Weighted Union Find Analysis

• Finds with weighted union are O(max up-tree height)

• But, an up-tree of height h with weighted union must have at least 2h nodes

• ?, 2max height = n and

max height = log n

• So, find takes O(log n)

Base case: h = 0, tree has 20 = 1 node

Induction hypothesis: assume true for h < h?

A merge can only increase tree height by one over the smaller tree. So, a tree of height h?-1 was merged with a larger tree to form the new tree. Each tree then has ? 2h?-1 nodes by the induction hypotheses for a total of at least 2h? nodes. QED.

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