The Unweighted and Weighted Reverse Shortest Path Problem for Disk Graphs

We study the reverse shortest path problem on disk graphs in the plane. In this problem, we consider the proximity graph of a set of \( n \) disks in the plane of arbitrary radii: In this graph, two disks are connected if the distance between them is at most some threshold parameter \( r \) . The case of intersection graphs is a special case with \(r=0\) . We give an algorithm that, given a target length \( k \) , computes the smallest value of \( r \) for which there is a path of length at most \( k \) between some given pair of disks in the proximity graph. Our algorithm runs in \(O^{*}(n^{5/4})\) randomized expected time, which improves to \(O^{*}(n^{6/5})\) for unit disk graphs, where all the disks have the same radius. (In this article, the \(O^{*}(\cdot)\) notation hides subpolynomial factors.) Our technique is robust and can be applied to many variants of the problem. One significant variant is the case of weighted proximity graphs, where edges are assigned real weights equal to the distance between the disks or between their centers, and \( k \) is replaced by a target weight \( w \) ; that is, we seek a path whose length is at most \( w \) . In other variants, we want to optimize a parameter different from \( r \) , such as a scale factor of the radii of the disks. The main technique for the decision version of the problem (determining whether the graph with a given \( r \) has the desired property) is based on efficient implementations of BFS (for the unweighted case) and of Dijkstra’s algorithm (for the weighted case), using efficient data structures for maintaining the bichromatic closest pair for certain bicliques and several distance functions. The optimization problem is then solved by combining the resulting decision procedure with enhanced variants of the interval shrinking and bifurcation technique.