Description
Problem 1. (10 marks) True or False? Justify your answer briefly.
a. n
3 + 2n
2 ∈ ω(0.1n
3.1 + n)
b. n log5 n ∈ Θ(n
5
4 )
c. 2
n log n ∈ Θ(4n
)
d. For any constant a > 1: a
n+1 ∈ o(n!).
e. √
log n ∈ o(log(√
n))
Problem 2. (10 marks) Order the following list of functions by increasing big-Oh growth rate. Group
together those functions that are big-Theta of one another.
3
√
n 2
2+1/n log n/ log log n log n eln n
2
2
n
4
n n
100 1/ log n 4n
3/2
3
√
log n 5(n + 1/n) n log3 n 2
n
2
log n
log(n!)
n! n
3 n
2
log n 4
log n √
log log n
Problem 3. (10 marks) Prove or disprove each of the following statements:
a. For any functions f, g : N −→ R+, either f ∈ O(g) or g ∈ O(f).
b. For any functions f, g : N −→ R+, if f(n) > g(n) for all n > 0 then f(n) + g(n) ∈ Θ(f(n)).
c. For any functions f, g : N −→ R+, if f(n) ∈ Θ(n) and g(n) ∈ Θ(n) then 2f(n) ∈ Θ(2g(n)
).
d. For any function f : N −→ R+, f(n) ∈ Θ((f(n))2
).
Problem 4. (10 marks) Suppose you are given a set of small boxes, numbered 1 to n, identical in every
aspect except that each of the first i contains a pearl whereas the remaining n − i are empty. In the first
part of this question, you are given two magic wands that can each test if a box is empty or not in a single
touch, except that a wand disappears if you test it on a box that is empty. Show that, without knowing
the value of i, you can use the two wands to determine all the boxes containing pearls using no more than
O(
√
n) wand touches.
a. We here sketch a solution for this part of the question.
We will fix number k later. The idea is first use one wand on boxes 1, k, 2k, 3k, . . . . The smallest i
for which the wand burns on box ik indicates that the first empty box is among (i − 1)k + 1, . . . , ik.
Now we use the second wand sequentially from (i − 1)k + 1 to ik − 1 to find it. The total number of
1
touches will be at most: n/k + k, where n/k is the number of boxes for the first wand and k for the
second.
Now, you are asked to complete this solution by describing how you choose the value of k and
explaining why your choice leads to an algorithm that has the desired efficiency (i.e., to determine
all the boxes containing pearls using no more than O(
√
n) wand touches).
b. Now suppose you are given three magic wands. How would you extend the above algorithm to obtain
an even more efficient one? Describe your algorithm, give an upper bound in running time (in terms
of the number of wand touches) using the big-O notation, and provide a brief analysis.
c. We should all be familiar with binary search: Given a sorted list of items, it works by repeatedly
dividing in half the portion of the list that could contain the item, until you’ve narrowed down the
possible locations to just one. In this question, the given list of boxes can be considered sorted:
non-empty boxes proceed empty boxes. Give an algorithm to find the first empty box in O(log n)
time independent of how many magic wards may be used. Then, answer the following questions: by
your algorithm, in the worst-case how many magic wards are required (express your answer using
the big-O notation)? What about the best-case? Justify your answers briefly.
2