Here are the steps to convert the given infix expressions to postfix: 1) Scan the infix expression from left to right 2) If the scanned character is an operand, output it 3) Else, if the scanned character is an '(', push it to the stack 4) If the scanned character is an ')', pop and output from the stack until an '(' is encountered 5) If the scanned character is an operator, pop and output operators from the stack until an operator with higher precedence or a '(' is encountered. Push the scanned operator to the stack 6) Repeat steps 2-5 until infix expression is scanned 7) Pop and output the rest of the elements from the stack

1. ADVANCED DATA
STRUCTURES
STACKS
CRISTIAN REY C. SECO, MIT

2. Objectives
At the end of the lesson, the student should be able
to:
 Explain the basic concepts and operations on the
ADT stack
 Implement the ADT stack using sequential and
linked representation
 Discuss applications of stack: the pattern
recognition problem and conversion from infix
to postfix
 Explain how multiple stacks can be stored
using one dimensional array
 Reallocate memory during stack overflow in
multiple-stack array using unit-shift policy and
Garwick's algorithm

3. Introduction
 Stack - linearly ordered set of elements having
the last-in, first-out (LIFO) discipline
 Operations are done on top of the stack only
and we have no access to other elements
 Basic operations: push and pop
 Representation: sequential or linked

4. Operations
 Insertion of new element onto the stack (push)
 Deletion of the top element from the stack (pop)
 Getting the size of stack
 Checking for empty stack
 Getting the top element without deleting it from the
stack

5. Operations
 Push

6. Operations
 Pop

7. Operations
procedure SPUSH(S,n,top,x)
array S(1:n)
if top = n then call STACKFULL
top <- top + 1
S(top) <- x
end SPUSH
procedure SPOP(S,n,top,x)
array S(1:n)
if top = 0 then call STACKEMPTY
else [ x <- S(top);
top <- top –1 ]
end SPOP

8. Operations
public interface Stack
{
public int size(); /* returns the size of the stack */
public boolean isEmpty(); /* checks if empty */
public Object top() throws StackException;
public Object pop() throws StackException;
public void push(Object item) throws StackException;
}
class StackException extends RuntimeException
{
public StackException(String err)
{
super(err);
}
}

9. interface
 interface – reserved word in java, a class
that contains only the method headings, and
each method heading is terminated with a
semicolon

10. inheritance
 Inheritance means that a new class can
be derived from or based on an already
existing class.
 The new class inherits features such as
methods from the existing class, which
saves a lot of time for programmers.
 Example: a newly define class
StackException that would extend the
definition of RuntimeException.

11. inheritance
 When you use inheritance, the class
containing your application program will
have more than one method.
 Constructor
 is a special type of method of a class that is
automatically executed when an object of the class
is created.
 It is used to initialize object. The name of the
constructor is always the same as the name of the
class

12. Implementations
 Sequential Representation
 Makes use of arrays
 Stack is empty if top=-1 and full if top=n-1
 Deletion from an empty stack causes an underflow
 Insertion onto a full stack causes an overflow

13. Implementations
public class ArrayStack implements Stack
{
public static final int CAPACITY = 1000;
public int capacity;
Object S[];
int top = -1;
public ArrayStack()
{
this(CAPACITY);
}
public ArrayStack(int c)
{
capacity = c;
S = new Object[capacity];
}
public int size()
{
return (top+1);
}
public boolean isEmpty()
{
return (top < 0);
}

14. Implementations
public Object top()
{
if (isEmpty()) throw new
StackException("Stack empty.");
return S[top];
}
public Object pop()
{
Object item;
if (isEmpty()) throw new
StackException("Stack underflow.");
item = S[top];
S[top--] = null;
return item;
}
public void push(Object item)
{
if (size()==capacity)
throw new StackException
("Stack overflow.");
S[++top]=item;
}

15. Implementations
•Linked Representation
– A linked list of stack nodes could be used to
implement a stack

16. Implementations
public class LinkedStack implements Stack
{
private Node top;
private int numElements = 0;
public int size()
{
return (numElements);
}
public boolean isEmpty()
{
return (top == null);
}
public Object top()
{
if (isEmpty())
throw new StackException
("Stack empty.");
return top.info;
}

17. Implementations
public Object pop()
{
Node temp;
if (isEmpty())
throw new StackException("Stack underflow.");
temp = top;
top = top.link;
return temp.info;
}
public void push(Object item)
{
Node newNode = new Node();
newNode.info = item;
newNode.link = top;
top = newNode;
}
}

18. Application: Infix to Postfix
 Expressions can be in:
 infix form - every subexpression is of the
form operand-operator-operand
 postfix form - every subexpression is of
the form operand-operand-operator, form
most appropriate for computers

19. Application: Infix to Postfix
 Theorem: A postfix expression is well-formed if the
rank of every proper head is greater than or equal to 1 and
the rank of the expression is 1.
Operator Priority Property Example
^ 3 right associative a^b^c = a^(b^c)
* / 2 left associative a*b*c = (a*b)*c
+ - 1 left associative a+b+c = (a+b)+c

20. Application: Infix to Postfix
 Examples:
Infix Expression Postfix Expression
a * b + c / d a b * c d / -
a ^ b ^ c - d a b c ^ ^ d -
a * ( b + ( c + d ) / e ) - f a b c d + e /+* f -
a * b / c + f * ( g + d ) / ( f – h ) ^ i a b * c / f g d + * f h – i ^ / +

21. Application: Infix to Postfix
 Exercises:
Infix Expression
1) a * b + c / d a b * c d / -
2) a ^ b ^ c - d a b c ^ ^ d -
3) a * ( b + ( c + d ) / e ) - f a b c d + e /+* f -
4) a * b / c + f * ( g + d ) / ( f – h ) ^ i a b * c / f g d + * f h – i ^ / +