Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil

Algorithms and Algorithms and Data Structures Data Structures

Lecture 5

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Applications Of Stack

Convert infix expression to postfix and prefixexpressions

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)

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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.

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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

Expression Evaluation

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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, >ANDOR

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The Operators and priorities

Data Structures in C++ by Dr. Varsha Patil Oxferd Univercity

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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

Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil8

…The example expression in various forms-infix, prefix and postfix

The postfix expressions can be evaluated easily hence infix expression is converted into postfix expression using stack.

Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil9

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 associativityThis problem can be overcome by writing or converting infix to alternate notation such as prefix or postfixThe postfix and prefix expressions possess many advantages as listed

Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil10

The need for parenthesis as in infix expression is overcome in postfix and prefix Notations

The priority of operators is no longer relevantOrder of evaluation depends on position of operator not on priority and

associativityThe expression evaluation process is much simpler than attempting a direct

evaluation from infix notation.

Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil11

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 stackThis evaluation process is much simpler than attempting a direct evaluation from infix notationProcess continues till stack is not empty or on occurrence of a character # which denotes the end of expression

Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil12

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 same3) Now move all operators so that they replace their corresponding right parenthesis;4)Finally delete all parentheses and we get the postfix expression