CSE221/ICT221CSE221/ICT221Analysis and Design of AlgorithmsAnalysis and Design of Algorithms

Analysis of Algorithm using Tree Data Structure

Asst.Prof. Dr.Surasak MungsingAsst.Prof. Dr.Surasak MungsingE-mail: Surasak.mu@spu.ac.th

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Trees

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Introduction to TreesIntroduction to Trees

General Trees Binary Trees Binary Search Trees AVL Trees

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Tree

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DefinitionDefinition

A tree t is a finite nonempty set of elements.

One of these elements is called the root. The remaining elements, if any, are

partitioned into trees, which are called the subtrees of t.

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Sub-trees

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Tree

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Level 3

Object

Number Throwable OutputStream

Integer Double Exception FileOutputStream

RuntimeException

Level 4

Level 2

Level 1

height = depth = number of levels

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Object

Number Throwable OutputStream

Integer Double Exception FileOutputStream

RuntimeException

2 1 1

0 0 1 0

0

Node Degree = Number Of Children

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Binary Tree

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Binary TreeBinary Tree

Finite (possibly empty) collection of elements.

A nonempty binary tree has a root element. The remaining elements (if any) are

partitioned into two binary trees. These are called the left and right subtrees

of the binary tree.

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Binary Tree

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A Tree vs A Binary TreeA Tree vs A Binary Tree

No node in a binary tree may have a degree more than 2, whereas there is no limit on the degree of a node in a tree.

A binary tree may be empty; a tree cannot be empty.

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A Tree vs A Binary TreeA Tree vs A Binary Tree

The subtrees of a binary tree are ordered; those of a tree are not ordered.

a

b

a

b

• Are different when viewed as binary trees.

• Are the same when viewed as trees.

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Forms of Binary Trees

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Complete Binary Trees

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Tree Traversal

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Processing and Walking OrderProcessing and Walking Order

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Depth First ProcessingDepth First Processing

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Preorder TraversalPreorder Traversal

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Breath First ProcessingBreath First Processing

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Height and number of nodesHeight and number of nodes

Maximum height of a binary tree Hmax = N

Minimum height of a binary treeHmin = logN + 1

Maximum and Minimum number of nodesNmin = H and Nmax = 2H - 1

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Expression Tree

การประยุ�กต์ใช้� Tree

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Arithmetic ExpressionsArithmetic Expressions

Expressions comprise three kinds of entities. Operators (+, -, /, *). Operands (a, b, c, d, e, f, g, h, 3.25, (a + b), (c + d),

etc.). Delimiters ((, )).

(a + b) * (c + d) + e – f/g*h + 3.25

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Infix FormInfix Form

Normal way to write an expression. Binary operators come in between their

left and right operands. a * b a + b * c a * b / c (a + b) * (c + d) + e – f/g*h + 3.25

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Operator PrioritiesOperator Priorities

How do you figure out the operands of an operator? a + b * c a * b + c / d

This is done by assigning operator priorities. priority(*) = priority(/) > priority(+) = priority(-)

When an operand lies between two operators, the operand associates with the operator that has higher priority.

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Tie BreakerTie Breaker

When an operand lies between two operators that have the same priority, the operand associates with the operator on the left. a + b - c a * b / c / d

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DelimitersDelimiters

Subexpression within delimiters is treated as a single operand, independent from the remainder of the expression. (a + b) * (c – d) / (e – f)

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Infix Expression Is Hard To ParseInfix Expression Is Hard To Parse

Need operator priorities, tie breaker, and delimiters.

This makes computer evaluation more difficult than is necessary.

Postfix and prefix expression forms do not rely on operator priorities, a tie breaker, or delimiters.

So it is easier for a computer to evaluate expressions that are in these forms.

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Postfix FormPostfix Form The postfix form of a variable or

constant is the same as its infix form. a, b, 3.25

The relative order of operands is the same in infix and postfix forms.

Operators come immediately after the postfix form of their operands. Infix = a + b Postfix = ab+

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Postfix ExamplesPostfix Examples

Infix = a + b * c Postfix =ab c * +

• Infix = a * b + c Postfix = a b * c +

• Infix = (a + b) * (c – d) / (e + f) Postfix = a b + c d - * e f + /

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Expression TreeExpression Tree

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Expression TreeExpression Tree

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Binary Tree FormBinary Tree Form a + b +

a b

• - a -

a

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Binary Tree FormBinary Tree Form

(a + b) * (c – d) / (e + f)

/

+

a b

-

c d

+

e f

*

/

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Expression Tree

Infix Expression =?

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Constructing an Expression TreeConstructing an Expression Treea b + c d * -

a bc da b

+

a b

+

c d

*

a b

+

c d

*

-

(a) (b)

(c) (d)

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Binary Search Trees

การประยุ�กต์ใช้� Tree

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Figure 8-1

Binary Search Tree

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Binary Search TreesBinary Search Trees

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Are these Binary Search Trees?

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Construct a Binary Search TreeConstruct a Binary Search Tree

เวลาที่��ใช้�ในการค้�นหาข้�อมู�ล Worst case?Average case?

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AVL Trees

Balance Binary Search Tree

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AVL TreesAVL Trees

Balanced binary tree structure, named after Adelson, Velski, and Landis

An AVL tree is a height balanced binary search tree.

|HL – HR| <= 1

where HL is the height of the left subtree and HR is the height of the left subtree

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Binary Search TreesBinary Search Trees

(b) AVL Tree

(a) An unbalanced BST

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Out of BalanceOut of Balance

Four cases of out of balance:

left of left (LL) - requires single rotation right of right (RR) - requires single rotation

Left of right (LR) - requires double rotation Right of left (RL) - requires double rotation

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Out of Balance (left of left)Out of Balance (left of left)

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Out of Balance (left of left)Out of Balance (left of left)

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Out of Balance (right of right)Out of Balance (right of right)

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Out of Balance (right of right)Out of Balance (right of right)

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Simple double rotation rightSimple double rotation right

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Complex double rotation rightComplex double rotation right

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Insert a node to AVL treeInsert a node to AVL tree

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Balancing BSTBalancing BST

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Deleting a node from AVL treeDeleting a node from AVL tree

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เวลาที่��ใช้�ในการค้�นหาข้�อมู�ลใน AVL Tree

Worst case?Average case?

Balance Binary Search Tree

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Priority QueuePriority Queue

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• Collection of elements.• Each element has a priority or key.• Supports following operations:

isEmpty size add/put an element into the priority queue get element with min/max priority remove element with min/max priority

Min Tree ExampleMin Tree Example

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2

4 9 3

4 8 7

9 9

Root is the minimum element

Max Tree ExampleMax Tree Example

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9

4 9 8

4 2 7

3 1

Root is the maximum element

Min Heap DefinitionMin Heap Definition

• complete binary tree• min tree

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Complete binary tree with 9 nodes that is also a min tree.

2

4

6 7 9 3

8 6

3

Max Heap With 9 NodesMax Heap With 9 Nodes

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9

8

6 7 2 6

5 1

7

Complete binary tree with 9 nodes that is also a max tree.

Heap HeightHeap Height

What is the height of an n node heap ?

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Since a heap is a complete binary tree, the height of an n node heap is log2 (n+1).

9 8 7 6 7 2 6 5 1

1 2 3 4 5 6 7 8 9 100

A Heap Is Efficiently Represented As An Array

9

8

6 7 2 6

5 1

7

Moving Up And Down A Heap

9

8

6 7 2 6

5 1

7

1

2 3

4 5 6 7

8 9

Putting An Element Into A Max Heap

Complete binary tree with 10 nodes.

9

8

6 7 2 6

5 1

7

7

Putting An Element Into A Max Heap

New element is 5.

9

8

6 7 2 6

5 1

7

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Putting An Element Into A Max Heap

New element is 20.

9

8

6

7

2 6

5 1

7

7

7

Putting An Element Into A Max Heap

New element is 20.

9

8

6

7

2 6

5 1

7

77

Putting An Element Into A Max Heap

New element is 20.

9

86

7

2 6

5 1

7

77

Putting An Element Into A Max Heap

New element is 20.

9

86

7

2 6

5 1

7

77

20

Putting An Element Into A Max Heap

Complete binary tree with 11 nodes.

9

86

7

2 6

5 1

7

77

20

Putting An Element Into A Max Heap

New element is 15.

9

86

7

2 6

5 1

7

77

20

Putting An Element Into A Max Heap

New element is 15.

9

8

6

7

2 6

5 1

7

77

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8

Putting An Element Into A Max Heap

New element is 15.

8

6

7

2 6

5 1

7

77

20

8

9

15

Complexity Of Put

Complexity is O(log n), where n is heap size.

8

6

7

2 6

5 1

7

77

20

8

9

15

Removing The Max Element

Max element is in the root.

8

6

7

2 6

5 1

7

77

20

8

9

15

Removing The Max Element

After max element is removed.

8

6

7

2 6

5 1

7

77 8

9

15

Removing The Max Element

Heap with 10 nodes.

8

6

7

2 6

5 1

7

77 8

9

15

Reinsert 8 into the heap.

Removing The Max Element

Reinsert 8 into the heap.

6

7

2 6

5 1

7

77

9

15

Removing The Max Element

Reinsert 8 into the heap.

6

7

2 6

5 1

7

77

9

15

Removing The Max Element

Reinsert 8 into the heap.

6

7

2 6

5 1

7

77

9

15

8

Removing The Max Element

Max element is 15.

6

7

2 6

5 1

7

77

9

15

8

Removing The Max Element

After max element is removed.

6

7

2 6

5 1

7

77

9

8

Removing The Max Element

Heap with 9 nodes.

6

7

2 6

5 1

7

77

9

8

Removing The Max Element

Reinsert 7.

6 2 6

5 1

79

8

Removing The Max Element

Reinsert 7.

6 2 6

5 1

7

9

8

Removing The Max Element

Reinsert 7.

6 2 6

5 1

7

9

8

7

Complexity Of Remove Max Element

Complexity is O(log n).

6 2 6

5 1

7

9

8

7

Complexity of Operations

Two good implementations are heaps and leftist trees.

isEmpty, size, and get => O(1) time

put and remove => O(log n) time where n is the size of the priority queue

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Practical ComplexitiesPractical Complexities

109 instructions/second

n n nlogn n2 n3

1000 1mic 10mic 1milli 1sec

10000 10mic 130mic 100milli 17min

106 1milli 20milli 17min 32years

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Impractical ComplexitiesImpractical Complexities109 instructions/second

n n4 n10 2n

1000 17min 3.2 x 1013 years

3.2 x 10283 years

10000

116 days

??? ???

106 3 x 107 years

?????? ??????

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Summary

n Insertion Sort

O(n2)

Shellsort

O(n7/6)

Heapsort

O(n log n)

Quicksort

O(n log n)

10 0.00044 0.00041 0.00057 0.00052

100 0.00675 0.00171 0.00420 0.00284

1000 0.59564 0.02927 0.05565 0.03153

10000 58864 0.42998 0.71650 0.36765

100000 NA 5.7298 8.8591 4.2298

1000000 NA 71.164 104.68 47.065

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Faster Computer Vs Better Faster Computer Vs Better AlgorithmAlgorithm

Algorithmic improvement more usefulthan hardware improvement.

E.g. 2n to n3

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