Efficient computation of implicit representations of sparse graphs

The problem of finding an implicit representation for a graph such that vertex adjacency can be tested quickly is fundamental to all graph algorithms. In particular, it is possible to represent sparse graphs on n vertices using O(n) space such that vertex adjacency is tested in O(1) time. We show here how to construct such a representation efficiently by providing simple and optimal algorithms, both in a sequential and a parallel setting. Our sequential algorithm runs in O(n) time. The parallel algorithm runs in O(log n) time using O(n/log n) CRCW PRAM processors, or in O(log n log* n) time using O(n/log n log* n) EREW PRAM processors. Previous results for this problem are based on matroid partitioning and thus have a high complexity.