Balanced Binary Search Tree

황승원Fall 2011

CSE, POSTECH

2

Balanced Binary Search Trees

height is O(log n), where n is the number of elements in the tree

AVL (Adelson-Velsky and Landis) trees red-black trees get, put, and remove take O(log n) time

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Balanced Binary Search Trees

Indexed AVL trees Indexed red-black trees Indexed operations also take

O(log n) time

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Balanced Search Trees

weight balanced binary search trees 2-3 & 2-3-4 trees AA trees B-trees BBST etc.

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AVL Tree

Definition Binary tree. If T is a nonempty binary tree with TL and TR as

its left and right subtrees, then T is an AVL tree iff1. TL and TR are AVL trees, and

2. |hL – hR| 1 where hL and hR are the heights of TL and TR, respectively

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Balance Factors

0 0

0 0

1

0

-1 0

1

0

-1

1

-1

This is an AVL tree.

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Height

The height of an AVL tree that has n nodes is at most 1.44 log2 (n+2).

Nh=Nh-1 + Nh-2+1 – reminds you anything?Fn=Fn-1 + Fn-2

Nh=Fh+2-1 when ~ / 5

(1 5) / 2

log ( 5( 1)) 2

hhF

h n

(by induction)

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AVL Search Tree

0 0

0 0

1

0

-1 0

1

0

-1

1

-110

7

83

1 5

30

40

20

25

35

45

60

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put(9)

0 0

0 0

1

0

-1 0

1

0

-1

1

-1

90

-1

0

10

7

83

1 5

30

40

20

25

35

45

60

10

put(29)

0 0

0 0

1

0

0

1

0

-1

1

-110

7

83

1 5

30

40

20

25

35

45

60

29

0

-1

-1-2

RR imbalance => new node is in right subtree of right subtree of blue node (node with bf = -2)

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put(29)

0 0

0 0

1

0

0

1

0

-1

1

-110

7

83

1 5

30

40

2535

45

600

RR rotation. 20

029

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AVL Rotations

RR LL RL LR

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Imbalance Types

After an insertion, when the balance factor of node A is –2 or 2, the node A is one of the following four imbalance types

1. LL: new node is in the left subtree of the left subtree of A

2. LR: new node is in the right subtree of the left subtree of A

3. RR: new node is in the right subtree of the right subtree of A

4. RL: new node is in the left subtree of the right subtree of A

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Rotation

Definition To switch children and parents among two or three

adjacent nodes to restore balance of a tree. A rotation may change the depth of some nodes,

but does not change their relative ordering.

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AVL Rotations

To balance an unbalanced AVL tree (after an insertion), we may need to perform one of the following rotations: LL, RR, LR, RL

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An LL Rotation

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An LR Rotation

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RR and RL?

Symmetric!

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Inserting into an AVL Search Tree

29

Insert(29)-1

1

0

0 0

0

1

1

-1 0

-1

0

0

10

40

30 45

20 35

25

60

7

3 8

1 5

• Where is 29 going to be inserted into?• After the insertion, is the tree still an AVL search tree? (i.e., still balanced?)

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Inserting into an AVL Search Tree

• What are the balance factors for 20, 25, 29?

-2

-1

0

Insert(29)

29

-1

1

0

0 0

0

1

1

0

-1

0

10

40

30 45

20 35

25

60

7

3 8

1 5

• RR imbalance new node is in the right subtree of right subtree of node 20 (node with bf = -2)

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After RR Rotation

-1

1

0

0 0

0

1

1

0

-1

0

10

40

30 45

35 60

7

3 8

1 5 0

00

25

20 29

• What would the left subtree of 30 look like after an RR rotation?• After the RR rotation, is the resulting tree an AVL search tree now?

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Deletion from an AVL Search Tree

Deletion of a node may also produce an imbalance

Imbalance incurred by deletion is classified intothe types R0, R1, R-1, L0, L1, and L-1

Rotation is also needed for rebalancing Read the observations after deleting a node from

an AVL search tree Read Section 11.2.6 for Deletion from an AVL

search tree

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An R1 (decided by B) Rotation=LL

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An R-1 Rotation=LR

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An R0 Rotation~LL

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Complexity of Dictionary Operations:Search, Insert, and Delete

Data Structure Worst Case Expected

Hash Table

Binary SearchTree

BalancedBinary SearchTree

O(n)

O(n)

O(log n)

O(1)

O(log n)

O(log n)

• n is the number of elements in dictionary.

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Complexity of Other OperationsAscend, Search(rank), and Delete(rank)

Data Structure ascend get and remove

Hash Table

Indexed BST

IndexedBalanced BST

O(D + n log n)

O(n)

O(n)

O(D + n log n)

O(n)

O(log n)

• D is the number of buckets.

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Red Black Trees

Colored Nodes Definition Binary search tree. Each node is colored red or black. Root and all external nodes are black. No root-to-external-node path has two consecutive

red nodes. All root-to-external-node paths have the same

number of black nodes

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Example Red Black Tree

10

7

8

1 5

30

40

20

25

35

45

60

3

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Properties

The height of a red black tree that has n (internal) nodes is between log2(n+1) and 2log2(n+1).

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Lemma 16.2

r : rank(=# black nodes-1) of root

2

2

2 1

2log ( 1)

r

h r

n

h n

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Insert

New pair is placed in a new node, which is inserted into the red-black tree.

New node color options.– Black node => one root-to-external-node path has an

extra black node (black pointer). Hard to remedy.

– Red node => one root-to-external-node path may have two consecutive red nodes (pointers).

May be remedied by color flips and/or a rotation.

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Classification Of 2 Red Nodes/Pointers

XYz– X => relationship between gp and pp.

pp left child of gp => X = L.– Y => relationship between pp and p.

p right child of pp => Y = R.– z = b (black) if d = null or a black node.– z = r (red) if d is a red node.

a b

cd

gppp

p

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XYr Color flip.

a b

cd

gppp

p

a b

cd

gppp

p

• Continue rebalancing if changing gp to red causes problems

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LLb Rotate.

• Done!

• Same as LL rotation of AVL tree.

y

x

a b

z

c d

a b

cd

gppp

p x

y

z

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LRb Rotate.

• Done!

• Same as LR rotation of AVL tree.

• RRb and RLb are symmetric.

y

x

a b

z

c d

b c

ad

gppp

py

x

z

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Red-black Tree Animation

http://www.ece.uc.edu/~franco/C321/html/RedBlack/redblack.html

java.util.TreeMap => red black tree

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Coming Up Next

READING: Ch 16.1~2 NEXT: Graphs (Ch 17.1~7)