 1 and j = 1,2,…, n - 1, we have

A[i, j] + A[i + 1, j + 1] ≤ A[i, j + 1] + A[i + 1, j].

(Hint: For the “only if” part, use induction separately on rows and columns.)

b. The following array is not Monge. Change one element in order to make it Monge. (Hint: Use part (a).)

c. Let f(i) be the index of the column containing the leftmost minimum element of row i. Prove that f(l) ≤ f (2) ≤…<f(m) for any m × n Monge array.

d. Here is a description of a divide-and-conquer algorithm that computes the leftmost minimum element in each row of an m × n Monge array A:

Construct a submatrix A' of A consisting of the even-numbered rows of A.

Recursively determine the leftmost minimum for each row of A'. Then compute the leftmost minimum in the odd-numbered rows of A.

Explain how to compute the leftmost minimum in the odd-numbered rows of A (given that the leftmost minimum of the even-numbered rows is known) in O(m + n) time.

e. Write the recurrence describing the running time of the algorithm described in part (d). Show that its solution is O(m + n log m).