Lecture No.04 Data Structures€¦ · Stack Using Linked List We can avoid the size limitation of a stack implemented with an array by using a linked list to hold the stack elements.

27-Sep-19

1

Lecture No.04

Data Structures

Stacks

Stacks in real life: stack of books, stack of plates

Add new items at the top

Remove an item at the top

Stack data structure similar to real life: collection of elements

arranged in a linear order.

Can only access element at the top

27-Sep-19

2

Stack Operations

Push(X) – insert X as the top element of the stack

Pop() – remove the top element of the stack and return it.

Top() – return the top element without removing it from the

stack.

Stack Operations

push(2)

top 2

push(5)

top

2

5

push(7)

top

2

5

7

push(1)

top

2

5

7

1

1 pop()

top

2

5

7

push(21)

top

2

5

7

21

21 pop()

top

2

5

7

7 pop()

2

5top

5 pop()

2top

27-Sep-19

3

Stack Operation

The last element to go into the stack is the first to come out:

LIFO – Last In First Out.

What happens if we call pop() and there is no element?

Have IsEmpty() boolean function that returns true if stack is

empty, false otherwise.

Throw StackEmpty exception: advanced C++ concept.

Stack Implementation: Array

Worst case for insertion and deletion from an array when

insert and delete from the beginning: shift elements to the left.

Best case for insert and delete is at the end of the array – no

need to shift any elements.

Implement push() and pop() by inserting and deleting at the

end of an array.

27-Sep-19

4

Stack using an Array

top

2

5

7

12 5 7 1

0 1 32 4

top = 3

Stack using an Array

In case of an array, it is possible that the array may “fill-up” if

we push enough elements.

Have a boolean function IsFull() which returns true is

stack (array) is full, false otherwise.

We would call this function before calling push(x).

27-Sep-19

5

Stack Operations with Array

int pop()

{

return A[current--];

}

void push(int x)

{

A[++current] = x;

}

Stack Operations with Array

int top(){

return A[current];} int IsEmpty(){

return ( current == -1 );}int IsFull(){

return ( current == size-1);}

A quick examination shows that all five operations take constant time.

27-Sep-19

6

Stack Using Linked List

We can avoid the size limitation of a stack implemented with

an array by using a linked list to hold the stack elements.

As with array, however, we need to decide where to insert

elements in the list and where to delete them so that push and

pop will run the fastest.


We can avoid the size limitation of a stack implemented with

an array by using a linked list to hold the stack elements.

As with array, however, we need to decide where to insert

elements in the list and where to delete them so that push and

pop will run the fastest.

27-Sep-19

7


For a singly-linked list, insert at start or end takes constant time using the head and current pointers respectively.

Removing an element at the start is constant time but removal at the end required traversing the list to the node one before the last.

Make sense to place stack elements at the start of the list because insert and removal are constant time.


No need for the current pointer; head is enough.

top

2

5

7

11 7 5 2

head

27-Sep-19

8

Stack Operation: List

int pop(){

int x = head->get();Node* p = head;head = head->getNext();delete p;return x;

}

top

2

5

71 7 5 2

head


void push(int x){

Node* newNode = new Node();newNode->set(x);newNode->setNext(head);head = newNode;

}

top

2

5

7

9

7 5 2

head

push(9)

9

newNode

27-Sep-19

9


int top(){

return head->get();} int IsEmpty(){

return ( head == NULL );}

All four operations take constant time.

Stack: Array or List

Since both implementations support stack operations in constant time, any reason to choose one over the other?

Allocating and deallocating memory for list nodes does take more time than preallocated array.

List uses only as much memory as required by the nodes; array requires allocation ahead of time.

List pointers (head, next) require extra memory.

Array has an upper limit; List is limited by dynamic memory allocation.

27-Sep-19

10

Use of Stack

Example of use: prefix, infix, postfix expressions.

Consider the expression A+B: we think of applying the

operator “+” to the operands A and B.

“+” is termed a binary operator: it takes two operands.

Writing the sum as A+B is called the infix form of the

expression.

Prefix, Infix, Postfix

Two other ways of writing the expression are

+ A B prefixA B + postfix

The prefixes “pre” and “post” refer to the position of the operator with respect to the two operands.

27-Sep-19

11


Consider the infix expression

A + B * C

We “know” that multiplication is done before addition.

The expression is interpreted as

A + ( B * C )

Multiplication has precedence over addition.


Conversion to postfix

A + ( B * C ) infix form

27-Sep-19

12




A + ( B C * ) convert multiplication





A ( B C * ) + convert addition

27-Sep-19

13





A ( B C * ) + convert addition

A B C * + postfix form



(A + B ) * C infix form

27-Sep-19

14




( A B + ) * C convert addition





( A B + ) C * convert multiplication

27-Sep-19

15





( A B + ) C * convert multiplication

A B + C * postfix form

Precedence of Operators

The five binary operators are: addition, subtraction,

multiplication, division and exponentiation.

The order of precedence is (highest to lowest)

Exponentiation

Multiplication/division *, /

Addition/subtraction +, -

27-Sep-19

16

Precedence of Operators

For operators of same precedence, the left-to-right rule

applies:

A+B+C means (A+B)+C.

For exponentiation, the right-to-left rule applies

A B C means A ( B C )

Infix to Postfix

Infix Postfix

A + B A B +

12 + 60 – 23 12 60 + 23 –

(A + B)*(C – D ) A B + C D – *

A B * C – D + E/F A B C*D – E F/+

27-Sep-19

17

Infix to Postfix

Note that the postfix form an expression does not require parenthesis.

Consider ‘4+3*5’ and ‘(4+3)*5’. The parenthesis are not needed in the first

but they are necessary in the second.

The postfix forms are:

4+3*5 435*+

(4+3)*5 43+5*

Evaluating Postfix

Each operator in a postfix expression refers to the previous two operands.

Each time we read an operand, we push it on a stack.

When we reach an operator, we pop the two operands from the top of the

stack, apply the operator and push the result back on the stack.

27-Sep-19

18

Evaluating Postfix

Stack s;

while( not end of input ) {

e = get next element of input

if( e is an operand )

s.push( e );

else {

op2 = s.pop();

op1 = s.pop();

value = result of applying operator ‘e’ to op1 and op2;

s.push( value );

}

}

finalresult = s.pop();

Evaluating Postfix

Evaluate 6 2 3 + - 3 8 2 / + * 2 3 +

Input op1 op2 value stack

6 6

27-Sep-19

19

Evaluating Postfix

Evaluate 6 2 3 + - 3 8 2 / + * 2 3 +


6 6

2 6,2

Evaluating Postfix

Evaluate 6 2 3 + - 3 8 2 / + * 2 3 +


6 6

2 6,2

3 6,2,3

27-Sep-19

20

Evaluating Postfix

Evaluate 6 2 3 + - 3 8 2 / + * 2 3 +


6 6

2 6,2

3 6,2,3

+ 2 3 5 6,5

Evaluating Postfix

Evaluate 6 2 3 + - 3 8 2 / + * 2 3 +


6 6

2 6,2

3 6,2,3

+ 2 3 5 6,5

- 6 5 1 1

27-Sep-19

21

Evaluating Postfix

Evaluate 6 2 3 + - 3 8 2 / + * 2 3 +


6 6

2 6,2

3 6,2,3

+ 2 3 5 6,5

- 6 5 1 1

3 6 5 1 1,3

Evaluating Postfix

Evaluate 6 2 3 + - 3 8 2 / + * 2 3 +


6 6

2 6,2

3 6,2,3

+ 2 3 5 6,5

- 6 5 1 1

3 6 5 1 1,3

8 6 5 1 1,3,8

27-Sep-19

22

Evaluating Postfix

Evaluate 6 2 3 + - 3 8 2 / + * 2 3 +


6 6

2 6,2

3 6,2,3

+ 2 3 5 6,5

- 6 5 1 1

3 6 5 1 1,3

8 6 5 1 1,3,8

2 6 5 1 1,3,8,2

Evaluating Postfix

Evaluate 6 2 3 + - 3 8 2 / + * 2 3 +


6 6

2 6,2

3 6,2,3

+ 2 3 5 6,5

- 6 5 1 1

3 6 5 1 1,3

8 6 5 1 1,3,8

2 6 5 1 1,3,8,2

/ 8 2 4 1,3,4

27-Sep-19

23

Evaluating Postfix

Evaluate 6 2 3 + - 3 8 2 / + * 2 3 +


6 6

2 6,2

3 6,2,3

+ 2 3 5 6,5

- 6 5 1 1

3 6 5 1 1,3

8 6 5 1 1,3,8

2 6 5 1 1,3,8,2

/ 8 2 4 1,3,4

+ 3 4 7 1,7

Evaluating Postfix

Evaluate 6 2 3 + - 3 8 2 / + * 2 3 +


6 6

2 6,2

3 6,2,3

+ 2 3 5 6,5

- 6 5 1 1

3 6 5 1 1,3

8 6 5 1 1,3,8

2 6 5 1 1,3,8,2

/ 8 2 4 1,3,4

+ 3 4 7 1,7

* 1 7 7 7

27-Sep-19

24

Evaluating Postfix

Evaluate 6 2 3 + - 3 8 2 / + * 2 3 +


6 6

2 6,2

3 6,2,3

+ 2 3 5 6,5

- 6 5 1 1

3 6 5 1 1,3

8 6 5 1 1,3,8

2 6 5 1 1,3,8,2

/ 8 2 4 1,3,4

+ 3 4 7 1,7

* 1 7 7 7

2 1 7 7 7,2

Evaluating Postfix

Evaluate 6 2 3 + - 3 8 2 / + * 2 3 +Input op1 op2 value stack

6 6

2 6,2

3 6,2,3

+ 2 3 5 6,5

- 6 5 1 1

3 6 5 1 1,3

8 6 5 1 1,3,8

2 6 5 1 1,3,8,2

/ 8 2 4 1,3,4

+ 3 4 7 1,7

* 1 7 7 7

2 1 7 7 7,2

7 2 49 49

27-Sep-19

25

Evaluating Postfix

Evaluate 6 2 3 + - 3 8 2 / + * 2 3 +


6 6

2 6,2

3 6,2,3

+ 2 3 5 6,5

- 6 5 1 1

3 6 5 1 1,3

8 6 5 1 1,3,8

2 6 5 1 1,3,8,2

/ 8 2 4 1,3,4

+ 3 4 7 1,7

* 1 7 7 7

2 1 7 7 7,2

7 2 49 49

3 7 2 49 49,3

Evaluating Postfix

Evaluate 6 2 3 + - 3 8 2 / + * 2 3 +


6 6

2 6,2

3 6,2,3

+ 2 3 5 6,5

- 6 5 1 1

3 6 5 1 1,3

8 6 5 1 1,3,8

2 6 5 1 1,3,8,2

/ 8 2 4 1,3,4

+ 3 4 7 1,7

* 1 7 7 7

2 1 7 7 7,2

7 2 49 49

3 7 2 49 49,3

+ 49 3 52 52

27-Sep-19

26

Evaluating Postfix

Evaluate 6 2 3 + - 3 8 2 / + * 2 3 +


6 6

2 6,2

3 6,2,3

+ 2 3 5 6,5

- 6 5 1 1

3 6 5 1 1,3

8 6 5 1 1,3,8

2 6 5 1 1,3,8,2

/ 8 2 4 1,3,4

+ 3 4 7 1,7

* 1 7 7 7

2 1 7 7 7,2

7 2 49 49

3 7 2 49 49,3

+ 49 3 52 52

Converting Infix to Postfix

Consider the infix expressions ‘A+B*C’ and ‘ (A+B)*C’.

The postfix versions are ‘ABC*+’ and ‘AB+C*’.

The order of operands in postfix is the same as the infix.

In scanning from left to right, the operand ‘A’ can be inserted into postfix

expression.

27-Sep-19

27


The ‘+’ cannot be inserted until its second operand has been scanned and

inserted.

The ‘+’ has to be stored away until its proper position is found.

When ‘B’ is seen, it is immediately inserted into the postfix expression.

Can the ‘+’ be inserted now? In the case of ‘A+B*C’ cannot because * has

precedence.


In case of ‘(A+B)*C’, the closing parenthesis indicates that ‘+’ must be

performed first.

Assume the existence of a function ‘prcd(op1,op2)’ where op1 and op2 are

two operators.

Prcd(op1,op2) returns TRUE if op1 has precedence over op2, FASLE

otherwise.

27-Sep-19

28


prcd(‘*’,’+’) is TRUE

prcd(‘+’,’+’) is TRUE

prcd(‘+’,’*’) is FALSE

Here is the algorithm that converts infix expression to its postfix form.

The infix expression is without parenthesis.

Lecture No.04 Data Structures€¦ · Stack Using Linked List We can avoid the size limitation of a stack implemented with an array by using a linked list to hold the stack elements.

Documents