Running Time

Now, CONNECTED is constant time and we repeat it |E| times.

JOIN is repeated |V|-1 times and can take up to |V|/2 time.

Too loose!

• Simple amortized analysis: Work is in relabeling nodes. How many times can a node u be relabeled?
• Each time it is relabeled, it joins a set at least twice as big as its old set. If it relabeled x times, what is the smallest set it could possibly be in?
• The final set size is |V|, so how many times can a node be relabeled?

Thus, for all calls to JOIN.

Total: SORT plus JOIN is .