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COMP3600/6466 Algorithms Assignment 2
COMP3600/6466 — Algorithms
Assignment 2
Due: Monday, 30 September 2019 23:59 GMT+10
Notes:
• This assignment consists of two parts. In the second part, you need to write computer programs. You should
write your programs in C/C++/Java. No other programming language is accepted. We will support only C++.
• This assignment comes with two scaffolding, main.cpp and rbbtree template.h, as well as testcases. All can be
downloaded from the class website.
• Please submit both the report and the source codes, following the format below, with [studentID] replaced by
your student ID, e.g., U1234567.
– Please write your report in a single .pdf file, named A2[studentID].pdf .
– Please name the source code that contains your main function for Part B Q2 to be A2[studentID].cpp (or
the suitable extension).
– Please put all your source codes in a single directory, named A2[studentID] .
– Please zip your report and the directory containing your source codes (please exclude the object files and
executables) into a single file, named A2[studentID].zip .
– Please submit the .zip file via Wattle. Please note that the maximum file size you can upload is 200MB.
– Please save your submission as draft, so that you can re-upload it until the last minute. Once the grace
period is over, the last saved draft in wattle will be locked as your submission.
• You can submit a scanned copy of a handwritten report. However, the handwriting must be neat enough for
the teaching staff to read them. If we can’t read them, we will not mark them and therefore, you will get a 0
mark for the report component. Our preference is you type your report. This is a good time to learn how to use
TeX/LaTeX.
• We provide 13 hours grace period. This means, there will be no penalty if you submit before Tuesday, 1 October
2019 13:00 Canberra time. However, we will NOT accept assignment submission beyond this time.
• Assignment marking:
– The total mark you can get in this assignment is 100 points.
– This assignment will contribute 20% to your overall mark.
– This assignment is redeemable. However, we strongly suggest that you do this assignment. The results of
this assignment will help you understand how much you have understood the materials.
• Discussion with your colleagues are allowed and encouraged. However, you still need to work on the assignment
on your own AND write the names you discussed this assignment with.
• You are allowed to use materials from books, slides, etc. as reference. You still need to write the solution
yourself and put a reference to the source. A reference need to be a full citation and specific, e.g., T.H. Cormen,
C.E. Leiserson, R.L. Rivest, and C. Stein. Introduction to Algorithms 3rd Ed. MIT Press. 2009. Sec. 2.2.
Clarifications:
• 10 Sep’19: Add clarification on the requirement to use red-black tree for Part B Q2c, in particular for this
question, your algorithm must:
– Only use red-black tree data structure
– Assume that the maximum time index is a variable (i.e., the complexity requirement should still hold if
the maximum time index is a variable k, which is given as input, rather than 145)
In the implementation of the algorithm (Part B Q2d), the implementation must follow the algorithm you have
written in Part B Q2c. You are only allowed to use other data structure to output the sorted list of session IDs
—that is, the second line of the output.
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COMP3600/6466 Algorithms Assignment 2
• 9 Sep’19: Clarifications in blue text, which consists of:
– Part A, Q1, clarify what k means.
– Part A, Q2, clarifications of x and y and the input to the function, as well as n as the number of nodes in
the tree T.
– Part B, Q2b, n is the number of nodes in the red-black tree that are already in the tree when a node is
inserted.
– Part B, Q2c, n refers to the number of lecture sessions.
– Part B, Program input: The session ID is sorted.
– You can add functionalities in the code we provided with this assignment.
• 29 Aug’19: The ancestor of a node x includes the node x itself.
• 29 Aug’19: Please set the output of your program (for Part B Q2d) to be sorted in ascending order.
[40 points] Part A.
1. [10 points] Mr Bug is known for developing software that are buggy. The latest software he delivers to Mr Fix’s
company is supposed to sort an array of n numbers in ascending order. However, upon using the software, Mr
Fix notices that the software does not sort the numbers exactly correct: It always places the numbers within k
slots of its proper position. This means, suppose the correct position of a number x in the sorted array is at indexi,
then in the output of Mr Bug s/w, the number x can be placed in any index between [max(0, i−k), min(i+k, n)].
Since Mr Bug is no longer reachable, Mr Fix decided to take the output of Mr Bug software and sort them
correctly. Since the data can be any type of data, Mr Fix would like to use comparison-based sorting to sort
the output of Mr Bug’s software. Mr Fix thinks for these “pre-sorted” numbers, the comparison-based sorting
procedure takes Ω(n log k) number of comparisons, rather than Ω(n log n),Is he correct? Please
provide the proof to support or refute Mr Fix argument.
2. [20 points] Let T be a full binary search tree of height h. And, let A(x) of a leaf node x in T be the ancestor set of
x (i.e., it is the union of x’s parent, grandparent, all the way to the root. Here, we follow a common convention
in algorithms that duplicates in union set of tree nodes is checked based on the key and that the ancestor of a
node includes the node itself. Suppose f(x, y) =
P
a∈A(x)∪A(y) a.key. Please:
(a) [10 points] Design an algorithm that can compute f(x, y) in O(n log n) time, where n is the number of
nodes in T.
(b) [10 points] Derive the time complexity of the algorithm you designed in 2a.
Note that when one implements a function that takes x and y as inputs, these inputs become input argument to
the implemented function. To simplify the problem, you can assume each node has a unique key. Of course, if
your algorithm can handle non-unique key, it should be fine too.
Figure 1: An illustration
of the luggage lock for
UNA’s students’ lockers.
3. [10 points] Every student enrolled at UNA is given a locker to store their books.
The locker has a lock mechanism akin to a luggage lock with 3-digit password lock
(Fig. 1 shows an illustration). Suppose the password each lock is set uniformly
at random and independently from the other locks by the locker administrator, and
suppose this password cannot be changed. The administrator will improve this lock
mechanism once the number of enrolled students at UNA is large enough that there’s
more than 50% chance that two UNA students have the same password lock. How
many enrolled students should UNA has before the administrator improve the lock
mechanism? Please explain your answer.
[60 points] Part B.
Finding a parking spot during semester time is a big problem at UNA, and Mr ParkingPls is determined to address
this problem. To this end, he observes that the difficulty in finding a parking spot is worst on a certain day of the
week and a certain time on that day. He hypothesises that the main culprit of the worst parking problem is that there
is an extremely large number of overlapping lecture sessions on that particular day and time. If this is true, he can
recommend that UNA either redistribute the lecture slots more evenly during the week or rent a nearby land on the
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COMP3600/6466 Algorithms Assignment 2
particular day and time each week to expand its parking capacity.
Of course first, Mr ParkingPls must test his hypothesis. To this end, he has obtained the schedule of all lectures
at UNA. Since all classes at UNA either starts or ends at the beginning of an hour or half an hour, Mr ParkingPls
simplifies the lecture sessions representation by creating a time index for every half-hour between 06:00 and 20:00
(inclusive) within a week —that is, time-1 refers to Monday 06:00, time-2 refers to Monday 06:30, · · ·, time-145
refers to Friday 20:00—. He then represents each lecture session as a half-open interval of time index, e.g., interval
[3, 5) means the lecture starts on Monday 07:00 end ends (at a negligible amount) before Monday 08:00. The question
is now to find the time index with the most number of overlapping lecture sessions. Now, since he has only limited
knowledge about Algorithms, he asks your help.
Your tasks
1. [15 pts] Mr ParkingPls is convinced that to find the time index with the most overlapping lecture sessions, one
only needs to find overlaps between time indices at the beginning or end of the intervals, rather the entire time
index. The question is:
(a) [5 pts] Is Mr ParkingPls correct?
(b) [10 pts] If Mr ParkingPls is correct, please support your answer with a proof. Otherwise, please provide a
counter-example.
2. [45 pts] Since Mr ParkingPls would like to use this work for other much larger projects where new sessions
can be added quite often, Mr ParkingPls would like to use Red-Black Tree and asked your help. To help him,
please:
(a) [10 pts] Design a suitable Red-Black Tree data structure for Mr ParkingPls’ problem. This means that
given a Red-Black Tree implementation as defined in [CLRS] ch. 13, how you would use/augment/modify
this underlying data structure for Mr ParkingPls problem. Hint: You might want to read [CLRS] sec. 14.3
first (provided as additional resources to this assignment).
(b) [5 pts] Please provide an algorithm with O(log n) complexity for inserting a node to the tree you have
designed in 2a, along with the derivation of its complexity results. The notation n refers to the number of
nodes that are already in the tree prior to insertion. For the algorithm, you can use the insertion procedure
of Red-Black Tree in [CLRS] ch. 13 as a basis and stated only the modifications needed, if alterations are
needed. If no alteration is needed, please explain why.
(c) [15 pts] Please provide an algorithm for finding the time index with the largest number of overlapping
lecture sessions in the red-black tree you designed. Please also provide the correctness proof (using loop
invariant) and the time complexity (with its derivation) of your algorithms. You will get full mark if the
time complexity of your algorithm is O(1) and at most 9 points if the time complexity of your algorithm
is O(log n), where n is the number of lecture sessions. Please note that in this question:
• You can only use red-black tree data structure
• You need to assume that the maximum time index is a variable (i.e., the complexity requirement
should still hold if the maximum time index is a variable k, which is given as input, rather than 145)
because, Mr ParkingPls would like to use the algorithm for other parking issues
(d) [15 pts] Please implement the above solution such that the user can find the time index by giving the
command “A2[studentID] input file name” OR “java A2[studentID] input file name” from the command
prompt. The input format is described in the next section. We provide you with a main.cpp and a scaffolding
for Red-Black Tree. If you use C/C++, you need to implement the node insertion function and the
algorithm you provided in 2c on top of the provided main function and Red-Black Tree scaffolding.
If you use java, you need to make sure that your java program accepts the same input as the provided
main.cpp . You also need to develop Red-Black Tree yourself. If you choose this path, you might want
to first check which operations of Red-Black Tree you need for this assignment and implement only those
operations. Note: You are not allowed to use TreeMap or other Java Red-Black Tree library.
Program Marking: If your program compiles and runs, you will get 3 points. We will then run your
program on 6 test cases: 2 cases would have up to 5,000 sessions, 2 cases would have 5,001 – 500,000
sessions, and 2 cases would have 500,001 – 5,000,000 sessions. For each test case, your program will be
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COMP3600/6466 Algorithms Assignment 2
given a total of (log n + 0.01)sec CPU time to find a solution. This time limit includes the time for reading
the file, inserting the data to the red-black tree, finding the solution, and printing the solution. The time
limit will be rounded up to 2 decimal digit. You can assume your program will have access to at most
12GB RAM. It will be run as a single thread process on a computer with Intel i7 3.4GHz processor. For
each test case that your program solves correctly within the given time and memory limit, you will get 2
points.
Examples of the test cases are available in
https://cs.anu.edu.au/courses/comp3600/a2-testCases.zip.
Input to the Program
The program will accept a single argument, which is the name of the input file. The input file contains N + 1 lines,
where N is the number of lecture session intervals.
The first line consists of a single number, which is N.
Each line in the next N lines consists of three numbers, separated by a white space. The first number is the session
ID, the second number is starting time index, and the third number is the ending time index.
The session ID is sorted, in the sense that ID-i would be in line-(i + 1).
Example:
Output of the Program
The program outputs two lines to the standard output stream (i.e., use cout if you use C++).
The first line contains two numbers, each separated by a white space. The first number is the time index with the most
number of overlapping lecture sessions. Let’s denote this time index as t. The second number (denoted as K) is the
number of lecture sessions that overlap with time index t.
The second line consists of K numbers sorted in ascending order, each separated by a white space. Each number in
this line is the ID of the session that overlaps with time index t.
Example output for the above input:

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