Hammock on Ears Decomposition A Technique for the E cient Parallel Solution of Shortest Paths and Other Problems

We show how to decompose e ciently in parallel any graph into a number of outerplanar subgraphs called hammocks satisfying certain separator properties Our work combines and extends the sequential hammock decomposition technique intro duced by G Frederickson and the parallel ear decomposition technique thus we call it the hammock on ears decomposition We mention that hammock on ears decomposi tion also draws from techniques in computational geometry and that an embedding of the graph does not need to be provided with the input We achieve this decomposition in O logn log logn time using O n m CREW PRAM processors for an n vertex m edge graph or digraph The hammock on ears decomposition implies a general frame work for solving graph problems e ciently Its value is demonstrated by a variety of applications on a signi cant class of di graphs namely that of sparse di graphs This class consists of all di graphs which have a between and n and includes planar graphs and graphs with genus o n We improve previous bounds for certain instances of shortest paths and related problems in this class of graphs These problems include all pairs shortest paths all pairs reachability and detection of a negative cycle