Question

1. Give an asymptotically tight bound to each of the following expressions:

3n^2 + 2n^3

3n log n + 2n^2

2^n + 3^n

2. Arrange the following asymptotic family from lower order to higher order. The first has been done for you.

O(n log n)

O(n^3)

O(log n)

O(n^2 log n)

O(n)

O(3^n)

O(2^n)

3. At work, Peter needs to solve a problem of different sizes. He has two algorithms available to solve the problem. Algorithm A can solve the problem with size n in time T(n) = n^3 +3n^2+507 milliseconds while Algorithm B can solve the same problem in time T(n)= 100n^2+300n+507 milliseconds. Find the minimum problem size n where Algorithm B runs faster than Algorithm A.

n=

4. True or False:

3n+n^2=O(n)

3n+n^2=O(n^2)

3n+n^2=O(n^3)

Answer 1

1. Give an asymptotically tight bound to each of the following expressions:
3n^2 + 2n^3     =>      O(n^3)
3n log n + 2n^2 =>      O(n^2)
2^n + 3^n       =>      O(3^n)

2. Arrange the following asymptotic family from lower order to higher order. The first has been done for you.
O(n log n)
O(n^3)
O(log n)
O(n^2 log n)
O(n)
O(3^n)
O(2^n)

Answer:
---------
O(log n)
O(n)
O(n log n)
O(n^2 log n)
O(n^3)
O(2^n)
O(3^n)

3. At work, Peter needs to solve a problem of different sizes. He has two algorithms available to solve the problem. Algorithm A can solve the problem with size n in time T(n) = n^3 +3n^2+507 milliseconds while Algorithm B can solve the same problem in time T(n)= 100n^2+300n+507 milliseconds. Find the minimum problem size n where Algorithm B runs faster than Algorithm A.
n=100

4. True or False:
3n+n^2=O(n)     ->  False
3n+n^2=O(n^2)   ->  True
3n+n^2=O(n^3)   ->  True