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Example of one of two shortest-path trees where the root vertex is the red square vertex. The edges in the tree are indicated with green lines while the two dashed lines are edges in the full graph but not in the tree. The numbers beside the vertices indicate the distance from the root vertex.
One example is the constrained shortest path problem, [16] which attempts to minimize the total cost of the path while at the same time maintaining another metric below a given threshold. This makes the problem NP-complete (such problems are not believed to be efficiently solvable for large sets of data, see P = NP problem ).
The minimum spanning tree for an entire graph is solvable in polynomial time.) Modularity maximization [5] Monochromatic triangle [3]: GT6 Pathwidth, [6] or, equivalently, interval thickness, and vertex separation number [7] Rank coloring; k-Chinese postman; Shortest total path length spanning tree [3]: ND3 Slope number two testing [8]
The following example shows how Suurballe's algorithm finds the shortest pair of disjoint paths from A to F. Figure A illustrates a weighted graph G. Figure B calculates the shortest path P 1 from A to F (A–B–D–F). Figure C illustrates the shortest path tree T rooted at A, and the computed distances from A to every vertex (u).
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 ...
Path (graph theory) Seven Bridges of Königsberg. Eulerian path; Three-cottage problem; Shortest path problem. Dijkstra's algorithm. Open Shortest Path First; Flooding algorithm; Route inspection problem; Hamiltonian path. Hamiltonian path problem; Knight's tour; Traveling salesman problem. Nearest neighbour algorithm; Bottleneck traveling ...
Shortest path problem. Bellman–Ford algorithm: computes shortest paths in a weighted graph (where some of the edge weights may be negative) Dijkstra's algorithm: computes shortest paths in a graph with non-negative edge weights; Floyd–Warshall algorithm: solves the all pairs shortest path problem in a weighted, directed graph
For example, given a graph G=(V,E), the shortest path p from a vertex u to a vertex v exhibits optimal substructure: take any intermediate vertex w on this shortest path p. If p is truly the shortest path, then it can be split into sub-paths p 1 from u to w and p 2 from w to v such that these, in turn, are indeed the shortest paths between the ...