## Efficient parallel graph algorithms for coarse grained multicomputers and BSP

E. Caceres, F. Dehne, A. Ferreira, P. Flocchini, I. Rieping, A. Roncato, N. Santoro, and S. W. Song

#### Abstract

In this paper, we present {\em deterministic} parallel algorithms for the coarse grained multicomputer (CGM) and bulk-synchronous parallel computer (BSP) models which solve the following well known graph problems: (1) list ranking, (2) Euler tour construction, (3) computing the connected components and spanning forest, (4) lowest common ancestor preprocessing, (5) tree contraction and expression tree evaluation, (6) computing an ear decomposition or open ear decomposition, (7) 2-edge connectivity and biconnectivity (testing and component computation), and (8) cordal graph recognition (finding a perfect elimination ordering). The algorithms for Problems 1-7 require $O(\log p)$ communication rounds and linear sequential work per round. Our results for Problems 1 and 2 hold for arbitrary ratios $\frac{n}{p}$, i.e. they are {\it fully scalable}, and for Problems 3-8 it is assumed that $\frac{n}{p} \geq p^{\epsilon}$, $\epsilon > 0$, which is true for all commercially available multiprocessors. We view the algorithms presented as an important step towards the final goal of $O(1)$ communication rounds. Note that, the number of communication rounds obtained in this paper is independent of $n$ and grows only very slowly with respect to $p$. Hence, for most practical purposes, the number of communication rounds can be considered as constant. The result for Problem 1 is a considerable improvement over those previously reported. The algorithms for Problems 2-7 are the first practically relevant deterministic parallel algorithms for these problems to be used for commercially available coarse grained parallel machines.

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