We consider the problem of computing shortest paths in weighted unit-disk graphs in constant dimension $d$. Although the single-source and all-pairs variants of this problem are well-studied in the plane case, no non-trivial exact distance oracles for unit-disk graphs have been known to date, even for $d=2$.
The classical result of Sedgewick and Vitter [Algorithmica '86] shows that for weighted unit-disk graphs in the plane the $A^*$ search has average-case performance superior to that of a standard shortest path algorithm, e.g., Dijkstra's algorithm. Specifically, if the $n$ corresponding points of a weighted unit-disk graph $G$ are picked from a unit square uniformly at random, and the connectivity radius is $r\in (0,1)$, $A^*$ finds a shortest path in $G$ in $O(n)$ expected time when $r=Ω(\sqrt{\log n/n})$, even though $G$ has $Θ((nr)^2)$ edges in expectation. In other words, the work done by the algorithm is in expectation proportional to the number of vertices and not the number of edges.
In this paper, we break this natural barrier and show even stronger sublinear time results. We propose a new heuristic approach to computing point-to-point exact shortest paths in unit-disk graphs. We analyze the average-case behavior of our heuristic using the same random graph model as used by Sedgewick and Vitter and prove it superior to $A^*$. Specifically, we show that, if we are able to report the set of all $k$ points of $G$ from an arbitrary rectangular region of the plane in $O(k + t(n))$ time, then a shortest path between arbitrary two points of such a random graph on the plane can be found in $O(1/r^2 + t(n))$ expected time. In particular, the state-of-the-art range reporting data structures imply a sublinear expected bound for all $r=Ω(\sqrt{\log n/n})$ and $O(\sqrt{n})$ expected bound for $r=Ω(n^{-1/4})$ after only near-linear preprocessing of the point set.
Holiest Minimum-Cost Paths and Flows in Surface Graphs
设$G$是一个边权有向图，其中$n$的顶点嵌入在亏格为$g$的可定向曲面上.我们描述了一个简单的确定性字典序扰动方案，该方案保证了$G$中最小成本流和最短路径的唯一性。扰动需要$O(gn)$时间来计算。我们使用我们的扰动计划在一个黑盒的方式来获得一个确定性的$O(n \log \log n)$时间算法的最小切割\emph{directed}边加权平面图和一个确定性的$O(g^2 n \log n)$时间预处理计划的多源最短路径问题的计算最短路径预言的所有顶点躺在一个表面嵌入图的公共面。后者的结果产生更快的确定性近线性时间算法的各种问题，在常亏格表面嵌入图。最后，我们打开黑盒子，以推广最近的线性时间算法的多源最短路径在无向平面图工作在任意定向表面。我们的算法运行在$O(g^2 n \log g)$时间在此设置，它可以用来给改进的线性时间算法的几个问题的无权无向面嵌入图的常亏格，包括计算最小割，最短的拓扑非平凡圈，最小的同源基。
