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.

翻译：我们研究计算访问一系列多边形 $P_1,\dots, P_k$ 的最短巡回路线问题，其中多边形总顶点数为 $n$。巡回路线是一条有向曲线，使得对于所有 $i$，存在点 $p_i\in P_i$ 且 $p_i$ 出现在 $p_{i+1}$ 之前。在开创性论文中，Dror、Efrat、Lubiw 和 Mitchell（STOC 2003）在 $L_2$ 距离下考虑了该问题，并针对不相交和相交的凸多边形分别给出了 $\widetilde O(nk)$ 和 $\widetilde O(nk^2)$ 算法。本文考虑正交情形，即输入多边形的边与坐标轴平行且距离度量为曼哈顿距离。我们获得以下结果：- 主要贡献是当序列中连续多边形不相交时，给出真正次二次的 $\widetilde O(n^{2-\frac{1}{48}})$ 算法；- 当连续多边形不相交时，针对正交凸多边形给出 $\widetilde O(n)$ 算法；- 针对轴对齐矩形给出 $O(n)$ 算法；- 无限制条件下给出 $\widetilde O(n^2)$ 和 $\widetilde O(n^{1.5}k^2)$ 算法。我们的算法建立在广泛技术之上，包括加性权重沃罗诺伊图、矩形分解、持久化数据结构以及加权平面图的动态距离预计算。