Stacks

A list data structure

Stacks -

• A list structure in which the elements are added and removed from the top only

• The last element to be added is the first one off the stack

• called a LIFO (last in, first out) data structure

Stack Applications

• Since the number of operations is restricted, the operations are very quick

· Reverse a string

· Test for balanced parentheses

· Evaluating postfix expressions

· Converting from infix to postfix

· The system stack pushes the return address in method calls

Checking for balanced parentheses

Push opening (, pop closing ); if there is anything left of the stack at the end of the string, they are unbalanced

(a + b * (c / (d - e) ) ) + (d / e)

Infix and Postfix

• Infix

– the operator is between the operands

– may require parentheses to specify order

– several ways to express the same thing (not unique)

• Postfix

– the operator follows the two operands

– eliminates the need for parentheses

• The compiler converts all infix expressions to postfix to evaluate them

Evaluating Postfix

• rules are much simpler than infix

– The operator applies to the two operands that immediately precede it

• no operator precedence rules

• no parentheses

• evaluate left to right

• the expression is unique

• The simplest way to evaluate is to use a stack

Postfix examples

• 6 2 / 5 +

• 4 5 + 7 2 - *

Simple algorithm to evaluate postfix

• if a number, push onto the stack

• if operator, pop two numbers, apply operator

• push result onto the stack

• Examples

4 5 + 7 2 - *

6 5 2 3 + 8 * + 3 + *

Converting infix to postfix

• The operator applies to the two operands that immediately precede it

• Convert a+b*c

• Convert (a+b)*c

• Notice

– The operands always stay in the same order with respect to each other

– An operator will move only “to the right” with respect to the operands

– All parentheses are removed

Converting to infix to postfix using a stack

• Read each character in the expression

• If the character is a operand, output it

• If the character is an operator or parentheses …4 cases:

– It is an opening parentheses

– It is of higher precedence than the top of the stack

– It is of lower precedence than the top of the stack

– It is a closing parentheses

What to do with the cases

– Push it onto the stack if:

• The character read is of higher precedence than the top of the stack

• It is an opening parentheses

– Otherwise pop

• If the top of the stack is a closing parentheses pop and output until you reach the opening parentheses

• If the character read is of lower precedence than the top of the stack

– Pop the stack and output until the top of stack the is lower precedence than the character read, then push the character read.