Bellman-Ford

Notes

• Bellman-Ford finds a shortest path in a weighted graph which has the possiblity of negative weights.
• It detects negative weight cycles
  • And reports a fialure if such a cycle exists and is reachable from the source.

• BELLMAN-FORD(G,w,s)
  
  // initialize single source shortest path
   for all v ∈ G.V 
      v.d = ∞
      v.π = NIL
   s.d = 0
   for i = 1 to |G.V| - 1 
     for each edge (u,v) ∈ G.V
  // RELAX
        if v.d < u.d + w(u,v)
           v.π = u
           v.d = u.d + w(u,v)
   for each edge (u,v) ∈ G.V
      if v.d > u.d + w(u,v)
        return FALSE
   return TRUE
  
  

• Try this on this graph.
• We understand that the longest path in a graph will be |V|-1
• So we can argue that if a path keeps decreasing after |V|-1 iterations, it contains a negative weight cycle.
• Consider this graph
  • In the previous graph x->t->z was -2 + -3 + 7 = 2 so the cost to go over that cyclic-path mutliple times was 2.
  • But in the new graph x->t->z is -5 + -3 + 7 or -1
  • So if we run the algorithm again, the cost from x to t will soon become