Question

Example 1:

Input: 5 4 3 2 1

Output: nil

Example 2:

Input: 5 4 3 2 6 1

Output: 0, 4 (indices)

Please suggest an algorithm to find such indices i, j that i < j and A[i] < A[j] in linear time and constant extra space. I have solved it in O(n^2) using 2 for loops.

Answer 1

Um… I would immediately make an assumption that if such i and j exist at all, then there also must exist i and j such that j == i + 1 and A[i] < A[j]. If so, the algorithm turns into a trivial single pass over the array.

In your second example that would be i = 3 and j = 4.

Indeed, let’s say we found i and j such that A[i] < A[j] and i + 1 < j. Let’s take a look at A[i + 1]. If A[i + 1] is greater than A[i], then just set j = i + 1 and we are done. Otherwise, if A[i + 1] is smaller or equal to A[i], then just set i = i + 1 and repeat. This will always lead us to a j == i + 1 pair that satisfies the A[i] < A[j] requirement.

In other words, just go over your array looking for A[i] < A[i + 1] situation. That’s all there is to it.