Up-Trees

3/4/01


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Table of Contents

Up-Trees

(Review) A Set of Elements

(Review) Equivalence Relation

(Review) Another Equivalence Relation

The FIND Operation

FIND -- an Example

The UNION Operation

Time Complexity

Up-Trees

Up-Tree Example

FIND using Up-Trees

UNION using Up-Trees

Time Complexity using Basic Up-Trees

Improving on the Basic Implementation

Enhanced UNION

Enhanced UNION (Cont)

Enhanced FIND

Path Compression

Worst-Case Analysis of Running Time for UNION-FIND using Path-Compressed Up-Trees

Ackermann’s Function

Inverse Ackermann’s Function

Theorem

The Argument for the Theorem

Argument, continued: “Pennies”)

Argument: Rank Groups

Argument: Payment Rules

Argument: Counting Pennies

Argument: Counting Canadian Pennies

Counting Canadian Pennies (Cont)

Total Pennies

Theorem Again

Applying the UNION-FIND ADT to Constructing Minimum Spanning Trees for Graphs

Example Graph + Minimum Spanning Tree

Kruskal’s Algorithm

Author: Steven L. Tanimoto

Email: tanimoto@cs.washington.edu

Home Page: www.cs.washington.edu/people/faculty/tanimoto.html

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