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The Dis oint Set Class

SAMREEN DHILLONCS W3137

SPRING 08

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Overview

E uivalence Relations

The Dynamic Equivalence Problem Simple algorithms for solving

Find

Union

Path Compression

Example

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Equivalence Relations (~)

Relation which satisfies the three ro erties:

Reflexive - a R a, for all a S Symmetric - a R b if and only if b R a

Trans t ve a R an R c mp es t at a R c

xamp es: Electrical Connectivity

country

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The dynamic Equivalence Problem

Given an e uivalence relation ~, decide ifa~b for

anya andb Solving via a 2-d array of booleans

What if relation is implicit?

Equivalence class of an element a S is a subset ofSwhich contains all the elements that are related to a

. Disjoint sets

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disjoint set union/find algorithm

find: returns the e uivalence class for a iven

element. union: adds relations Merges two equivalence classes

Dynamic : the sets change during the course of thea gor m

serva ons: Values doesnt matter ,location matters

==

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Data structure for Solving the problem

find in constant worst-time?

union in constant worst-time? Consider Each set as a tree

Initially all elements are disjoint

- - - - - - - -

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Union operation

union : make arent link of root of one tree to the

root of the other tree

Union(4,5)

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Union(6,7)

Union(4,6)

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DisjSets, find and union operations

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Smart Union Algorithms

union-b -size

Always make smaller tree subtree of the larger No extra space : instead of -1 set as -(size of tree) for root

Union(3,4)

- - - -

0 1 2 3 4 5 6 7

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Smart Union Algorithms

union-b -hei ht: Make a shallow tree a subtree of a

deep tree Update height only if two equally deep trees are joined

Instea o -1 , - e g t or root

-1 -1 -1 4 -3 4 4 6

0 1 2 3 4 5 6 7

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Smart find algorithm Path compression

Inde endent of union al orithm

Make every node on the path from x to root have itsparent changed to root.

Note that: union-by-height => union-by-rank

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Path Compression

A ter in (14)

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Path Compression

O(MlogN) in Worst case

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Example : Maze generator

http://www.math.com/students/puzzles/mazegen/mazegen.html

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Maze Generator: Al orithm

1. Start with walls everywhere, except for the entrance

and exit)

2. Continually, choose a wall randomly, and knock it

down if the cells that the wall separates are not.

. cells are connected

4. We have a maze.

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Maze Generator: Al orithm

1. Start with walls everywhere, except for the entrance

and exit).

Use the union /find data structur e

n t a y, eac ce s n ts ow n equ va enceclass

Initial State: All wa lls up, all cells in their ow nset.

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Maze Generator: Al orithm

2. Continually, choose a wall randomly, and knock it

down if the cells that the wall separates are notalready connected to each other.

At som e point in the algor ithm : several w alldown sets have mer ed. If at this oint thewall between squa res 8 and 13 is rando m lyselected, this w all is not knocked do wn ,because 8 an d 13 are a lready connected.

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Maze Generator: Al orithm

3. Repeat this process until the starting and ending

cells are connected

Per orm ng tw o f indoper at on s 18 an13 are in different sets. Therefore, performing u n i o n to combinethem.

W all betwe en squar es 18 and 13 is ran dom lyselected. The wa ll is kno cked dow n, because 18

an 13 are not a rea y connecte . T eir setsare m erged.

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Maze Generator: Al orithm

4. We have a maze

Run ning Time: O (N log N)

Eventually, 24 walls are knocked dow n. Allelemen ts are in the sam e set.

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Questions?