Abstract

We present a new algorithm for answering short path queries in planar graphs. For any fixed constant k and a given unweighted planar graph G = (V, E) one can build in O(|V |) time a data structure, which allows to check in O(1) time whether two given vertices are distant by at most k in G and if so a shortest path between them is returned. This significantly improves the previous result of D. Eppstein [5] where after a linear preprocessing the queries are answered in O(log |V |) time. Our approach can be applied to compute the girth of a planar graph and a corresponding shortest cycle in O(|V |) time provided that the constant bound on the girth is known. Our results can be easily generalized to other wide classes of graphs – for instance we can take graphs embeddable in a surface of bounded genus or graphs of bounded tree-width. Categories and Subject Descriptors G.2.2 [Discrete Mathematics]: Graph Theory—graph algorithms, path and circuit problems

Citations