Covering Points by Disjoint Boxes with Outliers
Abstract
For a set of n points in the plane, we consider the axis--aligned (p,k)-Box Covering problem: Find p axis-aligned, pairwise-disjoint boxes that together contain n-k points. In this paper, we consider the boxes to be either squares or rectangles, and we want to minimize the area of the largest box. For general p we show that the problem is NP-hard for both squares and rectangles. For a small, fixed number p, we give algorithms that find the solution in the following running times: For squares we have O(n+k log k) time for p=1, and O(n log n+k^p log^p k time for p = 2,3. For rectangles we get O(n + k^3) for p = 1 and O(n log n+k^{2+p} log^{p-1} k) time for p = 2,3. In all cases, our algorithms use O(n) space.
Cite
@article{arxiv.0910.1643,
title = {Covering Points by Disjoint Boxes with Outliers},
author = {Hee-Kap Ahn and Sang Won Bae and Erik D. Demaine and Martin L. Demaine and Sang-Sub Kim and Matias Korman and Iris Reinbacher and Wanbin Son},
journal= {arXiv preprint arXiv:0910.1643},
year = {2010}
}
Comments
updated version: - changed problem from 'cover exactly n-k points' to 'cover at least n-k points' to avoid having non-feasible solutions. Results are unchanged. - added Proof to Lemma 11, clarified some sections - corrected typos and small errors - updated affiliations of two authors