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. 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. 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, and has been applied to digital compositing, metabolic pathway analysis, and the computation of maximum flows. 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. 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.
Undirected graphs In an undirected graph, a widest path may be found as the path between the two vertices in the maximum spanning tree of the graph, and a minimax path may be found as the path between the two vertices in the minimum spanning tree. It follows immediately from this equivalence that all pairs widest paths in an n {\displaystyle n} -vertex undirected graph can be computed in time O ( n 2 ) {\displaystyle O(n^{2})} . In any graph, directed or undirected, there is a straightforward algorithm for finding a widest path once the weight of its minimum-weight edge is known: simply delete all smaller edges and search for any path among the remaining edges using breadth-first search or depth-first search. Based on this test, there also exists a linear time algorithm for finding a widest s-t path in an undirected graph, that does not use the maximum spanning tree. The main idea of the algorithm is to apply the linear-time path-finding algorithm to the median edge weight in the graph, and then either to delete all smaller edges or contract all larger edges according to whether a path does or does not exist, and recurse in the resulting smaller graph. Fernández, Garfinkel & Arbiol (1998) use undirected bottleneck shortest paths in order to form composite aerial photographs that combine multiple images of overlapping areas. In the subproblem to which the widest path problem applies, two images have already been transformed into a common coordinate system; the remaining task is to select a seam, a curve that passes through the region of overlap and divides one of the two images from the other. Pixels on one side of the seam will be copied from one of the images, and pixels on the other side of the seam will be copied from the other image. Unlike other compositing methods that average pixels from both images, this produces a valid photographic image of every part of the region being photographed. They weigh the edges of a grid graph by a numeric estimate of how visually apparent a seam across that edge would be, and find a bottleneck shortest path for these weights. Using this path as the seam, rather than a more conventional shortest path, causes their system to find a seam that is difficult to discern at all of its points, rather than allowing it to trade off greater visibility in one part of the image for lesser visibility elsewhere. A solution to the minimax path problem between the two opposite corners of a grid graph can be used to find the weak Fréchet distance between two polygonal chains. Here, each grid graph vertex represents a pair of line segments, one from each chain, and the weight of an edge represents the Fréchet distance needed to pass from one pair of segments to another. If all edge weights of an undirected graph are positive, then the minimax distances between pairs of points (the maximum edge weights of minimax paths) form an ultrametric; conversely every finite ultrametric space comes from minimax distances in this way. A data structure constructed from the minimum spanning tree allows the minimax distance between any pair of vertices to be queried in constant time per query, using lowest common ancestor queries in a Cartesian tree. The root of the Cartesian tree represents the heaviest minimum spanning tree edge, and the children of the root are Cartesian trees recursively constructed from the subtrees of the minimum spanning tree formed by removing the heaviest edge. The leaves of the Cartesian tree represent the vertices of the input graph, and the minimax distance between two vertices equals the weight of the Cartesian tree node that is their lowest common ancestor. Once the minimum spanning tree edges have been sorted, this Cartesian tree can be constructed in linear time.
Directed graphs In directed graphs, the maximum spanning tree solution cannot be used. Instead, several different algorithms are known; the choice of which algorithm to use depends on whether a start or destination vertex for the path is fixed, or whether paths for many start or destination vertices must be found simultaneously.
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