Notes

• Minimal spanning trees are trees that connect all vertices in a weighted graph with the minimal cost.
• They are an effective way to connect all points
  • IN a circuit
  • In an electric grid
  • In a network.
• The following discussion assumes a connected graph
  • Or a graph where you can get from every vertex to every other vertex.
• Prim's and Kruskal's algorithms both build minimal spanning trees using a greed approach.
• Prims takes a local approach, exploring out from a single node, adding in the least expensive edge that adds to the current tree.
• Kruskal's takes a global approach, adding in the least expensive edge that belongs in the tree.
• Both can be implemented in O(E log₂V) and Prim's can be implemented in O(E + V log₂V) with advanced data structures.
• w is the weight function (ie edge costs)
• Kruskal's
  • Kruskal's is stated in terms of sets.
  • We form a collection of sets, each containing a single vertex.
  • We then find the least expensive way to merge two sets by finding the lowest cost path between them.
  • The trick is to find a data structure that makes this all work.

  • MST-KRUSKAL(G,w)
    
     A ← {}
     for each vertex v ∈ G.V
        MAKE-SET(v)
     Sort the edges of G.E smallest to largest
     for each edge e=(u,v) ∈ G.E smallest to largest
         S = FIND-SET(u), T= FIND-SET(v)
         if S ≠ T
            A = A ∪{(u,v)}   // add the edge to the tree
            UNION(S, T)
     return A 
     

  • The performance is:
    • Line 1 is O(1)
    • Line 2-3 is at least O(V), depending on how we represent sets.
    • Line 4 O(E log₂E)
    • Line 5 is O(E)
      • In some discussions we have not had, we can do the set operations in nearly constant time, or O(α(V))
      • So the loop is O(E) for all intense and purposes.
    • Since the graph is connected, |E| ≤ |V| and |E| ≤ |V|².
      • If we use the latter, log₂(|E|) = 2 log₂ (|E|)
    • So we can state the performance of Kruskal's O(E log₂V)