Stabbing balls with line segments and polygonal paths
Alexander Neuhaus, Dennis Rohde

TL;DR
This paper investigates the problem of finding ordered line segments or polygonal paths intersecting a sequence of balls in high-dimensional space, providing decision algorithms, a randomized approximation scheme, and experimental validation.
Contribution
It introduces a deterministic decision algorithm for ordered stabbing of balls, a randomized approximation algorithm for a relaxed problem, and demonstrates practical applicability through experiments.
Findings
Deterministic algorithm decides ordered stabbing in $O(n^{4d-2} \log n)$ time.
Randomized algorithm approximates minimum-link paths with factor 2.
Experimental results show practical effectiveness of the proposed algorithms.
Abstract
We study the problem of ordered stabbing of balls (of arbitrary and possibly different radii, no ball contained in another) in , , with either a directed line segment or a (directed) polygonal curve. Here, the line segment, respectively polygonal curve, shall visit (intersect) the given sequence of balls in the order of the sequence. We present a deterministic algorithm that decides whether there exists a line segment stabbing the given sequence of balls in order, in time . Due to the descriptional complexity of the region containing these line segments, we can not extend this algorithm to actually compute one. We circumvent this hurdle by devising a randomized algorithm for a relaxed variant of the ordered line segment stabbing problem, which is built upon the central insights from the aforementioned decision algorithm. We further show…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Code & Models
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsComputational Geometry and Mesh Generation · Optimization and Packing Problems · Point processes and geometric inequalities
