Recall that if you have a sequence a0, a1, a2, ..., an, then a length k subsequence is of the form ai1, ai2, ai3, ..., aik. For example, if you had a sequence 1, 2, 3, 4, 5, 6, 7, then a subsequence could be 2, 5, 6 or 1, 2, 3, 5, 7. A monotone increasing subsequence is one where aij < aij+1 for 1 ≤ j ≤ k − 1, where k is the length of the subsequence. All this means is that each element in the subsequence is less than the next one, so 1, 2, 5, 6 is a monotone subsequence of 3, 1, 2, 7, 5, 6, but 3, 1, 7 is not because 3 > 1. Your Task: Modify the LCS algorithm to find an O(n^2) algorithm to find the longest monotonically increasing subsequence of n numbers.
