On k-enclosing slab problems

초록

Given a set P of n points in Rd for d⩾2 and an integer parameter 0⩽k⩽n, a slab, the volume between two parallel hyperplanes, is called k-enclosing if it encloses k points of P. We consider the problem of finding an optimal k-enclosing slab for P with minimum or maximum width. In this paper, we present two simple algorithmic approaches based on our new characterization of optimal slabs. Our first approach yields an O(n3logn)-time algorithm in R3 and an O(ndlog2n)-time algorithm in Rd, independently of k, for any constant d⩾3. These algorithms also work properly for the weighted variant of the problems in the same time bound. By our second approach, we present faster algorithms for d⩽3 when k is relatively small or large. More specifically, our algorithms solve both the minimum and maximum problems in R3 in O((k+1)1/3n3) time for small k<n/2 and in O(min{(n−k+1)1/3n3,(n−k+1)7/3n2}) time for large k>n/2. In particular, for large k>n/2, we further show that the minimization problem in R3 can be solved even faster in O((n−k+1)3n3/2+ϵ) time. We also discuss the planar case d=2 in which the minimum and maximum problems can be solved in O(nlogn+(n−k+1)n) time for any 0⩽k⩽n.

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