An optimal algorithm for L1 shortest paths in unit-disk graphs

Abstract

A unit-disk graph $G\left(P\right)$ of a set P of points in the plane is a graph with P as its vertex set such that two points of P are connected by an edge if the distance between the two points is at most 1 and the weight of the edge is equal to the distance of the two points. Given P and a source point $s\in P$, we consider the problem of finding shortest paths in $G\left(P\right)$ from s to all other vertices of $G\left(P\right)$. In the $L_2$ case where the distance is measured by the $L_2$ metric, the problem has been extensively studied and the current best algorithm runs in $O(n{\log }^{2}n)$ time, with $n=\left|P\right|$. In this paper, we study the $L_1$ case in which the distance is measured under the $L_1$ metric (and each disk becomes a diamond); we present an $O(n\log n)$ time algorithm, which matches the $\Omega (n\log n)$-time lower bound.