[SOLVED] Comp3121/9101 Assignment 1

Question 1
You have recently been hired to work for a large social media company. The company has sta↵
shortages due to a recent firing spree, and you have been tasked with designing the recommendation
algorithm used on the front page! After thinking for several days, you have figured out a way to
measure a user’s popularity, and now need to design an ecient algorithm to calculate it.For a given user with n posts, you have access to an integer array V = [v1, v2,…,vn], where vi
is the number of views the user’s i
th post has in thousands. Since these view counts are used to
order posts on a user’s profile page, they are already sorted, in descending order. The popularity
of a user is the largest value of p such that the user has at least p posts that each have at least p
thousand views.For example, suppose a user has n = 6 posts, and a view count array V = [7, 6, 5, 4, 4, 1]. Then
they have at least 4 posts each with at least 4,000 views (indeed, they have five such posts), so
p = 4 satisfies the criterion. You can confirm that the criterion is also satisfied for p = 3, but it is
not satisfied for p = 5 since there are only three qualifying posts. The user’s popularity index is
in fact 4, because p = 4 is the largest value for which the criterion holds.1.1 [7 marks] Before you are given full access to the codebase, you need to try and implement
your new popularity measure on top of the existing code. Given the integer n and the integer array
V for a user as well as an integer value p  n, you must determine whether that user’s popularity
is at least p. This snippet of code runs every single time the page gets loaded, so it needs to be
very ecient.Design an algorithm that runs in O(1) time and checks whether a user’s popularity is at least
p.1.2 [13 marks] The code you modified was written by a now-fired intern, and would simply try
every value of p until it found one that worked, for every user. Now that the company has grown
to serve millions of users, your boss has demanded that you develop a faster algorithm. Given the
integer n and the integer array V for a user, you must eciently compute the user’s popularity
index p.
Design an algorithm that runs in O(log n) time and computes p for a given user.Question 2
DAC Investments is a finance company that has a team of k bankers. The ith banker manages
M[i] dollars worth of investments, and has a performance rating of P[i]. The bankers are indexed
in decreasing order of performance rating.The profitability of DAC Investments is given by
(M[1] + … + M[k]) ⇥ min(P[1],…,P[k]).
For example, if M = [4, 8, 2, 7, 1] and P = [9, 6, 3, 3, 2], then the profitability is
(4 + 8 + 2 + 7 + 1) ⇥ min(9, 6, 3, 3, 2) = 22 ⇥ 2 = 44.2.1 [2 marks] DAC Investments has recently received a letter from the CEO, advising them
that his nephew is looking for work. Although the nephew’s performance rating is no better than
anyone in the team, DAC Investments have been strongly encouraged to hire him in place of one
of the current team members.You are given integer arrays M[1..(k + 1)] and P[1..(k + 1)], representing the amounts managed
and performance ratings respectively. In each array, the first k entries are for the current team
members and the last entry is for the CEO’s nephew. The performance ratings are listed in
decreasing order.In order to hire the CEO’s nephew, DAC Investments will have to remove one of their bankers.
How should they select which banker to remove, while maximising profitability?For the remaining subquestions, suppose DAC Investments has been acquired by a large bank.
There are now n employees available, so you are given integer arrays M[1..n] and P[1..n], the
latter of which is again in decreasing order. From these n candidates, DAC Investments would like
to select k team members.2.2 [10 marks] Design an algorithm that runs in O(nk) time and determines the set of k employees that maximise the profitability of DAC Investments.2.3 [8 marks] Design an algorithm that runs in O(n log k) time and determines the set of k
employees that maximise profitability.
You may choose to skip 2.2, in which case we will mark your submission for 2.3 as if it was
submitted for 2.2 also.Question 3
A traversal of a binary tree is an order in which one can visit all the nodes.
• In-order traversal first recurses on the left subtree, then visits the root node, and recurses on
the right subtree.• Pre-order traversal first visits the root node, then recurses on the left subtree, and finally
recurses on the right subtree.• Post-order traversal first recurses on the left subtree, then recurses on the right subtree, and
finally visits the root node.For example, the tree given below has:
• in-order traversal of [1, 8, 9, 5, 6, 3, 2],
• pre-order traversal of [5, 8, 1, 9, 3, 6, 2], and
• post-order traversal of [1, 9, 8, 6, 2, 3, 5].
5
8
1 9
3
6 23.1 [8 points] Given an array P that contains the pre-order traversal of a binary search tree
of n nodes with distinct integer values, design an algorithm that runs in O(n) time and computes
the post-order traversal of the tree.
Note that the tree pictured in the example above is not a binary search tree.3.2 [12 points] A binary tree is said to be height-balanced if:
• it has zero or one nodes, or• the heights of its left and right subtrees di↵er by at most one, and both subtrees are heightbalanced.
Given two arrays I and P which contain the in-order and pre-order traversals of a height-balanced
binary tree of n nodes with distinct integer values, design an algorithm that runs in O(n log n)
time and computes the post-order traversal of the tree in O(n log n).