Quiz on Divide And Conquer Answers with Explanation wherever possible
Q1) Let T(n)= T(n-1) + 1/n. Then T(n) is:
a. Ѳ (1)
b. Ѳ (log n)
c. Ѳ (log log n)
d. Ѳ (n)
Answer is b O(logN)
Explanation: By unfolding
T(n)=T(n−1)+1n=T(n−2)+1n+1n−1=⋯=T(0)+∑k=1n1kT(n)=T(n−1)+1n=T(n−2)+1n+1n−1=⋯=T(0)+∑k
=1n1k
Now we can easily approximate the sum on the RHS using that
∑k=1n1k≤1+∫n11xdx=1+logn−log1=1+logn∑k=1n1k≤1+∫1n1xdx=1+logn−log1=1+logn
Therefore T(n)≡O(logn)
Q2) Which of the following algorithms is NOT a divide & conquer algorithm by nature?
a. Heap Sort
b. Euclidean algorithm to compute the greatest common divisor
c. Cooley-Tukey fast Fourier transform
d. Quick Sort
Answer is a) Heap Sort
Q3) Consider the following function
find (int n)
{
if (n < 2 ) then return;
else
{
sum= 0;
for (i= 1; i ≤ 4; i++) find (n2);
for (i=1; i≤ n*n; i++) sum= sum + 1;
}
}
Assume that the division operation takes constant time and “sum” is global variable. What is the
time complexity of “find (n)” ?
a. n^2
b. n^3
c. n^2logn
d. None of these
