Kruskal's Algorithm

Kruskal's Algorithm
$\;\;\;\;\;$repeat
$\;\;\;\;\;$ $\;\;\;\;\;$find a light edge (u,v) between two unconnected components
$\;\;\;\;\;$ $\;\;\;\;\;$A = A $\cup$ {(u,v)}
$\;\;\;\;\;$until all edges have been considered

• Sort the edges by weight
• Use disjoint sets for speed (union by rank and path compression)

MST-Kruskal(G, w)		; G = (V, E)
1	A = {}
2	foreach v in V	; O(V)
3	$\;\;\;\;\;$MakeSet(v)
4	sort edges E by nondecreasing weight w	; O(E lg E)
5	foreach edge (u,v) in E, in order	; m = |E| operations
6	$\;\;\;\;\;$if FindSet(u) $\neq$ FindSet(v)	; n = |V| keys
7	$\;\;\;\;\;$then A = A $\cup$ {(u,v)}	; O(m $\alpha$(m,n))
8	$\;\;\;\;\;$ $\;\;\;\;\;$Union(u,v)	; O(E $\alpha$(E,V))
9	return A	; $\alpha$(E,V) = O(lg E)

T(V,E) = O(V) + O(E lg E) + O(E lg E), V = O(E)

= O(E lg E)