# Divide and Conquer, Sorting and Searching, and Randomized Algorithms Week 2 | Problem Set #2 Quiz Answer

## Divide and Conquer, Sorting and Searching, and Randomized Algorithms Week 2 | Problem Set #2 Quiz Answer

#### Problem Set #2

**Question 1)**

**This question will give you further practice with the Master Method. Suppose the running time of an algorithm is governed by the recurrence T(n) = 7 ✳ T(n / 3) +n2. What’s the overall asymptotic running time (i.e, the value of T(n))?**

- Ө(n2 log n)
- Ө(n2)
- Ө(n2.81)
- Ө(n log n)

**Question 2)**

**This question will give you further practice with the Master Method. Suppose the running time of an algorithm is governed by the recurrence T(n) = 9✳T(n / 3) +n2. What’s the overall asymptotic running time (i.e, the value of T(n))?**

- Ө(n2 log n)
- Ө(n2)
- Ө(n3.17)
- Ө(n log n)

**Question 3)**

**This question will give you further practice with the Master Method. Suppose the running time of an algorithm is governed by the recurrence T(n) = 5✳T(n / 3) +4n. What’s the overall asymptotic running time (i.e, the value of T(n))?**

- Ө(n2.59)
- Ө(n2)
- Ө(nlog 3 / log 5)
- Ө(n5/3)
- Ө(n log (n))
- Ө(nlog 3 / (5))

**Question 4)**

**Consider the following pseudocode for calculating a where a and bare positive integers) **

FastPower(a,b) :

if b = 1

return a

else

c : = a*a

ans := Fast Power(c,[b/2])

if b is odd

return a*ans

else return ans

end

**Here [x] denotes the floor function, that is, the largest integer less than or equal to x.**

**Now assuming that you use a calculator that supports multiplication and division (Le.. you can do multiplications and divisions in constant time), what would be the overall asymptotic running time of the above algorithm (as a function of b)? **

- Ө(b log(b))
- Ө(log(b))
- Ө(b)
- Ө(√b)

**Question 5)**

**Choose the smallest correct upper bound on the solution to the following recurrence T(1) = 1 and T (n) ≤ T( [√n]) +1 for n > 1. Here [x] denotes the “floor” function, which rounds down to the nearest integer. (Note that the Master Method does not apply.) **

- 0(log n)
- 0(√n)
- 0(1)
- 0(log log n )