An Improved Bound for Plane Covering Paths
🤖 AI Summary
This paper studies the planar covering path problem for $n$ points in the plane: find a non-self-intersecting polygonal chain such that every point lies on some segment of the chain, minimizing the number of segments. The previous best upper bound was $lceil 21n/22
ceil$. We propose a constructive geometric approach based on spatial partitioning and matching, improving the upper bound to $lceil 6n/7
ceil$—a bound potentially tight. Our proof is direct and yields a deterministic algorithm. Furthermore, we design an efficient $O(n log n)$-time implementation, enabling scalable generation of covering paths for large point sets. These results jointly advance the theoretical upper bound and algorithmic tractability of planar covering paths, strengthening the interplay between combinatorial geometry and computational geometry.
Application Category
📝 Abstract
A covering path for a finite set $P$ of points in the plane is a polygonal path such that every point of $P$ lies on a segment of the path. The vertices of the path need not be at points of $P$. A covering path is plane if its segments do not cross each other. Let $π(n)$ be the minimum number such that every set of $n$ points in the plane admits a plane covering path with at most $π(n)$ segments. We prove that $π(n)le lceil6n/7
ceil$. This improves the previous best-known upper bound of $lceil 21n/22
ceil$, due to Biniaz (SoCG 2023). Our proof is constructive and yields a simple $O(nlog n)$-time algorithm for computing a plane covering path.
Problem
Research questions and friction points this paper is trying to address.
Minimizing segments in plane covering paths
Improving upper bound for non-crossing paths
Efficient algorithm for constructing covering paths
Innovation
Methods, ideas, or system contributions that make the work stand out.
Constructs non-crossing polygonal paths
Improves segment bound to 6n/7
Provides O(n log n) time algorithm
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