Dijkstra's

• Given A graph $G=\{V,E\}$ and $s \in V, f \in V$, the least cost path from s to f.
• A greedy solution was proposed by Edsger Dijkstra in 1959.

  Set the distance to all vertexes to be ∞
  Set the cost to s to be 0
  while not finished
      Select the vertex with the lowest cost not yet explored
      If this vertex is f, stop
      else update the costs to all vertexes adjacent to this one
      

• The greedy choice is to select the current lowest cost vertex.
  • In the start, this is easy.
  • After this, you need to maintain a data structure to support rapid selection of the lowest cost path.
• An algorithm

  
  Dijkstra(G,s,f)
   For each v in V do
     v.parent = null
     v.cost = ∞
     Q.insert(v)
   EndFor
   s.cost =0
   Q.DecreaseKey(s)
   While Q.size > 0 and f has not been selected
      u = Q.ExtractMin
      For each v i u.Adj()
        if v.cost > u.cost + weight(u,v)
           v.cost = u.cost + weight(u,v)
           Q.DecreaseKey(v)
           v.parent = u
        EndIf
      EndFor
   EndWhile 
   

• Trace this on
• Note, what would happen if negative edges were allowed?