1.
Introduction to Algorithms
By
Hanumat Sastry. G
2.
Two Phases of Programming
• A typical programming task can be divided into two
phases:
• Problem solving phase
– produce an ordered sequence of steps that describe
solution of problem
– this sequence of steps is called an algorithm
• Implementation phase
– implement the program in some programming language
3.
Understand the problem
Decide on computational means
Exact vs approximate solution
Data structures
Algorithm design technique
Design an algorithm
Prove correctness
Analyze the algorithm
Code the algorithm
3
4.
Algorithms
• A sequence of unambiguous
instructions for solving a
problem, i.e. for obtaining the
required output for any legitimate
input in
a finite amount of time
5.
Algorithm Characteristics
• Input : Zero or more quantities or externally
supplied.
• Output: At least one quantity is produced.
• Definiteness: Each instruction is clear and
unambiguous.
• Finiteness: The algorithm should terminate
after a finite number of steps.
• Effectiveness:
6.
How to Represent an Algorithm
Algorithm can be represented in 3 ways.
1. In Natural Language (English etc…)
2. Pseudo Code.
3. Real Programming Language.
Popular representation is Pseudo Code.
7.
Pseudo Code Convention for Algorithm
• Pseudo code consists of keywords and Englishlike phrases which specify the flow control.
• Pseudo code representation highlights the
Computational aspects by abstracting the
implementation details.
8.
Sum of ‘n’ numbers
Algorithm in Natural
Language
Step 1: Select n number.
Step 2: Set sum S to Zero.
Step 3: Repeat from first
number to nth number
i.e S=S+A[i].
Step 4: Return sum (S).
Algorithm in Pseudo Code.
1.
2.
3.
4.
S<-0
for i<-1 to n
S<-S+A[i]
return S.
9.
gcd(m,n)
Algorithm-1
while n ≠0 do
r ← m mod n
m←n
n ←r
return m
Algorithm-2
1. t ← min (m ,n)
2. if m % t = 0 goto 3,
else goto 4
3. if n % t = 0 return t,
else goto 4
4. t ← t - 1
5. goto 2
10.
General approaches to
algorithm design
Divide and conquer
Greedy method
Dynamic Programming
Basic Search and Traversal Technique
Graph Theory
Linear Programming
Approximation Algorithm
NP Problem
11.
Some Applications
• Study problems these techniques can be
applied to
– sorting
– data retrieval
– network routing
– Games
– etc
12.
Computation Models
• Turing Machine Model
• Random Access Machine (RAM) Model.
13.
Analysis of Algorithms
• The present day algorithms are based on the
RAM (Random Access Machine) model.
• In RAM model, instructions execute one after
another with no, concurrent operations.
14.
Analysis of Algorithms
• Worst Case Complexity
• Average Case Complexity
• Best Case Complexity
15.
Worst Case Complexity
• The Worst Case Complexity of an algorithm is
the function defined by the maximum number
of steps taken on any instance size n.
16.
Best Case Complexity
• The best case complexity of the algorithm is
the function defined by the minimum number
of steps taken on any instance of size n.
17.
Average Case Complexity
• The average case complexity of the algorithm
is the function defined by an average number
of steps taken on an instance of size n.
18.
Graphical representation of Worst
Case, Average Case and Best Case
Complexity
19.
Exercises
• Write an algorithm for product of two matrix
• Design an algorithm for finding square of
given number
• Design an algorithm to find factorial of a
number
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