Stabbing Pairwise Intersecting Disks by Four Points

Paz Carmi; Matthew J. Katz; Pat Morin

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arXiv · 1812.06907

Stabbing Pairwise Intersecting Disks by Four Points

Paz Carmi·Matthew J. Katz·Pat Morin

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Abstract

In their seminal work, Danzer (1956, 1986) and Stach\'{o} (1981) established that every set of pairwise intersecting disks in the plane can be stabbed by four points. However, both these proofs are non-constructive, at least in the sense that they do not seem to imply an efficient algorithm for finding the stabbing points, given such a set of disks $D$. Recently, Har-Peled \etal (2018) presented a relatively simple linear-time algorithm for finding five points that stab $D$. We present an alternative proof (and the first in English) to the assertion that four points are sufficient to stab $D$. Moreover, our proof is constructive and provides a simple linear-time algorithm for finding the stabbing points. As a warmup, we present a nearly-trivial liner-time algorithm with an elementary proof for finding five points that stab $D$.

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BibTeX RIS

Paz Carmi, Matthew J. Katz, Pat Morin. 2018-12-17. Stabbing Pairwise Intersecting Disks by Four Points. https://arxiv.org/abs/1812.06907

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