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- W2950828712 abstract "The single-source shortest path problem (SSSP) with nonnegative edge weights is a notoriously difficult problem to solve efficiently in parallel---it is one of the graph problems said to suffer from the transitive-closure bottleneck. In practice, the $Delta$-stepping algorithm of Meyer and Sanders (J. Algorithms, 2003) often works efficiently but has no known theoretical bounds on general graphs. The algorithm takes a sequence of steps, each increasing the radius by a user-specified value $Delta$. Each step settles the vertices in its annulus but can take $Theta(n)$ substeps, each requiring $Theta(m)$ work ($n$ vertices and $m$ edges). In this paper, we describe Radius-Stepping, an algorithm with the best-known tradeoff between work and depth bounds for SSSP with nearly-linear ($otilde(m)$) work. The algorithm is a $Delta$-stepping-like algorithm but uses a variable instead of fixed-size increase in radii, allowing us to prove a bound on the number of steps. In particular, by using what we define as a vertex $k$-radius, each step takes at most $k+2$ substeps. Furthermore, we define a $(k, rho)$-graph property and show that if an undirected graph has this property, then the number of steps can be bounded by $O(frac{n}{rho} log rho L)$, for a total of $O(frac{kn}{rho} log rho L)$ substeps, each parallel. We describe how to preprocess a graph to have this property. Altogether, Radius-Stepping takes $O((m+nlog n)log frac{n}{rho})$ work and $O(frac{n}{rho}log n log (rho{}L))$ depth per source after preprocessing. The preprocessing step can be done in $O(mlog n + nrho^2)$ work and $O(rho^2)$ depth or in $O(mlog n + nrho^2log n)$ work and $O(rholog rho)$ depth, and adds no more than $O(nrho)$ edges." @default.
- W2950828712 created "2019-06-27" @default.
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- W2950828712 date "2016-02-11" @default.
- W2950828712 modified "2023-09-27" @default.
- W2950828712 title "Parallel Shortest-Paths Using Radius Stepping" @default.
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