Shortest path problem

In graph theory, the 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. An example is finding the quickest way to get from one location to another on a road map; in this case, the vertices represent locations and the edges represent segments of road and are weighted by the time needed to travel that segment.

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

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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:

• The single-source shortest path problem is a more general problem, in which we have to find shortest paths from a source vertex v to all other vertices in the graph.
• The all-pairs shortest path problem is an even more general problem, in which we have to find shortest paths between every pair of vertices v, v' in the graph.

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:

• 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* search algorithm solves for single source shortest paths using heuristics to try to speed up the search
• 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.
• Perturbation theory; finds (at worst) the locally shortest path

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 lighthearted 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".^[1]

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 (see P = NP problem) . The problem of finding the longest path in a graph is also NP-complete.

References

1. ^ Danny Z. Chen. Developing Algorithms and Software for Geometric Path Planning Problems. ACM Computing Surveys 28A(4), December 1996.

• Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Chapters 24: Single-Source Shortest Paths, and 25: All-Pairs Shortest Paths, pp.580–642.de:Kürzester Pfad

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