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why non-negative reweighted edges is a useful condition |
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#Finally, for each node ''s'', it runs [[Dijkstra's algorithm]] to find the shortest paths from ''s'' to each other vertex in the graph, using these modified weights to measure the length of each path.
In the reweighted graph, all paths between a pair ''s'' and ''t'' of nodes have the same quantity ''h(s)'' -''h(t)'' added to them, so a path that is shortest in the original graph remains shortest in the modified graph and vice versa. However, due to the way the values ''h(v)'' were computed, all modified edge lengths are non-negative, ensuring the optimality of the paths found by Dijkstra's algorithm. The distances in the original graph may be calculated from the distances calculated by Dijkstra's algorithm in the reweighted graph by reversing the reweighting transformation.
The [[time complexity]] of this algorithm, using [[Fibonacci heap]]s in the implementation of Dijkstra's algorithm, is O(''V''<sup>2</sup>log ''V'' + ''VE''): the algorithm uses O(''VE'') time for the Bellman-Ford stage of the algorithm, and O(''V'' log ''V'' + ''E'') for each of ''V'' instantiations of Dijkstra's algorithm. Thus, when the graph is sparse, the total time can be faster than the [[Floyd-Warshall algorithm]], which solves the same problem in time O(''V''<sup>3</sup>).
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