# Assignment 1 Solution

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## Description

Questions

Exercise 1 (80 points) We want to compare the performance of hash tables implemented using chaining and open addressing. In this assignment, we will consider hash tables implemented using the multipli-cation and linear probing methods. We will (respectively) call the hash functions h and g and describe them below. Note that we are using the hash function h to define g.

 Collisions solved by chaining (multiplication method): w >> w r h(k) = ((A k) mod 2 ) r ( ) Open addressing (linear probing): g(k; i) = (h(k) + i) mod 2

In the formula above, r and w are two integers such that w > r, and A is a random number such that 2w 1 < A < 2w. In addition, let n be the number of keys inserted, and m the number of slots in the hash tables. Here, we set m = 2r and r = dw=2e. The load factor is equal to mn .

We want to estimate the number of collisions in random sequences of insertions and deletions of keys with respect to the choice of values for w and . By default we will set

We provide you a set of three template files within COMP251HW1.zip that you will complete. This file contains three classes, a main class and one for each hash function. Those contain several helper functions, namely generateRandom that enables you to generate a random number within a specified range. Details on which functions are included, how to use them, and where to add in your code can be found as comments in the java files. Please read them with attention. In addition, we provide you a jar file to visualize your results named JavaPlotBuilder.jar.

Your first task is to complete the two java methods Open_Addressing.probe and Chaining.chain. These methods must implement the hash functions for (respectively) the linear probing and multiplica-

tion methods. They take as input a key k, as well as an integer 0 i < m for the linear probing method, and return a hash value in [0; m[. Note that the value of A must be updated when you change w.

Next, you will implement the method insertKey in both classes, which inserts a key k into the hash table and returns the number of collisions encountered before insertion. Note that for this exercise

as well as for the rest of the homework, we define the number of collisions as the number of keys en-countered, or “jumped over” before inserting or removing a key. You can assume the key is not negative.

You will also implement a method removeKey, this one only in Open_Addressing. This method should take as input a key k, and remove it from the hash table while visiting the minimum number of slots possible. Like insertKey, it should output the number of collisions. If the key is not in the hash table, the method should simply not change the hash table, and output the number of slots visited. You will notice from the code and comments that empty slots are given a value of 1. If appli-cable, you are allowed to use a different notation of your choice for slots containing a deleted element.

Finally, you will complete the method main.main, which calls the previous functions from the main class. There are three tasks to complete within this main method.

First, you will test the effect of increasing the number of keys on the average number of collisions for each hash function. You will be given an array of keys to insert, keysToInsert, and a list of values of n to test, nList inside the main method. Random seeds can be used in java in order to make ran-dom results reproducible (and eventually evaluate assignments). You will find the hash tables already initialized with such seed. For each value of n, insert the n first elements of keysToInsert into each hash table, and store the value, as well as the average number of collisions for that value of into the appropriate provided list. The program is already set up to output a CSV file for you to visualize.

Your second task is to test the removeKey method on the Open_Addressing table from task 1 with n=16. Initialize a new Open_Addressing hash table with the same seed as in Task 1, and in-sert the first 16 elements of keysToInsert. You will be given an array of keys to remove named keysToRemove. Call removeKey with each of these keys. Store the number of collisions associated with each removal operation in the arraylist removeCollisions, as well as the index of the key you just attempted to remove in removeIndex. Finally, use the provided method to output a CSV file.

Your third task is to evaluate the effect of varying w on the number of collisions for each method. For this exercise, the keys you insert will be generated randomly with generateRandom. Each key can be inserted only once (i.e. The random sequence of keys must have no duplicates). For this exercise, you will not be using a specific seed, which you can do by calling functions that require a seed argument with a seed of 1. Because your experiments will now have some variance, you will need to execute 10 simulations for each value of w to obtain representative averages. You will choose appropriate values of w, use the provided function to output a CSV file, and visualize your results. You will submit a pdf file, called Conclusions.pdf, including plots of your results, as well as a short explanation of what you observe, and an explanation for it.

To plot your results, you can use JavaPlotBuilder.jar to generate a plot with on the x-axis and the average number of collision on the y-axis. JavaPlotBuilder.jar is run from command line as follows:

java -jar JavaPlotBuilder.jar filename.csv

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where filename.csv is the name of the file that is generated by the main method.

For this assignment, you will need to submit a zip file containing the completed version of the three provided java files, n_comparison.csv, remove_collisions.csv, and w_comparison.csv (the three CSV files generated by the main method), as well as Conclusions.pdf, the PDF file with your observations and explanations.

Once you have submitted your files, you can proceed to the second part of the assignment.

Exercise 2 (20 points) This section is answerable through MyCourses. Note that you MUST use your own results to answer those questions. Answers to this quiz that would not match the results presented in your pdf file will be considered plagiarism (refer to course outline).

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