Questions:
1. [8 marks] Prove the following two identities using the limit method. Recall that lgn =
log2 n.
(i) [4 marks] n/lgn = o(n/lglgn).
(ii) [4 marks] √nlgn = o(n/lgn).
Give sufficient details in your solutions. For example, to show lim = 0, give
the steps of applying l’Hopital’s Rule.
2. [12 marks] For each of the following recurrences, use the “master theorem” and give
the solution using big-Θ notation. Explain your reasoning. If the “master theorem” does
not apply to a recurrence, show your reasoning, but you need not give a solution.
You are required to use the version of the master theorem taught in class (also posted
in the online notes), which is the same as the master theorem in the 4th edition of the
textbook. Do NOT use the master theorem in the third edition of the textbook,
which covers fewer cases.
(a) T(n) = 8T(n/2) + n2
(b) T(n) = 16T(n/2) + (n/lgn)
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(c) T(n) = 8T(n/3) + Θ(n2)
(d) T(n) = 3T(⌈n/3⌉) + nlgn
3. [10 marks] A problem database people are often interested in is the following: Consider
a set of hotels of acceptable ratings. For each hotel, we know how far away it is from the
beach and how much a room costs a night. Ideally, we want to book a room in the
cheapest hotel that is closest to the beach, but probably such a hotel will be impossible
to find. Therefore, we are willing to compromise. Whether we are willing to trade
distance from the beach for price depends very much on our personal preferences, but
we would like to avoid booking in a hotel A if there exists a hotel B that is both cheaper
and closer to the beach than hotel A.
For a hotel H, let dH be its distance from the beach and dH its price. Then, according to
the discussion above, we consider a hotel A a candidate for booking a room if there is
no hotel B that satisfies the following conditions:
(a) dB ≤ dA,
(b) pB ≤ pA, and
(c) at least one of these two inequalities is strict.
Given a set S of n hotels, your goal in this assignment is to develop an algorithm that
finds all the candidate hotels in O(nlgn) time using divide-and-conquer, listed in
increasing distance to the beach.
You are not allowed to using any sorting algorithms in your solution.
You can make the following assumptions about the input and the output:
Input: An array S[1..n] of pairs of integers (d,p), in which S[i].d stores the distance of
hotel i to the beach and S[i].p stores the price of hotel i. Note that two hotels may have
the same distances and/or prices.
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Output: An array C of pairs of integers such that its elements C[1],C[2],… give the
distances and prices of all the candidate hotels, listed from the candidate hotel closest
to the beach to the one farthest away from the beach.
Important: In all of the assignments of this course, when you are asked to give an
algorithm for a problem, you are (unless otherwise indicated) expected to
(i) describe the idea behind your algorithm in English;
(ii) provide pseudocode;
(iii) argue that your
algorithm is correct; and
(iv) analyze its running time.. Since these requirements apply
to all the assignments in this course, this reminder will not be repeated for future
assignment questions
