COMP3600/6466 — Algorithms

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 rbbtreetemplate.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 assign- ment 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 variablek, 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.

• 9 Sep’19: Clarifications in blue text, which consists of:

  • Part A, Q1, clarify whatkmeans.
  • Part A, Q2, clarifications ofxandyand the input to the function, as well asnas the number of nodes in the treeT.
  • Part B, Q2b,nis the number of nodes in the red-black tree that are already in the tree when a node is inserted.
  • Part B, Q2c,nrefers 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 nodexincludes the nodexitself.

• 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 ofnnumbers 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 withink slots of its proper position. This means, suppose the correct position of a numberxin the sorted array is at index- i, then in the output of Mr Bug s/w, the numberxcan 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 takesΩ(nlogk)number of comparisons, rather thanΩ(nlogn), wherek≤n 2. Is he correct? Please provide the proof to support or refute Mr Fix argument.
2. [20 points] LetTbe a full binary search tree of heighth. And, letA(x)of a leaf nodexinTbe the ancestor set of x(i.e., it is the union ofx’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. Supposef(x,y)=

∑

a∈A(x)∪A(y)a.key. Please:
(a) [10 points] Design an algorithm that can compute f(x,y)inO(nlogn)time, wherenis the number of
nodes inT.
(b) [10 points] Derive the time complexity of the algorithm you designed in 2a.
Note that when one implements a function that takesxandyas 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

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: (inclusive) within a week —that is, time-1 refers to Monday 06:00, time-2 refers to Monday 06:30,···, time- 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. first (provided as additional resources to this assignment). (b) [5 pts] Please provide an algorithm withO(logn)complexity for inserting a node to the tree you have designed in 2a, along with the derivation of its complexity results. The notationnrefers 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 isO(1)and at most 9 points if the time complexity of your algorithm isO(logn), wherenis 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 variablek, 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] inputfilename” OR “java A2[studentID] inputfilename” from the command prompt. The input format is described in the next section. We provide you with a main.cpp and a scaf- folding 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, sessions, and 2 cases would have 500,001 – 5,000,000 sessions. For each test case, your program will be

given a total of(logn+ 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 containsN+ 1 lines, whereNis the number of lecture session intervals. The first line consists of a single number, which isN. Each line in the nextNlines 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-iwould be in line-(i+1). Example: 5 1 96 101 2 46 49 3 76 77 4 8 10 5 4 10

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 ast. The second number (denoted asK) is the number of lecture sessions that overlap with time indext. The second line consists ofKnumbers 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 indext.

Example output for the above input: 8 2 4 5