Touring a Sequence of Orthogonal Polygons

2026-05-08

AI Summary

This study addresses the problem of finding the shortest path that sequentially visits a sequence of axis-aligned orthogonal polygons under the Manhattan metric. The work presents the first truly subquadratic-time algorithm for this problem in an orthogonal geometric setting, introducing near-linear (Õ(n)) and linear (O(n)) time solutions for orthogonally convex polygons and axis-aligned rectangles, respectively. Key technical contributions include the use of weighted Voronoi diagrams, rectangular decomposition, persistent data structures, and dynamic distance oracles on weighted planar graphs. For the general non-intersecting case, the algorithm achieves a running time of Õ(n^{2−1/48}), significantly improving upon previous results.

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. This paper considers the orthogonal setting, where the input polygons have axis-aligned edges and the distance metric is the Manhattan distance. We obtain the following results: - as our main contribution, 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.

Problem

• orthogonal polygons
• shortest tour
• Manhattan distance
• sequential visit
• computational geometry

Innovation

• orthogonal polygons
• Manhattan distance
• subquadratic algorithm
• Voronoi diagrams
• dynamic distance oracles