lecture 14

CS 332: Algorithms Red-Black Trees

Review: Red-Black Trees ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Review: Red-Black Properties ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Review: Black-Height ,[object Object],[object Object],[object Object],[object Object],[object Object]

Review: Proving Height Bound ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

RB Trees: Worst-Case Time ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Red-Black Trees: An Example ,[object Object],Red-black properties: 1. Every node is either red or black 2. Every leaf (NULL pointer) is black 3. If a node is red, both children are black 4. Every path from node to descendent leaf contains the same number of black nodes 5. The root is always black 7 5 9 12 12 5 9 7

[object Object],[object Object],Red-Black Trees: The Problem With Insertion 1. Every node is either red or black 2. Every leaf (NULL pointer) is black 3. If a node is red, both children are black 4. Every path from node to descendent leaf contains the same number of black nodes 5. The root is always black 12 5 9 7

[object Object],[object Object],[object Object],Red-Black Trees: The Problem With Insertion 12 5 9 7 8 1. Every node is either red or black 2. Every leaf (NULL pointer) is black 3. If a node is red, both children are black 4. Every path from node to descendent leaf contains the same number of black nodes 5. The root is always black

Red-Black Trees: The Problem With Insertion ,[object Object],[object Object],1. Every node is either red or black 2. Every leaf (NULL pointer) is black 3. If a node is red, both children are black 4. Every path from node to descendent leaf contains the same number of black nodes 5. The root is always black 12 5 9 7 8

Red-Black Trees: The Problem With Insertion ,[object Object],[object Object],[object Object],1. Every node is either red or black 2. Every leaf (NULL pointer) is black 3. If a node is red, both children are black 4. Every path from node to descendent leaf contains the same number of black nodes 5. The root is always black 12 5 9 7 8 11

Red-Black Trees: The Problem With Insertion ,[object Object],[object Object],[object Object],[object Object],1. Every node is either red or black 2. Every leaf (NULL pointer) is black 3. If a node is red, both children are black 4. Every path from node to descendent leaf contains the same number of black nodes 5. The root is always black 12 5 9 7 8 11

Red-Black Trees: The Problem With Insertion ,[object Object],[object Object],[object Object],[object Object],[object Object],1. Every node is either red or black 2. Every leaf (NULL pointer) is black 3. If a node is red, both children are black 4. Every path from node to descendent leaf contains the same number of black nodes 5. The root is always black 12 5 9 7 8 11

Red-Black Trees: The Problem With Insertion ,[object Object],[object Object],1. Every node is either red or black 2. Every leaf (NULL pointer) is black 3. If a node is red, both children are black 4. Every path from node to descendent leaf contains the same number of black nodes 5. The root is always black 12 5 9 7 8 11

Red-Black Trees: The Problem With Insertion ,[object Object],[object Object],[object Object],1. Every node is either red or black 2. Every leaf (NULL pointer) is black 3. If a node is red, both children are black 4. Every path from node to descendent leaf contains the same number of black nodes 5. The root is always black 12 5 9 7 8 11 10

Red-Black Trees: The Problem With Insertion ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],12 5 9 7 8 11 10

RB Trees: Rotation ,[object Object],[object Object],[object Object],y x C A B x A y B C rightRotate(y) leftRotate(x)

RB Trees: Rotation ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],rightRotate(y) y x C A B x A y B C

Rotation Example ,[object Object],12 5 9 7 8 11

Rotation Example ,[object Object],5 12 7 9 11 8

Red-Black Trees: Insertion ,[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],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Case 1 Case 2 Case 3

[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],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Case 1: uncle is RED Case 2 Case 3

RB Insert: Case 1 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],C A D  B     C A D  B     x y new x Change colors of some nodes, preserving #4: all downward paths have equal b.h. The while loop now continues with x’s grandparent as the new x case 1

RB Insert: Case 1 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],C A D    C A D   y new x Same action whether x is a left or a right child  case 1 B   x B   x

RB Insert: Case 2 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],C A  C B y  case 2  y  Transform case 2 into case 3 (x is left child) with a left rotation This preserves property 4: all downward paths contain same number of black nodes B   x A   x

RB Insert: Case 3 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],B A x  case 3 C B  y  C    Perform some color changes and do a right rotation Again, preserves property 4: all downward paths contain same number of black nodes A   x

RB Insert: Cases 4-6 ,[object Object],[object Object]

Red-Black Trees: Deletion ,[object Object],[object Object],[object Object],[object Object]

The End ,[object Object],[object Object],[object Object]

lecture 14

Recommandé

Recommandé

Contenu connexe

Tendances

Tendances (18)

Similaire à lecture 14

Similaire à lecture 14 (20)

Plus de sajinsc

Plus de sajinsc (20)

Dernier

Dernier (20)

lecture 14