Problem Set 2

Due before the start of lecture on October 6, 2025.

Preliminaries

Homework is due prior to the start of lecture. If it is submitted more than 10 minutes after the start of lecture, it will be considered a full day late. There will be a 10% deduction for late submissions that are made by 11:59 p.m. Eastern time on the Sunday after the deadline, and a 20% deduction for submissions that are made after that Sunday and before the start of the next lecture. We will not accept any homework that is more than 7 days late. Plan your time carefully, and don’t wait until the last minute to begin an assignment. Starting early will give you ample time to ask questions and obtain assistance.

In your work on this assignment, make sure to abide by our policies on academic conduct.

If you have questions while working on this assignment, please come to office hours, post them on Ed Discussion, or email cscie22-staff@lists.fas.harvard.edu

Part I

55-65 points total

Creating the necessary folder

1. If you haven’t already created a folder named e22 for your work in this course, follow these instructions to do so.

2. Then create a subfolder called ps2 within your e22 folder, and put all of the files for this assignment in that folder.

Creating the necessary file

The problems from Part I will all be completed in a single PDF file. To create it, you should do the following:

1. Access the template that we have created by clicking on this link and signing into your Google account as needed.

2. When asked, click on the Make a copy button, which will save a copy of the template file to your Google Drive.

3. Select File->Rename, and change the name of the file to ps2_partI.

4. Add your work for the problems from Part I to this file.

5. Once you have completed all of these problems, choose File->Download->PDF document, and save the PDF file on your machine. The resulting PDF file (ps2_partI.pdf) is the one that you will submit. See the submission guidelines at the end of Part I.

Problem 1 Sorting practice

14 points; 2 points for each part

Given the following array:

{10, 18, 4, 24, 33, 40, 8, 3, 12}

1. If the array were sorted using selection sort, what would the array look like after the second pass of the algorithm (i.e., after the second time that the algorithm performs a pass or partial pass through the elements of the array)?

2. If the array were sorted using insertion sort, what would the array look like after the fourth iteration of the outer loop of the algorithm?

3. If the array were sorted using Shell sort, what would the array look like after the initial phase of the algorithm, if you assume that it uses an increment of 3? (The method presented in lecture would start with an increment of 7, but you should assume that it uses an increment of 3 instead.)

4. If the array were sorted using bubble sort, what would the array look like after the third pass of the algorithm?

5. If the array were sorted using the version of quicksort presented in lecture, what would the array look like after the initial partitioning phase?

6. If the array were sorted using radix sort, what would the array look like after the initial pass of the algorithm?

7. If the array were sorted using the version of mergesort presented in lecture, what would the array look like after the completion of the fourth call to the merge() method—the method that merges two subarrays? Note: the merge method is the helper method; is not the recursive mSort method.

Problem 2 Counting comparisons

6 points total; 2 points each part

Given an already sorted array of 5 elements, how many comparisons of array elements would each of the following algorithms perform?

1. insertion sort

2. bubble sort

3. quicksort

Give an exact number of comparisons for each algorithm, and explain each answer briefly.

Problem 3 Comparing two algorithms

9 points total; 4.5 points each part

Suppose that you want to count the number of duplicates in an unsorted array of n elements. A duplicate is an element that appears multiple times; if a given element appears x times, x - 1 of them are considered duplicates. For example, consider the following array:

{10, 6, 2, 5, 6, 6, 8, 10, 5}

It includes four duplicates: one extra 10, two extra 6s, and one extra 5.

Below are two algorithms for counting duplicates in an array of integers:

Algorithm A:

public static int numDupsA(int[] arr) {
    Sort.mergesort(arr);
    int numDups = 0;
    for (int i = 1; i < arr.length; i++) {
        if (arr[i] == arr[i - 1]) {
            numDups++;
        }
    }
    return numDups;
}

Algorithm B:

public static int numDupsB(int[] arr) {
    int numDups = 0;
    for (int i = 0; i < arr.length - 1; i++) {
        for (int j = i + 1; j < arr.length; j++) {
            if (arr[j] == arr[i]) {
                numDups++;
                break;
            }
        }
    }
    return numDups;
}

1. What is the worst-case time efficiency of algorithm A in terms of the length n of the array? Make use of big-O notation and explain your answer briefly.

2. What is the worst-case time efficiency of algorithm B in terms of the length n of the array? Make use of big-O notation and explain your answer briefly.

Problem 4 Sum generator

9 points total; 4.5 points each part

Let’s say that you want to implement a method generateSums(n) that takes an integer n and generates and prints the following series of sums:

1
1 + 2
1 + 2 + 3
...
1 + 2 + ... + n.

For example, generateSums(4) should print the following:

1
3
6
10

One possible implementation of this method is:

public static void generateSums(int n) {
    for (int i = 1; i <= n; i++) {
        int sum = 0;
        for (int j = 1; j <= i; j++) {
            sum = sum + j;   // how many times is this executed?
        }
        System.out.println(sum);
    }
}

1. Derive an exact formula for the number of times that the line that increases the sum is executed, as a function of the parameter n.

2. What is the time efficiency of the method shown above as a function of the parameter n? Use big-O notation, and explain your answer briefly.

Problem 5 Improving generateSums()

10 points total; required for grad-credit students; may be completed for “partial” extra credit by others.

1. (7 points) Create an alternative, non-recursive implementation of the generateSums() method that has a better time efficiency than the method from Problem 4. Make your implementation as efficient as possible. Put your code in your ps2_partI file.

2. (3 points) What is the time efficiency of your alternative implementation as a function of the parameter n? Use big-O notation, and explain your answer briefly.

Problem 6 Practice with references

17 points total

As discussed in lecture, a doubly linked list consists of nodes that include two references: one called next to the next node in the linked list, and one called prev to the previous node in the linked list. The first node in such a list has a prev field whose value is null, and the last node has a next field whose value is null.

The top portion of the diagram below shows a doubly linked list of characters that could be used to represent the string "ran".

Each of the nodes shown is an instance of the following class:

public class DNode {
    private char ch;
    private DNode next;
    private DNode prev;
}

(In the diagram, we have labeled the individual fields of the DNode object that contains the character 'r'.)

In addition to the list representing "ran", the diagram shows an extra node containing the character 'i', and two reference variables: n, which holds a reference to the third node in the list (the 'n' node); and x, which holds a reference to the 'i' node. The diagram also shows memory addresses of the start of the variables and objects. For example, the 'r' node begins at address 0x360.

1. (12 points) Complete the table we have provided in ps2_partI, filling in the address and value of each expression from the left-hand column. You should assume the following:

   • the address of the ch field of a DNode is the same as the address of the DNode itself

   • the address of the next field of a DNode is 2 more than the address of the DNode itself

   • the address of the prev field of a DNode is 6 more than the address of the DNode itself, which means that it is also 4 more than the address of the next field.

2. (4 points) Write a Java code fragment that inserts the 'i' node between the 'a' node and the 'n' node, producing a linked list that represents the string "rain". Your code fragment should consist of a series of assignment statements. You should not make any method calls, and you should not use any variables other than the ones provided in the diagram. Make sure that the resulting doubly linked list has correct values for the next and prev fields in all nodes.

   Testing your code fragment
   You should start by convincing yourself on paper that your code is correct. Draw the necessary diagrams and trace through your code, making sure that it works correctly.

   Once you have checked your work on paper, you can make use of a simple version of the DNode class that we have created.

   To use it, you should click on this link and save the file in your ps2 folder.

   Open the ps2 folder in VSCodium, and you should see a simple DNode class that includes:

   • the definitions of the fields

   • a DNode constructor

   • a convert method that can be used to convert a Java String object to a doubly-linked list of DNode objects

   • a toString() method that allows us to print a DNode object and see the corresponding string

   • a main method with some initial test code – including code that sets up the initial diagram that we have provided above.

   To test your code fragment, you can add your lines of code to the specified section of the main method, and compile and run the program to see if you get the correct result.

   The output of the program that we have provided should be:

   before changes for 6-2: ran
   after changes for 6-2: rain
   

Submitting your work for Part I

Submit your ps2_partI.pdf file by taking the following steps:

1. If you still need to create a PDF file, open your file on Google Drive, choose File->Download->PDF document, and save the PDF file on your machine.

2. Click on the name of the assignment in the list of assignments on Gradescope. You should see a pop-up window labeled Submit Assignment. (If you don’t see it, click the Submit or Resubmit button at the bottom of the page.)

3. Choose the Submit PDF option, and then click the Select PDF button and find the PDF file that you created. Then click the Upload PDF button.

4. You should see a question outline along with thumbnails of the pages from your uploaded PDF. For each question in the outline:

   • Click the title of the question.
   • Click the page(s) on which your work for that question can be found.

   As you do so, click on the magnifying glass icon for each page and doublecheck that the pages that you see contain the work that you want us to grade.

5. Once you have assigned pages to all of the problems in the question outline, click the Submit button in the lower-right corner of the window. You should see a box saying that your submission was successful.

Part II

45 points total

Problem 7 Searching an array for pairs that sum to a specified value

25 points

Getting started

1. If you haven’t already done so, create a folder named ps2 for your work on this assignment. You can find instructions for doing so here.

2. Download our Sort class from lecture, making sure to save it in your ps2 folder:
   Sort.java

   Make sure to put the file in your ps2 folder. If your browser doesn’t allow you to specify where the file should be saved, try right-clicking on the link above and choosing Save as… or Save link as…, which should produce a dialog box that allows you to choose the correct folder for the file.

3. In VSCodium, select the File->Open Folder menu option, and use the resulting dialog box to find and open your ps2 folder. (Note: You must open the folder; it is not sufficient to simply open the file.)

4. Select File->New File, which will open up an empty editor window.

5. Select File->Save, and give the new file the name Problem7.java.

6. In the new file, create a class called Problem7.

Your tasks

1. In your Problem7 class, implement a method with the following header

   public static void pairSums(int k, int[] arr)
   

   It should take an integer k and an already sorted array of integers arr, and it should find and print all pairs of values in arr (if any) that sum to k. For example, if k is 12 and the array is {4, 5, 5, 5, 6, 7, 7, 8, 10, 15}, your code should output something like the following:

   4 + 8 = 12
   5 + 7 = 12
   5 + 7 = 12
   

   Note that we get two 5 + 7 sums because both 5 and 7 appear twice in the array. However, while the method that you write may print a given pair of values more than once in such cases, it is not necessary to do so. In addition, the order in which the sums (and the terms within each sum) are printed does not matter.

   For full credit, your method must be as efficient as possible. See below for more details.

   Important notes:

   • Your method should have a running time of O(n), where n is the length of the array. In addition, your method should use O(1) additional memory—i.e., it should not create and use a second array.

   • One inefficient approach would be to use nested loops that pair each element in the array with all elements that come after it. This approach would give the method a running time that is O(n²). Only half credit will be given for methods that have a time efficiency that is worse than O(n).

   • You might want to draw some inspiration from the approach quicksort takes in its partition() method – starting with two indices outside the array, and moving them gradually towards the center.

   • If arr is not null, you may assume that it is a sorted array of 0 or more integers.

   • If the parameter arr is null or is a reference to an array of length 0 or 1, your method should simply return.

2. Include a main method with code that tests your new method.

   In addition to the case shown above, you should include at least one other test case that you create.

Problem 8 Finding the union of two arrays

20 points

Getting started

1. As needed, in VSCodium, select the File->Open Folder menu option, and use the resulting dialog box to find and open your ps2 folder. (Note: You must open the folder; it is not sufficient to simply open the file.)

2. Select File->New File, which will open up an empty editor window.

3. Select File->Save, and give the new file the name Problem8.java.

4. In the new file, create a class called Problem8.

Your task

In your Problem8 class, implement a method with the following header

public static int[] union(int[] a1, int[] a2)

It should take two arrays of integers a1 and a2, and it should use an approach based on merging to create and return a new array containing the union of the values in a1 and a2 – i.e., all values that are found in one or both of the original arrays. The result should be in sorted order, and it should not include any duplicate values. For example, if you add the following test code to a main method in Problem8:

int[] a1 = {10, 5, 7, 5, 9, 4};
int[] a2 = {7, 5, 15, 7, 7, 9, 10};
int[] result1 = union(a1, a2);
System.out.println(Arrays.toString(result1));

int[] a3 = {0, 2, -4, 6, 10, 8};
int[] a4 = {12, 0, -4, 8};
int[] result2 = union(a3, a4);
System.out.println(Arrays.toString(result2));

it should display:

[4, 5, 7, 9, 10, 15, 0, 0, 0, 0, 0, 0, 0]
[-4, 0, 2, 6, 8, 10, 12, 0, 0, 0]

(Note that we end up with extra 0s at the end of both of these result arrays for reasons that are discussed below.)

For full credit, your method must be as efficient as possible. See below for more details.

Required approach

1. Begin by creating a new array for the union, giving it a length that is the sum of the lengths of the two original arrays.

2. Use one of the more efficient sorting algorithms from our Sort class to sort both of the original arrays. If you have downloaded our Sort class into the same folder as your Problem8 class (see the Getting started section at the start of Problem 7), you should be able to make method calls to an appropriate method in the Sort class from within your union method.

3. Find the union of the two arrays by employing an approach that is similar to the one that we used to merge two sorted subarrays (i.e., the approach taken by the merge method in Sort.java)—using indices to “walk down” the two original arrays and copying their values into the array that you created for the union.

   Important notes:

   • One important difference between merging and finding the union is that the union should not include any duplicates. In other words, a given number that appears in one or both of the original arrays should appear exactly once in the union. As a result, you will need to include code that skips over duplicate values as needed as you walk down the original arrays.

   • Your algorithm should be as efficient as possible. In particular, you should perform at most one complete pass through each of the arrays.

   • If either a1 or a2 is null, the method should throw an IllegalArgumentException.

4. At the end of the method, return a reference to the array containing the union.

5. Add test code for your method to a main method. Recall that that you can use the Arrays.toString() method to convert an array to a string; import the java.util. package to gain access to the Arrays class. In addition to the cases shown above, you should include at least one other test case that you create.

Other notes:

• In both of the tests shown above, the array containing the union has a length that is the sum of the lengths of the original arrays.

• Because we don’t include duplicates in the result, the number of values in the union can be smaller than the length of the results array (which should be the sum of the lengths of the original arrays). In such cases, you should end up with extra 0s at the end of the results array as shown in both of our test cases above.

• If 0 appears in one or both of the original arrays, it will also show up earlier in the results array as part of the union, as it does in our second example above.

Submitting your work for Part II

You should submit only the following files:

• Problem7.java
• Problem8.java

You do not need to submit Sort.java.

Here are the steps:

1. Click on the name of the assignment in the list of assignments. You should see a pop-up window with a box labeled DRAG & DROP. (If you don’t see it, click the Submit or Resubmit button at the bottom of the page.)

2. Add both files to the box labeled DRAG & DROP. You can either drag and drop the files from their folder into the box, or you can click on the box itself and browse for the files.

3. Click the Upload button.

4. You should see a box saying that your submission was successful. Click the (x) button to close that box.

5. The Autograder will perform some tests on your files. Once it is done, check the results to ensure that the tests were passed. If one or more of the tests did not pass, the name of that test will be in red, and there should be a message describing the failure. Based on those messages, make any necessary changes. Feel free to ask a staff member for help.

   Note: You will not see a complete Autograder score when you submit. That is because additional tests will be run later, after the final deadline for the submission has passed. For such problems, it is important to realize that passing all of the initial tests does not necessarily mean that you will ultimately get full credit on the problem. You should always run your own tests to convince yourself that the logic of your solutions is correct.

6. If needed, use the Resubmit button at the bottom of the page to resubmit your work. Important: Every time that you make a submission, you should submit all of the files for that Gradescope assignment, even if some of them have not changed since your last submission.

7. Near the top of the page, click on the box labeled Code. Then click on the name of each file to view its contents. Check to make sure that you see the code that you want us to grade.