On reverse shortest paths in geometric proximity graphs

Let S be a set of n geometric objects of constant complexity (e.g., points, line segments, disks, ellipses) in $R^2$, and let $\varrho :S\times S\to R_{\ge 0}$ be a distance function on S. For a parameter $r\ge 0$, we define the proximity graph $G\left(r\right)=(S,E)$ where $E=\left\{\right(e_1,e_2)\in S\times S|e_1\ne e_2,\varrho (e_1,e_2)\le r\}$. 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 $G\left(r^{⁎}\right)$ 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 $G\left(r\right)$ 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 $O\left(n^2\right)$ 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 $O^{⁎}\left(n^{4/3}\right)$ expected-time randomized algorithm (where $O^{⁎}(\cdot )$ hides $\mathrm{polylog}\left(n\right)$ factors) for the case where S is a set of (possibly intersecting) line segments in $R^2$ 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.