aad module 1 s2

7/29/2019 AAD Module 1 S2

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ALGORITHM

ANALYSIS AND DESIGN


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Definition of Algorithms

An algorithm is a finite set of instruction

that if followed accomplishes a particular

task.


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An algorithm should satisfy the following

criteria

Inputzero or more quantities are externally supplied.

Outputat least one quantity is produced

Definitenesseach instruction is clear and unambiguous

Finitenessthe algorithm terminates after a finite number of steps.

Effectivenessevery instruction must be very basic so that it can becarried out, by a person using only pencil and paper.


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A program is an expression of algorithm in

a programming language.


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How to devise Algorithm :-

Creating an algorithm is an art which may

never be fully automated. There are various

design techniques. By mastering thesedesign strategies, it will become easier for

you to devise new and useful algorithms.


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.

How to validate Algorithm-

Once an algorithm is devised, it is necessary

to show that it computes the correct answer

for all possible legal inputs independent of

issue concerning the programming language

it eventually be written in.


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How to Analyze Algorithms

Analysis of Algorithm or performance

analysis refers to the task of determining

how much computing time and storage analgorithm requires. This is a challenging

area which some times requires great

mathematical skill


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How to test a program

Testing a programs consist of two

phases

Debugging:- Debugging is the process

of executing the programs on sample

data sets to determine whether faultyresult occur and if so , to correct them.


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Profiling or performance measurement is

the process of executing a correct program

on data set and measuring the time andspace it take to compute the result.


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Algorithm Specification

We can use natural languages like English

Graphic representation is called flow chart

Another representation is Pseudocode that

resembles C and Pascal


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1. Comments begin with //2. Blocks are indicated with matching braces

{ }

3. An identifier begins with a letter4. Assignment of values to variables is done

using the assignment statement

< variable> :=5. Logical operators and ,or, notand the

relational operators , are

provided


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7. The following looping statements are

employed:

for, while, repeat-until

8. The following conditional statements can

be used

If. Then.

if .thenelse


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9. Inputs and outputs are done using the

instructions readand write

10. There is only one procedure: Algorithmthe heading takes the form

AlgorithmName ( parameter list)


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Recursive Algorithms

A function is said to be recursive if it is

defend in terms of itself. Similarly An

algorithm is said to be recursive if the samealgorithm is invoked in the body

A algorithm that calls itself is direct

recursive and an algorithm A is indirectrecursive if it calls another algorithm which

in turn calls A


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PERFORMANCE ANALYSIS

15

Space Complexity

Time Complexity

S C l it


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Space Complexity

S(P)=C+SP(I) Fixed Space Requirements (C)

Independent of the characteristics of the inputs

and outputs.

Typically includes the instruction space(space for code)

space for simple variables, fixed-size structuredvariable, constants and so on.

Variable Space Requirements (SP(I))

Space needed by component variables whose size

is dependent on the particular problem instancebeing solved.

- Space needed by referenced variables.

recursive stack space, return address


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float abc(float a, float b, float c)

{

return a + b + b * c + (a + b - c) / (a + b) + 4.00;

}

S(I) = 0


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Time complexity

The time T(P) taken by a program is the

sum of the compile time and runtime. We

may assume that a compiled program willrun several times without recompilation.

Consequently we concern ourselves with

just the run time of the program. Thisruntime is denoted as Tp. .


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Time Complexity

Compile time (C)

independent of instance characteristics

Run (execution) time TP

T(P)=C+TP(I)

TP(n)=caADD(n)+csSUB(n)+cmMUL(n)+cdDIV(n)+

Where n denotes the instance of characteristics


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Ca, cs, cm, cd , and so on respectively, denote the timeneeded for addition, subtraction, multiplication,division and so on, ADD, SUB, MUL,DIV and soon are functions whose values are number ofaddition, subtraction multiplication, division andso on, that are performed when the code for P isused on an instance with characteristic n.


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Asymptotic Notation (O, ,)


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Big oh:-

The function f(n) = O(g(n)) iff there exist

positive constants c and no such that f(n) c* g(n) for all n, n no

The function 3n + 2=O(n) as 3n +2 4n, forall n 2.

Here value of g(n) =n, c= 4 & no =2

3n+3 4n for all n 3.


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We write O(1) to mean a computing time

that is a constant

O(n) is called linear

O(n2) is called quadratic

O(n3) is called cubic

O(2n) is called Exponential


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f(n) is

(g(n)) if

positiveconstants c and n0such that 0

f(n) cg(n) n n0

Eg: 3n+3=(n) as 3n+3 3n for n 1


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f(n) is

(g(n)) if

positive constantsc1, c2, and n0 such that c1 g(n) f(n)

c2 g(n) n n0

f(n) = (g(n)) if and only if

f(n) = O(g(n)) AND f(n) = (g(n))


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Eg 3n+2 = (n)

As 3n+2 3n and 3n+2 4n for all n 3 and c1=3 and c2 =4 and

n0 =3


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Computation Complexity

Definition: Measure the efficiency of an

algorithm in terms of time or space required.

Time tends to be more important.

The efficiency of an algorithm is always stated

as a function of the input size

The Big-O notation: O(f(n))


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Time Complexity

We assume each operation in a program

take one time unit.

int sum (int n){

int partial_sum = 0;

for (int i = 1; i


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Basic Time Complexity

Functions In an increasing order of complexity:

Constant time: O(1)

Logarithmic time: O(logn) Linear time: O(n)

Polynomial time: O(n2)

Exponential time: O(2n


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Function Values

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log n n n logn n2 n3 2n

0 1 0 1 1 2

1 2 2 4 8 4

2 4 8 16 64 163 8 24 64 512 256

4 16 64 256 4096 65536

5 32 160 1024 32768 4294967296


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Basic Time Complexity Functions

0

5

10

15

20

0 5 10 15 20 25

R

unningTime

Input Size

O(n)

O(logn)

O(1)

O(n^2)O(2^n)


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PERFORMANCE MEASUREMENT

Performance measurement is concerned with

obtaining the space and time requirements of

a particular algorithm.

These quantities depend on the compiler and

options used as well as on the computer on

which the algorithm is run.

Do not concern with the space and time

needed for compilation. But this time is

important during program testing.


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We do not consider measuring runtime

spacerequirement of a program.

We focus on measuring the run time of a

program.

We assume the existence of a program

GetTime() that returns the current time in

milliseconds.

aad module 1 s2

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