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FAST- National University of
Computer and Emerging Sciences
CFD Campus
Data Structures and Algorithm
Application of Stack
Asma Sattar
[email protected]
Lecture No. 7
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Today’s Lecture
• Algebraic Expression
• Infix to Postfix
• Postfix Evaluation
• Infix to Prefix
• Prefix Evaluation
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Algebraic Expression
• An algebraic expression is a legal combination of operands and
the operators.
– Operand is the quantity on which a mathematical operation is
performed.
– Operator is a symbol which signifies a mathematical
operation
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Infix, Postfix and Prefix Expressions
• INFIX: expressions in which operands surround the
operator.
• POSTFIX: operator comes after the operands, also Known as
Reverse Polish Notation (RPN).
• PREFIX: operator comes before the operands, also Known as
Polish notation.
• Example
– Infix: A+B-C Postfix: AB+C- Prefix: +A-BC
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Why do we need PREFIX/POSTFIX?
• Appearance may be misleading, INFIX notations are not as
simple as they seem
• To evaluate an infix expression we need to consider
– Operators’ Priority – Associative property
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Why do we need PREFIX/POSTFIX?
• Infix Expression Is Hard To Parse and difficult to
evaluate.
• Postfix and prefix do not rely on operator priority and are
easier to parse.
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Why do we need PREFIX/POSTFIX?
• An expression in infix form is thus converted into prefix or
postfix form and then evaluated without considering the operators
priority and delimiters.
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Examples of infix to prefix and post
fix
Infix PostFix Prefix
A+B AB+ +AB
(A+B) * (C + D) AB+CD+* *+AB+CD
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Prefix example:
2+3*4-5/6^(2-1*3)
Prefix= -+2*34/5^6-2*13 (solve on board)
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Prefix Task:
2*5^6/3
Prefix=/*2^563
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Postfix example:
2+3*4-5/6^(2-1*3)
Postfix= 234*+56231*-^/-
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Postfix Task:
f^(g-h)+(j-k)
Postfix= F g h - ^ j k -+
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Algorithm Infix To Postfix
Assumptions:
1. Space delimited list of tokens represents a
infix expression
2. Operands are single characters.
3. Operators +,-,*,/ , ^
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Algorithm for Infix to postfix
conversionAlgorithm: Q is the given infix expression & we
want P.
1. Push “(“ on STACK and add “)” at the end of Q.
2. Scan Q from left to right and repeat steps 3 & 6 for each
element of Q until the STACK is empty.
3. If an operand is encountered, add it to P
4. If a left parenthesis is encountered, push it onto STACK.
5. If an operator X is encountered, then:a. Repeatedly pop from
STACK and add to P each operator which has same or higher
precedence than X
b. Push X on STACK
6. If a right parenthesis is encountered, then:a. Repeatedly pop
from STACK and add to P each operator until a left parenthesis
is
encountered
b. Remove the left parenthesis. [Do not add it to P]
7. Exit
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Infix to Postfix
( ( A + B ) * ( C - E ) ) / ( F + G )
stack: output: []
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Infix to Postfix
( ( A + B ) * ( C - E ) ) / ( F + G ) )
stack: (output: []
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Infix to Postfix
( A + B ) * ( C - E ) ) / ( F + G ) )
stack: ( (output: []
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Infix to Postfix
A + B ) * ( C - E ) ) / ( F + G ) )
stack: ( ( (output: []
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Infix to Postfix
+ B ) * ( C - E ) ) / ( F + G ) )
stack: ( ( (output: [A]
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Infix to Postfix
B ) * ( C - E ) ) / ( F + G ) )
stack: ( ( ( +output: [A]
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Infix to Postfix
) * ( C - E ) ) / ( F + G ) )
stack: ( ( ( +output: [A B]
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Infix to Postfix
* ( C - E ) ) / ( F + G ) )
stack: ( ( output: [A B + ]
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Infix to Postfix
( C - E ) ) / ( F + G ) )
stack: ( ( * output: [A B + ]
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Infix to Postfix
C - E ) ) / ( F + G ) )
stack: ( ( * (output: [A B + ]
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Infix to Postfix
- E ) ) / ( F + G ) )
stack: ( ( * (output: [A B + C ]
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Infix to Postfix
E ) ) / ( F + G ) )
stack: ( ( * ( -output: [A B + C ]
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Infix to Postfix
) ) / ( F + G ) )
stack: ( ( * ( -output: [A B + C E ]
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Infix to Postfix
) / ( F + G ) )
stack: ( ( *output: [A B + C E - ]
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Infix to Postfix
/ ( F + G ) )
stack: ( output: [A B + C E - * ]
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Infix to Postfix
( F + G ) )
stack: ( /output: [A B + C E - * ]
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Infix to Postfix
F + G ) )
stack: ( / (output: [A B + C E - * ]
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Infix to Postfix
+ G ) )
stack: ( / (output: [A B + C E - * F ]
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Infix to Postfix
G ) )
stack: ( / ( +output: [A B + C E - * F ]
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Infix to Postfix
) )
stack: ( / ( +output: [A B + C E - * F G ]
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Infix to Postfix
)
stack: ( /output: [A B + C E - * F G + ]
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Infix to Postfix
stack: output: [A B + C E - * F G + / ]
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Task
infix
postfixt
( a + b - c ) * d – ( e + f )
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Evaluating a postfix expression
Algorithm: P is the given postfix expression.
1. Add a right parenthesis “)” at the end of P. [acts as a
sentinel]
2. Scan P from left to right and repeat steps 3 & 4 for each
element of P until “)” is encountered.
3. If an operand is encountered, push it on STACK
4. If an operator is encountered, then:a. Pop two operands from
STACK: A & B
b. Evaluate: B operator A
c. Push result on STACK
5. Set value equal to the top element on STACK
6. Exit
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From Postfix to Answer
• The reason to convert infix to postfixexpression is that we
can compute theanswer of postfix expression easier byusing a
stack.
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From
Ex: 10 2 8 *
Postfix
+ 3 -
to Answer
• First, push(10) into thestack
10
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From
Ex: 10
2 8 *
Postfix
+ 3 -
to Answer
• Then, push(2) into thestack
2
10
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From Postfix to
Ex: 10 2 8 * + 3 -
Answer
• Push(8) into the stack
8
2
10
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From
Ex: 10
2 8 *
Postfix to
+ 3 -
Answer
• Now we see an operator *,that means we can get annew number by
calculation
8
2
10
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From Postfix to
+ 3 -
Answer
Ex: 10 2 8 *
• Now we see an operator *,that means we can get annew number by
calculation
Pop the first two numbers•
* = 1682
8
2
10
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From Postfix to
+ 3 -
Answer
Ex: 10 2 8 *
• Now we see an operator *,that means we can get annew number by
calculation
Push the new number back•
* = 168216
10
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From Postfix
+ 3 -
to Answer
Ex: 10 2 8 *
• Then we see the nextoperator + and performthe calculation
+ = 26161016
10
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From Postfix
+ 3 -
to Answer
Ex: 10 2 8 *
• Then we see the nextoperator + and performthe calculation
Push the new number back•
+ = 261610
26
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From
Ex: 10
2 8 *
Postfix
+ 3 -
to Answer
•
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We see the next number 3
Push (3) into the stack
3
26
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Compute the Answer
Ex: 10 2 8 * + 3 -
• The last operation
- = 23326
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From Postfix
+ 3 -
to Answer
Ex: 10 2 8 *
• The last operation
- = 23
answe
r!
326
23
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Task
623+-382/+*2^3+
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Task
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Quiz Time
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Reading Material
• Nell Dale – Chapter#4
• Schaum’s Outlines – Chapter#6
• D. S. Malik – Chapter#7
• http://www.cs.man.ac.uk/~pjj/cs2121/fix.html
• Examples
– animations\in2post\infixtopostfix.exe
– animations\evaluation\evaluation.exe
– animations\tower\tower.exe
CS 201 - Fall 2013
http://www.cs.man.ac.uk/~pjj/cs2121/fix.htmlanimations/in2post/infixtopostfix.exeanimations/evaluation/evaluation.exeanimations/tower/tower.exe
