Let S be a set of n geometric objects of constant complexity (e.g., points, line segments, disks, ellipses) in ${\mathbb{R}}^2$, and let $\varrho :S\times S\to {\mathbb{R}}_{\ge 0}$ be a distance function on S. For a parameter $r\ge 0$, we define the proximity graph $G(r)=(S,E)$ where $E=\{(e_1,e_2)\in S\times S|e_1\ne e_2,\varrho (e_1,e_2)\le r\}$. Given S, $s,t\in S$, and an integer $k\ge 1$, the reverse-shortest-path (RSP) problem asks for computing the smallest value $r^{⁎}\ge 0$ such that $G(r^{⁎})$ contains a path from s to t of length at most k.

In this paper we present a general randomized technique that solves the RSP problem efficiently for a large family of geometric objects and distance functions. Using standard, and sometimes more involved, semi-algebraic range-searching techniques, we first give an efficient algorithm for the decision problem, namely, given a value $r\ge 0$, determine whether $G(r)$ contains a path from s to t of length at most k. Next, we adapt our decision algorithm and combine it with a random-sampling method to compute $r^{⁎}$, by efficiently performing a binary search over an implicit set of $O(n^2)$ candidate ‘critical’ values that contains $r^{⁎}$.

We illustrate the versatility of our general technique by applying it to a variety of geometric proximity graphs. For example, we obtain (i) an $O^{⁎}(n^{4/3})$ expected-time randomized algorithm (where $O^{⁎}(\cdot )$ hides $\mathrm{polylog}(n)$ factors) for the case where S is a set of (possibly intersecting) line segments in ${\mathbb{R}}^2$ and $\varrho (e_1,e_2)={\min }_{x\in e_1,y\in e_2}\Vert x-y\Vert$ (where $\Vert \cdot \Vert$ is the Euclidean distance), and (ii) an $O^{⁎}(n+m^{4/3})$ expected-time randomized algorithm for the case where S is a set of m points lying on an x-monotone polygonal chain T with n vertices, and $\varrho (p,q)$, for $p,q\in S$, is the smallest value h such that the points $p^{\prime }:=p+(0,h)$ and $q^{\prime }:=q+(0,h)$ are visible to each other, i.e., all points on the segment $p^{\prime }{q}^{\prime }$ lie above or on the polygonal chain T.

设S是一组n 个具有恒定复杂度的几何对象（例如，点、线段、圆盘、椭圆）${\mathbb{右}}^2$， 然后让$\varrho :S\times S\to {\mathbb{右}}_{\ge 0}$是S上的距离函数。对于一个参数$r\ge 0$，我们定义邻近图 $G（r）=（S,乙）$在哪里$乙=\{（e_1,e_2）\epsilon S\times S|e_1\ne e_2,\varrho （e_1,e_2）\le r\}$。给定S，$s,t\epsilon S$，和一个整数$k\ge 1$，反向最短路径（RSP）问题要求计算最小值$r^{⁎}\ge 0$这样$G（r^{⁎}）$包含从s到t 的路径，长度至多为k。

在本文中，我们提出了一种通用的随机技术，可以有效地解决大量几何对象和距离函数的 RSP 问题。使用标准的、有时更复杂的半代数范围搜索技术，我们首先给出决策问题的有效算法，即给定一个值$r\ge 0$，判断是否$G（r）$包含从s到t 的路径，长度至多为k。接下来，我们调整决策算法并将其与随机采样方法相结合来计算$r^{⁎}$，通过对隐式集合有效地执行二分搜索$氧（n^2）$候选“关键”值包含$r^{⁎}$。

我们通过将其应用于各种几何邻近图来说明我们的通用技术的多功能性。例如，我们获得 (i)${氧}^{⁎}（n^{4/3}）$预期时间随机算法（其中${氧}^{⁎}（\cdot ）$隐藏$\mathrm{多对数}（n）$因子）对于S是一组（可能相交）线段的情况${\mathbb{右}}^2$和$\varrho （e_1,e_2）={\mathrm{分钟}}_{X\epsilon e_1,y\epsilon e_2}\Vert X-y\Vert$（在哪里$\Vert \cdot \Vert$是欧几里德距离），并且 (ii)${氧}^{⁎}（n+{米}^{4/3}）$预期时间随机算法，适用于S是位于具有n 个顶点的x单调多边形链T上的m个点的集合，以及$\varrho （p,q）$， 为了$p,q\epsilon S$，是最小值h使得点$p^{\prime }:=p+（0,H）$和$q^{\prime }:=q+（0,H）$彼此可见，即线段上的所有点$p^{\prime }{q}^{\prime }$位于多边形链T之上或之上。