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

Abstract

A unit-disk graph G(P) 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∈P, we consider the problem of finding shortest paths in G(P) from s to all other vertices of G(P). In the L2 case where the distance is measured by the L2 metric, the problem has been extensively studied and the current best algorithm runs in O(nlog2⁡n) time, with n=|P|. In this paper, we study the L1 case in which the distance is measured under the L1 metric (and each disk becomes a diamond); we present an O(nlog⁡n) time algorithm, which matches the Ω(nlog⁡n)-time lower bound.