1. COSC 2006 Chapter 7 Stacks III
April 12, 2017
CS Data Structures I
Chapter 7 Stacks III
Chapter 6: Stacks

2. Topics Applications Infix to postfix expression
Evaluate postfix expression

3. Infix Expressions Binary operators appear between operands:
COSC 2006
April 12, 2017
Infix Expressions
Binary operators appear between operands:
W - X / Y – Z (4+3)* (2+3)/(9-4)
Order of evaluation is determined by:
precedence rules
parentheses
association (L to R or R to L)
Chapter 6: Stacks

4. Application: Algebraic Expressions
Infix Expression Evaluation
To evaluate an infix expression
Convert the infix expression into postfix
Evaluate the postfix expression using stacks

5. Postfix Expressions Binary operators appear after both operands:
COSC 2006
April 12, 2017
Postfix Expressions
Binary operators appear after both operands:
X + Y in postfix form is: X Y +
Order of operations is completely determined by the expression
no parentheses or precedence required
for example:
A * B - C becomes A B * C -
Chapter 6: Stacks

6. Postfix Expression Infix Postfix 6-1 (4+3)*2 (2+3)/(9-4) COSC 2006
April 12, 2017
Postfix Expression
Infix Postfix
6-1
(4+3)*2
(2+3)/(9-4)
Chapter 6: Stacks

7. Postfix Expression Evaluation
Assumptions:
The string is syntactically correct postfix expression
No unary operators are present
No exponentiation operators are present

8. Postfix Expression Evaluation
COSC 2006
April 12, 2017
Postfix Expression Evaluation
Evaluation of postfix expressions makes use of an operand stack.
Algorithm:
Parse expression from left to right
When an operand is encountered, push it onto stack
when an operator is encountered, pop the last two operands off the stack and apply the operation, and push the result onto the stack
when the expression is completely scanned, the final result will be on the stack
Chapter 6: Stacks

9. Postfix Expression Evaluation
Pseudocode:
for ( each character ch in the string )
{ if (ch is an operand )
Push value that operand ch represents onto stack
else / / ch is an operator named op
{ / / evaluate and push the result
operand2 = top of stack
Pop the stack
operand1 = top of stack
result = operand1 op operand2
Push Result onto stack
} / / end if
} / / end for

10. Example
Evaluate * * + *

11. Converting Infix into Postfix
COSC 2006
April 12, 2017
Converting Infix into Postfix
Since postfix expressions are easily evaluated, the easiest way to evaluate infix is to convert from infix to postfix and then evaluate.
This conversion uses an operator stack since low precedence operators must be saved and applied after high precedence ones.
Chapter 6: Stacks

12. Converting Infix into Postfix
COSC 2006
April 12, 2017
Converting Infix into Postfix
The operands always stay in the same order with respect to one another
An operator will move only to “ the right” with respect to the operands
If in infix expression, the operand x precedes the operator op, in the postfix, the operand x also precedes the operator x
All parentheses are removed
Chapter 6: Stacks

13. Converting Infix into Postfix
Scan infix string from left to right
Each time an operand is encountered, copy it to output
When a bracket is encountered, check its orientation.
Push left brackets onto stack
If it is a right bracket, pop all operators on the stack and copy them to output until a matching left bracket is encountered. Discard the left bracket

14. Converting Infix into Postfix
When an operator is encountered, check the top item of the stack
If the priority is >= the current operator, pop the top operator and copy it to output
Continue until an operator of lower priority is encountered or until the stack is empty
Finally, push current operator onto the stack
When the end of the expression is reached, copy the remaining stack contents to output in the order popped

15. Converting Infix into Postfix
Precedence Table
Operator
Name
Precedence when on stack
Precedence when on input (
Opening parentheses
always stack 7 )
Closing parentheses
never stacked ^
Exponentiation 6 5
* /
Multiply & divide 4 3
+ -
Add & Subtract 2 1

16. Conversion Example W - X /Y + Z '-' is pushed onto operator stack
COSC 2006
April 12, 2017
Conversion Example
W - X /Y + Z
'-' is pushed onto operator stack
'/' has higher precedence than '-' on stack - it gets pushed
'+' has lower precedence than '/' on stack - pop '/' and output - then '-' is on top of stack - since subtraction associates left to right - pop '-' off stack and output - push '+' onto stack
when end of expression is reached pop remaining operator and output
resulting expression is: W X Y /- Z +
Chapter 6: Stacks

17. Converting Infix into Postfix
COSC 2006
April 12, 2017
Converting Infix into Postfix
stack.push(EOL marker)
for (each character ch in the infix expression) {
switch (ch) {
case operand:
//append ch to output postfix expression
postfixExpr = postfixExpr + ch;
case '(':
stack.push(ch); // push ch onto operator stack
case ')':
while (top of stack is not '(' ) {
postfixExpr = postfixExpr + stack.pop(); // pop and append expr. }
stack.pop();
case operator:
while (!stack.isEmpty() && inputPrec(ch) <= stackPrec(stack.top())) {
postfixExpr = postfixExpr + stack.pop();
stack.push(ch);
} // end for
while (!stack.isEmpty() { // pop all remaining operators off stack and append
Chapter 6: Stacks

18. Conversion Example 2 + 3 * [(7-5)/2] 2 3 7 5 2 copy into output
COSC 2006
April 12, 2017
Conversion Example
2 + 3 * [(7-5)/2]
2 copy into output
‘+' is pushed onto operator stack
3 copy into output
‘*' has higher precedence than ‘+' on stack - it gets pushed
‘[‘ is left bracket, push it onto stack
‘(‘is left bracket, push it onto stack
7 is copied into output
‘-' has higher precedence than ‘(‘, push onto stack
Chapter 6: Stacks

19. Conversion Example 2 + 3 * [(7-5)/2] COSC 2006 April 12, 2017
Chapter 6: Stacks

20. Prefix Expressions Infix Prefix 6-1 - 6 1 (4+3)*2 * + 4 3 2
COSC 2006
April 12, 2017
Prefix Expressions
Infix Prefix
(4+3)*2 *
(2+3)/(9-4) / – 9 4
Chapter 6: Stacks

21. Review
Which of the following is NOT true about converting infix expressions to postfix expressions?
the operands always stay in the same order with respect to one another
the operators always stay in the same order with respect to one another
an operator will move only “to the right” with respect to the operands
all parentheses are removed

22. Review
Which of the following is the postfix form of the infix expression: (a + b) * c / d
a b + c * d /
a b * c / d +
a + b * c / d
a b + c d * /

23. Review StackInterface provides the specifications for ______.
only the array-based implementation of a stack
only the reference-based implementation of a stack
only an implementation of a stack that uses the ADT list
all the implementations of a stack

24. Review
In the StackInterface class, the push method accepts as its parameter an item that is an instance of ______.
Integer
Double
String
Object