CSE222 Homework 1 Solved

Description

1. Given any two functions f(n) and g(n), show that f(n) + g(n) = Θ(max{f(n),g(n)}))

(10P).P).).

2. Show that f(n) =n²+ 2n+ 1 is Θ(n²) using induction. ( If you use l’Hopital, you will lose points.)(5P).).
3. P).rove the functions below (40P).P).).
4. a) If f(n) = 10P). log(n) + 5 (log(n))³ + 7n + 3n² + 6n³, then f(n) = O(n³) (5P).) b) 1 = O(n) (5P).)
   1. n = O(n²) (5P).)
   2. log(n) = O(n), 2 n + 1 = O(n) (5P).)
   3. n = Ω(1) (5P).)
   4. n² = Ω(n) (5P).)
   5. n² = Ω(n log(n)) (5P).)
   6. 2 n + 1 = Θ(n) (5P).)
5. Sort the following functions from fastest to slowest with respect to their growth rate. Do not use l’Hopital! P).rove all of them using induction (20P).P).).

n!,  n^k+n , n,  logn,  n(logn),  e⁷,  20P).19,  -7^n+m,  n⁴,  10P).0P).*n

*k and m are constants.

5. Explain the time complexity of the code snippets below (10P).P).).

a-)System.out.println = SOP void method4(int [] arr) { for(int i = 0; i < arr.length; i++) { for(int k = arr.length – 1; k > 0; k = k / 3 ) { SOP(arr[i]);

}

} }

b-)

void method3(int [] arr)

{ for(int i = 0; i < arr.length; i++)

{ method1(arr);

method2(arr);

}

}

void method1(int [] arr)

{ int n = arr.length;

for(int i = n – 1 ; i >= 0; i = i – 3)

{

SOP (arr[i]);

}

}

6. Calculate the time complexity of the following recurrence functions (Use the master theorem)
   • T(n) = T(n/7) + n⁴
   • T(n) = T(n/99) + n⁷⁵
   • T(n) = 2³T(n/12) + 6
7. Write mergesort with pseudo-code and analyze the algorithm’s worst case, best case and average case using asymptotic notations