All-pairs shortest paths for unweighted undirected graphs in o(mn) time
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
More algorithms for all-pairs shortest paths in weighted graphs
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
An O(n3 (loglogn/logn)5/4) time algorithm for all pairs shortest paths
ESA'06 Proceedings of the 14th conference on Annual European Symposium - Volume 14
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Journal of Experimental Algorithmics (JEA)
A Faster Shortest-Paths Algorithm for Minor-Closed Graph Classes
Graph-Theoretic Concepts in Computer Science
Discrete Applied Mathematics
More Algorithms for All-Pairs Shortest Paths in Weighted Graphs
SIAM Journal on Computing
Networks cannot compute their diameter in sublinear time
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Sensitivity analysis of minimum spanning trees in sub-inverse-ackermann time
ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
All-pairs shortest paths with real weights in O(n3/ log n) time
WADS'05 Proceedings of the 9th international conference on Algorithms and Data Structures
All-pairs shortest paths for unweighted undirected graphs in o(mn) time
ACM Transactions on Algorithms (TALG)
An O(n3 log log n/ log2 n) time algorithm for all pairs shortest paths
SWAT'12 Proceedings of the 13th Scandinavian conference on Algorithm Theory
Fast shortest-paths algorithms in the presence of few destinations of negative-weight arcs
Journal of Discrete Algorithms
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We present a new scheme for computing shortest paths on real-weighted undirected graphs in the fundamental comparison-addition model. In an efficient preprocessing phase our algorithm creates a linear-size structure that facilitates single-source shortest path computations in O(m log $\alpha$) time, where $\alpha$ = $\alpha$(m,n) is the very slowly growing inverse-Ackermann function, m the number of edges, and n the number of vertices. As special cases our algorithm implies new bounds on both the all-pairs and single-source shortest paths problems. We solve the all-pairs problem in O(mn log $\alpha$(m,n)) time and, if the ratio between the maximum and minimum edge lengths is bounded by n(log n)O(1), we can solve the single-source problem in O(m + n log log n) time. Both these results are theoretical improvements over Dijkstra's algorithm, which was the previous best for real weighted undirected graphs. Our algorithm takes the hierarchy-based approach invented by Thorup.