Directed graph

In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph, or set of vertices connected by edges, where the edges have a direction associated with them. In formal terms, a directed graph is an ordered pair G = (V, A) (sometimes G = (V, E)) where

It differs from an ordinary or undirected graph, in that the latter is defined in terms of unordered pairs of vertices, which are usually called edges, arcs, or lines.

A directed graph is called a simple digraph if it has no multiple arrows (two or more edges that connect the same two vertices in the same direction) and no loops (edges that connect vertices to themselves). A directed graph is called a directed multigraph or multidigraph if it may have multiple arrows (and sometimes loops). In the latter case the arrow set forms a multiset, rather than a set, of ordered pairs of vertices.

Dijkstra's algorithm

Dijkstra's algorithm is an algorithm for finding the shortest paths between nodes in a graph, which may represent, for example, road networks. It was conceived by computer scientist Edsger W. Dijkstra in 1956 and published three years later.

The algorithm exists in many variants; Dijkstra's original variant found the shortest path between two nodes, but a more common variant fixes a single node as the "source" node and finds shortest paths from the source to all other nodes in the graph, producing a shortest-path tree.

For a given source node in the graph, the algorithm finds the shortest path between that node and every other. It can also be used for finding the shortest paths from a single node to a single destination node by stopping the algorithm once the shortest path to the destination node has been determined. For example, if the nodes of the graph represent cities and edge path costs represent driving distances between pairs of cities connected by a direct road, Dijkstra's algorithm can be used to find the shortest route between one city and all other cities. As a result, the shortest path algorithm is widely used in network routing protocols, most notably IS-IS and Open Shortest Path First (OSPF). It is also employed as a subroutine in other algorithms such as Johnson's.

Floyd–Warshall algorithm

In computer science, the Floyd–Warshall algorithm is an algorithm for finding shortest paths in a weighted graph with positive or negative edge weights (but with no negative cycles). A single execution of the algorithm will find the lengths (summed weights) of the shortest paths between all pairs of vertices, though it does not return details of the paths themselves. Versions of the algorithm can also be used for finding the transitive closure of a relation , or (in connection with the Schulze voting system) widest paths between all pairs of vertices in a weighted graph.

History and naming

The Floyd–Warshall algorithm is an example of dynamic programming, and was published in its currently recognized form by Robert Floyd in 1962. However, it is essentially the same as algorithms previously published by Bernard Roy in 1959 and also by Stephen Warshall in 1962 for finding the transitive closure of a graph, and is closely related to Kleene's algorithm (published in 1956) for converting a deterministic finite automaton into a regular expression. The modern formulation of the algorithm as three nested for-loops was first described by Peter Ingerman, also in 1962.