stacks in algorithems and data structure

1. AAllggoorriitthhmmss aanndd DDaattaa
SSttrruuccttuurreess
Lecture 5
Oxford University Data Structures Using C++ by Dr Varsha Patil Press © 2012

2. 2
Applications Of Stack
 Convert infix expression to postfix and prefix
expressions
 Evaluate the postfix expression
 Reverse a string
 Check well-formed (nested) parenthesis
 Reverse a string
 Process subprogram function calls
 Parse (analyze the structure) of computer programs
 Simulate recursion
 In computations like decimal to binary conversion
 In Backtracking algorithms (often used in optimizations and
in games)

3. 3
Expression Evaluation And
Conversion
 The most frequent application of stacks is in evaluation of
arithmetic expressions.
 An arithmetic expression is made up of operands, operators,
and delimiters.
 When higher level programming language came into
existence one of the major difficulty faced by computer scientists
was to generate machine language instruction, which would
properly evaluate any arithmetic expression.

4. 4
Expression Evaluation
 To fix the order of evaluation, assign to each operator a priority
 Because even though we write expression in parenthesis, still
we have a question in mind, about whether to evaluate which
one bracket first
 Once the priorities are assigned then within any pairs of
parenthesis we can understand that operators with highest
priority are to be evaluated first
 While evaluating expression usually the following operation
precedence is used

5. 5
The following operators are written is in descending
order of their precedence:
Exponentiation ^, Unary +, Unary –, and not ~
Multiplication * and division /
Addition + and subtraction –
<, £ , =, ¹, ³, >
AND
OR

6. 6
The Operators and priorities
Data Structures in C++ by Dr. Varsha Patil Oxferd Univercity

7. 7
Polish Notation and Expression Conversion
Polish Mathematician Han Lukasiewicz suggested a notation called
pPolish notation, which gives two alternatives to represent an
arithmetic expression, the notation are postfix and prefix notations
The fundamental property of Polish notation is that the order in
which the operations are to be performed is determined by the
positions of the operators and operands in the expression.
Hence the advantage is parenthesis is not required while writing
expressions in pPolish notation

8. Oxford University Data Structures Using C++ by Dr Varsha Patil Press © 2012
8
The example ex…pression in various forms-infix,
prefix and postfix
The postfix expressions can be evaluated easily hence infix
expression is converted into postfix expression using stack.

9. Oxford University Data Structures Using C++ by Dr Varsha Patil Press © 2012
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Need of prefix and postfix expressions
Evaluation of infix expression using computer needs proper code
generation by compiler without any ambiguity and is difficult due to
various aspects such as operators priority and associativity
This problem can be overcome by writing or converting infix to alternate
notation such as prefix or postfix
The postfix and prefix expressions possess many advantages as listed

10. Oxford University Data Structures Using C++ by Dr Varsha Patil Press © 2012
10
The need for parenthesis as in infix expression is overcome in
postfix and prefix Notations
 The priority of operators is no longer relevant
Order of evaluation depends on position of operator not on priority and
associativity
The expression evaluation process is much simpler than attempting a direct
evaluation from infix notation.

11. Oxford University Data Structures Using C++ by Dr Varsha Patil Press © 2012
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Postfix Expression Evaluation process
The postfix expression may be evaluated by making a left to right scan,
stacking operands, and evaluating operators using as operands the correct
number from the stack and again placing the result again onto stack
This evaluation process is much simpler than attempting a direct evaluation
from infix notation
Process continues till stack is not empty or on occurrence of a character #
which denotes the end of expression

12. Oxford University Data Structures Using C++ by Dr Varsha Patil Press © 2012
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Steps for manually converting expression from one
notation to other
1)Initially fully parenthesize the given infix expression.
2) Use operator precedence and associability rules for the same
3) Now move all operators so that they replace their corresponding right
parenthesis;
4)Finally delete all parentheses and we get the postfix expression