Problem Solving Techniques Slides

8/6/2019 Problem Solving Techniques Slides

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PROBLEM SOLVING TECHNIQUES

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UNIT I

INTRODUCTION TO COMPUTER PROBLEM SOLVING

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INTRODUCTION

Problem solving is an intricate process that

requires,

Much thought

Careful planning Logical planning

Persistence and

Attention to detail

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PROGRAMS AND ALGORITHMS

Algorithm:

A set of explicit and unambiguous finite steps which, whencarried out for a given set of initial conditions, produce thecorresponding output and terminate in a finite time.

A solution to a problem that is independent of anyprogramming language.

Is capable of being implemented as a correct and efficientcomputer program.

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INTRODUCTION


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REQUIREMENTS FOR SOLVING PROBLEMS BY

COMPUTER

Depth of understanding

Data organization

Input / Output: Supply the program with input or data so that

the program manipulates the data according toits instructions and produces an output whichrepresents the computer solution to the problem.

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THE PROBLEM-SOLVING ASPECT

Problem Definition Phase:

Understand what it is we are trying to solve

Work out what must be done rather than how to

do it Extract from the problem statement a set of

precisely defined tasks.

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Getting started on a problem:

Many ways to solve most problems

Many solutions to most problems.

Difficult to find quickly which paths are fruitlessand which are productive.

Dont concern with details of the implementationbefore understanding a problem.

The sooner you start coding your program the

longer it is going to take

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The use of specific examples:

Pick a specific example of the general problem tosolve

Try to work out the mechanism to solve thisparticular problem.

Employ geometrical or schematic diagramsrepresenting certain aspects of the problem.

Examine the specifications or test cases for theproblem carefully

Check whether or not the proposed algorithm canmeet those requirements. 9


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Similarities among problems:

See if there are any similarities between the currentproblem and other problems solved in the past.

More tools and techniques can be brought in tackling aproblem.

It is wise to solve the problem independently.

View the problem from a variety of angles upsidedown, backwards, forwards, etc..

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Working backwards from the solution:

Try to work backwards to the starting conditions if in

some cases we have the solution to the problem.

Write down the attempts whatever made along thevarious steps and explorations to systematize ourinvestigations and avoid duplication of effort.

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General Problem-Solving Strategies:

Divide-and-conquer strategy:

The idea behind this strategy is to split the problem into smaller andsmaller sub-problems until the sub-problems are enough to solve.

Wide application in sorting, searching and selection algorithms.

Dynamic programming:

To build a solution to a problem via a sequence of intermediate steps.

Good solution to a large problem can sometimes be built up from goodor optimal solutions to smaller problems.

Greedy search, backtracking and branch-and-bound techniques.

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TOP-DOWN DESIGN

Breaking a problem into subproblems:

Before applying top-down design to a problem, do thegroundwork that gives the outlines of a solution.

The general outline may consist of a single statement or a setof statements.

Top-down design suggests that,

Take the general statements one at a time

Break them into a set of more precisely defined subtasks.Subtasks should describe how the final goal is to be reached.

Subtasks need to interact with each other and be preciselydefined.

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TOP-DOWN DESIGN

Choice of a suitable data structure:

All programs operate on data and the way the data is organized canhave a effect on every aspect of the solution.

Inappropriate choice of data structure leads to inefficient and difficultimplementations.

Data structures and algorithms are linked to one another.

A change in data organization can have a influence on the algorithm

required to solve the problem.

There are no rules stating for this class of problems this choice of datastructure is appropriate.

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WHILE SETTING UP DATA STRUCTURES

Be aware of

Can the data structure be easily updated / searched?

Does the data structure involve the excessive use ofstorage?

Is it possible to impose some data structure on aproblem that is not initially apparent?

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TOP-DOWN DESIGN

Construction of loops:

The initial conditions that need to apply before theloop begins to execute.

The invariant relation that must apply after eachiteration of the loop and

The conditions under which the iterative processmust terminate.

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TOP-DOWN DESIGN

Establishing initial conditions for loops:

Set the loop variables to the values that they wouldhave to assume in order to solve the smallest problemassociated with the loop.

Usually the number of iterations that must be made bya loop are in the range 0


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TOP-DOWN DESIGN

Finding the iterative construct:

Once we have the condition for solving the smallestproblem the next step is to extend it to the next smallestproblem.

For e.g. in the summation of n numbers, the solution for

n=1 is,i=1

s=a[1]

This solution for n=1 is built from the solution for n=0using the values for i and s when n=0 and the twoexpressions

i=i+1

s=s+a[i] ----------------------> generalized solutionfor n>0

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TOP-DOWN DESIGN

Termination of loops:

The simplest condition for terminating a loop occurs when itis known in advance how many iterations need to be made.

for i=1 to n{

This loop terminates unconditionally after n iterations.

}

A second way in which loops can terminate is when some

conditional expression becomes false.E.g. while (x>0 and x


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IMPLEMENTATION OF ALGORITHMS

Choice of variable names:

Choose appropriate variable and constant names tomake programs more meaningful and easier tounderstand.

This practice make programs more self-documenting.

Each variable should only have one role in a program.

A clear definition of all variables and constants at thestart of each procedure can also be very useful.

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Documentation of programs:

At the start of each modular part, give a brief accuratecomment.

Write programs with good documentation so that they canbe executed and used by other people unfamiliar with theworkings and input requirements of the program.

The program must specify during execution exactly whatresponses and their format it requires from the user.

The program should catch incorrect responses to itsrequests and inform the user in an appropriate manner. 24


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Debugging programs:

Carry out a number of tests to ensure that the program isbehaving correctly according to its specifications.

There may be logical errors in the program that is not shown inthe compilation phase.

To detect logical errors, build into the programs a set of statements that will print out information at strategic points in

the computation. These statements can be made conditionally executable.

If debug then /* debug is a boolean variable that is set to true when */

{ /* debugging output is required for the program */

printf();

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Always work the program by hand beforeattempting to execute it.

Draw up a two-dimensional table consisting of

steps executed against all the variables used inthe section of the program under consideration.

Update the variables in the table as each variableis changed when the statements in the program

section gets executed.

A good rule when debugging is not to assumeanything.

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Program testing: Design to solve that it will cope with the limiting and unusual

cases.

For E.g., Tests for binary search algorithm.

Will the algorithm handle the search of array of one element? Will the algorithm handle the case where the value sought is

at an odd or even array location?

Will it handle the case where all array values are equal?

Programs should be accompanied by input and output assertions.

Build into the program mechanisms that informatively respond tothe user when it receives input conditions it was not designed tohandle.

Design algorithms to be very general so that they will handle awhole class of problems rather than just one specific case.

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PROGRAM VERIFICATION

The application of mathematical proof techniques to establishthat the results obtained by the execution of a program witharbitrary inputs are in accord with formally defined outputspecifications.

1. Computer model for program execution.

2. Input assertion specify any constraints that have beenplaced on the values of the input variables used by theprogram.

3. Output assertion specify symbolically the results that theprogram is expected to produce for input data that satisfiesthe input assertion.

4. Implications and symbolic execution.

5. Verification of straight-line program segments.

6. Verification of program segments with branches / loops /arrays.

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THE EFFICIENCY OF ALGORITHMS

Design algorithms that are economical in the use ofCPU time and memory because of high cost of computing resources.

Suggestions that are useful in designing efficient

algorithms. Redundant computations.

Referencing array elements.

Inefficiency due to late termination.

Early detection of desired output conditions.

Trading storage for efficiency gains.

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FUNDAMENTAL ALGORITHMS

EXCHANGING THE VALUES OF TWO

VARIABLES

Problem Statement:

Given two variables a and b,

interchange the values of thevariables.

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ALGORITHM:

1. Save the value of variable a in variable t.2. Assign to variable a the value of variable b.

3. Assign to variable b the value of variable a

stored in variable t.

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EXCHANGING THE VALUES OF TWO VARIABLES


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IMPLEMENTATION:

void exchange (int a, int b)

{

int t; //temporary variable

t=a;

a=b;

b=t;

}

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EXCHANGING THE VALUES OF TWO VARIABLES

Applications:Used in SORTING


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COUNTING

Problem Statement:

Given a set of n studentsexamination marks (in the range0 to 100), make a count of numberof students passed theexamination. A pass is awardedfor all marks of 50 and above.

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COUNTING

Example:

Given following are the marks obtained by

5 students in a particular subject.

45,67,89,34,82

O/P: Number of students passed the subject is 3.

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ALGORITHM:

1. Prompt and read the number of marks to be processed.2. Initialize count to zero.

3. While there are still marks to be processed repeatedly doa. Read next mark.

b. If it is a pass then add one to count.

4. Print total number of passed students.

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COUNTING


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SUMMATION OF A SET OF N NUMBERS

Problem Statement:

Given a set of n numbers,design an algorithm that adds these

numbers and returns the resultantsum. Assume n is >=0.

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Example:

Given following are the set of numbers to

find the resultant sum.

45,67,89,34,82

Resultant sum = 45+67+89+34+82 = 317

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ALGORITHM:

1. Prompt and read total numbers to sum.

2. Initialize sum for zero numbers.3. While less than n numbers have been summed repeatedly do

a. Read next mark.

b. Compute current sum by addin

4. Print total number of passed students.

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The algorithm uses n additions to sum n

numbers.

Applications:Average calculations, variance and least

square calculations.

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FACTORIAL COMPUTATION

Problem Statement:

Given a number n,compute n factorial (written asn!) where n>=0.

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Example:

5!= 5*4*3*2*1=120

0!=1

1!=1*1In general,

n!=n*(n-1)! for n>=1

(or)

n!=1*2*3**(n-1)*n for n>=1

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ALGORITHM:

1. Establish n, the factorial required where n>=0.2. Set product p for 0! Also set product count to zero.

3. While less than n products have been calculated repeatedly

do,

a. Increment product count.

b. Compute the ith product p by multiplying I by the mostrecent product.

4. Return the result n!.

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IMPLEMENTATION:

/* computes and returns n! for n >= 0 */int nfactorial(int n)

{int i; /* loop index representing the ith factorial */int factor; /* i! */{assert (n >= 0)}factor = 1;{invariant: factor = i! after the ith iteration and i


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The algorithm uses n multiplications to compute

n!.

Possible to express n! in terms of (n/2)! (Similar

to calculating nth fibonacci number)

Applications:

Probability, Statistical and mathematical

computations.

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SINE FUNCTION COMPUTATION

Problem Statement:

Design an algorithm to evaluate thefunction sin(x) as defined by the infiniteseries expansion

sin(x) = x/1! x3/3! + x5/5! x7/7! + .

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ALGORITHM:

1. Set up initial conditions for the first term that cannot becomputed iteratively.

2. While the absolute value of current term is greater than the

acceptable error do

a. Identify the current ith term.

b. generate current term from its predecessor.c. add current term with the appropriate sign to the

accumulated sum for the sine function.

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IMPLEMENTATION:

const float error = 1.0e-6;

float Abs(float x){

if(x>0.0)

return x;

else

return -x;

}

/* function returns sin(x) with an accuracy of


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{invariant: after the jth iteration, i=2j+1 and term = (-1)^j * (x^i)/ i! and tsin is

sum of first (j+1) terms}while(Abs(term) > error) /* generate and accumulate successive terms

of sine expression */

{

i = i + 2;

term = -term * x2/(i*(i-1));

tsin = tsin + term;

}return tsin;

{assert: tsin~= sine(x) and abs(term)


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GENERATION OF THE FIBONACCI SEQUENCE

Problem Statement:

Generate and print the first n terms of thefibonacci sequence where n>=1.

The first few terms are: 0,1,1,2,3,5,8,13.

Each term beyond the first two is derivedfrom the sum of its two nearest predecessors.

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Example:

If n=5, the generated fibonacci sequence

is 0,1,1,2,3

New term = preceding term + term before

preceding term

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void Fibonacci()

{int a,b; /* Fibonacci number variables */

int i; /* number of Fibonacci numbers generated */

int n; /* number of Fibonacci numbers to be generated */

a = 0;

b = 1;

i = 2;

printf("Enter the number of Fibonacci numbers to be generated : ");

scanf("%d",&n);

{assert( n > 0)}{invariant: after the jth iteration I = 2j+2 and first I fibonacci numbers have been

generated and a=(i-1)th Fib. No. and b= ith fib. No}

while(i


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To generate n fibonacci numbers, n steps are required. The algorithm

works correctly for all values of n >=1

Throughout the computation, the variables a and b always contain

the two most recently generated fibonacci numbers.

Applications:

Botany, Electrical network theory, sorting

and searching.

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REVERSING THE DIGITS OF AN INTEGER

Problem Statement:

Design an algorithm thataccepts a positive integer andreverses the order of its digits.

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ALGORITHM:

1. Establish n, the positive integer to be reversed.

2. Set the initial condition for the reversed integer dreverse.3. While the integer being reversed is greater than zero do,

a. Use the remainder function to extract the rightmost digitof the number being reversed.

b. Increase the previous reversed integer representation

dreverse by a factor of 10 and add to it the most recentlyextracted digit to give the current dreverse value.

c. Use integer division by 10 to remove the rightmost digit fromthe number being reversed.

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REVERSING THE DIGITS OF AN INTEGER


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Dreverse(int n)

{

int reverse;

{assert: n>=0 and n contains k digits a(1), a(2), a(3), ., a(k)}

reverse=0;

{invariant: after jth iteration, n=a(1), a(2), a(3), .a(k-j) and reverse = a(k), a(k-1) .a(k-j+1)}

while (n>0)

{

reverse=reverse*10+((n % 10));n=n/10;

}

{assert: reverse = a(k), a(k-1),..a(1)}

printf(Reversed number=%d,reverse);

}

The number of steps to reverse the digits in an integer is directlyproportional to the number of digits in the integer.

Applications:

Hashing and information retrieval, database applications. 60

REVERSING THE DIGITS OF AN INTEGERImplementation:


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BASE CONVERSION

Problem Statement:

Convert a decimal integerto its corresponding octal

representation.

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EXAMPLE:

The octal representation of decimal 275 is

423.

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BASE CONVERSION


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BASE CONVERSION


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void basechange( int n, int newbase)

{

int i; /* index for new digit output array */

int ascii; /* ascii value of current digit */int ndigit; /* current counter of new digits computed*/

int q; /* current quotient */

int r; /* current digit for newbase representation*/

int zero; /* ascii value of zero character*/

char newrep(100); /* output array */

{assert: n>0 and 20;i--) printf(%c,newrep[i]);

}

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Implementation:


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Applications:

Interpretation of stored computer data and

instructions.

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BASE CONVERSION


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CHARACTER TO NUMBER CONVERSION

Problem Statement:

Given the characterrepresentation of an integer convert

it to its conventional decimal format.

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Example:

Character representation 125 when

converted to decimal format results in

125.

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Algorithm:

1. Establish the character string for conversion to decimaland its length n.

2. Initialize decimal value to zero.3. Set base0 value to the ascii or ordinal value of 0.

4. While less than n characters have been examined do

a. Convert next character to corresponding decimaldigit.

b. Shift current decimal value to the left one digit andadd in digit for current character.

5. Return decimal integer corresponding to input characterrepresentation.

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Implementation:

int chrtodec(char str[],int n)

{

int i; /*Index for count of characters converted*/

int dec /*Used to build converted decimal integer*/

int base0/*ascii of character 0*/

{assert: n>=0 and string [1..n] represents a non-negative number}

dec=0;

base0=0;

{invariant: after the ith iteration, dec contains the I leftmostdigits of the string in integer form and i

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Problem Solving Techniques Slides