LECTURE 39:THE ROLLING STONES, MASTERS OF THE BALANCED TREE

CSC 213 – Large Scale Programming

(2,4) Trees: Pro & Con

Cons: Cannot reuse BST code, since it is not

binary treePros: (2,4) Trees balance without rotations Fewer balancing cases than AVL or splay

trees

(2,4) Trees: Pro & Con

Cons: Cannot reuse BST code, since it is not

binary treePros: (2,4) Trees balance without rotations Fewer balancing cases than AVL or splay

trees

Sick, twisted, & wrong: n-node naming scheme is crime against

humanity

Red-Black Trees

Represents a (2,4) tree using binary tree Red node when Entry & parent’s Entry share

node (2,4) tree Black node’s Entry also in child of parent’s node

in (2,4) tree 2 6 73 54

4 62 7

53

35OR

Red-Black Trees

Represents a (2,4) tree using binary tree Red node when Entry & parent’s Entry share

node (2,4) tree Black node’s Entry also in child of parent’s node

in (2,4) tree

Maximizes code reuse, since subclass of BST

2 6 73 54

4 62 7

53

35OR

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 (equivalent to (2,4) tree split)

3 4 6 86

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

As in splitting, double red may propagate up

2 4 6 7

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

As in splitting, double red may propagate up

2 4 6 76 7

… 4 …

2

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

As in splitting, double red may propagate up

2 4 6 76 7

… 4 …

2

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

As in splitting, double red may propagate up

2 4 6 76 7

… 4 …

2

4

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

As in splitting, double red may propagate up

2 4 6 76 7

… 4 …

2

4

672

4

672

Double Red With Black Aunt

Poorly balanced 4-node causes this double red Restore tree balance to use AVL tree

restructuring Preserves overall balance of the tree

3 4 6683

48

Double Red With Black Aunt

Rebalance tree using AVL tree restructuring

Recolor nodes, but no changes (2,4) tree 4

672

Double Red With Black Aunt

Rebalance tree using AVL tree restructuring

Recolor nodes, but no changes (2,4) tree 4

672 4

67

2

Double Red With Black Aunt

Rebalance tree using AVL tree restructuring

Recolor nodes, but no changes (2,4) tree 4

672

4 6 7

.. 2 ..

46

72

Double Red With Black Aunt

Rebalance tree using AVL tree restructuring

Recolor nodes, but no changes (2,4) tree 4

672

4 6 7

.. 2 ..

46

72

4 6 7

.. 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 Equal to (2,4) tree underflow, so recolor

nodes Case 3: sibling is red

Adjust subtree to better represent 3-node Once complete apply case 1 or case 2

Black Sibling With Red Niece Solve double black using (2,4) tree

transfer…9…

6 8 10

9

6 10

8

Black Sibling With Red Niece Solve double black using (2,4) tree

transfer…9…

6 8

9

6

8

Black Sibling With Red Niece Solve double black using (2,4) tree

transfer …8…

6 9

…9…

6 8

9

6

8

Black Sibling With Red Niece Solve double black using (2,4) tree

transfer …8…

6 9

…9…

6 8

9

6

8

8

6 9

Black Sibling With Red Niece Solve double black using AVL

restructuring …8…

6 9

…9…

6 8

9

6

8

8

6 9

Sibling & Children are Black Solve double black using (2,4) tree

fusion5 9

6 10

5

10

9

6

…

…

Sibling & Children are Black Solve double black using (2,4) tree

fusion5 9

6

5

9

6

…

…

Sibling & Children are Black Solve double black using (2,4) tree

fusion5 9

6

5

9

6

…

…

5

… 6 9

Sibling & Children are Black Solve double black recoloring

parent & sibling5 9

6

5

9

6

…

…

5

… 6 9

5

9

6

…

Sibling & Children are Black Solve double black recoloring

parent & sibling5 9

6

5

9

6

…

…

5

… 6 9

5

9

6

…

Sibling & Children are Black If parent already black, it becomes

double black 4

1 10 1 4

41 10

41

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 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 next Tuesday Talk to me if struggling on problems

Look at what all this means in real world code What are B-trees? How are they related to (2,4) & red-black

trees? Where are they used?

Reminder: project #3 tests due today