Infix to postfix.pptx

8/9/2019 Infix to postfix.pptx

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Stack ApplicationsLecture #9

Infix Notation , Prefix Notation

& Postfix Notation

Data Structure & AlgorithmsSE-203


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Infix, Postfix and Prefix Notations

The usual way of expressing the sum of two numbers

A and B is :A+B

The operator ‘+’ is placed between the two operands Aand B.

This is called the “ Infix Notation” Consider a bit more complex example:

(13 – 5) / (3 + 1) When the parentheses are removed the situation becomes ambiguous.

13 – 5 / 3 + 1

is it (13 – 5) / (3 + 1)

or 13 – (5 / 3) + 1 To cater for such ambiguity, you must have operator

precedence rules to follow (as in C++).


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Infix, Postfix and Prefix Notations

In the absence of parentheses

13 – 5 / 3 + 1

Will be evaluated as 13 – (5 / 3) + 1.

Operator precedence isby-passed with thehelp of parentheses as in (13 – 5) / (3 + 1).

The infix notation is therefore cumbersome

due to Operator Precedence rules and

Evaluation of Parentheses


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Postfix Notation

It is a notation for writing arithmetic expressions

in which operands appear before the operator.

E.g. A + B is written as A B + in postfix notation.

There are no precedence rules to be learnt in it.

Parentheses are neer needed..

!ue to its simplicit", some calculators use postfix

notation. This is also called the #$eerse Polish Notation or

$PN%.


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Postfix Notation & 'ome examples


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(onersion from Infix to Postfix

Notation

We have to accommodate the presence of operator precedence rules andParentheses while converting from infix to postfix.

Data objects required for the conversion are

◦ An operator / parentheses stack.

◦ A Postfix expression string to store the resultant.

◦ An infix expression string read one item at a time.


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(onersion from Infix to Postfix

The Algorithm

◦ What are possible items in an input Infix expression◦ Read an item from input infix expression

◦ If item is an operand append it to postfix string

◦ If item is “(“ push it on the stack

◦ If the item is an operator If the operator has higher precedence than the one already ontop of the stack then push it onto the operator stack

If the operator has lower precedence than the one already ontop of the stack then

pop the operator on top of the operator stack and append itto postfix string, and

push lower precedence operator onto the stack

◦ If item is “)” pop all operators from top of the stack one-by-one,

until a “(“ is encountered on stack and removed◦ If end of infix strin o the stack one-b-one and a end to


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Tr" it "ourself

'how a trace of algorithm that conerts the infix expression

a + ) b * c & d e


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Exercise


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'olution


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Ealuation of Postfix Expression

After an infix expression is conerted to postfix, its ealuation is a simple

affair.

'tac- comes in hand", AAIN

The Algorithm

◦ $ead the postfix expression one item at/a/time

◦

If item is an operand push it on to the stac-.◦ If item is an operator pop the top two operands from stac- and appl" the

operator.

◦ Push the result bac- on top of the stac-, which will become an operand for nextoperation.

◦ 0inal result will be the onl" item left on top of the stac-.


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top

Stack in Action ….

5 7 +

6 2 - *

Postfix Expression


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5top


5 7 +

6 2 - *

Postfix Expression


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5

top


5 7 + 6 2 - *

Postfix Expression7


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5

top


5 7 6 2 - *

Postfix Expression7

+

$esult 1 Pop) #+% Pop) Push )$esult

top


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12top


5 7 + 6 2-

*

Postfix Expression


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12

top


5 7 + 6 2 - *

Postfix Expression 6


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12

top


5 7 + 6 2 - *


2


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12

top


5 7 + 6 2 - *


2

$esult 1 Pop) #/% Pop) Push )$esult

12

top4


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5 7 + 6 2 - *

Postfix Expression

12

top4


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12

top4

$esult 1 Pop) # * % Pop) Push )$esult

top

485 7 + 6 2 - *

Postfix Expression


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$esult 1 Pop)

top48

5 7 + 6 2 - *

Postfix Expression

top

$esult 1 23


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Exercise

(onert the following infix expression into postfix

A + B * ( + ) ! * E + 0 *

Now assume the gien alues for the postfix expression

A14, B15, (16, !17, E12, 013, 18

And ealuate the postfix expression


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'olution

Postfix expression is

A B C * + D E * F + G * +

And eva!ation o" e#$%e&&ion !&in' 'iven va!e& i&

1(7


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Ealuation of Postfix Expression

Conversion from infix to postfix is difficult because :◦ Rules governing the precedence of operators are to becatered for

◦ Many possibilities for incoming characters

◦ To cater for parentheses

◦ To cater for error conditions / checks

Evaluation of postfix expression is very simple to

implement because operators appear in precisely theorder in which they are to be executed.


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9otiation for the conersion

Motivation for this conversion is the need to have theoperators in the precise order for execution.

While using paper and pencil to do the conversion wecan “foresee” the expression string and the depth of allthe scopes (if the expressions are not very long andcomplicated).

When a program is required to evaluate an expression,it must be accurate.

At any time during scanning of an expression we cannot be sure that we have reached the inner most scope.

Encountering an operator or parentheses may requirefrequent “backtracking”.Rather than backtracking, we use the stack to“remember” the operators encountered previously.

Documents

Infix to postfix.pptx