Kruskal's and Prim's

• These two algorithms build a minimal spanning tree
  • This is a tree that connects all nodes of a tree such that
  • $ w(T) = \sum\limits_{(u,v)\in T} w(u,v) $ is minimized.
• Review homework 4 (Prims)
• Kruskal's
  • For undirected graphs.

  • 
    KRUSKAL(G = (V,E) )
      A = {} 
     For each $v \in V $ do
       v.set = v
     Sort E lowest to highest cost
     For each $e=(u,v) \in E$ do
       if Find(u)  $\ne$ Find(v) 
         A = A ∪ {(u,v)}
         Union(u, v)
     Return A 
     

  • It is possible to implment Union and Find in $O(\log v)$
  • And to implement line 6 in the same time.
  • Therefore since $|V|^2 \le |E|$, the running time is $O(|E| \log |E|)$
•