Data Structure and Algorithms Stacks

Introduction: List and Array representations,
Operations on stack (traversal, push and pop)
Arithmetic expressions: polish notation, evaluation
and transformation of expressions.

2
Introduction to Stacks
Consider a card game with a discard pile
Discards always placed on the top of the pile
Players may retrieve a card only from the top
We seek a way to represent and manipulate this
in a computer program
This is a stack
What other examples
can you think of that
are modeled by a stack?
What other examples
can you think of that
are modeled by a stack?

3
Introduction to Stacks
A stack is a last-in-first-out (LIFO) data structure
Adding an item
Referred to as pushing it onto the stack
Removing an item
Referred to as
popping it from
the stack

4
Stack
Definition:
An ordered collection of data items
Can be accessed at only one end (the top)
Operations:
construct a stack (usually empty)
check if it is empty
Push: add an element to the top
Top: retrieve the top element
Pop: remove the top element

Stack
items[MAX-1]
. .
. .
. .
items[2] C
items[1] B
items[0] A
Top=2

Insert an item A
A new item (A) is inserted at the Top of the stack
items[MAX-1]
. .
. .
items[3]
items[2]
items[1]
items[0] A Top=0

Insert an item B
A new item (B) is inserted at the Top of the stack
items[MAX-1]
. .
. .
items[3]
items[2]
items[1] B Top=1
items[0] A

Insert an item C
A new item (C) is inserted at the Top of the stack
items[MAX-1]
. .
. .
items[3]
items[2] C Top=2
items[1] B
items[0] A

Insert an item D
A new item (D) is inserted at the Top of the stack
items[MAX-1]
. .
. .
items[3] D Top=3
items[2] C
items[1] B
items[0] A

Insert Operation(Array)
PUSH(STACK, N, TOP, ITEM)
1. If TOP=N:
write OVERFLOW, and Return.
2. Set TOP := TOP + 1.
3. Set STACK[TOP]:= ITEM.
4. Return.

Delete D
an item (D) is deleted from the Top of the stack
items[MAX-1]
. .
. .
items[3] D
items[2] C Top=2
items[1] B
items[0] A

Delete C
an item (C) is deleted from the Top of the stack.
items[MAX-1]
. .
. .
items[3] D
items[2] C
items[1] B Top=1
items[0] A

Delete B
an item (B) is deleted from the Top of the stack
items[MAX-1]
. .
. .
items[3] D
items[2] C
items[1] B
items[0] A Top=0

Delete A
an item (A) is deleted from the Top of the stack.
items[MAX-1]
. .
items[4]
items[3] D
items[2] C
items[1] B
items[0] A Top=-1

Delete Operation(Array)
POP(STACK, N, TOP, ITEM)
1. If TOP = NULL then :
write: UNDERFLOW, and Return.
2. Set ITEM := STACK [TOP].
3. Set TOP := TOP - 1.
4. Return.

Insert Operation(LL)
PUSH(INFO, LINK, TOP, AVAIL, ITEM)
1. If AVAIL = NULL, then :
write OVERFLOW, and Exit.
2. Set NEW := AVAIL and AVAIL := LINK[AVAIL]
3. Set INFO[NEW]:= ITEM
4. Set LINK[NEW] := TOP
5. Set TOP := NEW
6.Exit

Delete Operation(LL)
POP(INFO, LINK, TOP, AVAIL, ITEM)
1. If TOP = NULL, then :
write UNDERFLOW, and Exit.
2. Set ITEM :=INFO[TEMP]
3. Set TEMP := TOP and TOP :=LINK[TOP]
4. LINK[TEMP] = AVAIL and AVAIL = TEMP
5. Exit

18
Postfix Notation(RPN)
Polish notation, also known as prefix notation.
It is a symbolic logic invented by Polish mathematician
Jan Lukasiewicz in the 1920's.
Most compilers convert an expression in infix
notation to postfix
the operators are written after the operands
So a * b + c becomes a b * c +
Advantage:
expressions can be written without parentheses

19
Postfix and Prefix Examples
INFIX POSTFIX PREFIX
A + B
A * B + C
A * (B + C)
A - (B - (C - D))
A - B - C - D
A B * C +
A B C + *
A B C D---
A B-C-D-
+ * A B C
* A + B C
-A-B-C D
---A B C D
A B + + A B
Prefix : Operators come
before the operands
Prefix : Operators come
before the operands

20
→ 2 7 5 6 - - *
→ 2 7 5 6 - - *
→ 2 7 -1 - *
→ 2 7 -1 - * →
"By hand" (Underlining technique):
1. Scan the expression from left to right to find an operator.
2. Locate ("underline") the last two preceding operands
and combine them using this operator.
3. Repeat until the end of the expression is reached.
Example:
2 3 4 + 5 6 - - *
→ 2 3 4 + 5 6 - - *
2 8 * → 2 8 * → 16

Evaluating RPN Expressions
 P is an arithmetic expression in Postfix Notation.
1. Add a right parenthesis “)” at the end of P.
2. Scan P from left to right and Repeat Step 3 and 4 for each element of P
until the sentinel “)” is encountered.
3. If an operand is encountered, put it on STACK.
4. If an operator @ is encountered, then:
(a) Remove the two top elements of STACK, where A
is the top element and B is the next to top element.
(b) Evaluate B @ A.
(c) Place the result of (b) back on STACK.
[End of if structure.]
[End of step 2 Loop.]
5. Set VALUE equal to the top element on STACK.
6. Exit.
21

22
Evaluating
RPN
Expressions
Note the
changing
status of the
stack

23
By hand: "Fully parenthesize-move-erase" method:
1. Fully parenthesize the expression.
2. Replace each right parenthesis by the corresponding
operator.
3. Erase all left parentheses.
Examples:
A * B + C →
→ ((A B * C +
→ A B * C +
A * (B + C) →
→ (A (B C + *
→ A B C + *
((A * B) + C) (A * (B + C) )

24
Stack Algorithm
POLISH (Q, P)
1. PUSH “(” on to STACK and add “)” to the end of Q.
2. Scan Q from left to right and Repeat steps 3 to 6 for each element of Q until
the STACK is empty:
3. If an operand is encountered, add it to P.
4. If a left parenthesis is encountered, push it onto STACK.
5. If an operator is encountered, then:
(a) Repeatedly POP from STACK and add to P each
operator (On the TOP of STACK) which has the same precedence as or
higher precedence than @.
(b) Add @ to STACK.
[End of If structure.]
6. If a right parenthesis is encountered, then:
(a) Repeatedly POP from STACK and add to P each operator (On the
TOP of STACK.) until a left parenthesis is encountered.
(b) Remove the left parenthesis. [Don’t add the left parenthesis to P.]
[End of If Structure.]
[End of step 2 Loop.]
7. Exit.

25
By hand: "Fully parenthesize-move-erase" method:
1. Fully parenthesize the expression.
2. Replace each left parenthesis by the corresponding
operator.
3. Erase all right parentheses.
Examples:
A * B + C →
→ + * A B) C)
→ + * A B C
A * (B + C) →
→ * A + B C ))
→ * A + B C
((A * B) + C) (A * (B + C))

Data Structure and Algorithms Stacks

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