PROBLEM SOLVING TECHNIQUES

SECTION - A

1. Answer any ten of the following

a. Define an algorithm.

An algorithm is a finite set of instructions that

if followed, accomplishes a particular task.

b. Define a program.

A program is a set of instructions to solve a

particular problem by the computer language.

c. What is a verification condition (VC)?

Symbolic executing enables us to transform

the verification procedure into proving that

the input assertion with symbolic values

substituted for all input variables implies the

output assertion with final symbolic values

substituted for all variables. A proposition

phrased in this way is referred to as a

Verification Condition (VC).

d. What is Fibonacci series?

The Fibonacci Sequence is the series of

numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... The

next number is found by adding up the two

numbers before it.

e. Given a number 18910 convert it to octal

equivalent.

2758

f. What is the error condition (fixed error) that

might be used to terminate square root

algorithm?

1 x 10-6

or 0.0000001

g. What are prime numbers? Write first 10

prime numbers.

Prime number is a number which is not

divisible by any other number other then 1 and itself.

First 10 prime numbers are 2,3,5,7,11,13,17,19,23,29.

h. Find the GCD of 18 and 30.

6

i. What is longest monotone subsequence?

A monotone increasing subsequence is a

subset of numbers which are strictly

increasing from left to right. This definition

does not require that the numbers be

adjacent in the original set or that the longest

sequence is unique. For example:

j. In which applications removing duplicates

from an array algorithm is used?

Data Compression and text processing

problems

k. What is sorting?

Sorting is any process of arranging items

systematically

l. What is hashing?

Hash searching is used to index and retrieve items

in a list using a shorter fixed-length value because

it is faster to find the item using the shorter

hashed key than to find it using the original value.

SECTION - B

Answer any four of the following

2. What is a flowchart? Draw and explain various

symbols used in flowchart.

Flowchart can be defined as a diagrammatic

representation of and algorithm.

Symbol Used Geometrical

Name

Uses

Oval Start and Stop

Parallellogram Input and

Output

Rectangle Processing

Arrows Connectors

Small Circle Connector

Rhombus Decision

Making

Hexagon Repetition or

Looping

3. Write the algorithm to exchange the values of two

variables.

PROBLEM STATEMENT: Given two

variables, a and b, exchange the

values assigned to them.

INPUT: a as integer, b as integer

OUTPUT: new value of a must be old

value of b, new value of b must be

old value of a.

ALGORITHM exchange(a,b)

{

t <-- a;

a <-- b;

b <-- t;

}

4. Write and explain the expression and parameters

used for generating random numbers using Linear

Congruential Method.

where the parameters a, b, m and x0 must be carefully

chosen in advance according to certain criteria. The

parameters a, b, and m are referred to as the

multiplier, increment, and modulus respectively.

All parameters should be integers greater than or equal

to zero and m should be greater than x0,a and b.

Parameter x0: The parameter x0 can be chosen

arbitrarily within the range 0--x0<m.

Parameter m: The value of m should e greater than or

equal to the length of the random sequence required.

In addition it must be possible to do the computation

(a*x)+b mod m without roundoff.

Further requirements on a are that it should be larger

than √m and less than m-√m, (a-1) should be a

multiple of every prime dividing into m, and if m is a

multiple of 4 then (a-1) should also be a multiple of 4.

These conditions together with the requirements that

b should relatively prime to m are needed to guarantee

that the sequence has a period of m.

Parameter a: The choice of a depends on the choice of

m. If m is a power of 2 then a should satisfy the

condition: a mod 8 = 5 If m is a power of 10, then a

should be chosen such that: a mod 200 = 21

Paramter b:

The constant b should be odd and not a multiple of 5.

When a, b, and m are chosen according to the

conditions outlined above a sequence of m pseudo-

random numbers in the range 0 to (m-1) can be

generated before the sequence begins to repeat.

5. Write the algorithm to compute nth Fibonacci

number.

ALGORITHM fibonacci(n)

//PROBLEM STATEMENT: Algorithm to

print the first n fibonacci numbers.

//INPUT: n - numbers to be generated

of the series.

a - as number before the

preceding term

b - as number preceding

term

c - new term

// OUTPUT: fibonacci series

{

a <-- 0;

b <-- 1;

i <-- 2;

write (a);

write (b);

while i < n do

{

a <-- a + b;

b <-- a + b;

write (a);

write (b);

i <-- i + 2;

}

return;

}

6. Explain how you rearrange the elements in an array

so that they appear in reverse order.

The problem of reversing the order of an array of

numbers appears to be completely

straightforward. We start the design of this

algorithm by careful examination of the array

before and after it has been reversed; for example,

What we observe from our diagram is that the first

element ends up in the last position. Carrying this

process through we get the following set of

exchanges.

In terms of suffixes the exchanges are :

Examining the effects of these exchanges we

discover that after step 3 the array is completely

reversed. In setting up our algorithm we need a

pair of suffixes that model this increasing-

decreasing behavior. Our increasing suffix can be

the variable i which is incremented by1.For our

decreasing suffix we might try n-i since this

decreases by 1 with each increase in i by 1.This

means that when i=1 we find that n-i is equal to n-

1 rather than n. We can correct this by adding 1.

So the suffix n-i-1 can be used at other end rather

than using n-i.

7. Write a short note on Two –way merge.

Merging two or more sets of data is a task that is

frequently performed in computing. It is simpler than

sorting because it is possible to take advantage of the

partial order in the data.

Examination of two ordered arrays should help to

discover the essential of a suitable merging procedure.

Consider the two arrays:

A little though reveals that the merged result should be

as indicated below:

The origins are written above each element in the c

array. What we see here is c is longer than a and b. In

fact c must contain a number of elements

corresponding to the sum of the elements in a and b

(i.e., a+b). To see how this might be done let us

consider the smallest merging problem. To merge the

two one dimensional array all we need to do is select

the smaller of the a and b elements and place it in

c.The larger element is then placed in c.

8 is less than 15 so 8 will take c[1] place and 15 c[2]

place. In the same way we start merging arrays of

lengths m and n.The comparison between a[1] and b[1]

allows us to set c[1]. After placing 8 in c[1] we need a

way of deciding which element must be placed next in

the c array. In the general case the next element to be

placed into c is always going to be the smaller of the

first elements in the unmanaged parts of arrays a and

b. To keep track of the “yet to be merged” parts of

both the a and b arrays two index pointers i and j will

be needed. As an element is selected from either a or b

the appropriate pointer must be incremented

/decremented by 1.

SECTION - C

Answer any four of the following

8. a. Explain various characteristics of an algorithm.

• INPUT: An algorithm may accept zero or more

quantities.

• OUTPUT : An algorithm must produce atleast

one quantity.

• DEFINITENESS: Each instruction is clear,

precise and well defined. There should not be

any ambiguity.

• FINITENESS: The algorithm must have finite

number of instructions. Thus it must

terminate after a finite number of steps.

• EFFECTIVENESS: Every instruction must be

basic so that it must be carried out, in

principle, by a person using only pencil and

paper.

b. What is top-down design? Explain how you

break a problem in sub problems.

A technique for algorithm design that tries to

accommodate this human limitation is known as

top-down design or stepwise refinement. Top-

down design provides us with a way of handling

the inherent logical complexity and details,

frequently encountered in computer algorithms.

Breaking a problem into subproblems

Top-down design suggests that we take the

general statements that we have about the

solution, one at a time, and break them down into

a set of more precisely defined subtasks. These

subtasks should more accurately describe how the

final goal is to be reached. With each splitting of a

task into subtasks it is essential that the way in

which the subtasks need to interact with each

other be precisely defined. Only in this way is it

possible to preserve the overall structure of the

solution to the problem. Preservation of the

overall structure in the solution to a problem is

important both for making the algorithm

comprehensible and also for making it possible to

prove the correctness of the solution. The process

of repeatedly breaking a task down into subtasks

and then each subtask into still smaller subtasks

must continue until we eventually end up with

subtasks that can be implemented as program

statements.

9. Write the algorithm and description of sine

function computation.

Algorithm Description

Algorithm

ALGORITHM sin(x)

//PROBLEM STATEMENT : Given a

value x generate an algorithm to

evaluate the function sin(x) as

defined by the series expansion

sin(x) = x/1!- x3/3! + x5/5! -

x7/7!+ ....

//INPUT : x as real

//OUTPUT: to generate sin(x)

{

//error is termination

condition

error <-- 1.0* e-6;

i <-- 1;

x2 <-- x * x;

term <-- x;

tsin <-- x;

while abs(term) > error do

{

i <-- i + 2;

term <-- -term * x2 /

(i* (i-1));

tsin <-- tsin + term;

}

return tsin;

}

10. a. Write the algorithm to find GCD of two

numbers.

ALGORITHM gcd(n,m)

PROBLEM STATEMENT: Given two

positive non-zero integers n and m

design an algorithm for finding

their greatest common divisor

(GCD).

INPUT: n, m as integers whose gcd

is to be sought

OUTPUT: gcd of n, m integers

{

//r :integer variable

to store the remainder

repeat

r <--n mod m;

n <-- m;

m <-- r;

until r=0

return m;

}

b. Write the algorithm to find smallest divisor of

an integer.

ALGORITHM smallestdivisor(n)

{

if NOT odd(n) then

{

sdivisor <- 2;

}

else

{

r <- trun(sqrt(n));

d <- 3;

while (n mod d = 0) AND (d < r)

do

{

d <- d + 2;

if n mod d = 0 then

sdivisor <- d;

else

sdivisor <- 1;

}

}

}

11. Write short notes on

a. Finding maximum in a set.

b. Array Counting or Histogramming.

12. a. Given a set of numbers below , sort them into

non decreasing order using insertion sort

20 35 18 8 14 41 3 39

b. Given an element x and a set of data that is

in ascending order, write an Binary Search

algorithm to find whether or not x is present

in the set.

//PROBLEM STATEMENT: Algorithm to

find an element using binary search

method.

//INPUT: a: integer array

n : integer variable to hold

upper limit

key: integer variable, key

element

//OUTPUT: position of the element

found.

Algorithm BinarySearch(a,n,key)

{

bottom = 1;

top = n;

do {

mid = (bottom + top) / 2;

if (item < a[mid])

top = mid - 1;

else if (item > a[mid])

bottom = mid + 1;

} while (item != a[mid] && bottom <=

top);

if (item == a[mid]) {

return mid+1;

} else {

return -1;

}

}