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Encyclopedia > Shortest path

In graph theory, the single-source shortest path problem is the problem of finding a path between two vertices such that the sum of the weights of its constituent edges is minimized. More formally, given a weighted graph (that is, a set V of vertices, a set E of edges, and a real-valued weight function f : E → R), and given further two elements n, n' of N, find a path P from n to n' so that A diagram of a graph with 6 vertices and 7 edges. ... In mathematics, a path in a graph is a sequence of vertices such that from each of its vertices there is an edge to the successor vertex. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...

is minimal among all paths connecting n to n' . The all-pairs shortest path problem is a similar problem, in which we have to find such paths for every two vertices n to n' .

A solution to the shortest path problem is sometimes called a pathing algorithm. The most important algorithms for solving this problem are:

• Dijkstra's algorithm — solves single source problem if all edge weights are greater than or equal to zero. Without worsening the run time, this algorithm can in fact compute the shortest paths from a given start point s to all other nodes.
• Bellman-Ford algorithm — solves single source problem if edge weights may be negative.
• A* algorithm (or A* pathing algorithm) — a heuristic for single source shortest paths.
• Floyd-Warshall algorithm — solves all pairs shortest paths.
• Johnson's algorithm — solves all pairs shortest paths, may be faster than Floyd-Warshall on sparse graphs.

A related problem is the traveling salesman problem, which is the problem of finding the shortest path that goes through every node exactly once, and returns to the start. That problem is NP-hard, so an efficient solution is not likely to exist. Dijkstras algorithm, named after its inventor, Dutch computer scientist Edsger Dijkstra, is an algorithm that solves the single-source shortest path problem for a directed graph with nonnegative edge weights. ... The Bellman-Ford algorithm computes single-source shortest paths in a weighted graph (where some of the edge weights may be negative). ... The A* search algorithm (pronounced A star) is a graph search algorithm that finds a path from a given initial node to a given goal node (or one passing a given goal test). ... Heuristic is the art and science of discovery and invention. ... In computer science, the Floyd-Warshall algorithm is an algorithm for solving the all-pairs shortest path problem in a weighted, directed graph (V, E) by multiplying an adjacency matrix representation of the graph multiple times. ... Johnsons algorithm is a way to solve the all pairs shortest path problem in a sparse weighted, directed graph. ... In the mathematical subfield of numerical analysis a sparse matrix is a matrix populated primarily with zeros. ... The traveling salesman problem or travelling salesman problem (TSP), also known as the traveling salesperson problem, is a problem in discrete or combinatorial optimization. ... In computational complexity theory, NP-hard (Non-deterministic Polynomial-time hard) refers to the class of decision problems that contains all problems H such that for all decision problems L in NP there is a polynomial-time many-one reduction to H. Informally this class can be described as containing...

In a networking or telecommunications mindset, this shortest path problem is sometimes called the min-delay path problem and usually tied with a widest path problem. e.g.: Shortest (min-delay) widest path or Widest shortest (min-delay) path. Results from FactBites:

 Shortest path problem - Wikipedia, the free encyclopedia (388 words) In graph theory, the single-source shortest path problem is the problem of finding a path between two vertices such that the sum of the weights of its constituent edges is minimized. The all-pairs shortest path problem is a similar problem, in which we have to find such paths for every two vertices n to n'. In a networking or telecommunications mindset, this shortest path problem is sometimes called the min-delay path problem and usually tied with a widest path problem.
 tutORial's Shortest Path Module (1423 words) In this framework a path is a sequence of cities and a feasible path is a path with the property that from any city on this path there is a direct link to the next city on the path. The length of a path is assumed to be equal to the sum of the lengths of the links between consecutive cities on the path. Accordingly, should we conclude that the length of the shortest path from node i to node j is equal to infinity, the implication would be that there is no (feasible) path from node i to node j.
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