Shortest paths in intersection graphs of unit disks

Let G be a unit disk graph in the plane defined by n disks whose positions are known. For the case when G   is unweighted, we give a simple algorithm to compute a shortest path tree from a given source in O(nlog⁡n)O(nlog⁡n) time. For the case when G   is weighted, we show that a shortest path tree from a given source can be computed in O(n1+ε)O(n1+ε) time, improving the previous best time bound of O(n4/3+ε)O(n4/3+ε).