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This document from portland state university covers parallel strategies for traversing graphs, finding connected components using hirshberg's algorithm, and solving single-source shortest paths problems using moore's algorithm. The document also discusses the parallelization of these algorithms.
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Jingke Li
Portland State University
Jingke Li (Portland State University) CS 415/515 Graph Algorithms 1 / 25
Graph algorithms are frequently used in non-numerical applications. We’d like to examine parallel algorithms for a few common graph problems.
Graph Basics:
Common strategies include
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Traversing a graph with p processors.
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Common Approaches:
Based on the 3rd approach.
It iterates over three stages:
Complexity: O(log^2 n) with n^2 processors.
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Iteration 1:
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forall (i in V) R(i) = i for (t = 0; t < log n; t++) { forall (i in V) { T(i) = min_j { R(j) | A(i,j)=1 && R(j) != R(i) } if none then R(i) } forall (i in V) { T(i) = min_j { T(j) | R(j)=i, T(j) != i } if none then R(i) } forall (i in V) R(i) = T(i) for (k = 0; k < log n; k++) { forall (i in V) T(i) = T(T(i)) } forall (i in V) R(i) = min { R(T(i)), T(i) } }
Given an n-node weighted graph G , the goal is to produce an n × n matrix A, where aij is the length of the shortest path from node i to node j.
Matrix-Multiplication Based Algorithm:
A = An−^1 = A^1 ◦ A^1 ◦ · · · ◦ A^1
With optimization, the number of matrix multiplications can be reduced to O(log n).
Complexity: O(log^2 n) with O(n^3 ) processors.
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The problem is to find the shortest path from a specified node, s, to all other nodes in a weighted directed graph.
Moore’s Algorithm:
Find the min-cost spanning tree for a weighted, undirected, connected graph.
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Start the tree with any node, then repeat the following step until all nodes are on the tree: Find the edge of lowest weight that connects a node already on the tree to a node not on the tree; then add the edge and the node to the tree.
Complexity: O(n^2 )
The step of finding the edge of lowest weight can be parallelized:
Parallel Complexity: Θ(
n^2 p
) + Θ(n log p)
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Start with a forest of n trees (each containing a single node); then iterate the following step until there is only one tree remaining:
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The algorithm’s complexity is O(n^2 log n).
Comments: Can be parallelized with domain decomposition.
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