Algorithms for dense graphs and networks on the random access computer

We improve upon the running time of several graph and network algorithms when applied to dense graphs. In particular, we show how to compute on a machine with word size λ=Ω (logn) a maximal matching in ann-vertex bipartite graph in timeO(n2+n2.5/λ)=O(n2.5/logn), how to compute the transitive closure of a digraph withn vertices andm edges in timeO(n2+nm/λ), how to solve the uncapacitated transportation problem with integer costs in the range [O.C] and integer demands in the range [−U.U] in timeO ((n3 (log log/logn)1/2+n2 logU) lognC), and how to solve the assignment problem with integer costs in the range [O.C] in timeO(n2.5 lognC/(logn/loglogn)1/4).Assuming a suitably compressed input, we also show how to do depth-first and breadth-first search and how to compute strongly connected components and biconnected components in timeO(nλ+n2/λ), and how to solve the single source shortest-path problem with integer costs in the range [O.C] in time0 (n2(logC)/logn). For the transitive closure algorithm we also report on the experiences with an implementation.