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 )