2. UNIT I
Introduction to Algorithm Design and Data Structure: Design & analysis of algorithm,
Top-down and Bottom-up approaches to algorithm design, Analysis of Algorithm,
Frequency count, Complexity measures in terms of time and space.
Arrays, Stacks and Queues : Representation of Array (Single & Multi Dimensional
Arrays), Address Calculation using column & row major Ordering, Array and linked
representation and implementation of queues. Applications of Arrays, Stacks & Queues;
Conversion from Infix to Postfix & Prefix and Evaluation of Prefix expressions using
Stack, Array and linked representation and implementation of queues, Operations on
Queue: Create, Add, Delete, Full and Empty. Circular queue, Deque and Priority Queue
3. Data and Information
Data is the collection of facts and statistics put together for reference or analysis.
Data means any value or set of values for an item/entity . All the real world things are
represented by some values(data).
Note : Data may not be understandable
Ex 23, Prakash, 34, Umesh, 23000.
Ex 4, 8, 16, 32, 64,
4. Data and Information
Information is processed Data
Information is a meaningful arrangement of data such that it makes sense.
Common definition is :- facts provided or learned about something or someone is information
Information is able to answer what, why, where, how and when
Ex Prakash age 23 and Umesh age 34 work in a company and earn Rs 23000 per month.
Who? Where? What?
5.
6. Data Structures
This organization of data into fields, records and files is not efficient
enough to process certain collection of data. Thus data are organized in
more complex structures.
Study of this organization of data is called Data Structure.
Def: - Data structure is a data organization, management, and storage
format that enables efficient access and modification. More precisely, a data
structure is a collection of data values, the relationships among them, and
the functions or operations that can be applied to the data.
8. Algorithm
A process or set of rules to be followed in
calculations or other problem-solving
operations, especially by a computer.
Algorithm is a step-by-step finite sequence of
instruction, to solve a well-defined
computational problem.
Ex driving a car, making tea, taking a bus,
hiring a cab, writing a program
9. Data Structure is the structural representation of logical relationships between
elements of data.
Data Structure = Organized Data + Operations
Data Structure + Algorithm = Program
Algorithm is a step-by-step finite sequence of instruction, to solve a well-defined
computational problem.
• Data Structure (In Memory) = Storage Structure
• Data Structure (In Auxiliary Memory) = File Structure
Complexity Analysis of Algorithm
Time Complexity Space Complexity
11. Time Complexity
How much time does a program takes to run?
How we can decide the time taken by a algorithm?
Is the time dependent on the processing speed of the computer system?
Is the time dependent on the variable used in the program?
Or is the running time is dependent on the way the program is written
Yes, the time complexity is inversely proportional to the space, the more elaborated the program is the less time
it will take to run.
12. Example (Time Complexity)
⚫ Let us now rewrite the above algorithm in the form of
a flow-chart using very basic operations (see Fig. 1.7).
We will pretend that each of these operations takes
the same time to execute on a computer. This is not
precisely true but neither is it wildly false.
⚫ Now we can start timing the algorithm. We just want
to count how many times each box is executed. Boxes
A and B are only executed once. Boxes C, E and F are
on a loop that the program goes round n - 1 times, so
each of these is executed n - 1 times. That leaves box
D. This is where all our problems come.
⚫ Consider the list (4,5,3,6,1,2). As the algorithm scans
along this it will start with min=4. It will then do a
`detour' on the third element to make min=3 and it
will make a final detour on the fifth element to give
min its final value of 1. In all the box D gets executed
twice.
⚫ On the other hand, if we apply the algorithm to the list
(1,4,3,6,2,5) the detour box D will never be executed.
min will get its correct value right at the start and will
never need to be updated.
13. Cont..
This is our problem. There is no fixed value for the number of detours. It depends entirely on the
list being processed. We can make three obvious statements about the value d of the number of
detours:
First, the minimum value of d is zero. This will occur whenever the list has its smallest element
in first place.
Second, the maximum value of d for a list of n elements is n - 1. This will occur when the list is
in strictly decreasing order, so that the current value of the minimum has to be updated on each
step.
Thirdly, for lists of n elements d can take any value between these extremes (exercise).
These comments allow us to make the following statement. Suppose each box in the flow chart
takes unit time to execute. Then the time T(n) taken by the algorithm to find the minimum of n
numbers lies in the range
3(n - 1) + 2 <=T(n)<= 3(n - 1) + 2 + (n - 1)
or
3n - 1 <=T(n) <=4n - 2
For many purposes this might be all that you need to know. The timing is roughly proportional
to the number of elements in the list, allowing for some variation due to the different number of
detours made for different lists.
14. Time Complexity
⚫ Time complexity of an algorithm signifies the
total time required by the program to run till its
completion.
⚫ It includes, loading, linking, jumping time of variables
along with execution time of program.
⚫ It is generally represented in Big Oh (O) notation
15. Asymptotic Notations
⚫ Asymptotic notations are the mathematical notations
used to describe the running time of an algorithm when
the input tends towards a particular value or a limiting
value.
16. Omega Notation (Ω-notation)
Omega notation represents the
lower bound of the running
time of an algorithm. Thus, it
provides best case complexity
of an algorithm.
17. Big-O Notation (O-notation)
Big-O notation represents the upper
bound of the running time of an
algorithm. Thus, it gives the worst
case complexity of an algorithm.
18. Theta notation encloses the
function from above and below.
Since it represents the upper and
the lower bound of the running
time of an algorithm, it is used
for analyzing the average case
complexity of an algorithm.
Theta Notation (Θ-notation)
19. Space Complexity
⚫ Space complexity is a measure of the amount of working
storage an algorithm needs. That means how much memory, in
the worst case, is needed at any point in the algorithm.
⚫ It includes the memory occupied by the variables, constants,
and other pre included files.
⚫ It is represented in the memory units eg. Bytes
20. Time and Space Tradeoff
⚫ The time and space complexity are inversely proportional to each other that
means if the time taken by a program is to be reduced the space occupied by
it will increase.
T∝
1
𝑆
22. Arrays
Definition: A contiguous storage structure having same type
of elements.
a[1] a[2] ………………………………………………..a[n]
• Name of an array represents the address of the array called base address.
•There may be more than ONE Dimensional array, such as TWO Dimensional,
Three Dimensional etc.
•Two dimension array is an example of Matrices. They are represented as a[m][n],
where m is number of rows and n is number of column.
23. Array
⚫ Address Calculation of an Array:
->Single Dimension:
Address of first element is Base address which is also name of the array.
Address of kth element i.e a[k]
:=base address+(k-1)=addr(a)+(k-1)
->Two Dimension:Multi Dimesions array are also stored sequentially in memory,i.e in a linear fashion not like
matrices as we assume.
There are two ways to store them depending on the programming Language:
1)Row Major:In this way first all the elements of first row is stored ,then elements of second row and so on in a
linear fashion.
Eg : a[m][n]
Address of a[ j ][ k ]=addrs(a)+((j-i)n+(k-1))
2)Column Major:
Address of a[ j ][ k ]=addrs(a)+((k-1)m + (j-1))
24. Arrays
⚫ An array is a collection of data elements that are of the
same type (e.g., a collection of integers, collection of
characters, collection of doubles).
26. Array Applications
⚫ Given a list of test scores, determine the
maximum and minimum scores.
⚫ Read in a list of student names and rearrange
them in alphabetical order (sorting).
⚫ Given the height measurements of students in a
class, output the names of those students who are
taller than average.
27. Array Declaration
⚫ Syntax in c:
<type> <arrayName>[<array_size>]
Ex. int Ar[10];
⚫ The array elements are all values of the type <type>.
⚫ The size of the array is indicated by <array_size>, the
number of elements in the array.
⚫ <array_size> must be an int constant or a constant
expression. Note that an array can have multiple dimensions.
29. Subscripting
⚫ Declare an array of 10 integers:
int Ar[10]; // array of 10 ints
⚫ To access an individual element we must apply a subscript to array
named Ar.
⚫ A subscript is a bracketed expression.
⚫ The expression in the brackets is known as the index.
⚫ First element of array has index 0.
Ar[0]
⚫ Second element of array has index 1, and so on.
Ar[1], Ar[2], Ar[3],…
⚫ Last element has an index one less than the size of the array.
Ar[9]
⚫ Incorrect indexing is a common error.
33. Initializing arrays with random values
The following loop initializes the array myList with random values
between 0 and 99:
for (int i = 0; i < ARRAY_SIZE; i++)
{
myList[i] = rand() % 100;
}
34. Program with Arrays
int main()
{
int values[5]= {0,1,3,6,10};
for (int i = 1; i < 5; i++)
{
values[i] = values[i] + values[i-1];
}
values[0] = values[1] + values[4];
}
35. Printing arrays
To print an array, you have to print each element in the array using a
loop like the following:
for (int i = 0; i < ARRAY_SIZE; i++)
{
printf(“%d”,myList[i]);
}
37. Summing All Elements
Use a variable named total to store the sum. Initially total is 0. Add
each element in the array to total using a loop like this:
double total = 0;
for (int i = 0; i < ARRAY_SIZE; i++)
{
total += myList[i];
}
38. Finding the Largest Element
Use a variable named max to store the largest element. Initially max
is myList[0]. To find the largest element in the array myList,
compare each element in myList with max, update max if the
element is greater than max.
double max = myList[0];
for (int i = 1; i < ARRAY_SIZE; i++)
{
if (myList[i] > max) max = myList[i];
}
39. Finding the smallest index of the largest
element
double max = myList[0];
int indexOfMax = 0;
for (int i = 1; i < ARRAY_SIZE; i++)
{
if (myList[i] > max)
{
max = myList[i];
indexOfMax = i;
}
}
40. Shifting Elements
double temp = myList[0]; // Retain the first element
// Shift elements left
for (int i = 1; i < myList.length; i++)
{
myList[i - 1] = myList[i];
}
// Move the first element to fill in the last position
myList[myList.length - 1] = temp;
42. STACK
The word “stack” is chosen by analogy with a stack of plates, where the last
one placed on the top of the stack is usually the first one to be used. Thus a
stack implements a “last in first out” ordering on a set of items.
43. Stack : Definition
Def 1 : A Stack is a list of elements in which elements may be
inserted or deleted only at one end, called top of the stack. Thus
elements are removed from a stack in reverse order in which they
are inserted
Def 2 : A Stack is a non-primitive linear data structure (LIFO).
It is an ordered collections of items where insertion and deletion
take place at one end only called top of stack (TOS).
44. Operations on Stack
A stack is a collection of elements or items, for which the
following operations are defined:
create(S) creates an empty stack S;
isEmpty(S) is a predicate that returns ``true'' if S exists and is
empty, and ``false'' otherwise;
push(S,item) adds the given item to the stack S
pop(S) removes the most recently added item from the stack S
and returns it as the value of the function.
45. Push and Pop
Push:Insert an element
at Top of the stack.
Pop:Removes an
element from top of the
stack.
50. Push and Pop
Push(Algo);
Step1:If top=Max-1,then print
Overflow and return.
Step 2:Top=Top+1
Step 3;Stack[Top]=item
Pop(Algo):
Step1:If Top=-1 then print
underflow and return
Step 2:item=stack[Top]
Step 3:Top=Top-1
51. Peep and Change
Occasionally stack is permitted to have two more options
namely, peep() and change().
In peep() function we are allowed to copy value of any
position in the stack
In change() function we are allowed to change a value of
any position in the stack.
But in both the cases the stack size remains intact.
52. Applications of Stack
Calling of a function/Recursion
Compiler: Parsing
Evaluation of algebraic expressions
Undo/redo in a software
Backward /forward in browser
Calculating factorial
Reversing a list
Counter in a program
Number of turn in a game
53. Queue
Definition: Queue is a list of elements in which elements are
inserted at one end called REAR and deleted from other end
called FRONT.
55. Queue
A queue is a collection of elements, or items, for which the following operations are
defined:
create(Q) creates an empty queue Q;
isEmpty(Q) is a predicate that returns ``true'' if Q exists and is empty, and ``false''
otherwise;
insert(Q,item) adds the given item to the queue Q;
delete(Q) removes from the queue Q the least recently added item that remains in
the queue, and returns it as the value of the function
56. ⚫The data structure queue can be considered as the processing by
FIRST IN FIRST OUT technique, commonly known as FIFO.
⚫Insertion Algo:
if(rear==max-1)
then print(“Overflow)
else
rear=rear+1;
queue[rear]=item;
Queue
58. Problem
⚫The major problem in the above implementation is that
whenever we remove an element from the queue, front
will increment and the location used by the element
cannot be used again. This problem can be solved if we
shift all the elements to the left by one location on every
delete operation. This will be very time consuming and is
not the effective way of solving the problem.
59. Circular Queue
⚫The above problem can be solved only when the first
position in the array will be logically the next position
of the last position of the array. By this way we can
say that the array is circular in nature because every
position in the array will have logical next position in
the array. The queue, which we are going to handle,
using this approach is called the circular queue.
Remember that it is not the infinite queue but we
reuse the empty locations effectively.
63. Infix to post fix
Rule
Write operand , push operator
When opening bracket ( is met - push
When closing bracket ) is met – pop until opening bracket
is met
Do above steps until expression is finished
Note : expressions are evaluated using BODMAS rule
64. Infix to pre fix
Rule
Reverse the string
Find post fix notation
Reverse the string
66. Infix to Postfix
b - ( a + b ) * { ( c – d ) / ( a + a ) }
1. b is operand, write b.
2. On -, Stack being empty, push.
3. On (, push.
4. On a, operand, Hence write a.
5. On +, Stack being (, push. +
6. On b, operand, Hence write b.
7. On ) , pop till (, and then print.
8. Therefore pop +, print +.
)
+
(
-
bab+
-
67. b - ( a + b ) * { ( c – d ) / (a + a) }
9. On *, Stack being -, push *
10. On {, push {
11. On (, push (
12. On C, operand, Hence print C.
13. On -, push-.
14. On d, operand, Hence print d.
15. On ) , pop till (, and then print.
16. Therefore pop -, print -.
17. On /, push /
18. On (, Push(
19. On a, operand, Hence print a.
)
-
(
{
*
-
bab+cd-a
(
/
{
*
-
68. b - ( a + b ) * { ( c – d ) / (a + a) }
20. On +, stack being (, push +
21. On a, operand, Hence print a.
22. On ), pop till (, and then print +
23. On }, pop till {, and then print /
24. End of the Infix expression.
25. pop all and print, Hence print *. And then
print -.
26. Therefore the generated postfix expression
is
)
+
(
/
{
*
-
bab+cd-aa+/*-
}
/
{
*
-
69. Infix to prefix
b - ( a + b ) * { ( c – d ) / (a + a) }
Reverse
{ ( a + a ) / ( d – c ) } * ( b + a ) - b )
+
(
{
aa+dc-/ba+b-*
Reverse
*-b+ab/-cd+aa
}
/
{
)
-
(
/
{
)
+
(
*
I II III IV V
-
*
72. More examples
S.n. Infix Notation Prefix Notation Postfix Notation
1 a + b
2 (a + b) * c
3 a * (b + c)
4 a / b + c / d
5 (a + b) * (c + d)
6 ((a + b) * c) - d
73. More examples
S.n. Infix Notation Prefix Notation Postfix Notation
1 a + b + a b a b +
2 (a + b) * c * + a b c a b + c *
3 a * (b + c) * a + b c a b c + *
4 a / b + c / d + / a b / c d a b / c d / +
5 (a + b) * (c + d) * + a b + c d a b + c d + *
6 ((a + b) * c) - d - * + a b c d a b + c * d -
74. Deque
⚫ a double-ended queue (dequeue, often abbreviated
to deque, pronounced deck) 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, though
properly this refers to a specific data structure
implementation
75. Deque
⚫ Double-Ended Queue = stack + queue hybrid, supports
adding and removing elements at the front and at the end.
⚫ Implemented using a circular queue
? P Q R ?
FRONT
REAR
77. Operations
Operation C
insert element at back push_rear
insert element at front push_front
remove last element pop_rear
remove first element pop_front
81. Applications of Deque
⚫ Undo and Redo Operations
⚫ Forward and backward in browser
⚫ Palindrome checking
⚫ Scheduling in processor A steal job scheduling algorithm
⚫ Manufacturing multiple items in industry
⚫ Multiprocessing
82. Palindrome checker
R A D A R
PUSH FRONT
R A D A R
POP FRONT POP REAR
If value on POP FRONT == value on POP REAR
Then true
Else
False
If front == rear
Then return
83.
84. Explanation
⚫ One example where a deque can be used is the A-Steal job scheduling
algorithm. This algorithm implements task scheduling for several processors.
A separate deque with threads to be executed is maintained for each
processor. To execute the next thread, the processor gets the first element
from the deque (using the "remove first element" deque operation). If the
current thread forks, it is put back to the front of the deque ("insert element
at front") and a new thread is executed. When one of the processors finishes
execution of its own threads (i.e. its deque is empty), it can "steal" a thread
from another processor: it gets the last element from the deque of another
processor ("remove last element") and executes it. The steal-job scheduling
algorithm is used by Intel's Threading Building Blocks (TBB) library for
parallel programming.
85. Priority Queue
A priority queue stores a collection of entries
Each entry is a pair of (key, value)
Keys in a priority queue can be arbitrary objects
on which an order is defined
Two distinct entries in a priority queue can have
the same key
1 2 3 4 5
20 10 40 30 50
KEY
VALUE
FRONT REAR
86. Priority Queue
Main methods of the Priority Queue ADT
insert(k, x)
inserts an entry with key k and value x
removeMin()
removes and returns the entry with smallest key
Additional methods
min()
returns, but does not remove, an entry with
smallest key
size(), isEmpty()
87. Priority Queue
Applications:
Shortest job first scheduling
Stand by flyers
Auctions
Stock market
Bandwidth management
Dijkstra's algorithm
Huffman coding
Best-first search algorithms
Prim's algorithm for minimum spanning tree
89. Tutorial
Write the short note on
Data structure and Algorithm including definition, analysis
and example
Time and space complexity explain with example
Circular queue with diagram and array representation
Priority queue
90. Quiz Question
Harry battels with Voldemort
Voldemort has a big army, so he has maintained his
people in a queue of size N to fight Harry's army.
The queue contains the fighters with different power
and they are called in the first in first out order.
Harry maintains a stack of fighters and calls them when
his previous fighter is down.
The stack contains fighters with different power and
they are called in last in first out order.
When ever a fighter from Voldemort army strike the
harry’s fighter will fight against it
91. Quiz Question
Harry battels with Voldemort
The value at top of the stack denotes the power of the
fighter of harry’s army
The value at the front of the queue denotes the power of
the fighter of Voldemort’s army
The duel has three results if power of harry’s fighter is
greater than the power of Voldemort’s fighter then 1
point for harry otherwise 1 point for Voldemort, if they
have equal power the fight will be a draw and both will
not get any point.
As they are busy fighting, can you help Dumbledore to
keep the record who wins the battel
