Unformatted text preview: Finally, for each of these (ﬁnal) trees, perform a ﬁnd with path compression on the deepest node, and show the resulting tree. 3. (10 pts) Suppose we start with n singleton nodes, and perform an arbitrary sequence of unions using the union by height rule—that is, always make the shallower tree the child. Then, give a proof (reasoning) that the worstcase depth of any tree is O (log n ). 4. (10 pts) Give a proof for the following assertion: if all of the unions precede all the ﬁnds, then the disjoint set algorithm with path compression requires O ( n ) time, even if unions are done arbitrarily....
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 Winter '11
 SURI
 Computational complexity theory, Disjoint sets, Disjointset data structure

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