Compact oracles for reachability and approximate distances in planar digraphs

It is shown that a planar digraph can be preprocessed in near-linear time, producing a near-linear space distance oracle that can answer reachability queries in constant time. The oracle can be distributed as an O(log n) space label for each vertex and then we can determine if one vertex can reach another considering their two labels only. The approach generalizes to approximate distances in weighted planar digraphs where we can then get a (1+ε) approximation distance in O(log log Δ+1/ε) time where Δ is the longest finite distance in the graph and weights are assumed to be non-negative integers. Our scheme can be extended to find and route along the short dipaths. Our technique is based on a novel dipath decomposition of planar digraphs that instead of using the standard separator with O(√n) vertices, in effect finds a separator using a constant number of dipaths.