3-PIERCING OF d-DIMENSIONAL BOXES AND HOMOTHETIC TRIANGLES

Abstract

In the p-piercing problem, a set [Formula: see text] of nd-dimensional objects is given, and one has to compute a piercing set for [Formula: see text] of size p, if such a set exists. We consider several instances of the 3-piercing problem that admit linear or almost linear solutions: (i) If [Formula: see text] consists of axis-parallel boxes in ℛd, then a piercing triplet for [Formula: see text] can be found (if such a triplet exists) in O(n log  n) time, for 3 ≤ d ≤ 5, and in [Formula: see text] time, for d ≥ 6. Based on the solutions for 3 ≤ d ≤ 5, efficient solutions are obtained to the corresponding 3-center problem — Given a set [Formula: see text] of n points of ℛd, compute the smallest edge length λ such that [Formula: see text] can be covered by the union of three axis-parallel cubes of edge length λ. (ii) If [Formula: see text] consists of homothetic triangles in the plane, or of 4-oriented trapezoids in the plane, then a piercing triplet can be found in O(n log  n) time.