5 Rules 1 Red Black Tree Properties - A 1. Every Node Is Either - - PowerPoint PPT Presentation

Red Black Tree 5 Rules

1

Red Black Tree Properties - A

• 1. Every Node Is Either RED or BLACK
• 2. Every NILL Node Is BLACK  sometimes referred to as a "Null Leaf"

Red Black Tree Properties - B

• 3. Every RED Node Has Two Black Child Nodes (Or Is Childless)  There

Can't Be Two RED Nodes In A Row

• 4. Every Distance Path From Node X Down To A Leaf Node Has The Same

Number Of BLACK Nodes

• 5. The Root Node Is Always BLACK

Rotation Generalization Right Rotation

4

Right Rotation - 1/2

• There Exists Some Square Node

(somewhere in the tree)

• There Exists Some Circle Node

(Where Value In Circle Is Less Than The Value In Square)

• There Exists Some Subtrees:

A, B, & C A B C

• Putting Values In Nodes Is Not

Necessary, But Since I Did It To Demonstrate The AVL Rotations, I Will Do It For The Red Black.

Right Rotation - 2/2

• Simple Way To Rotate The Valid Binary

Search Tree Without Altering The Search Capabilities.

• Single Rotate Right  AVL

A B C 100 90 90 A C 100 B

Rotation Generalization Left Rotation

7

Left Rotation - 1/2

• There Exists Some Square Node

(somewhere in the tree)

• There Exists Some Circle Node

(Where Value In Circle Is Greater Than The Value In Square)

• There Exists Some Subtrees:

A, B, & C A B C

• Putting Values In Nodes Is Not

Necessary, But Since I Did It To Demonstrate The AVL Rotations, I Will Do It For The Red Black.

Left Rotation - 2/2

• Simple Way To Rotate The Valid Binary Search

Tree Without Altering The Search Capabilities.

• Single Rotate Left  AVL

A 90 100 C B A B C 90 100

Violation 1L RED UNCLE Case

(On Left)

10

Easy Case 1L - RED UNCLE Case - 1 (P On Left)

• The Color Distance Will Not

Change The Number of Black Down Any Branch

• If We Make The New Node RED, It

Minimizes The Number Of Bad Things That Can Happen:

• Only Thing That Can Go Wrong Is

Two Red Nodes In A Row

50 150 125 100

• Label P  Pf  Pg  Pu

P Pf Pg Pu

• This RED UNCLE Case Is

Occurring As We Walk Our Way Up The Tree.

• Label The Distance Paths

c d e f b

a

Easy Case 1L - RED UNCLE Case - 2 (P On Left)

• Easy Case 1L: UNCLE is RED  No Rotation Required!

50 150 125 100

c

• ac = 1

d

• ad = 1

e

• ae = 1

f

• af = 1
• ab = 1

b

a

• Record The Relative Distances Down

The Various Paths In The Tree:

Easy Case 1L - RED UNCLE Case - 3 (P On Left)

• Easy Case 1L: UNCLE is RED  No Rotation Required!

50 150 125 100

c d e f b

a

• How Fix?

P Pf Pg Pu

• ab = 1
• ac = 1
• ad = 1
• ae = 1
• af = 1
• Pf  Color = BLACK Pu  Color = BLACK Pg  Color = RED

50 150 125 100

c d e f b

a

P Pf Pg Pu

• Double Check The Distances ab=1, ac=1, ad=1, ae=1, af=1  YEA!

Case 1L Rule Summary

50 150 125 100

c d e f b

a

P Pf Pg Pu

50 150 125 100

c d e f b

a

P Pf Pg Pu

Violation 1R RED UNCLE Case

(On Right)

15

Easy Case 1R - RED UNCLE Case - 1 (P On Right)

• The Color Distance Will Not

Change The Number of Black Down Any Branch

• If We Make The New Node RED, It

Minimizes The Number Of Bad Things That Can Happen:

• Only Thing That Can Go Wrong Is

Two Red Nodes In A Row

50 200 150 125

• Label P  Pf  Pg  Pu

P Pf Pg Pu

• This RED UNCLE Case Is

Occurring As We Walk Our Way Up The Tree.

• Label The Distance Paths

c d e f b

a

Easy Case 1R - RED UNCLE Case - 2 (P On Right)

• Easy Case 1R: UNCLE is RED  No Rotation Required!
• ac = 1
• ad = 1
• ae = 1
• af = 1
• ab = 1
• Record The Relative Distances Down

The Various Paths In The Tree:

50 200 150 125

• Label P  Pf  Pg  Pu

P Pf Pg Pu

• Label The Distance Paths

c d e f b

a

Easy Case 1R - RED UNCLE Case - 3 (P On Right)

• Easy Case 1L: UNCLE is RED  No Rotation Required!
• How Fix?
• ab = 1
• ac = 1
• ad = 1
• ae = 1
• af = 1
• Pf  Color = BLACK Pu  Color = BLACK Pg  Color = RED
• Double Check The Distances ab=1, ac=1, ad=1, ae=1, af=1  YEA!

50 200 150 125

c d e f b

a

P Pf Pg Pu

50 200 150 125

c d e f b

a

P Pf Pg Pu

Case 1R Rule Summary

50 200 150 125

c d e f b

a

P Pf Pg Pu

50 200 150 125

c d e f b

a

P Pf Pg Pu

RED Uncle Can Occur With Insertion RED UNCLE Case

(On Left)

20

• Distance Property = 2 For All - If Count Null

Easy Case 1L - RED UNCLE Case - 1 (P On Left)

• Easy Case 1L: UNCLE is RED  No Rotation Required!

200 250

• Add 50 to Tree - Make it Red
• Will Not Change The Number of

Black Down Any Branch

• Minimize The Number Of Bad

Things That Can Happen:

• Only Thing That Can Go Wrong Is

Two Red Nodes In A Row

50 150 125 100

• Label P  Pf  Pg  Pu

P Pf Pg Pu

Easy Case 1L - RED UNCLE Case - 2 (P On Left)

• Easy Case 1L: UNCLE is RED  No Rotation Required!

200 250 50 150 125 100

• How Fix?
• THIS IS EASY CASE - Pu Is Red

(125)

• Might Become Easier If We Don't

Try To Fix Everything At Once Start With This Portion

• Make Pg RED

Make Pf BLACK Make Pu BLACK

• This Shifts The Problem Up Two

Generations! P Pg Pf Pu

Easy Case 1L - RED UNCLE Case - 3 (P On Left)

200 250 50 150 125 100

• How Fix?
• How Fix?
• Change Pf To BLACK
• 2 BLACK UNCLE Cases: One Requires A Rotation

One Requies A Preliminary Rotation And A Second Rotation

• Distance Property = 3 For All

P Pf

RED Uncle Can Occur With Insertion Add To Tree Violation 1R RED UNCLE Case

(On Right)

24

• Distance Property = 2 For All If Count Null

Easy Case 1R - RED UNCLE Case - 1 (P On Right)

• Easy Case 1R: UNCLE is RED  No Rotation Required!

100 50 150 200

• Add 225 to Tree - Make it Red
• Will Not Change The Number of

Black Down Any Branch

• Minimize The Number Of Bad

Things That Can Happen:

• Only Thing That Can Go Wrong Is

Two Red Nodes In A Row

125 225

• Label P  Pf  Pg  Pu

P Pf Pg Pu

Easy Case 1R - RED UNCLE Case - 2 (P On Right)

100 50 150 200

• How Fix?
• THIS IS EASY CASE - Uncle Of P

Is Red (125)

• Might Become Easier If We Don't

Try To Fix Everything At Once Start With This Portion

• Make Pg RED

Make Pf BLACK Make Pu BLACK

125 225

• This Shifts The Problem Up Two

Generations! Pg Pf P Pu

Easy Case 1R - RED UNCLE Case - 3 (P On Right)

100 50 150 200

• How Fix?
• Change Pf To BLACK

125 225

• 2 BLACK UNCLE Cases: One Requires A Rotation

One Requies A Preliminary Rotation And A Second Rotation

• Distance Property = 3 For All

Pf P

Violation 3L BLACK UNCLE Case

With One Rotation

(On Left)

28

Case 3L - BLACK UNCLE One Rotation Class- 1 (P On Left)

200 250

• This BLACK UNCLE Case Is

Occurring As We Walk Our Way Up The Tree.

• We Have Already Established That

The Distance Property Is OK

125 100

• ab = 1

a b c

• ac = 1

d

• ad = 1

e

• ae = 2

f

• af = 2
• GOAL: Fix The RED CHILD With RED

Father Problem Without Messing Up The Distance Properties.

Case 3L - BLACK UNCLE One Rotation Class- 2 (P On Left)

200 250

• Solve RED RED Problem?

125 100

a b c d e f

• ab = 1
• ac = 1
• ad = 1
• ae = 2
• af = 2
• Label P  Pf  Pg  Pu

P Pf Pg Pu b c P Pf 125

100

• Promote Pf Up

a

200 250

e f Pg Pu

• Hook Pg Right of Pf

d

• Hook up 'd'
• Swap Colors Pg Pf
• Double Check The Distances ab=1, ac=1, ad=1, ae=2, af=2  YEA!

Case 3L Rule Summary f

200 250 125 100

a b c d e P Pf Pg Pu P Pu b c Pf 125

100

a

250

e f Pg d

200

Violation 3R BLACK UNCLE Case

With One Rotation

(On Right)

32

Case 3R - BLACK UNCLE One Rotation Class- 1 (P On Right)

200 250

• This BLACK UNCLE Case Is

Occurring As We Walk Our Way Up The Tree.

• We Have Already Established That

The Distance Property Is OK

125 300

• ab = 2

a b c

• ac = 2

d

• ad = 1

e

• ae = 1

f

• af = 1
• GOAL: Fix The RED CHILD With RED

Father Problem Without Messing Up The Distance Properties.

• Hook Pg Left of Pf

200 125

b c Case 3R - BLACK UNCLE One Rotation Class- 2 (P On Right)

200

• Solve RED RED Problem?

125

a b c

250 300

d e f

• ab = 2
• ac = 2
• ad = 1
• ae = 1
• af = 1
• Label P  Pf  Pg  Pu

P Pf Pg Pu a

• Double Check The Distances ab=2, ac=2, ad=1, ae=1, af=1  YEA!
• Promote Pf Up

250 300

e f P Pf

• Hook up 'd'

d

• Swap Colors Pg Pf

Case 3R Rule Summary

200 125

a b c

250 300

d e f P Pf Pg Pu

200 125

b c a

250 300

e f P Pf

Violation 2L BLACK UNCLE Case

With Two Rotations

(On Left)

36

Case 2L - BLACK UNCLE Two Rotations Class- 1 (P On Left)

200 250 125 150

• Label P  Pf  Pg  Pu

P

Pf Pg Pu

• This BLACK UNCLE Case Is

Occurring As We Walk Our Way Up The Tree.

• We Have Already Established That

The Distance Property Is OK c

• ac = 1

d

• ad = 1

e

• ae = 2

f

• af = 2

a b

• ab = 1
• Case 2L is going to turn into Case 3L.
• This Is The Worst Possible Case

Case 2L - BLACK UNCLE Two Rotations Class- 2 (P On Left)

200 250 100 125

• ab = 1
• ac = 1
• ad = 1
• ae = 2
• af = 2

P

Pf Pg Pu c d e f a b

• Might Become Easier If We Don't

Try To Fix Everything At Once Start With This Portion

• Hook P To The Left Of Pg

125 P

d

200 250

Pg Pu e f a

1

100

Pf b

• Hook Pf To The Left Of P

c

• Hook up 'c'
• This Is Case 3L

Case 2L - BLACK UNCLE Two Rotations Class- 3 (P On Left)

• ab = 1
• ac = 1
• ad = 1
• ae = 2
• af = 2
• Swap Pointers Pf  P

1

125 Pf

d

200 250

Pg Pu e f a

100

P b c

Case 2L - BLACK UNCLE Two Rotations Class- 4 (P On Left)

• ab = 1
• ac = 1
• ad = 1
• ae = 2
• af = 2
• Swap Pointers Pf & P

2

125 Pf

d

200 250

Pg Pu e f a

100

P b c b c P Pf 125

100

• Do Case 3L Rotation

200 250

e f Pg Pu d a

• Double Check The Distances ab=1, ac=1, ad=1, ae=2, af=2  YEA!

Case 2L Rule Summary

200 250 125 150 P

Pf Pg Pu c d e f a b b c P Pf 125

100 200 250

e f Pg Pu d a

Violation 2R BLACK UNCLE Case

With Two Rotations

(On Right)

42

Case 2R- BLACK UNCLE Two Rotations Class- 1 (P On Right)

• This BLACK UNCLE Case Is

Occurring As We Walk Our Way Up The Tree.

• We Have Already Established That

The Distance Property Is OK c

• ac = 2

d

• ad = 1

e

• ae = 1

f

• af = 1

a b

• ab = 2
• Case 2R is going to turn into Case 3R.
• This Is The Worst Possible Case

200 125

a

300 250 P

Pf Pg Pu

Case 2R - BLACK UNCLE Two Rotations Class- 2 (P On Right)

• ab = 2
• ac = 2
• ad = 1
• ae = 1
• af = 1
• Might Become Easier If We Don't

Try To Fix Everything At Once Start With This Portion

1

• This Is Case 3L

c d e f a b

200 125

a

300 250 P

Pf Pg Pu

• Hook P To The Right Of Pg

d

250 P

c a b

200 125

Pg Pu

• Hook up 'e'

e

• Hook Pf To The Right Of P

f

300

Pf

Case 2R - BLACK UNCLE Two Rotations Class- 3 (P On Right) d

250 P

c a b

200 125

Pg Pu e f

300

Pf

• Swap Labels Pf  P
• This Is Case 3L
• ab = 2
• ac = 2
• ad = 1
• ae = 1
• af = 1

Case 2R - BLACK UNCLE Two Rotations Class- 4 (P On Right)

• Swap Labels Pf  P

2

• Double Check The Distances ab=2, ac=2, ad=1, ae=1, af=1  YEA!
• ab = 2
• ac = 2
• ad = 1
• ae = 1
• af = 1

d

250 P

c a b

200 125

Pg Pu e f

300

Pf

• Do Case 3R Rotation

200 125

b c

250 300

e f P Pf d a

Case 2R Rule Summary c d e f a b

200 125

a

300 250 P

Pf Pg Pu

200 125

b c

250 300

e f P Pf d a

Rotation Generalization Right Rotation

48

Practice Problem #1

50

#1 Practice Red Black Insertion  Add 10 to Empty Tree Some folks draw in the NULL Leaves - I do not! Some folks code this with an empty node to represent the Leaves

• I do not!

Label P  Pf  Pg  Pu

P Pf = / Pg = / Pu = /

Practice Problem #2

52

#2 Practice Red Black Insertion  Add 5 Tee Is There A Need To Do Anything To Preserve The Red Black Tree? No Label P  Pf  Pg  Pu

P Pf Pg = / Pu = /

Practice Problem #3

54

#3 Practice Red Black Insertion  Add 15 Tee Is There A Need To Do Anything To Preserve The Red Black Tree? No Label P  Pf  Pg  Pu

P Pf Pg = / Pu = /

Practice Problem #4

56

#4 Practice Red Black Insertion  Add 15 Tree

 Is There A Need To Do Anything To Preserve The Red Black Tree?

No

Label P  Pf  Pg  Pu

P Pf Pg = / Pu = /

Practice Problem #5

58

#5 Practice Red Black Insertion  Add 5 Tree

 Is There A Need To Do Anything To Preserve The Red Black Tree?

No

Label P  Pf  Pg  Pu

P Pf Pg = / Pu = /

Practice Problem #6

60

#6 Practice Red Black Insertion  Add 3 - 1

 Is There A Need To Do Anything To Preserve The Red Black Tree?

Yes

Label P  Pf  Pg  Pu

P Pf Pg Pu = /

Simple Rotate Right [ Pg Around Pf ]

P Pf Pg

What Do Now?

#6 Practice Red Black Insertion  Add 3 - 2

P Pf Pg

Color Change (Pf & Pg)

Pg Pf P

• AVL  Simple Rotate Right

What Do Now?

Practice Problem #7

63

#7 Practice Red Black Insertion  Add 20 - 1

 Is There A Need To Do Anything To Preserve The Red Black Tree?

Yes

Label P  Pf  Pg  Pu

P Pf Pg Pu = /

Simple Rotate Left [ Pg Around Pf ]

P Pf Pg

What Do Now?

#7 Practice Red Black Insertion  Add 20 - 2

P Pf Pg P Pf Pg

Color Change (Pf & Pg)

• AVL  Simple Rotate Left

What Do Now?

Practice Problem #8

66

#8 Practice Red Black Insertion  Add 8 - 1

 Is There A Need To Do Anything To Preserve The Red Black Tree? Yes

Label P  Pf  Pg  Pu

P Pf Pg Pu = /

Rotate Pf & P To The Left Around Pf

Pg Pf P

What Do Now?

#8 Practice Red Black Insertion  Add 8 - 2 Color Change (Pf & Pg)

Pg Pf P

Simple Rotate Right [ Pg Around Pf ]

Pg Pf P Pg Pf P

What Do Now? What Do Now?

• AVL  Simple Double Rotate Right

Pg P Pf

Swap Pointers Pf & P What Do Now?

Practice Problem #9

69

#9 Practice Red Black Insertion  Add 8 - 1

 Is There A Need To Do Anything To Preserve The Red Black Tree? Yes

Label P  Pf  Pg  Pu

P Pf Pg Pu = /

Rotate Pf & P To The Right Around Pf

Pg P Pf

What Do Now?

#9 Practice Red Black Insertion  Add 8 - 2 Color Change (Pf & Pg) Simple Rotate Left[ Pg Around Pf ]

P Pf Pg P Pf Pg

What Do Now? What Do Now?

• AVL  Simple Double Rotate Left

Pg P Pf Pg Pf P

What Do Now? Swap Pointers Pf & P

Practice Problem #10

72

#10 Practice Red Black Insertion  Add 5 - 1

 Is There A Need To Do Anything To Preserve The Red Black Tree?

Yes

Label P  Pf  Pg  Pu

P Pf Pg Pu

What Do Now? Violation 1L

#10 Practice Red Black Insertion  Add 5 - 2

P Pg Pf Pu

Fix Violation 1L

P Pg Pf Pu

Make Tree Root BLACK What Do Now?

P Pg Pf Pu

Practice Problem #11

75

#11 Practice Red Black Insertion  Add 15 - 1

 Is There A Need To Do Anything To Preserve The Red Black Tree?

Yes

Label P  Pf  Pg  Pu

P Pf Pg Pu

What Do Now? Violation 1R

Fix Violation 1R #11 Practice Red Black Insertion  Add 15 - 2

P Pf Pg Pu P Pf Pg Pu

Make Tree Root BLACK

P Pf Pg Pu

What Do Now?

Practice Problem #12

78

#12 Practice Red Black Insertion  Add 25

 Is There A Need To Do Anything To Preserve The Red Black Tree?

No

P Pg Pf Pu

Label P  Pf  Pg  Pu

Practice Problem #13

80

#13 Practice Red Black Insertion  Add 8 - 1

 Is There A Need To Do Anything To Preserve The Red Black Tree?

Yes

P Pg Pf Pu = /

Label P  Pf  Pg  Pu

Pf Pg P

Simple Rotate Right Pg Around Pf What Do Now?

#13 Practice Red Black Insertion  Add 8 - 2

Pf Pg P Pf Pg P

Swap Pointers Pf & P What Do Now?

#13 Practice Red Black Insertion  Add 8 - 3 Simple Rotate Right PB Around Pf

Pf Pg P

What Do Now?

Pf Pg P

#13 Practice Red Black Insertion  Add 8 - 4

Pf Pg P

What Do Now? Color Change (Pf & Pg)

Pf Pg P

Practice Problem #14

85

#14 Practice Red Black Insertion  Add 8 - 1

 Is There A Need To Do Anything To Preserve The Red Black Tree? Yes P Pg Pf Pu = /

Label P  Pf  Pg  Pu

Pf Pg P

Rotate P & Pf To The Left Around Pf What Do Now?

#14 Practice Red Black Insertion  Add 8 - 2

Pf Pg P P Pg Pf

Swap Pointers Pf & P What Do Now?

#14 Practice Red Black Insertion  Add 8 - 3 Simple Rotate Right Pg Around Pf Color Change (Pf & Pg)

P Pg Pf

What Do Now?

P Pg Pf

What Do Now?

P Pg Pf

Practice Problem #15

89

#15 Practice Red Black Insertion  Add 9 - 1

 Is There A Need To Do Anything To Preserve The Red Black Tree?

Yes

P Pg Pf Pu

Label P  Pf  Pg  Pu

#15 Practice Red Black Insertion  Add 9 - 2

 Is There A Need To Do Anything To Preserve The Red Black Tree?

Yes

P Pg Pf Pu

Label P  Pf  Pg  Pu Violation 1R

Practice Problem #16

92

#16 Practice Red Black Insertion  Add 13 - 1

P Pg Pf Pu

Label P  Pf  Pg  Pu Violation 1R

 Is There A Need To Do Anything To Preserve The Red Black Tree?

Yes

What Do Now?

#16 Practice Red Black Insertion  Add 13 - 2

P Pg Pf Pu

Apply Violation 1R

#16 Practice Red Black Insertion  Add 13 - 3 What Do Now?

P Pg Pf Pu P Pg Pf Pu

Move Potential Problem (focus) Up Two Levels

#16 Practice Red Black Insertion  Add 13 - 4 What Do Now?

P Pg Pf Pu

Rotate P & Pf To The Left Around Pf

P Pg Pf Pu

#16 Practice Red Black Insertion  Add 13 - 5 What Do Now? Swap Pointers P & Pf

P Pg Pf Pu P Pg Pf Pu

#16 Practice Red Black Insertion  Add 13 -6 What Do Now? Fix Violation 3L

P Pg Pf Pu P Pg Pf Pu

Note That Pf-R Goes To Pg-L

Rotation Generalization Right Rotation

100