Question

Data Structures and Algorithms

1. For each of the following program fragments, give an analysis of the running time using Big-Oh notation. Do not give formulas, but analyze every line to calculate the running time, e.g. sum = 0 is equal to 1 unit time ...

b.

sum = 0;

   for( i = 0; i < n; i++)

           for( j = 0; j < i*i; j++)

                 for( k = 0; k < j; k++)    

                                      sum++;

c.

sum = 0;          
for( i = 0; i < n²; i++)  
for( j = 0; j < i; j++)
sum++;

d,

sum = 0;

for( i = 0; i < 2n; i++ )

    for( j = 0; j < i; j++ )

         sum++;

Answer 1

b)answer)

sum = 0;

for( i = 0; i < n; i++)

for( j = 0; j < i*i; j++)

for( k = 0; k < j; k++)

sum++;
sum=0 take one unit of time.so it is written as O(1)
for i=0 ,j loop runs 0 times so k loop runs 0 times
for i=1,j loop runs 1 time.so k loop runs 1 time
for i=2,j values are 0,1,2,3,4.
when j=0 ,k loop runs for 0 times
when j=1 ,k loop runs for 1 times
when j=2 ,k loop runs for 2 times
when j=3 ,k loop runs for 3 times
when j=4 ,k loop runs for 4 times
here total k loop runs for 0+1+2+3+4=4*(4+1)/2 times
..........
for i=n,j values are 0,1,2,3,4,.....n^2
when j=0 ,k loop runs for 0 times
when j=1 ,k loop runs for 1 times
when j=2 ,k loop runs for 2 times
when j=3 ,k loop runs for 3 times
when j=4 ,k loop runs for 4 times
....
when j=m*m ,k loop runs for n*n times
here total k loop runs for 0+1+2+3+4+....+m*m=m²(m²+1)/2 times

total work is

time complexity=O(n⁵)

c)answer)

sum = 0;
for( i = 0; i < n^2; i++)
   for( j = 0; j < i; j++)
       sum++;
for i=0, j loop runs for 0 times
for i=1, j loop runs for 1 times
for i=2, j loop runs for 2 times
...
for i=n^2, j loop runs for n^2 times
  
sum=0 and sum++ take one unit of time.

total j loop runs for 1+2+3+4+.....+n^2
=n^2 *(n^2 +1)/2
total time complexity is O(n^4)

d)answer)

sum = 0;
for( i = 0; i < 2^n; i++ )
   for( j = 0; j < i; j++)
       sum++;
for i=0, j loop runs for 0 times
for i=1, j loop runs for 1 times
for i=2, j loop runs for 2 times
...
for i=2^n, j loop runs for 2^n times
  
sum=0 and sum++ take one unit of time.

total j loop runs for 1+2+3+4+.....+2^n
=2^n *(2^n +1)/2
total time complexity is O(2²ⁿ)