spTreeBellmanFord: Shortest path tree with Bellman-Ford algorithm

Description

The spTreeBellmanFord function computes the shortest path tree of an undirected or directed graph with the Bellman-Ford algorithm.

Usage

spTreeBellmanFord(nodes, arcs, source.node = 1, directed = TRUE)

Arguments

nodes

vector containing the nodes of the graph, identified by a number that goes from $1$ to the order of the graph.

arcs

matrix with the list of arcs of the graph. Each row represents one arc. The first two columns contain the two endpoints of each arc and the third column contains their weights.

source.node

number pointing the source node of the graph. It's node $1$ by default.

directed

logical value indicating whether the graph is directed (TRUE) or not (FALSE).

Value

tree.nodes: vector containing the nodes of the shortest path tree.
tree.arcs: matrix containing the list of arcs of the shortest path tree.
stages: number of stages required.
distances: vector with distances from source to other nodes

Details

The Bellman-Ford algorithm gets its name for two of the developers, Richard Bellman y Lester Ford Jr., that published it in 1958 and 1956 respectively. The same algorithm also was published independently in 1957 by Edward F. Moore.

The Bellman-Ford algorithm can compute the shortest path from a source node to the rest of nodes that make a connected graph, directed or not, with weights that can be negatives. If the graph is connected and there isn't negative cycles, the algorithm always finds a shortest path tree.

References

Bellman, Richard (1958). "On a routing problem". Quarterly of Applied Mathematics 16, 87-90.

Ford Jr., Lester R. (1956). Network Flow Theory. Paper P-923. Santa Monica, California: RAND Corporation.