LECTURE 22:THE ROLLING STONES, MASTERS OF THE BALANCED TREE

CSC 213 – Large Scale Programming

Red-Black Tree Properties

Root Property: Root node painted black External Property: Leaves are painted

black Internal Property: Red nodes’ children are

black Depth Property: Leaves have identical

black depth Number of black ancestors for the node

9

154

62 12

7

21

Insertion

Begins as BST insertion (just like splay & AVL)

New node’s initial color set by where it is If node is root, paint it black Other nodes colored red when insertion

completes

Insertion

Begins as BST insertion (just like splay & AVL)

New node’s initial color set by where it is If node is root, paint it black Other nodes colored red when insertion

completes Example: insert(3)

68

Insertion

Begins as BST insertion (just like splay & AVL)

New node’s initial color set by where it is If node is root, paint it black Other nodes colored red when insertion

completes Example: insert(3)

683

Insertion

Begins as BST insertion (just like splay & AVL)

New node’s initial color set by where it is If node is root, paint it black Other nodes colored red when insertion

completes Example: insert(3)

683

Insertion

New node’s initial color set by where it is If node is root, paint it black Other nodes colored red when insertion

completes If node’s parent is red, violates internal

property Must reorganize tree to remove double

red Example: insert(4)

683

4

Insertion

New node’s initial color set by where it is If node is root, paint it black Other nodes colored red when insertion

completes If node’s parent is red, violates internal

property Must reorganize tree to remove double

red Example: insert(4)makes tree

unbalanced

683

4

Double Red With Red Aunt

Double red represents creation of 5-node

Perform recoloring to find if balance really off

6

83

4

Double Red With Red Aunt

Parent & uncle painted black, grandparent red When grandparent is root, must paint it

black When easier, promote 2nd Entry to parent

Must continue check, double red can propagate4

672

Double Red With Red Aunt

Parent & uncle painted black, grandparent red When grandparent is root, must paint it

black When easier, promote 2nd Entry to parent

Must continue check, double red can propagate4

672

4

672

Double Red With Red Aunt

Parent & uncle painted black, grandparent red When grandparent is root, must paint it

black When easier, promote 2nd Entry to parent

Must continue check, double red can propagate4

672

4

672

Double Red With Black Aunt

Poorly balanced tree causes this double red Restore tree balance to use AVL tree

restructuring Preserves overall balance of the tree6

83

4

Double Red With Black Aunt

Rebalance tree using AVL tree restructuring

Recolors nodes also, but changes are very small

4

672

Double Red With Black Aunt

Rebalance tree using AVL tree restructuring

Recolors nodes also, but changes are very small

4

672 4

67

2

Double Red Restructuring

4 different restructures needed to remedy Differ in how node, parent, & grandparent

related Identical result no matter where we start4

67

7

46

7

64

4

76

Double Red Restructuring

4 different restructures needed to remedy Differ in how node, parent, & grandparent

related Identical result no matter where we start4

67

7

46

7

64

4

76

4 76

Deletion

Start with normal BST deletion If Entry in red node or leafs’s sibling

red Leaf’s sibling is painted black

Example: remove(1)6

3 8

41

Deletion

Start with normal BST deletion If Entry in red node or leafs’s sibling

red Leaf’s sibling is painted black

Example: remove(1)6

3 8

41

Deletion

Start with normal BST deletion If Entry in red node or leafs’s sibling

red Leaf’s sibling is painted black

Example: remove(1)6

3 8

4

Deletion

Start with normal BST deletion If Entry in red node or leafs’s sibling

red Leaf’s sibling is painted black

Example: remove(1)6

3 8

4

What’s Blacker Than Black?

If removed Entry & leaf’s sibling already black Paint sibling double black This is an illegal state – violates internal

property Example: remove(8)6

3 8

4

What’s Blacker Than Black?

If removed Entry & leaf’s sibling already black Paint sibling double black This is an illegal state – violates internal

property Example: remove(8)6

3 8

4

What’s Blacker Than Black?

If removed Entry & leaf’s sibling already black Paint sibling double black This is an illegal state – violates internal

property Example: remove(8)6

3

4

What’s Blacker Than Black?

If removed Entry & leaf’s sibling already black Paint sibling double black This is an illegal state – violates internal

property Example: remove(8) causes double

black6

3

4

Remedying Double Black

Case 1: sibling is black with red child Reorder nodes using AVL tree restructure

Case 2: sibling and its children are black Equal to (2,4) tree underflow, so recolor

nodes

Remedying Double Black

Case 1: sibling is black with red child Reorder nodes using AVL tree restructure

Case 2: sibling and its children are black Just a bad balance, so just recolor nodes

Case 3: sibling is red Adjust subtree to better balance tree Once complete apply case 1 or case 2

Black Sibling With Red Niece Solve double black using AVL

restructuring

9

6 10

8

Black Sibling With Red Niece Solve double black using AVL

restructuring

9

6

8

Black Sibling With Red Niece Solve double black using AVL

restructuring

9

6

8

8

6 9

Sibling & Children are Black Solve double black recoloring

parent & sibling

5

10

9

6

…

Sibling & Children are Black Solve double black recoloring

parent & sibling

5

9

6

…

Sibling & Children are Black Solve double black recoloring

parent & sibling

5

9

6

…

5

9

6

…

Sibling & Children are Black Solve double black recoloring

parent & sibling

5

9

6

…

5

9

6

…

Sibling is Red

Adjusting double black stalls for time Transforms situation into something we can

fix

9

5 10

4

Sibling is Red

Adjusting double black stalls for time Transforms situation into something we can

fix

9

5 10

4

Sibling is Red

Adjusting double black stalls for time Transforms situation into something we can

fix

9

5

4

Sibling is Red

Adjusting double black stalls for time Transforms situation into something we can

fix

9

5

4

Sibling is Red

Adjusting double black stalls for time Transforms situation into something we can

fix Once completed, re-examine double black

node9

5

4

54 9

Do the Activity

For Next Lecture

Weekly assignment available to test skills Due at regular time Tuesday after break Talk to me if struggling on problems

Last idea from Map/Dictionary after break What are Sets? Why are we getting to

them now? It may not appear related, but how are they

used? Reminder: lab phase #2 due Monday

after break