Kruskal's

• A minimal spanning tree is a minimal subset of edges in a weighted, undirected graph that connects every node to every other node without cycles.
• Kruskal's

• KRUSKAL(G=(V,E), w) 
  
   A = {}
   for each vertex $v \in V$
      MAKE-SET(v)
   sort the edges into nondecreasing order by weight w
   for each edge $(u,v) \in E$ in non-decreasing order
      if FIND-SET(u) ≠ FIND-SET(v)
        $A = A \cup \{u,v\}$
        UNION(FIND-SET(u), FIND-SET(v))
   return A
  
     

• 
• Running Time
  • Line 1 is O(1)
  • Line 2-3 depends on the data structure for a set.
  • Line 4 is $O(E\; log_2\; E)$
  • Line 5 is O(E)
  • Lines 6-8 again depend on the data structure for a set.
  • The set data structure
    • We will not have time to explore this.
    • But with careful implementation, we can achieve these in $O((V+E)\alpha (V))$
      • Where $\alpha(V)$ is a very slow growing function.
      • Specifically $\alpha(V) = O(log_2 V)
  • So the running time can be $O(E\; log_2\; E)$.
  • But observing that $E \le V^2$, $log_2(E) = 2 log_2 V$
  • So it can be written that it is $O(E\; log_2\; V)$