Disjoint sets

Disjoint Sets Data Structure (Chap. 21) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Multiple Operations ,[object Object],[object Object],[object Object],[object Object]

An Application of Disjoint-Set ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Linked-List Implementation ,[object Object],[object Object],[object Object],[object Object]

Linked-lists for two sets head tail g head tail c head Set { c , h , e } Set { f , g } UNION of two Sets e tail c h e f f g h

UNION Implementation ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Weighted-Union Heuristic ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Disjoint-set Implementation: Forests ,[object Object],d d h e c c Set { c , h , e } Set { f , d } UNION c f h e c c c f

Straightforward Solution ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Union by Rank & Path Compression ,[object Object],[object Object]

Path Compression f e d c f e d c

Algorithm for Disjoint-Set Forest ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Worst case running time for m MAKE-SET, UNION, FIND-SET operations is: O ( m  ( n )) where  ( n )  4. So nearly linear in m . UNION( x , y ) 1. LINK(FIND-SET( x ),FIND-SET( y ))

Analysis of Union by Rank with Path Compression (by amortized analysis) ,[object Object],[object Object],[object Object],[object Object]

A very quickly growing function and its inverse ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Quickness of Function A k ( j )’s Increase ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

How Quick A k ( j ) Increase ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Inverse of A k ( n ):  ( n ) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

O ( m  ( n )) bound: Property of Ranks ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

O ( m  ( n )) bound proof ,[object Object],[object Object],[object Object],[object Object],[object Object]

Potential Function ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

level( x ) and iter( x ) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Relations among rank [ p [ x ]], level( x ) and iter( x ) ,[object Object],[object Object]

Properties for Potential Function  q ( x ) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Potential Changes of Operations ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Amortized Costs of Operations ,[object Object],[object Object],[object Object]

Amortized Costs of Operations (cont.) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Amortized Costs of Operations (cont.) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Proof of Lemma 21.12 (cont.) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Proof of Lemma 21.12 (cont.) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Upper bound for Disjoint-sets ,[object Object],[object Object]

Summary ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

A typical example using Disjoint Set ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Disjoint sets

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Disjoint sets