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
COCOON'99 Proceedings of the 5th annual international conference on Computing and combinatorics
Efficient algorithms for solving shortest paths on nearly acyclic directed graphs
CATS '05 Proceedings of the 2005 Australasian symposium on Theory of computing - Volume 41
Solving shortest paths efficiently on nearly acyclic directed graphs
Theoretical Computer Science
Improved shortest path algorithms for nearly acyclic directed graphs
ACSC '07 Proceedings of the thirtieth Australasian conference on Computer science - Volume 62
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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Dijkstra's algorithm solves the single-source shortest path problem on any directed graph in O(m + n log n) time when a Fibonacci heap is used as the frontier set data structure. Here n is the number of vertices and m is the number of edges in the graph. If the graph is nearly acyclic, other algorithms can achieve a time complexity lower than that of Dijkstra's algorithm. Abuaiadh and Kingston gave a single-source shortest path algorithm for nearly acyclic graphs with O(m + n log t) time complexity, where the new parameter, t, is the number of delete-min operations performed in priority queue manipulation. If the graph is nearly acyclic, then t is expected to be small, and the algorithm out-performs Dijkstra's algorithm. Takaoka, using a different definition for acyclicity, gave an algorithm with O(m + n log k) time complexity. In this algorithm, the new parameter, k, is the maximum cardinality of the strongly connected components in the graph.The generalised single-source (GSS) problem allows an initial distance to be defined at each vertex in the graph. Decomposing a graph into r trees allows the GSS problem to be solved within O(m + r logr) time. This paper presents a new all-pairs algorithm with a time complexity of O(mn + nr log r), where r is the number of acyclic parts resulting when the graph is decomposed into acyclic parts. The acyclic decomposition used is setwise unique and can be computed in O(mn) time. If the decomposition has been pre-calculated, then GSS can be solved within O(m + r log r) time whenever edge-costs in the graph change. A second new all-pairs algorithm is presented, with O(mn + nr2) worst-case time complexity, where r is the number of vertices in a pre-calculated feedback vertex set for the nearly acyclic graph. For certain graphs, these new algorithms offer an improvement on the time complexity of the previous algorithms.