Parallel algorithms for graph problems.

Abstract

In this thesis we examine three problems in graph theory and propose efficient parallel algorithms for solving them. We also introduce a number of parallel algorithmic techniques. Computing the connected components of an undirected graph $G=(V,E)$ is the first problem we examine. We propose an efficient algorithm which runs in $O({\rm log}\sp{3/2}\vert V\vert)$ parallel time using $\vert V\vert+\vert E\vert$ CREW PRAM processors. This result settles a question that remained unresolved for many years: a connectivity algorithm for this model with running time $o({\rm log}\sp2\vert V\vert)$ was a challenge that had thus far eluded researchers. The second problem examined is the vertex updating for a minimum spanning tree of a graph $G=(V,E)$. We give optimal parallel algorithms (as well as linear-time sequential ones) which run in $O({\rm log}\vert V\vert)$ time using ${\vert V\vert\over{\rm log}\vert V\vert}$ EREW PRAM processors. This result is then used in the solution of the multiple updates of a minimum spanning tree problem. We present an algorithm for this problem which runs in $O({\rm log} k{\cdot}{\rm log}\vert V\vert)$ parallel time using ${k{\cdot}\vert V\vert\over{\rm log} k{\cdot}{\rm log}\vert V\vert}$ EREW PRAM processors, where k is the number of updates. In the process of solving these problems, we introduce a number of parallel algorithmic techniques. In particular, we give an algorithm for the pseudotree contraction problem having running time $O({\rm log} h)$, where h is the length of the longest simple path in the pseudotree. Also, we propose the edge-plugging scheme which solves the edge-list augmentation problem in constant time without concurrent writing. Finally, we introduce the growth-control scheduling technique. This technique balances carefully the work-performed versus progress-made ratio of an algorithm giving a better running time.