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The Floyd–Warshall algorithm is an example of dynamic programming, and was published in its currently recognized form by Robert Floyd in 1962. [3] However, it is essentially the same as algorithms previously published by Bernard Roy in 1959 [4] and also by Stephen Warshall in 1962 [5] for finding the transitive closure of a graph, [6] and is closely related to Kleene's algorithm (published ...
In mathematics, the transitive closure R + of a homogeneous binary relation R on a set X is the smallest relation on X that contains R and is transitive.For finite sets, "smallest" can be taken in its usual sense, of having the fewest related pairs; for infinite sets R + is the unique minimal transitive superset of R.
The Floyd–Warshall algorithm [5] can be used to compute the transitive closure of any directed graph, which gives rise to the reachability relation as in the definition, above. The algorithm requires (| |) time and (| |) space in the worst case. This algorithm is not solely interested in reachability as it also computes the shortest path ...
In all of these transitive closure algorithms, it is possible to distinguish pairs of vertices that are reachable by at least one path of length two or more from pairs that can only be connected by a length-one path. The transitive reduction consists of the edges that form length-one paths that are the only paths connecting their endpoints ...
Floyd–Warshall algorithm: solves the all pairs shortest path problem in a weighted, directed graph; Johnson's algorithm: all pairs shortest path algorithm in sparse weighted directed graph; Transitive closure problem: find the transitive closure of a given binary relation; Traveling salesman problem. Christofides algorithm; Nearest neighbour ...
Shortest path (A, C, E, D, F), blue, between vertices A and F in the weighted directed graph. In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized.
The k shortest path routing problem is a generalization of the shortest path routing problem in a given network. It asks not only about a shortest path but also about next k−1 shortest paths (which may be longer than the shortest path). A variation of the problem is the loopless k shortest paths.
Cut locus C(P) of a point P on the surface of a cylinder. A point Q in the cut locus is shown with two distinct shortest paths , connecting it to P.. In the Euclidean plane, a point p has an empty cut locus, because every other point is connected to p by a unique geodesic (the line segment between the points).