Covering Paths for Planar Point Sets

TL;DR

This paper studies the minimum number of straight segments needed to connect all points in a planar set with a covering path, providing bounds for crossing and noncrossing paths and analyzing computational complexity.

Contribution

It introduces new bounds on the number of segments for covering paths and trees, and analyzes the complexity of computing noncrossing covering paths.

Findings

Every set admits a covering path with about n/2 + O(n/log n) segments.

Noncrossing paths can be achieved with (1 - ε)n segments for small ε.

Computing a noncrossing covering path requires Ω(n log n) time.

Abstract

Given $n$ points in the plane, a \emph{covering path} is a polygonal path that visits all the points. If no three points are collinear, every covering path requires at least $n/2$ segments, and $n-1$ straight line segments obviously suffice even if the covering path is required to be noncrossing. We show that every set of $n$ points in the plane admits a (possibly self-crossi ng) covering path consisting of $n/2 +O(n/\log{n})$ straight line segments. If the path is required to be noncrossing, we prove that $(1-\eps)n$ straight line segments suffice for a small constant $\eps>0$, and we exhibit $n$-element point sets that require at least $5n/9 -O(1)$ segments in every such path. Further, the analogous question for noncrossing \emph{covering trees} is considered and similar bounds are obtained. Finally, it is shown that computing a noncrossing covering path for $n$ points in the plane requires $\Omega(n \log{n})$ time in the worst case.