stack applications Infix to Postfix, Prefix & Evaluation of Post Fix, Prefix

1. DATA STRUCTURES
Applications of Stacks
Dr. P. Subathra
subathrakishore@yahoo.com
Professor
Dept. of Information Technology
KAMARAJ College of Engineering & Technology
(AUTONOMOUS)
Madurai
Tamil Nadu
India

2. CS8391 – DATA STRUCTURES
ONLINE CLASSES – CLASS NO. 14
25.09.2020
(10:00 AM – 12:00 PM)

3. Applications of Stacks

4. Prefix, Postfix, Infix Notation

5. Infix Notation
• To add A, B, we write
A+B
• To multiply A, B, we write
A*B
• The operators ('+' and '*') go in between the
operands ('A' and 'B')
• This is "Infix" notation.

6. Prefix Notation
• Instead of saying "A plus B", we could say "add
A,B " and write
+ A B
• "Multiply A,B" would be written
* A B
• This is Prefix notation.

7. Postfix Notation
• Another alternative is to put the operators
after the operands as in
A B +
and
A B *
• This is Postfix notation.

8. • The terms infix, prefix, and postfix tell us
whether the operators go between, before, or
after the operands.
Pre A In B Post

9. Parentheses
• Evaluate 2+3*5.
• + First:
(2+3)*5 = 5*5 = 25
• * First:
2+(3*5) = 2+15 = 17
• Infix notation requires Parentheses.

10. What about Prefix Notation?
• + 2 * 3 5 =
= + 2 * 3 5
= + 2 15 = 17
• * + 2 3 5 =
= * + 2 3 5
= * 5 5 = 25
• No parentheses needed!

11. Postfix Notation
• 2 3 5 * + =
= 2 3 5 * +
= 2 15 + = 17
• 2 3 + 5 * =
= 2 3 + 5 *
= 5 5 * = 25
• No parentheses needed here either!

12. Conclusion:
• Infix is the only notation that requires
parentheses in order to change the order in
which the operations are done.

13. CS314 Stacks 13
Mathematical Calculations
• What does 3 + 2 * 4 equal?
2 * 4 + 3? 3 * 2 + 4?
• The precedence of operators affects the order of
operations.
• A mathematical expression cannot simply be evaluated left
to right.
• A challenge when evaluating a program.
• Lexical analysis is the process of interpreting a program.
What about 1 - 2 - 4 ^ 5 * 3 * 6 / 7 ^ 2 ^ 3

14. CONVERSION OF
INFIX
TO
PREFIX AND POSTFIX
EXPRESSIONS

15. Fully Parenthesized Expression
• A FPE has exactly one set of Parentheses
enclosing each operator and its operands.
• Which is fully parenthesized?
( A + B ) * C
( ( A + B) * C )
( ( A + B) * ( C ) )

16. Infix to Prefix Conversion
Move each operator to the left of its operands &
remove the parentheses:
( ( A + B) * ( C + D ) )

17. Infix to Prefix Conversion
Move each operator to the left of its operands &
remove the parentheses:
( + A B) * ( C + D ) )

18. Infix to Prefix Conversion
Move each operator to the left of its operands &
remove the parentheses:
( + AB) * ( C + D ) )

19. Infix to Prefix Conversion
Move each operator to the left of its operands &
remove the parentheses:
( + AB) * +CD ) )

20. Infix to Prefix Conversion
Move each operator to the left of its operands &
remove the parentheses:
( + AB) * + CD ) )

21. Infix to Prefix Conversion
Move each operator to the left of its operands &
remove the parentheses:
* + AB) + CD ) )

22. Infix to Prefix Conversion
Move each operator to the left of its operands &
remove the parentheses:
* + AB) + CD ) )
* + AB)+ CD ) )

23. Infix to Prefix Conversion
Now Remove all the Closing Parenthesis:
* + AB)+ CD ) )
* + AB+ CD

24. Infix to Prefix Conversion
Move each operator to the left of its operands &
remove the parentheses:
* + A B + C D
Order of operands does not change!

25. Infix to Postfix
( ( ( A + B ) * C ) - ( ( D + E ) / F ) )

26. Infix to Postfix
( ( ( A + B ) * C ) - ( ( D + E ) / F ) )
( ( ( AB + * C ) - ( ( D E + / F ) )

27. Infix to Postfix
( ( ( AB + * C ) - ( ( D E + / F ) )
( ( ( AB + C * - ( ( D E + / F ) )
( ( ( AB + C * - ( ( D E + F / )
( ( ( AB +C * - ( ( D E + F / )

28. Infix to Postfix
( ( ( AB +C * - ( ( D E + F / )
( ( ( AB +C * - ( ( D E + F / )
( ( ( AB +C * ( ( D E + F / -
( ( ( AB +C * ( ( D E + F / -

29. Infix to Postfix
( ( ( AB +C * ( ( D E + F / -
( ( ( AB +C * ( ( D E + F / -
Remove all open parenthesis
AB +C * D E + F / -
Postfix Expression: AB +C * D E + F / -
• Operand order does not change!
• Operators are in order of evaluation

30. Infix to Postfix
• Initialize a Stack for operators, output list
• Split the input into a list of tokens.
• for each token (left to right):
if it is operand: append to output
if it is '(': push onto Stack
if it is ')': pop & append till '('
if it in '+-*/':
while peek has precedence ≥ it:
pop & append
push onto Stack
pop and append the rest of the Stack.

31. Operators Precedence & Associativity Table
Operator Precedence

32. InfixVect
postfixVect
( a + b - c ) * d – ( e + f )
Infix to postfix conversion

33. infixVect
postfixVect
a + b - c ) * d – ( e + f )
(
stackVect
Infix to postfix conversion

34. infixVect
postfixVect
+ b - c ) * d – ( e + f )
(
a
Infix to postfix conversionstackVect

35. infixVect
postfixVect
b - c ) * d – ( e + f )
(
a
+
Infix to postfix conversionstackVect

36. infixVect
postfixVect
- c ) * d – ( e + f )
(
a b
+
Infix to postfix conversionstackVect

37. infixVect
postfixVect
c ) * d – ( e + f )
(
a b +
-
Infix to postfix conversionstackVect

38. infixVect
postfixVect
) * d – ( e + f )
(
a b + c
-
Infix to postfix conversionstackVect

39. infixVect
postfixVect
* d – ( e + f )
a b + c -
Infix to postfix conversionstackVect

40. infixVect
postfixVect
d – ( e + f )
a b + c -
*
Infix to postfix conversionstackVect

41. infixVect
postfixVect
– ( e + f )
a b + c - d
*
Infix to postfix conversionstackVect

42. infixVect
postfixVect
( e + f )
a b + c – d *
-
Infix to postfix conversionstackVect

43. infixVect
postfixVect
e + f )
a b + c – d *
-
(
Infix to postfix conversionstackVect

44. infixVect
postfixVect
+ f )
a b + c – d * e
-
(
Infix to postfix conversionstackVect

45. infixVect
postfixVect
f )
a b + c – d * e
-
(
+
Infix to postfix conversionstackVect

46. infixVect
postfixVect
)
a b + c – d * e f
-
(
+
Infix to postfix conversionstackVect

47. infixVect
postfixVect
a b + c – d * e f +
-
Infix to postfix conversionstackVect

48. infixVect
postfixVect
a b + c – d * e f + -
Infix to postfix conversionstackVect

49. Infix to Postfix Expression
https://www.youtube.com/watch?v=OVFwgYrM
Shw

50. Infix to Prefix Expression
• https://www.youtube.com/watch?v=sJ0VhIbv
Ctc

51. EVALUATION OF POSTFIX EXPRESSION
SIMPLE POSTFIX EXPRESSION EVALUATION

52. Evaluation Rule of a Postfix
Expression States
• While reading the expression from left to
right, push the element in the stack if it is an
operand.
• Pop the two operands from the stack, if the
element is an operator and then evaluate it.
• Push back the result of the evaluation.
• Repeat it till the end of the expression.
• Finally the result is in the Stack

53. Example : Evaluate the Postfix
Expression
• 456*+

54. Evaluation of Postfix Expression
https://www.youtube.com/watch?v=uh7fD8WiT
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