Question

Suppose that I have an array of random reals, how do I the indices pair (i, j) where j<i and

maximise (a[j] - a[i]) in O(n ln(n)) time

n ln n sugguest recursions. So I was thinking of sorting the array in n ln n sort. but then the min and mix of the sorted array might not satisfy j<i

The difference a[j]-a[i] depends on all i , j so scanning all the possible permutation is O(n^2). Does anybody have a suggestion on how I can divide the problem space?

Answer 1

If I understand you correctly, you want to find max(a[j] - a[i]) among all pairs (i, j) where j < i.

You can do it in O(n) without much problem.

For each index i, to maximize a[j] - a[i] expression we need to find max(a[j]) on interval [0 .. i - 1]. So, let’s move from left to right (increase i) and keep current maximum value of a[j].

int maxa = a[0];
for (int i = 1; i < n; ++i) {
    int current = maxa - a[i];
    if (current > best) {
        best = current;
    }
    maxa = max(maxa, a[i]);
}