Smallest k-Enclosing Rectangle Revisited

2019/03/15 by Chan, Timothy M., Har-Peled, Sariel · 2 citations
#Computational Geometry (cs.CG) #FOS: Computer and information sciences

paper · doi:10.48550/arxiv.1903.06785

Abstract

Given a set of n points in the plane, and a parameter k, we consider the problem of computing the minimum (perimeter or area) axis-aligned rectangle enclosing k points. We present the first near quadratic time algorithm for this problem, improving over the previous near-O(n^5/2)-time algorithm by Kaplan etal [KRS17]. We provide an almost matching conditional lower bound, under the assumption that (min,+)-convolution cannot be solved in truly subquadratic time. Furthermore, we present a new reduction (for either perimeter or area) that can make the time bound sensitive to k, giving near O(n k) time. We also present a near linear time (1+ε)-approximation algorithm to the minimum area of the optimal rectangle containing k points. In addition, we study related problems including the 3-sided, arbitrarily oriented, weighted, and subset sum versions of the problem.

Cited by

• A Bouquet of Results on Maximum Range Sum: General Techniques and Hardness Reductions
• Computing Largest Subsets of Points Whose Convex Hulls have Bounded Area and Diameter

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