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<p><b>Abstract</b>—Let <tmath>${\cal G}(V,\ E)$</tmath> be a directed graph in which each vertex has a nonnegative weight. The cost of a path between two vertices in <tmath>$\cal G$</tmath> is the sum of the weights of the vertices on that path. In this paper, we show that, for such graphs, the time complexity of Dijkstra's algorithm, implemented with a binary heap, is <tmath>${\cal O}(|E| + |V|\ \log\ |V|).$</tmath></p>
Analysis of algorithms, combinatorial problems, data structures
Michael Barbehenn, "A Note on the Complexity of Dijkstra's Algorithm for Graphs with Weighted Vertices", IEEE Transactions on Computers, vol. 47, no. , pp. 263, February 1998, doi:10.1109/12.663776
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