Question

Hello, I would like to get help with the following algorithms and their respective analysis of runtime along with their recurrence equations, thanks in advance.

1.       Analyze each of the following algorithms by providing a tight big-Oh bound on the run time with respect to the value n. Create a recurrence equation.

a.    

void padawan(int n)

{

      if( n <= 10 )

            time++;

      else

      {

            for(int i=0; i<n; i++)

                        time++;

            padawan(n/3);

            padawan(n/3);

            padawan(n/3);

      }

}

b.   

void nerfHerder(int n)

{

      if( n <= 15 )

            time++;

      else

      {

            for(int i=0; i<n; i++)

                  time++;

            nerfHerder(n-1);

      }

}

c.   

void squire(int n)

{

      if( n <= 12 )

            time++;

      else

      {

            squire(n/2);

            squire(n/2);

            for(int i=0; i<n; i++)

                  for(int j=i; j<n; j++)

                        time++;

            squire(n/2);

            squire(n/2);

            squire(n/2);

            squire(n/2);

            squire(n/2);

            squire(n/2);

      }

}

Answer 1

Master method

T(n) = aT(n/b) + f(n) where a>=1 and b>1

There are following three cases:
1. If f(n) = Θ(n^c) where c < Log_ba then T(n) = Θ(n^Log_ba)

2. If f(n) = Θ(n^c) where c = Log_ba then T(n) = Θ(n^cLog n)

3.If f(n) = Θ(n^c) where c > Log_ba then T(n) = Θ(f(n))

a) Given equation runtime

Runtime for if condition is O(1)

In else condition runtime for loop is O(n) and for recurrence relation runtime can be written as

By using Master method as as c=log^a_b it falls under case 2, T(n)=

b)runtime for loop is O(n)

recurrence relation is

and c>log^a_b as c=1,a=1,b=1 it falls under case 3,T(n)=

3)runtime for for loops is O(n²)

As there are 6 recursive functions

c<log^a_b because a=6,b=2 and c=2 this falls under case 1,Therefore