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M. Kainer, J. Träff:
"More Parallelism in Dijkstra's Single-Source Shortest Path Algorithm";
Report for CoRR - Computing Research Repository; Report No. arXiv:1903.12085, 2019; 29 pages.

Dijkstra's algorithm for the Single-Source Shortest Path (SSSP) problem is notoriously hard to parallelize in o(n) depth, n being the number of vertices in the input graph, without increasing the required parallel work unreasonably. Crauser et al.\ (1998) presented observations that allow to identify more than a single vertex at a time as correct and correspondingly more edges to be relaxed simultaneously. Their algorithm runs in parallel phases, and for certain random graphs they showed that the number of phases is O(n^(1/3)) with high probability. A work-efficient CRCW PRAM with this depth was given, but no implementation on a real, parallel system.
In this paper we strengthen the criteria of Crauser et al., and discuss tradeoffs between work and number of phases in their implementation. We present simulation results with a range of common input graphs for the depth that an ideal parallel algorithm that can apply the criteria at no cost and parallelize relaxations without conflicts can achieve. These results show that the number of phases is indeed a small root of n, but still off from the shortest path length lower bound that can also be computed.
We give a shared-memory parallel implementation of the most work-efficient version of a Dijkstra's algorithm running in parallel phases, which we compare to an own implementation of the well-known Δ-stepping algorithm. We can show that the work-efficient SSSP algorithm applying the criteria of Crauser et al. is competitive to and often better than Δ-stepping on our chosen input graphs. Despite not providing an o(n) guarantee on the number of required phases, criteria allowing concurrent relaxation of many correct vertices may be a viable approach to practically fast, parallel SSSP implementations.