Fibonacci heaps and their uses in improved network optimization algorithms
Journal of the ACM (JACM)
Shortest path algorithms for nearly acyclic directed graphs
Theoretical Computer Science - Special issue: graph theoretic concepts in computer science
ISAAC '94 Proceedings of the 5th International Symposium on Algorithms and Computation
Improved shortest path algorithms for nearly acyclic graphs
Theoretical Computer Science
A new approach to all-pairs shortest paths on real-weighted graphs
Theoretical Computer Science - Special issue on automata, languages and programming
COCOON'99 Proceedings of the 5th annual international conference on Computing and combinatorics
Sharing information in all pairs shortest path algorithms
CATS '11 Proceedings of the Seventeenth Computing: The Australasian Theory Symposium - Volume 119
Sharing information in all pairs shortest path algorithms
CATS 2011 Proceedings of the Seventeenth Computing on The Australasian Theory Symposium - Volume 119
Sharing information for the all pairs shortest path problem
Theoretical Computer Science
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Shortest path problems can be solved very efficiently when a directed graph is nearly acyclic. Earlier results defined a graph decomposition, now called the 1-dominator set, which consists of a unique collection of acyclic structures with each single acyclic structure dominated by a single associated trigger vertex. In this framework, a specialised shortest path algorithm only spends delete-min operations on trigger vertices, thereby making the computation of shortest paths through non-trigger vertices easier. A previously presented algorithm computed the 1-dominator set in O(mn) worst-case time, which allowed it to be integrated as part of an O(mn+nrlogr) time all-pairs algorithm. Here m and n respectively denote the number of edges and vertices in the graph, while r denotes the number of trigger vertices. A new algorithm presented in this paper computes the 1-dominator set in just O(m) time. This can be integrated as part of the O(m+rlogr) time spent solving single-source, improving on the value of r obtained by the earlier tree-decomposition single-source algorithm. In addition, a new bidirectional form of 1-dominator set is presented, which further improves the value of r by defining acyclic structures in both directions over edges in the graph. The bidirectional 1-dominator set can similarly be computed in O(m) time and included as part of the O(m+rlogr) time spent computing single-source. This paper also presents a new all-pairs algorithm under the more general framework where r is defined as the size of any predetermined feedback vertex set of the graph, improving the previous all-pairs time complexity from O(mn+nr^2) to O(mn+r^3).