Bellman–Ford algorithm

In computer science, Bellman-Ford is an algorithm used to compute the shortest distances and paths from a single node within a graph to all other nodes.

Usage
It is more robust than Dijkstra's algorithm because it is able to handle graphs containing negative edge weights. It may be improved by noting that, if an iteration of the main loop of the algorithm completes without making any changes, the algorithm can be immediately completed, as future iterations will not make any changes. Its runtime is  time, where  and  are the number of nodes and edges respectively.

Psuedocode

 function BellmanFord(list nodes, list edges, node source) is

     distance := list
     predecessor := list              // Predecessor holds shortest paths

     // Step 1: initialize 
     for each node n in nodes do
         distance[n] := inf           // Initialize the distance to all nodes to infinity
         predecessor[n] := null       // And having a null predecessor
          distance[source] := 0            // The distance from the source to itself is zero

     // Step 2: relax edges repeatedly     repeat |nodes|−1 times:
         for each edge (u, v) with weight w in edges do
             if distance[u] + w < distance[v] then
                 distance[v] := distance[u] + w
                 predecessor[v] := u

     // Step 3: check for negative-weight cycles''
     for each edge (u, v) with weight w in edges do
         if distance[u] + w < distance[v] then
             error "Graph contains a negative-weight cycle"

     return distance, predecessor

Category:Algorithms