Stacks, Queues, types of queues and standard operations on queues and stacks

1. Stacks and Queues
Trupti Agrawal 1

2. Stacks
 Linear list.
 One end is called top.
 Other end is called bottom.
 Additions to and removals from the top end only.
Trupti Agrawal 2

3. Stack Of Cups
top
bottom
E
D
C
B
A
 Add a cup to the stack.
top
bottom
F
E
D
C
B
A
• Remove a cup from new stack.
• A stack is a LIFO list.
Trupti Agrawal 3

4. Towers Of Hanoi/Brahma
1 2 3 4
A B C
 64 gold disks to be moved from tower A to tower C
 each tower operates as a stack
 cannot place big disk on top of a smaller one
Trupti Agrawal 4

5. Towers Of Hanoi/Brahma
1 2 3
A B C
 3-disk Towers Of Hanoi/Brahma
Trupti Agrawal 5

6. Towers Of Hanoi/Brahma
1 2
A B C
 3-disk Towers Of Hanoi/Brahma
3
Trupti Agrawal 6

7. Towers Of Hanoi/Brahma
1 2 3
A B C
 3-disk Towers Of Hanoi/Brahma
Trupti Agrawal 7

8. Towers Of Hanoi/Brahma
1 2 3
A B C
 3-disk Towers Of Hanoi/Brahma
Trupti Agrawal 8

9. Towers Of Hanoi/Brahma
1 2 3
A B C
 3-disk Towers Of Hanoi/Brahma
Trupti Agrawal 9

10. Towers Of Hanoi/Brahma
3 2 1
A B C
 3-disk Towers Of Hanoi/Brahma
Trupti Agrawal 10

11. Towers Of Hanoi/Brahma
A B C
 3-disk Towers Of Hanoi/Brahma
1 2
3
Trupti Agrawal 11

12. Towers Of Hanoi/Brahma
A B C
 3-disk Towers Of Hanoi/Brahma
1 2 3
• 7 disk moves Trupti Agrawal 12

13. Recursive Solution
1
A B C
 n > 0 gold disks to be moved from A to C using B
 move top n-1 disks from A to B using C
Trupti Agrawal 13

14. Recursive Solution
1
A B C
 move top disk from A to C
Trupti Agrawal 14

15. Recursive Solution
A B C
 move top n-1 disks from B to C using A
1
Trupti Agrawal 15

16. Recursive Solution
1
A B C
 moves(n) = 0 when n = 0
 moves(n) = 2*moves(n-1) + 1 = 2n-1 when n > 0
Trupti Agrawal 16

17. Towers Of Hanoi/Brahma
 moves(64) = 1.8 * 1019 (approximately)
 Performing 109 moves/second, a computer would take about
570 years to complete.
 At 1 disk move/min, the monks will take about 3.4 * 1013 years.
Trupti Agrawal 17

18. Rat In A Maze
Trupti Agrawal 18

19. Rat In A Maze
 Move order is: right, down, left, up
 Block positions to avoid revisit.
Trupti Agrawal 19

20. Rat In A Maze
 Move order is: right, down, left, up
 Block positions to avoid revisit.
Trupti Agrawal 20

21. Rat In A Maze
 Move backward until we reach a square from which a forward
move is possible.
Trupti Agrawal 21

22. Rat In A Maze
 Move down.
Trupti Agrawal 22

23. Rat In A Maze
 Move left.
Trupti Agrawal 23

24. Rat In A Maze
 Move down.
Trupti Agrawal 24

25. Rat In A Maze
 Move backward until we reach a square from which a forward
move is possible.
Trupti Agrawal 25

26. Rat In A Maze
 Move backward until we reach a square from which a
forward move is possible.
• Move downwTarurptid Ag.rawal 26

27. Rat In A Maze
 Move right.
• Backtrack. Trupti Agrawal 27

28. Rat In A Maze
 Move downward.
Trupti Agrawal 28

29. Rat In A Maze
 Move right.
Trupti Agrawal 29

30. Rat In A Maze
 Move one down and then right.
Trupti Agrawal 30

31. Rat In A Maze
 Move one up and then right.
Trupti Agrawal 31

32. Rat In A Maze
 Move down to exit and eat cheese.
 Path from maze entry to current position operates as a stack.
Trupti Agrawal 32

33. Stacks
 Standard operations:
 IsEmpty … return true iff stack is empty
 IsFull … return true iff stack has no remaining capacity
 Top … return top element of stack
 Push … add an element to the top of the stack
 Pop … delete the top element of the stack
Trupti Agrawal 33

34. Stacks
 Use a 1D array to represent a stack.
 Stack elements are stored in stack[0] through stack[top].
Trupti Agrawal 34

35. Stacks
a b c d e
0 1 2 3 4 5 6
 stack top is at element e
 IsEmpty() => check whether top >= 0
 O(1) time
 IsFull() => check whether top == capacity – 1
 O(1) time
 Top() => If not empty return stack[top]
 O(1) time
Trupti Agrawal 35

36. Derive From arrayList
a b c d e
0 1 2 3 4 5 6
 Push(theElement) => if full then either error or
increase capacity and then add at stack[top+1]
 Suppose we increase capacity when full
 O(capacity) time when full; otherwise O(1)
Pop() => if not empty, delete from stack[top]
 O(1) time
Trupti Agrawal 36

37. Push
a b c d e
0 1 2 3 4
top
 void push(element item)
 {/* add an item to the global stack */
 if (top >= MAX_STACK_SIZE - 1)
 StackFull();
 /* add at stack top */
 stack[++top] = item;
 }

Trupti Agrawal 37

38. Pop
a b c d e
0 1 2 3 4
top
 element pop()
 {
 if (top == -1)
 return StackEmpty();
 return stack[top--];
 }
Trupti Agrawal 38

39. StackFull()
void StackFull()
{
fprintf(stderr, “Stack is full, cannot add
element.”);
exit(EXIT_FAILURE);
}
Trupti Agrawal 39

40. StackFull()/Dynamic Array
 Use a variable called capacity in place of
MAX_STACK_SIZE
 Initialize this variable to (say) 1
 When stack is full, double the capacity using REALLOC
 This is called array doubling
Trupti Agrawal 40

41. StackFull()/Dynamic Array
void StackFull()
{
REALLOC(stack, 2*capacity*sizeof(*stack);
capacity *= 2;
}
Trupti Agrawal 41

42. Complexity Of Array
Doubling
 Let final value of capacity be 2k
 Number of pushes is at least 2k-1+ 1
 Total time spent on array doubling is S1<=i=k2i
 This is O(2k)
 So, although the time for an individual push is
O(capacity), the time for all n pushes remains O(n)!
Trupti Agrawal 42

43. Queues
 Linear list.
 One end is called front.
 Other end is called rear.
 Additions are done at the rear only.
 Removals are made from the front only.
Trupti Agrawal 43

44. Bus Stop Queue
Bus
Stop
front
rear
rear rear rear rear
Trupti Agrawal 44

45. Bus Stop Queue
Bus
Stop
front
rear
rear rear
Trupti Agrawal 45

46. Bus Stop Queue
Bus
Stop
front
rear
rear
Trupti Agrawal 46

47. Bus Stop Queue
Bus
Stop
front
rear
rear
Trupti Agrawal 47

48. Revisit Of Stack Applications
 Applications in which the stack cannot be replaced
with a queue.
 Parentheses matching.
 Towers of Hanoi.
 Switchbox routing.
 Method invocation and return.
 Try-catch-throw implementation.
 Application in which the stack may be replaced with a
queue.
 Rat in a maze.
 Results in finding shortest path to exit.
Trupti Agrawal 48

49. Queue Operations
 IsFullQ … return true iff queue is full
 IsEmptyQ … return true iff queue is empty
 AddQ … add an element at the rear of the queue
 DeleteQ … delete and return the front element of the queue
Trupti Agrawal 49

50. Queue in an Array
 Use a 1D array to represent a queue.
 Suppose queue elements are stored with the front element in
queue[0], the next in queue[1], and so on.
Trupti Agrawal 50

51. Queue in an Array(Linear Queue)
a b c d e
0 1 2 3 4 5 6
 DeleteQ() => delete queue[0]
 O(queue size) time
 AddQ(x) => if there is capacity, add at right end
 O(1) time
Trupti Agrawal 51

52. Circular Array(Circular Queue)
 Use a 1D array queue.
queue[]
• Circular view of array.
[0]
[1]
[2] [3]
[4]
[5]
Trupti Agrawal 52

53. Circular Array
• Possible configuration with 3 elements.
[0]
[1]
[2] [3]
[4]
[5]
A B
C
Trupti Agrawal 53

54. Circular Array
• Another possible configuration with 3
elements.
[0]
[1]
[2] [3]
[4]
[5]
B A
C
Trupti Agrawal 54

55. Circular Array
• Use integer variables front and rear.
– front is one position counterclockwise from first
element
– rear gives position of last element
[0]
[1]
[2] [3]
[4]
[5]
A B
C
front
rear
[0]
[1]
[2] [3]
[4]
[5]
B A
C
front
rear
Trupti Agrawal 55

56. Add An Element
[0]
[1]
[2] [3]
[4]
[5]
A B
C
front
rear
• Move rear one clockwise.
Trupti Agrawal 56

57. Add An Element
• Move rear one clockwise.
• Then put into queue[rear].
[0]
[1]
[2] [3]
[4]
[5]
A B
C
front
D
rear
Trupti Agrawal 57

58. Delete An Element
[0]
[1]
[2] [3]
[4]
[5]
A B
C
front
rear
• Move front one clockwise.
Trupti Agrawal 58

59. Delete An Element
• Then extract from queue[front].
[0]
[1]
[2] [3]
[4]
[5]
A B
C
front
rear
• Move front one clockwise.
Trupti Agrawal 59

60. Moving rear Clockwise
• rear++;
if (rear = = capacity) rear = 0;
front rear
[0]
[1]
[2] [3]
[4]
[5]
A B
C
• rear = (rear + 1) % capacity;
Trupti Agrawal 60

61. Empty That Queue
[0]
[1]
[2] [3]
[4]
[5]
B A
C
front
rear
Trupti Agrawal 61

62. Empty That Queue
[0]
[1]
[2] [3]
[4]
[5]
B
C
front
rear
Trupti Agrawal 62

63. Empty That Queue
[0]
[1]
[2] [3]
[4]
[5]
C
front
rear
Trupti Agrawal 63

64. Empty That Queue
[0]
rear
[1]
[2] [3]
[4]
[5]
front
Trupti Agrawal 64

65. Dequque
 In computer science, a double-ended queue (dequeue,
often abbreviated to deque) is an abstract data type that
generalizes a queue, for which elements can be added to
or removed from either the front (head) or back (tail).
 It is also often called a head-tail linked list,
Trupti Agrawal 65

66. Dequque [Cont…]
 This differs from the queue abstract data type or First-In-
First-Out List (FIFO), where elements can only be added to
one end and removed from the other. This general data
class has some possible sub-types:
 An input-restricted deque is one where deletion can be
made from both ends, but insertion can be made at one end
only.
 An output-restricted deque is one where insertion can be
made at both ends, but deletion can be made from one end
only
Trupti Agrawal 66

67. Dequque [Cont…]
 Both the basic and most common list types in
computing, queues and stacks can be considered
specializations of deques, and can be implemented using
deques.
Trupti Agrawal 67

68. Applications of stacks: Conversion form
infix to postfix and prefix
expressions
 There are a number of applications of stacks such as:
 To print characters/string in reverse order.
 Check the parentheses in the expression.
 To evaluate the arithmetic expressions such as, infix, prefix
and postfix.
Trupti Agrawal 68

69.  Arithmetic expression:
An expression is defined as a number of operands or data
items combined using several operators. There are
basically three types of notations for an expression:
1) Infix notation
2) Prefix notation
3) Postfix notation
(Main advantage of postfix and prefix notation are no need
to use brackets)
Trupti Agrawal 69

70.  Infix notation: It is most common notation in which,
the operator is written or placed in-between the two
operands. For eg. The expression to add two numbers
A and B is written in infix notation as:
 A+ B
• A and B : Operands
• + : Operator
 In this example, the operator is placed in-between the
operands A and B. The reason why this notation is
called infix.
Trupti Agrawal 70

71.  Prefix Notation: It is also called Polish notation,
named after in the honor of the mathematician Jan
Lukasiewicz, refers to the notation in which the
operator is placed before the operand as: +AB
 As the operator ‘+’ is placed before the operands A and
B, this notation is called prefix (pre means before).
Trupti Agrawal 71

72.  Postfix Notation: In the postfix notation the
operators are written after the operands, so it is
called the postfix notation (post means after), it is
also known as suffix notation or reverse polish
notation. The above postfix if written in postfix
notation looks like follows; AB+
Trupti Agrawal 72

73.  Notation Conversions: Let us take an expression A+B*C
which is given in infix notation. To evaluate this expression
for values 2, 3, 5 for A, B, C respectively we must follow
certain rule in order to have right result. For eg. A+B*C =
2+3*5 = 5*5 = 25!!!!!!!!
 Is this the right result? No, this is because the multiplication
is to be done before addition, because it has higher
precedence over addition. This means that for an expression
to be calculated we must have the knowledge of precedence
of operators.
Trupti Agrawal 73

74. Operator Precedence
Operator Symbol Precedence
Exponential $ Highest
Multiplication/Division *,/ Next highest
Addition/Substraction +,- Lowest
Trupti Agrawal 74

75. THANK YOU…!!!
Trupti Agrawal 75