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.

翻译：本文研究计算最短路径的问题，该路径需依次访问一系列多边形 $P_1,\dots, P_k$（总顶点数为 $n$），其中路径是一条有向曲线，存在点 $p_i\in P_i$ 使得 $p_i$ 不晚于 $p_{i+1}$ 出现。在开创性工作中，Dror、Efrat、Lubiw 和 Mitchell（STOC 2003）在 $L_2$ 距离下考虑了该问题，并针对不交和相交的凸多边形分别给出了 $\widetilde O(nk)$ 和 $\widetilde O(nk^2)$ 算法。本文考虑正交场景（正交多边形与曼哈顿距离），取得以下成果：- 当序列中相邻多边形不交时，得到真正次二次的 $\widetilde O(n^{2-\frac{1}{48}})$ 算法；- 对于相邻多边形不交的正交凸多边形，给出 $\widetilde O(n)$ 算法；- 对于轴对齐矩形，给出 $O(n)$ 算法；- 在无限制条件下，给出 $\widetilde O(n^2)$ 和 $\widetilde O(n^{1.5}k^2)$ 算法。我们的算法基于多种技术，包括加性权重沃罗诺伊图、矩形分解、持久化数据结构以及带权平面图的动态距离查询结构。