longest nondecreasing subsequence in O(nlgn)

Question

I have the following algorithm which works well

I tried explaining it here for myself http://nemo.la/?p=943 and it is explained here http://www.geeksforgeeks.org/lon

Answer 1

A completely different solution to this problem is the following. Make a copy of the array and sort it. Then, compute the minimum nonzero difference between any two elements of the array (this will be the minimum nonzero difference between two adjacent array elements) and call it δ. This step takes time O(n log n).

The key observation is that if you add 0 to element 0 of the original array, δ/n to the second element of the original array, 2δ/n to the third element of the array, etc., then any nondecreasing sequence in the original array becomes a strictly increasing sequence in the new array and vice-versa. Therefore, you can transform the array this way, then run a standard longest increasing subsequence solver, which runs in time O(n log n). The net result of this process is an O(n log n) algorithm for finding the longest nondecreasing subsequence.

For example, consider 30, 20, 20, 10, 10, 10, 10. In this case δ = 10 and n = 7, so δ / n ≈ 1.42. The new array is then

40, 21.42, 22.84, 14.28, 15.71, 17.14, 18.57

Here, the LIS is 14.28, 15.71, 17.14, 18.57, which maps back to 10, 10, 10, 10 in the original array.

Hope this helps!