Keywords and Synonyms
Shortest route; Quickest route
Problem Definition
The single source shortest path problem (SSSP) is, given a graph \( { G = (V,E,\ell) } \) and a source vertex \( { s\in V } \), to find the shortest path from s to every \( { v\in V } \). The difficulty of the problem depends on whether the graph is directed or undirected and the assumptions placed on the length function ℓ. In the most general situation \( { \ell\colon E\rightarrow \mathbb{R} } \) assigns arbitrary (positive & negative) real lengths. The algorithms of Bellman-Ford and Edmonds [1,4] may be applied in this situation and have running times of roughly O(mn),Footnote 1 where \( { m = |E| } \) and \( { n=|V| } \) are the number of edges and vertices. If ℓ assigns only non-negative real edge lengths then the algorithms of Dijkstra and Pettie‐Ramachandran [4,14] may be applied on directed and undirected graphs, respectively. These algorithms include a sorting bottleneck and, in the worst case, take \( {...
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Notes
- 1.
Edmonds's algorithm works for undirected graphs and presumes that there are no negative length simple cycles.
- 2.
The [14] algorithm actually runs in \( { O(m + n\log\log n) } \) time if the ratio of any two edge lengths is polynomial in n.
- 3.
There is some flexibility in the definition of shortest path since floating-point addition is neither commutative nor associative.
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Pettie, S. (2008). Single-Source Shortest Paths. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-30162-4_377
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