Parallel ear decomposition search (EDS) and st-numbering in graphs

Yael Maon, Baruch Schieber, Uzi Vishkin
1986 Theoretical Computer Science  
The linear time serial algorithm of Lempel et al. (1967) for testing planarity of graphs uses the linear time serial algorithm of Even and Tarjan (1976) for st-numbering. This st-numbering algorithm is based on depth-first search (DFS). A known conjecture states that DFS, which is a key technique in designing serial algorithms, is not amenable to poly-log time parallelism using 'around linearly' (or even polynomially) many processors. The first contribution of this paper is a general method for
more » ... searching efficiently in parallel undirected graphs, called ear decomposition search (EDS). The second contribution demonstrates the applicability of this search method. We present an efficient parallel algorithm for st-numbering in a biconnected graph. The algorithm runs in logarithmic time using a linear number of processors on a concurrent-read concurrent-write (CRCW) PRAM. An efficient parallel algorithm for the problem did not exist before. The problem was not even known to be in NC. Input: an undirected graph G( V, E) and some specified edge e = (s, t) in E. (Denote n=lV I and re=lEt.) Let Po be the path that consists of the edge e. An ear decomposition of G starting with Po is a decomposition E = PoU P1 u. • • u Pk, where Pi+l is a simple path whose * endpoints belong to Pou" • • u Pi, but its interval vertices do not. A simple path Pi is called an ear. It is called an open ear if the two endpoints of P~ do not coincide, and a closed ear otherwise. An ear decomposition is called open if all its ears are open. The ear decomposition problem: find an ear decomposition starting with Po-The open ear decomposition problem: find an open ear decomposition starting with Po. A one-to-one function f from V to {1,..., n} is called an st-numbering if it satisfies (i) f(s) = 1 and f(t) = n, and (ii) for each v e V-{s, t} there exist adjacent vertices vl and v2 such that f(vl) <f(v) <f(v2).
doi:10.1016/0304-3975(86)90153-2 fatcat:y65vame6gbdebj2qkqvh6dkbsi