Question

I was reading wikipeida and found Kruskal’s Pseudocode as the following:

KRUSKAL(G):    
    foreach v ∈ G.V:    
        MAKE_SET(v)    
    G.E = sort(G.E)    
    i = 0    
    while (i != |V|-1):      
        pick the next (u, v) edge from sorted list of edges G.E        
        if (FIND_SET(u) != FIND_SET(v)):          
            UNION(u, v)        
            i = i + 1 

I’m not quiet sure what FIND_SET() does, and Wikipedia has the follow description:

if that edge connects two different trees, then add it to the forest, combining two trees into a single tree.

So I guess it checks if two different trees are connected, but what does this really mean?

Answer 1

Initially, each vertex is in a set all by itself: There is a singleton set {v} for every vertex v. In the pseudo-code, these sets are the result of make_set(v).

For a given vertex v, the function find_set(v) gives you the set containing v.

The algorithm merges sets iteratively, so if {u}, {v} are singleton sets initially and there is an edge (u, v), then the algorithm replaces the two sets by their union {u, v}. Now both find_set(u) and find_set(v) will return that set.

The algorithm terminates after you’ve added |V| - 1 non-trivial edges, which is precisely the number of edges in a tree.