1. Introduction to Algorithms

2. Solving Problems (1)
When faced with a problem:
1. We first clearly define the problem
2. Think of possible solutions
3. Select the one that we think is the best
under the prevailing circumstances
4. And then apply that solution
5. If the solution works as desired, fine;
else we go back to step 2

3. Solving Problems (2)
 It is quite common to first solve a problem for a particular case
 Then for another
 And, possibly another
 And watch for patterns and trends that emerge
 And to use the knowledge form those patterns and trends in coming up with
a general solution

4. Solving Problems (3)
 It helps if we have experienced that problem or similar ones before
 Generally, there are many ways of solving a given problem; the best problem-
solvers come-up with the most appropriate solution more often than not!
 The process that can be used to solve a problem is termed as the “algorithm”

5. Examples
 Addition
 Conversion from decimal to binary
 The process of boiling an egg
 The process of mailing a letter
 Sorting
 Searching

6. Let us write down the algorithm for a problem
that is familiar to us
Converting a decimal number into binary

7. Convert 75 to Binary
752
37 12
18 12
9 02
4 12
2 02
1 02
0 1
1001011
remainder

8. Algorithm for Decimal-to-Binary Conversion
1. Write the decimal number
2. Divide by 2; write quotient and remainder
3. Repeat step 2 on the quotient; keep on repeating until the quotient becomes
zero
4. Write all remainder digits in the reverse order (last remainder first) to form the
final result

9. Points to Note:
1. The process consists of repeated application of simple steps
2. All steps are unambiguous (clearly defined)
3. We are capable of doing all those steps
4. Only a limited no. of steps needs to be taken
5. Once all those steps are taken according to the prescribed sequence, the
required result will be found
6. Moreover, the process will stop at that point

10. Algorithm (Better Definition)
1st Definition:
Sequence of steps that can be taken to solve a problem
Better Definition:
A precise sequence of a limited number of unambiguous, executable
steps that terminates in the form of a solution

11. Three Requirements:
1. Sequence is:
a. Precise
b. Consists of a limited number of steps
2. Each step is:
a. Unambiguous
b. Executable
3. The sequence of steps terminates in the form of a solution

12. Why Algorithms are Useful?
 Once we find an algorithm for solving a problem, we do not need to re-
discover it the next time we are faced with that problem
 Once an algorithm is known, the task of solving the problem reduces to
following (almost blindly and without thinking) the instructions precisely
 All the knowledge required for solving the problem is present in the algorithm

13. Analysis of Algorithms
 Analysis in the context of algorithms is concerned with predicting the resources
that are requires:
 Computational time
 Memory
 Bandwidth
 Logic functions
 However, Time – generally measured in terms of the number of steps required to
execute an algorithm - is the resource of most interest
 By analyzing several candidate algorithms, the most efficient one(s) can be
identified

14. Selecting Among Algorithms
When choosing among competing, successful solutions to a problem, choose the
one which is the least complex
This principle is called the “Ockham’s Razor,” after William of Ockham - famous
13-th century English philosopher

15. Syntax & Semantics
An algorithm is “correct” if its:
 Semantics are correct
 Syntax is correct
Semantics:
The concept embedded in an
algorithm (the soul!)
Syntax:
The actual representation of an
algorithm (the body!)
WARNINGS:
1. An algorithm can be
syntactically correct, yet
semantically incorrect –
very dangerous situation!
2. Syntactic correctness is
easier to check as
compared with semantic

16. Now onto Algorithm Representation
 We have said enough about algorithms – their definition, their types, etc.
 But, how do we actually represent them?
 Generally, SW developers represent them in one of three forms:
 Pseudo code
 Flowcharts
 Actual code

17. Pseudo Code
 Language that is typically used for writing algorithms
 Similar to a programming language, but not as rigid
 The method of expression most suitable for a given situation is used:
 At times, plain English
 At others, a programming language like syntax

18. Flowchart
 A graphical representation of a process (e.g. an algorithm), in which graphic
objects are used to indicate the steps & decisions that are taken as the process
moves along from start to finish
 Individual steps are represented by boxes and other shapes on the flowchart, with
arrows between those shapes indicating the order in which the steps are taken

19. Start or stop
Process
Input or output
Connector
Decision
Flow line
Off-page connector
Flowchart
Elements

20. Algorithm Building Blocks
All problems can be solved by employing any one of the following
building blocks or their combinations
1. Sequences
2. Conditionals
3. Loops

21. Sequences
A sequence of instructions that are executed in the precise order they
are written in:
statement block 1
statement block 2
statement block 3
statement block 1
statement block 2
statement block 3

22. Conditionals
Select between alternate courses of action depending upon the evaluation of a
condition
If ( condition = true )
statement block 1
Else
statement block 2
End if statement
block 1
condition
True False
statement
block 2

23. Loops
Loop through a set of statements as long as a condition is true
Loop while ( condition = true )
statement block
End Loop
condition
True
False
statement
block

24. Problem Statement
Convert a decimal number into binary

25. Convert 75 to Binary
752
37 12
18 12
9 02
4 12
2 02
1 02
0 1
1001011
remainder

26. Solution in Pseudo Code
1. Let the decimal number be an integer x, x > 0
2. Let the binary equivalent be an empty string y
3. Repeat while x > 0 {
Determine the quotient & remainder of x ÷ 2
y = CONCATENATE( remainder, y )
x = quotient
}
4. Print y
5. Stop

27. Start
Find quotient
& remainder
of x ÷ 2
Get x
x>0 ?
Print y
Stop
y = CONC(remainder, x)
x = quotient
x is the decimal number
y is the binary equivalent
Flowchart of Decimal
to Binary Conversion
Yes
No

28. Another Example: Sorting
Sort the following objects w.r.t. their heights

29. Expected Result

30. Strategy
There are many strategies for solving this problem. We demonstrate a simple one:
Repeat the following steps while the list is un-
sorted:
Start with the first object in the list
Swap it with the one next to it if they are in the wrong
order
Repeat the same with the next to the first object
Keep on repeating until you reach the last object in the
list

31. Back to the Objects to be Sorted

32. Sorting: Step A1

33. Sorting: Step A1
Swap? Yes

34. Sorting: Step A2

35. Sorting: Step A2
Swap? Yes

36. Sorting: Step A3

37. Sorting: Step A3
Swap? No

38. Sorting: After Step A7

39. Q: Is the list sorted?
A: No

40. Sorting: Step B1

41. Sorting: Step B1
Swap? Yes

42. Sorting: Step B2

43. Sorting: Step B2
Swap? No

44. Sorting: After Step B7

45. Q: Is the list sorted?
A: No

46. Sorting: Step C1

47. Sorting: Step C1
Swap? No

48. Sorting: After Step C7

49. Q: Is the list sorted?
A: Yes

52.  A number is even if it can be divided by 2 without remainder. Such numbers
are 2, 4, 6, 8.. and so on. The numbers that leave a remainder are called
odd. They are 1, 3, 5, 7.. and so on.
 In programming we find the remainder of a division with the operator %. Also
we use the double equals “==” to compare values for equality.

54.  Summing two numbers was easy – the calculation was one block from the flow
chart. But how about 50? Do you have to write 50 blocks to solve this task?
Happily – no.
 You can automate this process by repeatedly incrementing the value of a
variable and checking it every time if it exceeds the last value – 50. Then sum
that number every step and... there you go! This construction is called loop.

56. Find the biggest of 100 prices and reduce it
by 10%