trees-i prof. muhammad saeed analysis of algorithms

Trees-I

Prof. Muhammad Saeed

Analysis of Algorithms

ASH

h cd shcd ahs cah sdcs c

Analysis Of Algorithms 2

Tree Representation …..

Tree


Tree

….. Tree Representation


Nodes (13)Size (13)Degree of a nodeDepth of a tree (3)Height of a tree (3)Level of a nodeLeaf (terminal)NonterminalParentChildrenSiblingAncestor

K L

E F

B

G

C

M

H I J

D

A

Level

1

2

3

4

3

2 1 3

1 0 0 1 0 0

0 0 0

1

2 2 2

3 3 3 3 3 3

4 4 4

Degree Level

Nomenclature

Tree


Tree

Types

Binary TreeBinary Search TreeB-TreeAVL TreeRed-Black TreeSplay TreeBinomial Tree


Tree

A forest is a set of n >= 0 disjoint trees

A E G

B C D F H I G

H

I

A

B

C

D

F

E

Forest

Analysis Of Algorithms

1

2 3

75

9

4

8

6

1

2 3

75

11

4

10

6

98 15141312

Complete binary tree Full binary tree of depth 4

Tree

7


Tree

A binary tree can be traversed using four different algorithms 1.Pre-order: Root-Left-Right, It employs

Depth First Search.

2. Inorder: Left-Root-Right.

3. Post-order: Left-Right-Root

4. Level-by-level.

Binary Tree Traversal


inorder traversalA / B * C * D + Einfix expressionpreorder traversal+ * * / A B C D Eprefix expressionpostorder traversalA B / C * D * E +postfix expressionlevel order traversal+ * E * D / C A B

+

*

A

*

/

E

D

C

B

Arithmetic Expression Using Binary Tree

Tree


Tree

Property:The root of max heap (min heap) contains the largest (smallest).

Heaps


Tree

Priority queue representations

Representation Insertion Deletion

Unordered array

(1) (n)

Unordered linked list

(1) (n)

Sorted array O(n) (1) Sorted linked list

O(n) (1)

Max heap O(log2n) O(log2n)


56

26 200

18 28 190 213

12 24 27

Binary Search Tree …..

Stored keys must satisfy the binary search tree property.if y is in left subtree of x,

then key[y] key[x].If y is in right subtree of x,

then key[y] key[x].

The binary-search-tree property guarantees that: The minimum is located at the left-most node. The maximum is located at the right-most

node.

Tree


Tree

All BST operations are O(d), where d is tree depthminimum d is d=log2N for a binary tree with N nodes What is the best case tree? What is the worst case tree?

So, best case running time of BST operations is O(log N)

….. Binary Search Tree - Best Time …..


…..Binary Search Tree - Worst Time …..

Tree

Worst case running time is O(N) What happens when you Insert elements in ascending

order?• Insert: 2, 4, 6, 8, 10, 12 into an empty BST

Problem: Lack of “balance”: • compare depths of left and right subtree

Unbalanced degenerate tree


Balanced and unbalanced BST

Tree

4

2 5

1 3

1

5

2

4

3

7

6

4

2 6

5 71 3

Is this “balanced”?


Tree

Rotations: Single Rotation …..


0

1

0

2

0

6

4

9

8

1 5

1

0

7

2

10

2

0

6

4 9

81 5

1

0

7

Tree

…..Rotations: Single Rotation …..

j

k

XY

Zh

h+1 h

j

k

XY

Zh

h+1 h


Tree

……. Rotations …..


Tree


AVL trees are balanced.An AVL Tree is a binary

search tree such that for every internal node v of T, the heights of the children of v can differ by at most 1.

88

44

17 78

32 50

48 62

2

4

1

1

2

3

1

188

44

17 78

32 50

48 62

2

4

1

1

2

3

1

1

An example of an AVL tree where the heights are shown next to the nodes:

AVL(Adelson-Velskii-Landis) trees

TreeTree

ASH

Hello


Height of an AVL Tree …..

Tree

Proposition: The height of an AVL tree T storing n keys is O(log n).Justification: The easiest way to approach this problem is to find n(h): the minimum number of internal nodes of an AVL tree of height h.We see that base case is n(0) = 1 and n(1) = 2For n ≥ 3, an AVL tree of height h contains the root node, one AVL subtree of height n-1 and the other AVL subtree of height n-2.i.e. n(h) = 1 + n(h-1) + n(h-2)


Knowing n(h-1) > n(h-2), we get n(h) > 2n(h-2)n(h) > 2n(h-2)n(h) > 4n(h-4)n(h) > 8n(h-6)…n(h) > 2in(h-2i)For any integer I such that h-2i 1

Solving the base case we get: n(h) ≥ 2 h/2-1

Taking logarithms: h < 2log n(h) +2Thus the height of an AVL tree is O(log n)

………. Height of an AVL Tree

Tree


Rotation: Double rotation (inside case) …..

3

0

3

20

10 30

25

1

40

2

50

20

10 35

30

1

405

45

0 1

2

3

Imbalance

45

0

1

Insertion of 34

35

34

0

0

1 25 340

Tree


Tree

…..Rotations …..Double rotation

j

k

X

V

Z

W

i


…..Rotations

Tree

Double or Single

Tree



Red and Black Trees

Tree

Every node is red or black The root is blackEvery leaf is NIL and is blackIf a node is red, then both its children are blackFor each node, all paths from the node to descendant leaves contain the same number of black nodes.

A Red–Black tree is a binary search tree that inserts and deletes in such a way that the tree is always reasonably balanced.


A Red and Black Tree with n internal nodes has height at most 2log(n+1).

Tree


Case 1 – U is Red

Just recolor and move up

X

P

G

U

PG

U

X

Tree


XP

G

U

S X

P G

S UCase 2 – Zig-Zag

Double rotate X around P and X around G

Recolor G and X

Tree


Tree

X

P

G

US P

X G

S UCase 3 – Zig-Zig

Single Rotate P around G

Recolor P and G


Tree

11

14

152

1 7

5 8

Insert 4 into this R-B Tree


Red–Black trees offer worst-case guarantees for insertion time, deletion time, and search time.

Tree

Completely Fair Scheduler used in current Linux kernels uses Red–Black trees.

Red–Black trees are also particularly valuable in functional programming, where they are one of the most common persistent data structures, used to construct associative arrays and sets which can retain previous versions after mutations.

The persistent version of Red–Black trees requires O(log n) worst-case for each insertion or deletion, in addition to time whereas oher BST’s require O(n).

trees-i prof. muhammad saeed analysis of algorithms

Documents

binary tree tree slide

avl tree tree slide

redblack tree

tree proposition

tree representation

analysis of algorithms6

analysis of algorithms13

analysis of algorithms8