Enclosing k points in the smallest axis parallel rectangle

Abstract

We consider the following clustering problem. Given a set S of n points in the plane, and given an integer k, n 2 < k ⩽ n we want to find the smallest axis parallel rectangle (smallest perimeter or area) that encloses exactly k points of S. We present an algorithm which runs in time O( n + k( n − k) 2) improving previous algorithms which run in time O( k 2 n) and do not perform well for larger k values. We present an algorithm to enclose k of n given points in an axis parallel box in d-dimensional space which runs in time O( dn + dk( n − k) 2( d − 1) and occupies O( dn) space. We slightly improve algorithms for other problems whose runtimes depend on k.