×

The quickest path problem. (English) Zbl 0698.90083

The authors consider the following modification of shortest path problems: let a digraph with n nodes and m arcs be given. For any arc (u,v) we have its lead time l(u,v) and its capacity c(u,v). The transmission time for \(\sigma\) units of data along this arc is defined as \(t(u,v):=l(u,v)+\sigma /c(u,v)\). The quickest path problem asks for a path from the source to the sink along which the sum of transmission times is minimum. For fixed \(\sigma\), the authors state an algorithm of time complexity \(O(m^ 2+mn \log m)\) to solve the quickest path problem. If a quickest path for any arbitrary \(\sigma\) is looked for, then the network can be preprocessed in \(O(m^ 2+mn \log m)\) time such that finding a quickest path requires only O(log m) additional steps.
Reviewer: R.E.Burkard

MSC:

90C35 Programming involving graphs or networks
90C27 Combinatorial optimization
68Q25 Analysis of algorithms and problem complexity
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ahuja, R. K., Minimum cost-reliability ratio problem, Computers Opns Res., 16, 83-89 (1988) · Zbl 0643.90088
[2] Bodin, L. D.; Golden, B. L.; Assad, A. A.; Ball, M. O., Routing and scheduling of vehicles and crews. The state of the art, Computers Opns Res., 10, 63-211 (1982)
[3] Deo, N.; Pang, C-Y., Shortest-path algorithms: taxonomy and annotation, Networks, 14, 275-323 (1984) · Zbl 0542.90101
[4] Golden, B. L.; Magnanti, T. L., Deterministic network optimization: a bibliography, Networks, 7, 149-183 (1977) · Zbl 0362.90116
[5] Pierce, A. R., Bibliography on algorithms for shortest path, shortest spanning tree, and related circuit routing problems, Networks, 5, 129-149 (1975) · Zbl 0307.90078
[6] Hu, T. C., Combinatorial Algorithms, ((1982), Addison-Wesley: Addison-Wesley New York), 4-5 · Zbl 0505.68022
[7] Fredman, M. L.; Tarjan, R. E., Fibonacci heaps and their uses in improved network optimization algorithms, J. Am. C. M., 34, 596-615 (1987) · Zbl 1412.68048
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.