Prim's Algorithm

Notes

• Prim's takes a starting point and explores out from there.
• Each vertex is marked with the "cost" to reach that vertex.
• These need to be stored in a structure (heap) that allows us to find the minimal edge, vertex to add to our expanding MST
• We will use a min-heap (priority queue) for Q.

• MST-PRIM(G,w,r)
  
   For each u ∈ G.V
       u.key = ∞
       u.π = NIL
   r.key = 0
   Q = G.V
   while not Q.Empty()
      u = Q.Extract-Min()
      for each v ∈ G.Adj(u)
          if v ∈ Q and w(u,v) < v.key
             v.π = u
             v.key = w(u,v)
   

• Performance
  • Line 1-3 are O(V)
  • Line 5 is O(V log₂ V) as we need to maintain the priority queue as we build it.
  • Since everything is added to the heap in line 5, line 6 is clearly O(V)
  • Line 7 is O(log₂V), as is line 11?
  • Line 8 is our old pal, O(E)
  • So 8-11 is O(E log₂V)
• The entire thing is O(E log₂ V)
• There is a specialized data structure, fibonacci heap, that can make line 11 O(1)
• So the algorithm can be implemented in O(E + V log₂V)