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comp2123 Assignment 5 s1 2021
This assignment is due on May 27 and should be submitted on Gradescope. All
submitted work must be done individually without consulting someone else’s solutions
in accordance with the University’s“Academic Dishonesty and Plagiarism”
policies.
As a first step go to the last page and read the section: Advice on how to do the
assignment.
Problem 1. (10 points)
We are given a rooted tree T = (V, E) and its root u, i.e., a simple connected undirected
acyclic graph and a vertex u of this tree that we consider to be its root. We
want to compute the smallest set of vertices S ⊆ V such that every edge in E is
incident to some vertex in S.
Example:
u
v
w x
Vertex u is the root of the above tree. If we choose S = {v}, we indeed have the
property that every edge is incident to a vertex in S. Furthermore, there’s no smaller
set with this property. Picking S = {u, w, x} would also yield a set where every edge
is incident to a vertex in S, but it isn’t the smallest set. Picking S = {u, w} would
mean that edge (v, x) isn’t incident to any vertex in S.
We are given two algorithms for this problem:
degreeAlgorithm sorts the vertices of T by their degree and adds a vertex to S
if it’s incident to some edge that isn’t incident to any vertex in S.
BFSAlgorithm runs a breadth first search from the root u. Then it compares the
size of the union of all vertices in even layers to the size of the union of all vertices
in odd layers and returns the smaller of the two sets.
1: function degreeAlgorithm(M)
2: S ← ∅
3: Sort vertices in T by their degree and rename such that deg(v1) ≥ deg(v2)
≥ … ≥ deg(vn)
4: for each vi
in T do
5: if vi
is incident to some edge e
6: and e isn’t incident to any vertex in S then
7: S ← S ∪ {vi}
8: return S
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comp2123 Assignment 5 s1 2021
1: function BFSAlgorithm(T, u)
2: layers, parent ← BFS(T, u)
3: even ← the union of all vertices in layers[i], where i is even
4: odd ← the union of all vertices in layers[i], where i is odd
5: if |even| < |odd| then
6: return even
7: else
8: return odd
Argue whether degreeAlgorithm always returns the correct answer by either
arguing its correctness (if you think it’s correct) or by providing a counterexample
(if you think it’s incorrect).
a)
Argue whether BFSAlgorithm always returns the correct answer by either
arguing its correctness (if you think it’s correct) or by providing a counterexample
(if you think it’s incorrect).
b)
Problem 2. (25 points)
Our publishing company has accepted contracts to publish a set of n books B. For
each book bi
(0 ≤ i < n) we know the number of copies ci
that we agreed to
print, the time pi
that printing a single copy of this book takes, and the deadline
di by which all copies of the book should be printed. All ci
, pi
, and di are positive
integers. We can start printing at time 0 and all deadlines are given relative to this,
i.e., if di = x this means that printing all copies of book bi should be completed by
time x.
As we’re a small printing company, we only have a single printing press, so we
can only work on printing one book at a time. We’re allowed to change which book
we’re printing after we completed a full copy (i.e., we can’t switch halfway into
printing a book). We’re worried that we agreed to publish too much and you’ve
been hired to develop an algorithm to decide if this is indeed the case (and we
might need to start apologising to our clients). In other words, you need to design
an algorithm that returns true if there is an order to print the copies of the books
such that all deadlines are met, and false otherwise.
Example:
ci pi di
b1 3 1 5
b2 1 9 18
b3 2 2 7
In this case we can indeed print all copies of all books, for example by first
printing 1 copy of b1, then 1 copy of b3, 2 more copies of b1, 1 more copy of b3, and
finally 1 copy of b2. When printing starts at time 0, the above schedule implies that
b1 finishes printing at time 5, b2 is done at time 16, and b3 is completed at time 7.
In other words all books are printed by their deadline and so our algorithm should
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comp2123 Assignment 5 s1 2021
return true. If we were required to print more than 1 copy of b2, there would no
longer exist a way to print all books by their respective deadlines and thus our
algorithm should return false.
Your task is to give a greedy algorithm for this problem that runs in O(n log n)
time. Remember to:
a) Describe your algorithm in plain English.
b) Prove the correctness of your algorithm.
c) Analyze the time complexity of your algorithm.
Problem 3. (25 points)
We are given an array A of length n with the following property: the first k integers
occur in increasing sorted order, the next 2n/3 integers occur in decreasing sorted
order, and the final n/3 − k + 2 integers occur in increasing sorted order. Note that
the last integer of the increasing sequence is also the first integer of the decreasing
sequence (hence, k ≥ 1), and that the last integer of the decreasing sequence is also
the first integer of the second increasing sequence. You are asked to determine the
minimum and maximum integer stored in this array. For simplicity you can assume
that n is a multiple of 3 and all integers are distinct.
Example:
A = [4, 5, 2, −2, −3, 0]
The first increasing sequence is [4, 5], the decreasing sequence is [5, 2, −2, −3], and
the second increasing sequence is [−3, 0]. The minimum integer in the array is −3
and the maximum is 5.
Your task is to give a divide and conquer algorithm for this problem that runs
in O(log n) time. Remember to:
a) Describe your algorithm in plain English.
b) Prove the correctness of your algorithm.
c) Analyze the time complexity of your algorithm.
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comp2123 Assignment 5 s1 2021
Advice on how to do the assignment
• Assignments should be typed and submitted as pdf (no handwriting).
• Start by typing your student ID at the top of the first page of your submission.
Do not type your name.
• Submit only your answers to the questions. Do not copy the questions.
• When designing an algorithm or data structure, it might help you (and us)
if you briefly describe your general idea, and after that you might want to
develop and elaborate on details. If we don’t see/understand your general
idea, we cannot give you points for it.
• Be careful with giving multiple or alternative answers. If you give multiple
answers, then we will give you marks only for “your worst answer”, as this
indicates how well you understood the question.
• Some of the questions are very easy (with the help of the lecture notes or
book). You can use the material presented in the lecture or book without proving
it. You do not need to write more than necessary (see comment above).
• When giving answers to questions, always prove/explain/motivate your answers.
• When giving an algorithm as an answer, the algorithm does not have to be
given as (pseudo-)code.
• If you do give (pseudo-)code, then you still have to explain your code and
your ideas in plain English.
• Unless otherwise stated, we always ask about worst-case analysis, worst case
running times etc.
• As done in the lecture, and as it is typical for an algorithms course, we are
interested in the most efficient algorithms and data structures.
• If you use further resources (books, scientific papers, the internet,…) to formulate
your answers, then add references to your sources.
• If you refer to a result in a scientific paper or on the web you need to explain
the results to show that you understand the results, and how it was proven.