Touring a Sequence of Orthogonal Polygons

TL;DR

This paper develops efficient algorithms for computing shortest tours visiting a sequence of orthogonal polygons under Manhattan distance, improving on prior work with new techniques and various special case solutions.

Contribution

It introduces new subquadratic and linear-time algorithms for orthogonal polygon sequences, extending previous results to the orthogonal and Manhattan distance setting.

Findings

Subquadratic $ ilde{O}(n^{2-rac{1}{48}})$ algorithm for disjoint polygons

Linear $ ilde{O}(n)$ algorithm for ortho-convex polygons

Linear $O(n)$ algorithm for axis-aligned rectangles

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.