Improved Algorithms for Computing k-Sink on Dynamic Flow Path Networks
We address the problem of locating k sinks on dynamic flow path networks with n vertices in such a way that the evacuation completion time to them is minimized. Our two algorithms run in \(O(n\log n + k^2\log ^4 n)\) and \(O(n\log ^3 n)\) time, respectively. When all edges have the same capacity, we also present two algorithms which run in \(O(n + k^2\log ^2n)\) time and \(O(n\log n)\) time, respectively. These algorithms together improve upon the previously most efficient algorithms, which have time complexities \(O(kn\log ^2n)\)  and O(kn) , in the general and uniform edge capacity cases, respectively. The above results are achieved by organizing relevant data for subpaths in a strategic way during preprocessing, and the final results are obtained by extracting/merging them in an efficient manner.
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