Covering points by disjoint boxes with outliers

Authors:
Hee-Kap Ahn;Sang Won Bae;Erik D. Demaine;Martin L. Demaine;Sang-Sub Kim;Matias Korman;Iris Reinbacher;Wanbin Son
Affiliations:
Department of Computer Science and Engineering, POSTECH, South Korea;Department of Computer Science, Kyonggi University, Suwon, South Korea;MIT Computer Science and Artificial Intelligence Laboratory, United States;MIT Computer Science and Artificial Intelligence Laboratory, United States;Department of Computer Science and Engineering, POSTECH, South Korea;Computer Science Department, Université Libre de Bruxelles (ULB), Belgium;Department of Computer Science and Engineering, POSTECH, South Korea;Department of Computer Science and Engineering, POSTECH, South Korea
Venue:
Computational Geometry: Theory and Applications
Year:
2011

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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 at least 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+klogk) time for p=1, and O(nlogn+k^plog^pk) time for p=2,3. For rectangles we get O(n+k^3) for p=1 and O(nlogn+k^2^+^plog^p^-^1k) time for p=2,3. In all cases, our algorithms use O(n) space.