1. Stacks
Chapter 5
Adapted from Pearson Education, Inc.

2. Contents Specifications of the ADT Stack
Using a Stack to Process Algebraic Expressions
A Problem Solved: Checking for Balanced Delimiters in an Infix Algebraic Expression
A Problem Solved: Transforming an Infix Expression to a Postfix Expression
A Problem Solved: Evaluating Postfix Expressions
A Problem Solved: Evaluating Infix Expressions
The Program Stack
Java Class Library: The Class Stack
Adapted from Pearson Education, Inc.

3. Objectives Describe operations of ADT stack
Use stack to decide whether delimiters in an algebraic expression are paired correctly
Use stack to convert infix expression to postfix expression
Use stack to evaluate postfix expression
Use stack to evaluate infix expression
Use a stack in a program
Describe how Java run-time environment uses stack to track execution of methods
Adapted from Pearson Education, Inc.

4. Specifications of a Stack
Organizes entries according to order added
All additions added to one end of stack
Added to “top”
Called a “push”
Access to stack restricted
Access only top entry
Remove called a “pop”
Adapted from Pearson Education, Inc.

5. ADT Stack
Adapted from Pearson Education, Inc.

6. ADT Stack
Adapted from Pearson Education, Inc.

7. Adapted from Pearson Education, Inc.
A Stack Interface
public interface StackInterface < T > { /** Adds a new entry to the top of this newEntry an object to be added to the stack */ public void push (T newEntry); /** Removes and returns this stacks top either the object at the top of the stack or, if the stack is empty before the operation, null */ public T pop (); /** Retrieves this stacks top either the object at the top of the stack or null if the stack is empty */ public T peek (); /** Detects whether this stack is true if the stack is empty */ public boolean isEmpty (); /** Removes all entries from this stack */ public void clear (); } // end StackInterface
Adapted from Pearson Education, Inc.

8. Adapted from Pearson Education, Inc.
Stack Usage Example
Adapted from Pearson Education, Inc.

9. Adapted from Pearson Education, Inc.
Stack Usage Example
Adapted from Pearson Education, Inc.

10. Algebraic Expressions
Algebraic expressions are composed of
Operands (variables, constants)
Operators (+, -, /, *, ^)
Operators can be unary or binary
Different precedence notations
Infix a + b
Prefix + a b
Postfix a b +
Precedence must be maintained
Order of operators
Use of parentheses (must be balanced)
Use stacks to evaluate parentheses usage
Scan expression
Push symbols
Pop symbols
Adapted from Pearson Education, Inc.

11. Algebraic Expressions
Algebraic expressions are composed of
Operands (variables, constants)
Operators (+, -, /, *, ^)
Operators can be unary or binary
Precedence must be maintained
According to order of operators
According to parentheses
When parentheses are used, they must be balanced
Different notations
Infix a + b
Prefix + a b
Postfix a b +
Adapted from Pearson Education, Inc.

12. Parentheses Balance Checker
Scan expression from left to right
When you encounter:
( { [  Push opening parenthesis symbols onto stack
) } ]  Pop symbol from stack and check if it matches
Figure The contents of a stack during the scan of an expression that contains the balanced delimiters { [ ( ) ] }
Adapted from Pearson Education, Inc.

13. Parentheses Balance Checker
Figure 5-4
The contents of a stack during the scan of an expression that contains the unbalanced delimiters { [ ( ] ) }
Adapted from Pearson Education, Inc.

14. Parentheses Balance Checker
Figure 5-5
The contents of a stack during the scan of an expression that contains the unbalanced delimiters [ ( ) ] }
Can not pop to match }
Adapted from Pearson Education, Inc.

15. Parentheses Balance Checker
Figure 5-6
The contents of a stack during the scan of an expression that contains the unbalanced delimiters { [ ( ) ]
Reached end of string
Adapted from Pearson Education, Inc.

16. Infix vs. Postfix Expressions
In-fix expressions:
Operator comes between two operands  a + b
Precedence of operators determined as follows:
+ and -  have lowest precedence, are evaluated left-to-right
* and /  have higher precedence, are evaluated left-to-right
^  has highest precedence, is evaluated right-to-left
Parentheses  override precedence
Post-fix expressions:
Operator comes after two operands  a b +
Precedence is determined by order of operators in the expression
No parentheses are used in postfix expressions
Adapted from Pearson Education, Inc.

17. Infix to Postfix Conversion - Manual
Steps:
Put parentheses according to correct operator precedence
Move operator to right inside parentheses
Remove parentheses
Example: (a + b) * c
( ( a + b ) * c )
( ( a b + ) c * )
a b + c *
Notice the following:
Postfix expressions do NOT need parentheses to specify precedence
The order of operands in the postfix expression remains the same as in the infix expression
Example: a + b * c
( a + ( b * c ) )
( a ( b c * ) + )
a b c * +
Adapted from Pearson Education, Inc.

18. Infix to Postfix Conversion - Algorithm
Algorithm basics
Scan infix expression left to right
When operand found, place it at end of new postfix expression
When operator found, save it until new position is determined
Why?
The 2nd operand must be output before the operator
Maybe there will be an operator with higher precedence, then it needs to be output first
Adapted from Pearson Education, Inc.

19. Infix to Postfix Conversion - Algorithm
Operand
Append to end of output expression
Operator ^
Push ^ onto stack
Operators +, -, *, /
Pop from stack, append to output expression, until stack empty or top operator has lower precedence than new operator
Push new operator onto stack
Open parenthesis
Push ( onto stack
Close parenthesis
Pop operators from stack and append to output
Until open parenthesis is popped.
Discard both parentheses
Keep popping all operators that have higher or equal precedence
Adapted from Pearson Education, Inc.

20. Infix to Postfix Conversion – Example (1)
Figure 5-7
Converting a + b * c to postfix form
Next Char
Postfix
Operator Stack
(bottom to top) a
(empty) + b
a b *
+ * c
a b c
(end of infix)
a b c *
a b c * +
Adapted from Pearson Education, Inc.

21. Infix to Postfix Conversion – Example (1)
Figure 5-7
Converting a + b * c to postfix form
Next Char
Postfix
Operator Stack
(bottom to top) a
(empty) + b
a b *
+ * c
a b c
(end of infix)
a b c *
a b c * +
Adapted from Pearson Education, Inc.

22. Infix to Postfix Conversion – Example (2)
Converting the following to postfix form
7 + 2 ^ ( ) / ( * 3 – 5 )
The only thing that can pop an open parenthesis is a matching closing parenthesis
Next Char
Postfix
Operator Stack
(bottom to top) 7
(empty) + 2 72 ^ +^ (
+^( 1
721
+^(+ 3
7213
Adapted from Pearson Education, Inc.

23. Infix to Postfix Conversion – Example (2)
Converting the following to postfix form
7 + 2 ^ ( ) / ( * 3 – 5 )
Next Char
Postfix
Operator Stack
(bottom to top) 3
7213
+^(+ )
7213+ +^ /
7213+^ +/ (
+/( 6
7213+^6 +
+/(+ 5
7213+^65 *
+/(+*
Adapted from Pearson Education, Inc.

24. Infix to Postfix Conversion – Example (2)
Converting the following to postfix form
7 + 2 ^ ( ) / ( * 3 – 5 )
Are we done?
We are only done when we reach the end of the infix expression AND the stack is empty.
Next Char
Postfix
Operator Stack
(bottom to top) *
7213+^65
+/(+* 3
7213+^653 -
7213+^653*
+/(+
7213+^653*+
+/(- 5
7213+^653*+5 )
7213+^653*+5- +/
(end of infix)
7213+^653*+5-/ +
7213+^653*+5-/+
(empty)
Adapted from Pearson Education, Inc.

25. Infix to Postfix Conversion – Example (2)
Converting the following to postfix form
7 + 2 ^ ( ) / ( * 3 – 5 )
Can you evaluate this?
7 + 2 ^ 4 / (6 + 5 * 3 – 5) =
/ ( – 5) =
/ (21 – 5) =
/ 16 =
7 + 1 = 8
Next Char
Postfix
Operator Stack
(bottom to top) *
7213+^65
+/(+* 3
7213+^653 -
7213+^653*
+/(+
7213+^653*+
+/(- 5
7213+^653*+5 )
7213+^653*+5- +/
(end of infix)
7213+^653*+5-/ +
7213+^653*+5-/+
(empty)
Adapted from Pearson Education, Inc.

26. Postfix Evaluation - Algorithm
Any operator operates on its two previous operands
Example - manual
^ * / +
7 2 4 ^ * / +
/ +
/ +
/ +
7 1 + 8
Algorithm basics
Scan postfix expression left to right
When operand found, push it onto operand stack
When operator found:
Pop stack for 2nd operand
Pop stack for 1st operand
Evaluate
Push result onto operand stack
Adapted from Pearson Education, Inc.

27. Evaluating Postfix Expression – Example (1)
^ * / +
1 + 3
2 ^ 4 7 2 7 1 2 7 3 1 2 7
427
167 6
167
56167
5 * 3
6 + 15
21 - 5
16 / 16
7 + 1
356167
156167
21167 5 21
167
16167 17 8
Adapted from Pearson Education, Inc.

28. Evaluating Infix Expressions
evaluation of a + b * c when a is 2, b is 3, and c is 4
after reaching the end of the expression
while performing the multiplication;
while performing the addition
Adapted from Pearson Education, Inc.

29. The Program Stack when main begins execution
when methodA begins execution
when methodB begins execution
Adapted from Pearson Education, Inc.

30. Java Class Library: The Class Stack
Has a single constructor
Creates an empty stack
Remaining methods – differences from our StackInterface are highlighted
public T push(T item);
public T pop();
public T peek();
public boolean empty();
Adapted from Pearson Education, Inc.

31. Adapted from Pearson Education, Inc.
End
Chapter 5
Adapted from Pearson Education, Inc.