In this graph, the widest path from Maldon to Feering has bandwidth 29, and passes through Clacton, Tiptree, Harwich, and Blaxhall.
In graph algorithms, the widest path problem is the problem of finding a path between two designated vertices in a weighted graph, maximizing the weight of the minimum-weight edge in the path. The widest path problem is also known as the maximum capacity path problem. It is possible to adapt most shortest path algorithms to compute widest paths, by modifying them to use the bottleneck distance instead of path length.[1] However, in many cases even faster algorithms are possible.
For instance, in a graph that represents connections between routers in the Internet, where the weight of an edge represents the bandwidth of a connection between two routers, the widest path problem is the problem of finding an end-to-end path between two Internet nodes that has the maximum possible bandwidth.[2] The smallest edge weight on this path is known as the capacity or bandwidth of the path. As well as its applications in network routing, the widest path problem is also an important component of the Schulze method for deciding the winner of a multiway election,[3] and has been applied to digital compositing,[4] metabolic pathway analysis,[5] and the computation of maximum flows.[6]
A closely related problem, the minimax path problem or bottleneck shortest path problem asks for the path that minimizes the maximum weight of any of its edges. It has applications that include transportation planning.[7] Any algorithm for the widest path problem can be transformed into an algorithm for the minimax path problem, or vice versa, by reversing the sense of all the weight comparisons performed by the algorithm, or equivalently by replacing every edge weight by its negation.
^Pollack, Maurice (1960), "The maximum capacity through a network", Operations Research, 8 (5): 733–736, doi:10.1287/opre.8.5.733, JSTOR 167387
^Shacham, N. (1992), "Multicast routing of hierarchical data", IEEE International Conference on Communications (ICC '92), vol. 3, pp. 1217–1221, doi:10.1109/ICC.1992.268047, hdl:2060/19990017646, ISBN 978-0-7803-0599-1, S2CID 60475077; Wang, Zheng; Crowcroft, J. (1995), "Bandwidth-delay based routing algorithms", IEEE Global Telecommunications Conference (GLOBECOM '95), vol. 3, pp. 2129–2133, doi:10.1109/GLOCOM.1995.502780, ISBN 978-0-7803-2509-8, S2CID 9117583
^Schulze, Markus (2011), "A new monotonic, clone-independent, reversal symmetric, and Condorcet-consistent single-winner election method", Social Choice and Welfare, 36 (2): 267–303, doi:10.1007/s00355-010-0475-4, S2CID 1927244
^Cite error: The named reference fga was invoked but never defined (see the help page).
^Ullah, E.; Lee, Kyongbum; Hassoun, S. (2009), "An algorithm for identifying dominant-edge metabolic pathways", IEEE/ACM International Conference on Computer-Aided Design (ICCAD 2009), pp. 144–150
^Cite error: The named reference amo was invoked but never defined (see the help page).
^Berman, Oded; Handler, Gabriel Y. (1987), "Optimal Minimax Path of a Single Service Unit on a Network to Nonservice Destinations", Transportation Science, 21 (2): 115–122, doi:10.1287/trsc.21.2.115
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taking home more than half of all income and giving the U.S. one of the widest income distributions among OECD members. The U.S. ranks first in the number...
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