Simple linear time algorithms for piercing pairwise intersecting disks

Abstract

A set $D$ of disks in the plane is said to be pierced by a point set P if each disk in $D$ contains a point of P. Any set of pairwise intersecting unit disks can be pierced by 3 points (Hadwiger and Debrunner (1955) [7]). Stachó and independently Danzer established that any set of pairwise intersecting arbitrary disks can be pierced by 4 points (Stachó (1981–1984) [16]. Danzer (1986) [4]). Existing linear-time algorithms for finding a set of 4 or 5 points that pierce pairwise intersecting disks of arbitrary radius use the LP-type problem as a subroutine. We present simple linear-time algorithms for finding 3 points for piercing pairwise intersecting unit disks, and 5 points for piercing pairwise intersecting disks of arbitrary radius. Our algorithms use simple geometric transformations and avoid heavy machinery. We also show that 3 points are sometimes necessary for piercing pairwise intersecting unit disks.