Speeding up dynamic programming in the line-constrained k-median

Abstract

In the planar k-median problem we are given a set of demand points and want to open up to k facilities as to minimize the sum of the transportation costs from each demand point to its nearest facility.In the line-constrained version the medians are required to lie on a given line.We present a new dynamic programming formulation for this problem, based on constructing a weighted DAG over a set of median candidates.We prove that, for any convex distance metric and any line, this DAG satisfies the concave Monge property.This allows us to construct efficient algorithms in L∞ and L1 and any line, while the previously known solution (Wang and Zhang, ISAAC 2014) works only for vertical lines.We also provide an asymptotically optimal O(n) solution for the case of k = 1.