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arXiv · 2605.07882

Touring a Sequence of Orthogonal Polygons

Katrin Casel·Sándor Kisfaludi-Bak·Linda Kleist·Jeroen S. K. Lamme·Eunjin Oh·Yanheng Wang
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Abstract

We study the problem of computing a shortest tour that visits a sequence of $k$ polygons $P_1,\dots, P_k$ with a total number of $n$ vertices. A tour is an oriented curve such that there exist points $p_i\in P_i$ for all $i$ where $p_i$ appears not after $p_{i+1}$. In a seminal paper, Dror, Efrat, Lubiw and Mitchell (STOC 2003) considered the problem under $L_2$ distance, and gave $\widetilde O(nk)$ and $\widetilde O(nk^2)$ algorithms for disjoint and intersecting convex polygons, respectively. In this paper, we consider the orthogonal setting (with orthogonal polygons and Manhattan distance) and obtain the following results: - a truly subquadratic $\widetilde O(n^{2-\frac{1}{48}})$ algorithm when consecutive polygons in the sequence are disjoint; - an $\widetilde O(n)$ algorithm for ortho-convex polygons when consecutive polygons are disjoint; - an $O(n)$ algorithm for axis-aligned rectangles; - $\widetilde O(n^2)$ and $\widetilde O(n^{1.5}k^2)$ algorithms without restrictions. Our algorithms build on a wide range of techniques, including additively weighted Voronoi diagrams, rectangle decompositions, persistent data structures, and dynamic distance oracles for weighted planar graphs.

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Katrin Casel, Sándor Kisfaludi-Bak, Linda Kleist, Jeroen S. K. Lamme, Eunjin Oh, Yanheng Wang. 2026-05-08. Touring a Sequence of Orthogonal Polygons. https://arxiv.org/abs/2605.07882

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