Lecture Module3 Prof Saroj Kaushik, IIT Delhisaroj/IITJ/Lect3.pdfProf Saroj Kaushik, IIT Delhi 3 `...

Lecture_Module3Prof Saroj Kaushik, IIT Delhi

Problem definitionMethod (how to solve it)◦ Algorithm◦ Data structure◦ Verification for correctness◦ Analysis for efficiencyCoding in given “programming language”Understanding of computer “architecture”“Compilation”, “testing”, “de-bugging”Documentation

2Prof Saroj Kaushik, IIT Delhi

Most difficultRequires interaction between “programmer” and userSpecs include:◦ Input data

Type, accuracy, units, range, format, location, sequence◦ Special symbols to signal end of data◦ Output data (results)

type, accuracy, units, range, format, location, “headings”◦ How is output related to input◦ Any special constraintExample: “find the phone no. of a person”Problems get revised often


It is a finite set of instructions which, iffollowed accomplish a particular task.

It is basically used to describe a problemsolving method suitable forimplementation as a computer program.

Algorithm is independent of the machineand language used for implementation.


Input◦ Zero or more quantities are supplied externallyOutput◦ At least one quantity is producedDefiniteness◦ Each instruction is clear & unambiguousFiniteness◦ It terminates after finite stepsEffectiveness◦ Each instruction is simple to be carried out manually.


An “unambiguous specification of a method”Is characterized by:◦ Ordered sequence of well-defined, effective

operations that, when executed, will produce a result after “terminating” within a finite no of steps


Well-defined and effective◦ No ambiguity, a method must exist◦ Good Examples:

Add 1 to xcompute largest prime no. < 100compute square root of x to 4 decimal places

◦ Bad examples:divide 10 by xcompute largest primecompute square root of x


Always terminate, and be sure about itProduce correct results◦ this may require some hard thinking◦ testing helps, but is not adequate


Basic differences are:

◦ Program is written in programminglanguage whereas algorithm is inEnglish like pseudo language.

◦ Program may be non terminating (OS)whereas algorithm should terminate infinite steps.


Study of algorithm can be classifiedin four distinct areas namely how to◦ devise◦ express◦ validate◦ analyze algorithms


Devising good algorithm:◦ Requires study of various design techniques◦ Top down, bottom up and object oriented

approachesExpressing an algorithm:◦ Good algorithms are expressed using principle of

structured programmingValidation of algorithm:◦ It should be validated for correctness of all possible

legal inputs.(correct, incorrect, exceptions)

◦ Note that algorithm need not yet be expressed as acomputer program.


Analysis of an algorithm:◦ Study of behavior pattern or performance

profile.◦ It can be calculated in terms of computing

time and space requirement in the machine.Time Complexity: Running time of the programas a function of the size of input.Space Complexity: Amount of computer memoryrequired during the program execution.


In top-down model, an overview of thesystem is formulated, without goinginto detail for any part of it.

Each part of the system is then refinedin more details.

Each new part may then be refinedagain, defining it in yet more detailsuntil the entire specification is detailedenough to validate the model.


This design model can also be appliedwhile developing algorithm.

It basically refers to successiverefinement of the problem (task) intosub problems (subtasks).

Refinement is applied until we reach tothe stage where the subtasks can bedirectly carried out.


subtask1 subtask2 subtask3

Main Task


In bottom-up design individual parts ofthe system are specified in details.

The parts are then linked together toform larger components, which are inturn linked until a complete system isformed.

Object-oriented languages such as C++or JAVA use bottom-up approach whereeach object is identified first.


It is a technique using which one can write algorithms (programs) in top down fashion.

This technique should be used with every level of refinement.

In SP, one entry and one exit principle is adopted in all the constructs.

Basically there are three structures in SP


Sequential (sequence)entry

T1

T2

exit


If cond then task1Y

Cond task1N

If cond then task1 else task2

task2 N Cond Y task1


WhileWhile (cond) do{

}

RepeatRepeat

Until (cond)


for ordered sequenceT1, while C then T2,T3

T1

T2

T3

CN


Y

for ordered sequenceT1, Repeat T2 until C,T3

T1

T2

T3

CN


Y

What is the difference between:

S1

S2

S3

C

S1

S2

S3

CN

Y


S1, Repeat S2 until C, S3 S1, While C { S2 }, S3

N

Y

Computation of income taxGiven a tax table as below, compute the tax, T, on an income, X.

Here is an algorithm:

Step 1: Input INC;Step 2: Compute tax, T;Step 3: Output T

INCOME TAX0

Step 1: Input INC;

Step 2a: if INC > 300000 then T ← 30000 + 0.30*(INC‐300000);

Step 2b: if INC > 200000 and INC ≤ 300000then T ← 10000 + 0.20*(INC‐200000);

Step 2c: if INC > 100000 and INC ≤ 200000then T ← 0 + 0.10*(INC‐100000);

Step 2d: if INC ≤ 100000 then T ← 0;

Step 3: Output T


Table look-up: Consider: ◦ special key K,◦ length of list, 5 or N, more generally◦ unsorted L = [(x1, y1), (x2, y2), (x3, y3), (x4, y4), (x5, y5)],

where xi is key and yi is corresponding output valueProblem is to search whether key K is in the list and output relevant pair if it existHere is an algorithm:

Step 1: Input all data, K, and list L;Step 2: Search for K in L;Step 3: Output results


Here is refined version of the algorithm:

Input K;Input (x1, y1);Input (x2, y2);Input (x3, y3);Input (x4, y4);Input (x5, y5);If K = x1 then output (x1, y1);If K = x2 then output (x2, y2);If K = x3 then output (x3, y3);If K = x4 then output (x4, y4);If K = x5 then output (x5, y5)


This algorithm does the same thing, except that it is compact:

Input K;Repeat these operations 5 times:

{ Input (x, y);If K = x then output (x, y)

}


An even better algorithm:

Input n;If n > 0 then

{ Input K;Repeat these operations n times:{ Input (x, y);

If K = x then output (x, y)}

}


Formula: x1 = [(-b) + √(b2 – 4ac)]/2a x2 = [(-b) - √(b2 – 4ac)]/2a Algorithm: input a, b, c; d = b*b – 4*a*c; if (d ≥ 0) then { e = sqrt(d); r1 = (-b + e) / (2*a); r2 = (-b - e) / (2*a); i1=0; i2 =0 } else

{ e = sqrt(-d); r1 = (-b) / (2*a); r2 = r1; i1= e / (2*a); i2 = -e/(2*a); } output r1,r2,i1,i2

Formula:fact = n * n-1 * …* 2 * 1

Algorithm:input n;i = 0; initializationfact = 1;

loopingwhile (i < n) { i = i + 1;

fact = i * fact;}output fact

Prof Saroj Kaushik, IIT Delhi 32

ExecutionLet n = 4;i = 0; fact = 1;while loop0 < 4; i = i +1=1; fact = 1 * 1;1 < 4; i = i +1=2; fact = 1 * 2;2 < 4; i = i +1=3; fact = 2 * 3;3 < 4; i = i +1=4; fact = 6 * 44 < 4 is false so exit the while loop; output fact = 24

Algorithm:input num;rev = 0;while (num > 0)

{ rev = rev*10 + num mod 10;

num = num div 10;}

output rev

Executionnum = 245;rev = 0;while looprev = 0 * 10 + 5 = 5;num = 24;rev = 5 * 10 + 4 = 54;num = 2;rev = 54 * 10 + 2 =

542;num = 0;exit whileoutput rev as 542


Algorithm: Here number is read in the loop. Atthe end of loop, read elements are notavailable.

max = 0;i = 0;while (i < 100){ i = i+1;

input num;if (num > max) then max = num;

}output max


Algorithm:input num(i), i = 1,100;max = 0;i = 0;while (i < 100){ i = i+1;

if (num(i) > max) then max = num(i);}output max, num(i),

i = 1,100

ExecutionLet numbers are: 4, 2,7,1max = 0; i = 0;while loopi = 1; 4 > max so max = 4;i = 2; 2 < max so no

changei = 3; 7 > max so max = 7;i = 4; 1 < max so no

changeexit of whileoutput max as 7 and

numbers as 4, 2, 7, 1


Series: f = 0! + 1! + 2! + …+ n! (n ≥ 0)Algorithm: Here we are given the value of n. Series

sum is calculated by finding sum after addingprevious sum with a term at each step.

input n;f = 0; term = 1; i = 0;while (i < n){ f = f + term;

i = i + 1;term = term * i;

}output f

Problem Solving using ComputerProblem solvingProblem definition AlgorithmCharacteristics of an AlgorithmAlgorithms(1) Algorithms (2)Algorithms (3)Algorithm verses Program Study of AlgorithmContd…Analysis of AlgorithmTop Down Design ModelTop Down Concept in Problem SolvingTop Down DesignBottom-up DesignBottom up DesignStructured Programming (SP)Three types of control flow in SPSelection (test) RepetitionFlow diagram Ordered SequencesFlow diagramRepeat and While ordered sequencesAlgorithms: an exampleAlgorithms: (refined)Algorithms: another exampleTable look-up algorithms (contd.)Table look-up algorithms (contd.)Table look-up algorithms (contd.)Problem 1: Find the roots of a quadratic equation of the form a*x2 + b*x + c = 0Problem 2: Write algorithm to find factorial of nProblem 3: Reversing integer digits (3542 2453)�Problem4: Find maximum out of 100 numbers (+ve integers)�Input all 100 numbers in an array (vector). Then compute maximum out of 100 numbers. At the end of loop we have all the numbers.Problem5: Design an algorithm to find the sum of first n terms of the series

Lecture Module3 Prof Saroj Kaushik, IIT Delhisaroj/IITJ/Lect3.pdfProf Saroj Kaushik, IIT Delhi 3 `...

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