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In graph theory and theoretical computer science, the longest path problem is the problem of finding a simple path of maximum length in a given graph.A path is called simple if it does not have any repeated vertices; the length of a path may either be measured by its number of edges, or (in weighted graphs) by the sum of the weights of its edges.
Dijkstra's algorithm finds the shortest path from a given source node to every other node. [7]: 196–206 It can be used to find the shortest path to a specific destination node, by terminating the algorithm after determining the shortest path to the destination node. For example, if the nodes of the graph represent cities, and the costs of ...
Edmonds' algorithm (also known as Chu–Liu/Edmonds' algorithm): find maximum or minimum branchings; Euclidean minimum spanning tree: algorithms for computing the minimum spanning tree of a set of points in the plane; Longest path problem: find a simple path of maximum length in a given graph; Minimum spanning tree. Borůvka's algorithm ...
The algorithm described so far only gives the length of the shortest path. To find the actual sequence of steps, the algorithm can be easily revised so that each node on the path keeps track of its predecessor. After this algorithm is run, the ending node will point to its predecessor, and so on, until some node's predecessor is the start node.
Pointer jumping or path doubling is a design technique for parallel algorithms that operate on pointer structures, such as linked lists and directed graphs. Pointer jumping allows an algorithm to follow paths with a time complexity that is logarithmic with respect to the length of the longest path.
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.
Breadth-first search can be used to solve many problems in graph theory, for example: Copying garbage collection, Cheney's algorithm; Finding the shortest path between two nodes u and v, with path length measured by number of edges (an advantage over depth-first search) [14] (Reverse) Cuthill–McKee mesh numbering
Find the path of minimum total length between two given nodes and . We use the fact that, if R {\displaystyle R} is a node on the minimal path from P {\displaystyle P} to Q {\displaystyle Q} , knowledge of the latter implies the knowledge of the minimal path from P {\displaystyle P} to R {\displaystyle R} .