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) C  There Exists Some Subtrees: A, B, & C A B  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 - 2/2 100 90 C 90 A 100 A  Simple Way To Rotate The Valid Binary B C B Search Tree Without Altering The Search Capabilities.  Single Rotate Right  AVL

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) A  There Exists Some Subtrees: A, B, & C C B  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 90 100 A 100 C 90 C B A B  Simple Way To Rotate The Valid Binary Search Tree Without Altering The Search Capabilities.  Single Rotate Left  AVL

Violation 1L RED UNCLE Case (On Left) 10

Easy Case 1L - RED UNCLE Case - 1 (P On Left)  This RED UNCLE Case Is a Occurring As We Walk Our Way Up Pg 125 The Tree.  If We Make The New Node RED, It Minimizes The Number Of Bad Pf 100 150 Pu Things That Can Happen: e d f  The Color Distance Will Not 50 P Change The Number of Black Down Any Branch b c  Only Thing That Can Go Wrong Is Two Red Nodes In A Row  Label P  Pf  Pg  Pu  Label The Distance Paths

Easy Case 1L - RED UNCLE Case - 2 (P On Left)  Record The Relative Distances Down a The Various Paths In The Tree: 125  ab = 1  ac = 1 100 150  ad = 1  ae = 1 d e f 50  af = 1 b c  Easy Case 1L:  UNCLE is RED  No Rotation Required!

Easy Case 1L - RED UNCLE Case - 3 (P On Left)  ab = 1 a  ac = 1 a  ad = 1 Pg  How Fix? 125 Pg 125  ae = 1  af = 1 Pf 100 150 Pu Pf 100 150 Pu d e f 50 d e P f 50 P b c b c  Pf  Color = BLACK Pu  Color = BLACK Pg  Color = RED  Easy Case 1L:  UNCLE is RED  No Rotation Required!  Double Check The Distances ab=1, ac=1, ad=1, ae=1, af=1  YEA!

Case 1L Rule Summary a a Pg Pg 125 125 Pf Pf 100 100 150 150 Pu Pu e d e d f f 50 50 P P b b c c

Violation 1R RED UNCLE Case (On Right) 15

Easy Case 1R - RED UNCLE Case - 1 (P On Right)  This RED UNCLE Case Is a Occurring As We Walk Our Way Up Pg 150 The Tree.  If We Make The New Node RED, It Minimizes The Number Of Bad Pu 125 Pf 200 Things That Can Happen: b c P  The Color Distance Will Not d 50 Change The Number of Black Down Any Branch e f  Only Thing That Can Go Wrong Is Two Red Nodes In A Row  Label P  Pf  Pg  Pu  Label The Distance Paths

Easy Case 1R - RED UNCLE Case - 2 (P On Right)  Record The Relative Distances Down a The Various Paths In The Tree: Pg 150  ab = 1  ac = 1 Pu 125 Pf 200  ad = 1  ae = 1 b c P d 50  af = 1 e f  Label P  Pf  Pg  Pu  Label The Distance Paths  Easy Case 1R:  UNCLE is RED  No Rotation Required!

Easy Case 1R - RED UNCLE Case - 3 (P On Right)  ab = 1 a  ac = 1 a  ad = 1  How Fix? Pg 150 Pg 150  ae = 1  af = 1 125 Pu 200 Pf 125 Pu 200 Pf b P c 50 d b c 50 P d e f e f  Pf  Color = BLACK Pu  Color = BLACK Pg  Color = RED  Easy Case 1L:  UNCLE is RED  No Rotation Required!  Double Check The Distances ab=1, ac=1, ad=1, ae=1, af=1  YEA!

Case 1R Rule Summary a a Pg Pg 150 150 125 125 Pu 200 Pf Pu 200 Pf b b c P 50 P c 50 d d e e f f

RED Uncle Can Occur With Insertion RED UNCLE Case (On Left) 20

Easy Case 1L - RED UNCLE Case - 1 (P On Left)  Distance Property = 2 For All - If Count Null  Add 50 to Tree - Make it Red  Minimize The Number Of Bad 200 Things That Can Happen:  Will Not Change The Number of Pg 250 125 Black Down Any Branch  Only Thing That Can Go Wrong Is 100 Pf 150 Two Red Nodes In A Row Pu 50 P  Label P  Pf  Pg  Pu  Easy Case 1L:  UNCLE is RED  No Rotation Required!

Easy Case 1L - RED UNCLE Case - 2 (P On Left)  How Fix?  Might Become Easier If We Don't 200 Try To Fix Everything At Once Start With This Portion Pg 125 250  THIS IS EASY CASE - Pu Is Red (125) 100 Pf Pu 150  Make Pg RED Make Pf BLACK Make Pu BLACK 50 P  This Shifts The Problem Up Two Generations!  Easy Case 1L:  UNCLE is RED  No Rotation Required!

Easy Case 1L - RED UNCLE Case - 3 (P On Left)  How Fix?  How Fix?  Change Pf To BLACK 200 Pf  Distance Property = 3 For All 125 250 P 100 150 50  2 BLACK UNCLE Cases: One Requires A Rotation One Requies A Preliminary Rotation And A Second Rotation

RED Uncle Can Occur With Insertion Add To Tree Violation 1R RED UNCLE Case (On Right) 24

Easy Case 1R - RED UNCLE Case - 1 (P On Right)  Distance Property = 2 For All If Count Null  Add 225 to Tree - Make it Red 100  Minimize The Number Of Bad Things That Can Happen: 50 150 Pg  Will Not Change The Number of Black Down Any Branch  Only Thing That Can Go Wrong Is 125 Pu 200 Pf Two Red Nodes In A Row P 225  Label P  Pf  Pg  Pu  Easy Case 1R:  UNCLE is RED  No Rotation Required!

Easy Case 1R - RED UNCLE Case - 2 (P On Right)  How Fix? 100  Might Become Easier If We Don't Try To Fix Everything At Once 50 150 Pg Start With This Portion  THIS IS EASY CASE - Uncle Of P 125 Is Red (125) Pu 200 Pf  Make Pg RED Make Pf BLACK P 225 Make Pu BLACK  This Shifts The Problem Up Two Generations!

Easy Case 1R - RED UNCLE Case - 3 (P On Right)  How Fix? Pf 100  Change Pf To BLACK 50 P 150  Distance Property = 3 For All 125 200 225  2 BLACK UNCLE Cases: One Requires A Rotation One Requies A Preliminary Rotation And A Second Rotation

Violation 3L BLACK UNCLE Case With One Rotation (On Left) 28

Case 3L - BLACK UNCLE One Rotation Class- 1 (P On Left)  This BLACK UNCLE Case Is a Occurring As We Walk Our Way Up 200 The Tree.  We Have Already Established That 250 125 The Distance Property Is OK  ab = 1 f e d  ac = 1 100  ad = 1  ae = 2 b c  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)  ab = 1 a  ac = 1 a  ad = 1 Pg 200 Pf 125  ae = 2  af = 2 Pu 250 Pf 125 Pg 100 200 P e f d b c 100 d P 250 Pu b c e f  Solve RED RED Problem?  Promote Pf Up  Hook up 'd'  Label P  Pf  Pg  Pu  Hook Pg Right of Pf  Swap Colors Pg Pf  Double Check The Distances ab=1, ac=1, ad=1, ae=2, af=2  YEA!

Case 3L Rule Summary a a Pf 125 Pg 200 250 Pu 125 Pf 100 Pg P 200 e d f b c d Pu 250 100 P e f b c

Violation 3R BLACK UNCLE Case With One Rotation (On Right) 32

Case 3R - BLACK UNCLE One Rotation Class- 1 (P On Right)  This BLACK UNCLE Case Is a Occurring As We Walk Our Way Up 200 The Tree.  We Have Already Established That 250 The Distance Property Is OK 125  ab = 2 d c b 300  ac = 2  ad = 1  ae = 1 e f  af = 1  GOAL: Fix The RED CHILD With RED Father Problem Without Messing Up The Distance Properties.

Case 3R - BLACK UNCLE One Rotation Class- 2 (P On Right)  ab = 2 a  ac = 2 a  ad = 1 Pg 200 Pf 250  ae = 1  af = 1 Pu 250 200 Pf 125 P 300 d e b c f 300 P 125 d e f b c  Solve RED RED Problem?  Promote Pf Up  Hook up 'd'  Label P  Pf  Pg  Pu  Hook Pg Left of Pf  Swap Colors Pg Pf  Double Check The Distances ab=2, ac=2, ad=1, ae=1, af=1  YEA!

Case 3R Rule Summary a a Pg Pf 200 250 200 P Pu 300 250 Pf 125 e d f b c 125 300 P e b f c

Violation 2L BLACK UNCLE Case With Two Rotations (On Left) 36

Case 2L - BLACK UNCLE Two Rotations Class- 1 (P On Left)  This BLACK UNCLE Case Is a Occurring As We Walk Our Way Up Pg 200 The Tree.  We Have Already Established That Pu 250 The Distance Property Is OK Pf 125  ab = 1  ac = 1 b e f P 150  ad = 1  ae = 2 c d  af = 2  Label P  Pf  Pg  Pu  Case 2L is going to turn into Case 3L.  This Is The Worst Possible Case