Stabbing balls with line segments and polygonal paths

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 $n$ balls (of arbitrary and possibly different radii, no ball contained in another) in $\mathbb{R}^d$, $d \geq 3$, 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 $O(n^{4d-2} \log n)$. 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 that this algorithm can be plugged into an algorithmic scheme by Guibas et al., yielding an algorithm for a relaxed variant of the minimum-link ordered stabbing path problem that achieves approximation factor 2 with respect to the number of links. We conclude with experimental evaluations of the latter two algorithms, showing practical applicability.