| Shortest Path Problem |
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| CATEGORIES ABOUT SHORTEST PATH PROBLEM | |
| graph theory | |
| polynomial-time problems | |
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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 one element ''v'' of ''V'', find a path ''P'' from ''v'' to each ''v' '' of ''V'' so that : is minimal among all paths connecting ''v'' to ''v' ''. Sometimes it is called the single-pair shortest path problem, to distinguish it from the following generalizations:
Both these generalizations have significantly more performant algorithms in practice than simply running a single-pair shortest path algorithm on all relevant pairs of vertices. ALGORITHMS The most important algorithms for solving this problem are:
APPLICATIONS Shortest path algorithms are applied in an obvious way to automatically find directions between physical locations, such as driving directions on Web Mapping websites like Mapquest . If one represents a nondeterministic Abstract Machine as a graph where vertices describe states and edges describe possible transitions, shortest path algorithms can be used to find an optimal sequence of choices to reach a certain goal state, or to establish lower bounds on the time needed to reach a given state. For example, if vertices represents the states of a puzzle like a Rubik's Cube and each directed edge corresponds to a single move or turn, shortest path algorithms can be used to find a solution that uses the minimum possible number of moves. 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. A more spurious application is the games of " Six Degrees Of Separation " that try to find the shortest path in graphs like movie stars appearing in the same film. Other applications cited by Danny Z. Chen include " Operations Research , plant and facility layout, Robotics , Transportation , and VLSI design".Danny Z. Chen. Developing Algorithms and Software for Geometric Path Planning Problems. ''ACM Computing Surveys'' 28A(4), December 1996, http://www.acm.org/surveys/1996/ChenDeveloping/ RELATED PROBLEMS For shortest path problems in Computational Geometry , see Euclidean Shortest Path . The Traveling Salesman Problem is the problem of finding the shortest path that goes through every vertex exactly once, and returns to the start. Unlike the shortest path problem, this problem is NP-complete and, as such, Is Believed Not to be efficiently solvable. The problem of Finding The Longest Path in a graph is also NP-complete. REFERENCES
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