Line-Constrained $k$-Median, $k$-Means, and $k$-Center Problems in the Plane

Abstract

The (weighted) $k$-median, $k$-means, and $k$-center problems in the plane are known to be NP-hard. In this paper, we study these problems with an additional constraint that requires the sought $k$ facilities to be on a given line. We present efficient algorithms for various distance metrics such as $L_1,L_2,L_{\infty }$. Assume all $n$ weighted points are given sorted by their projections on the given line. For $k$-median, our algorithms for $L_1$ and $L_{\infty }$ metrics run in $O(\min \{nk,n\sqrt{k\log n}\log n, n2^{O(\sqrt{\log k\log \log n})}\log n\})$ time and $O(\min \{nk\log n,n\sqrt{k\log n}\log ^2 n, n2^{O(\sqrt{\log k\log \log n})}\log ^2 n\})$ time, respectively. For $k$-means, which is defined only on the $L_2$ metric, we give an $O(\min \{nk,n\sqrt{k\log n}, n2^{O(\sqrt{\log k\log \log n})}\})$ time algorithm. For $k$-center, our algorithms run in $O(n\log n)$ time for all three metrics, and in $O(n)$ time for the unweighted version under $L_1$ and $L_{\infty }$ metrics.

This research was supported in part by NSF under Grant CCF-1317143.