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Abstract:
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.
