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Computer Algorithms

Submitted by:

Rishi Jethwa

Suvarna Angal


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Contents Red-Black Trees

Basics

roperties Rotations !nsertions

"nion #ind Algorithms $inked $ist Representation "nion By Rank ath compression


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Red-Black Trees RB tress is a binary tree with one

e%tra bit o& storage per node' its

color( which can be either R)* orB$AC+,

!ts data structure &or binary search

tree with only dierence that thetrees are appro%imately balanced,


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Red-Black Trees A binary tree is a red-black tree i& it

satis.es &ollowing rules &or red-black tree,

)very node is either red or black, The root is always black, $ea& nodes are black, !& a node is red( then both its children are

black, The number o& black nodes on every path are

same,


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Red-Black Trees roperties o& red-black trees

Suppose number o& black nodes are /0(

then the minimum height can be /0 andma%imum height o& the tree can be at most/1, 2ence the ma%imum can be at most /less than twice o& its minimum height,

3a%imum path length is 45log n6, $ookup &or searches are good( 45log n6, !nsertion and deletion are not an overhead

e%actly( comple%ity is 45log n6,


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Red-Black Trees Rotation o& red-black trees,

A structural change to the red-black trees, !nsertion and deletion modi&y the tree( the

result may violate the properties o& red-black trees, To restore this propertiesrotations are done,

7e can have either o& le&t rotation or rightrotation,


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Red-Black Trees

b

c

d e

a

The above diagram depicts left and right rotations

c

eb

a d

Left rotation

Right rotation

Here in right diagram a < b < d < c < e


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Red-Black TreesInsertion in red black trees.

11

2 14

151 7

5

4

8

riginal tree

!"mber 4 added


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Red-Black Trees The idea to insertion is that we traverse

the tree to see where it .ts( assume it

.ts at end ( so the idea is to traverse upagain,

Coloring rule while insertion, $ook at the &ather node( i& it is red and the

uncle node is red too and i& the grand&athernode is black ( then make &ather and uncleas black and grand&ather as red,


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Red-Black TreesDiagram depicting rule for

insertion mentioned in the

previous slide.


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Red-Black TreesInsertion example

11

2 14

151 7

5 8

4 #iolation of r"le$ after 4 added to the tree%


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11

2 14

151 7

5 8

4 &ase 1


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11

7 14

152 8

4

1 5

&ase 2


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2

4

1 5

77

11

14

15

8

&ase '


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"nion 4peration

1 2

43

Union: Merge 2 sets

and create a new set

Initially each number is a set by itself.

From n singleton sets gradually merge to form a set.

After n1 union o!erations we get a single set of n

numbers.

Union o!eration is used for merging sets in "rus#al$s

algorithm


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#ind operation )very set has a name

Thus #ind5number6 returns name o&the set,

er&ect application in +ruskal8salgorithm when there is a new

edge added, *iscard the alreadyaccounted &or edge,


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$inked $ist Representation

Represent each set using a linkedlist

#irst ob9ect in each linked listserves as the set8s name

)ach list maintains pointers head(

to the representative( and tail( tothe last ob9ect in the list,


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$inked $ist Representation

2 ' (

5 7

)%trapointers

pointing tothe head

7hen unitingthese sets(

pointers &or nodes; and < will have

to be madepointing to ,


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*rawbacks :

By using this representation( #indwill take constant time 45/6,

But "nion takes linear time as a&terunion all the pointers have to beredirected to the head,


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*evelop new datastructure

2

%

3

&

' Add the !ointer

!ointing from

new set$s head

to the old old

one.

1

4Union of 1 and 4


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Combination o& the sets

2

%

3

&

'

1

4

(hile combining) we can ha*e

a !ointer from 1 to 2 or from 2to1.

+ut we choose the one from 1

to 2.

,his gi*es us a balanced

structure. ,he highest ho!

remains 2.


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"nion by Rank Algorithm

The root o& the tree with &ewer nodes ismade to point to the root o& the tree

with more nodes, #or each node( a rank is maintained that

is an upper bound on the height o& thenode,

!n union by rank( the root with smallerrank is made to point to the root withlarger rank during a "=!4= operation,


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$ongest path lengthunions:

7hen we always take singleton

sets and keep merging the sets weget a star structure,

Time taken: n .nds and n-/ unions,

This is best case,


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7orst case

3erge sets o& e>ual path lengths,

32

1

4

&'

%

-

ere) the !ath length becomes 3.

For n1 unions and n finds:

Union: /0n

Finds: !ath lengths can get biggerso) /0n log n.

,otal sum: /0n log n.


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ath Compression

#or union by rank(

1

Log n

)est &ase

*orst &ase


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Algorithm &or athCompression

/stwalk: #ind the name o& the set ,Take a walk until we reach the root,

ndwalk: Retrace the path and 9oinall the elements along the path tothe root using another pointer,

This enables &uture .nds to takeshorter paths,


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ath compression

/

? root / ?

root

Be&ore #ind: )achnode has pointer to

its parent

A&ter #ind: )ach

node points directlyto the root


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Comparisons o& &unctions

#5n6Dn

$og n

$ogn

5n6


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Calculations

A?56 D A5A5A5666

7hen kD/( A/596 D 9H/

#or k D( A596 D A/5A/5I, A/59666

A596 D 9H/59H/6 -/

"sing the above analysis: A?5/6 D 0

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