Isotonic Regression | Algorithms Notes

Isotonic Regression | Algorithms Notes

Given a sequence A of n integers, design an algorithm to find an non-decreasing sequence B of n integers, such that is minimized.

Solution:

This problem can be solved by minimum cost flow. Here we use dynamic programming.

Let F(k, z) be the optimal objective value for the first k elements with B[k] = z. Then, we have a recurrence relation: F(k, z) = min F(k-1, x) + |z – A[k]|. Thus, the problem becomes how to efficiently find the z minimizing F(k, z) for all fixed k. It can be showed that F(k, z) is a piecewise linear convex function in z and the minimizing value must be in A. Thus, the dynamic programming can be speed up by using slope optimization and find the solution in O(n lg n) time.

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