1. Queue Applications
Lecture 31
Tue, Apr 11, 2006

2. Topics
Evaluating infix expressions
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3. Infix Expression Evaluation
An infix expression with one (binary) operator is written in the order: left-operand, operator, right-operand.
Example:
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4. Disadvantages of Infix Notation
Parentheses are often needed to indicate order of operation.
Example: (3 + 4) * (5 + 6)
Operators follow a precedence hierarchy.
Example: * 5 – 6 / 7
Operators have left or right associativity.
Example: 100 – 50 – 10 – 5 – 1
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5. Infix Expression Evaluation
To evaluate an infix expression, we first convert it to postfix.
Then begin with an empty stack and an empty queue.
Process the tokens from left to right according to the following rules.
If the token is a number,
Enqueue the token.
If the token is a left parenthesis,
Push the token onto the stack.
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6. Infix Expression Evaluation
If the token is a right parenthesis,
Pop tokens off the stack and enqueue them until a left parenthesis is popped.
Discard the right and left parentheses.
If the token is an operator,
Pop tokens off the stack and enqueue them until
An operator of lower precedence is on top of the stack, or
A left parenthesis is on top of the stack, or
The stack is empty.
Push the operator onto the stack.
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7. Infix Expression Evaluation
After processing the last token
Pop all tokens off the stack and enqueue them.
The queue now contains the expression in post-fix notation.
Process the queue as a post-fix expression.
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8. Example Convert the expression 1 + 2*(3*4 + 5)/6 – 7
from infix to postfix notation.
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9. Example Read 1 : Enqueue 1. Read + : Push +. Read 2 : Enqueue 2.
Stack Queue 1
Read + : Push +.
Stack + Queue 1
Read 2 : Enqueue 2.
Stack + Queue 2 1
Read * : Push *.
Stack + * Queue 2 1
Read ( : Push (.
Stack + * ( Queue 2 1
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10. Example Read 3 : Enqueue 3. Read * : Push *. Read 4 : Enqueue 4.
Stack + * ( Queue 3 2 1
Read * : Push *.
Stack + * ( * Queue 3 2 1
Read 4 : Enqueue 4.
Stack + * ( * Queue
Read + : Pop and enqueue *, push +.
Stack + * ( + Queue *
Read 5 : Enqueue 5.
Stack + * ( + Queue 5 *
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11. Example Read ) : Pop and enqueue +, pop (.
Stack + * Queue + 5 *
Read / : Pop and enqueue *, push /.
Stack + / Queue * + 5 *
Read 6 : Enqueue 6.
Stack + / Queue 6 * + 5 *
Read – : Pop and enqueue / and +, push –.
Stack – Queue + / 6 * + 5 *
Read 7 : Enqueue 7, pop and enqueue –.
Stack Queue – 7 + / 6 * + 5 *
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12. Example The queue contains – 7 + / 6 * + 5 * 4 3 2 1.
These tokens will be dequeued in the order
* 5 + * 6 / + 7 –,
which is the postfix order for the original expression.
Read 1 : Push 1.
Stack 1
Read 2 : Push 2.
Stack 2
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13. Example Read 3 : Push 3. Read 4 : Push 4.
Stack 1 2 3
Read 4 : Push 4.
Stack
Read * : Pop 4 and 3, push 12.
Stack
Read 5 : Push 5.
Stack
Read + : Pop 5 and 12, push 17.
Stack
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14. Example Read * : Pop 17 and 2, push 34. Read 6 : Push 6.
Stack 1 34
Read 6 : Push 6.
Stack
Read / : Pop 6 and 34, push
Stack
Read + : Pop and 1, push
Stack 6.667
Read 7 : Push 7.
Stack
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15. Example Read – : Pop 7 and 6.667, push -0.333. Stack -0.333. 4/18/2019
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