## Description

1. Purpose The purpose of this assignment is to implement priority queues. 2. Description 2.1. Implementation (70 points) Your main task is to implement two priority queues. All two implementations should implement the provided PriorityQueue interface (include implements PriorityQueue in your Java code), which means they should work with priorities that have type double and there are no corresponding items attached to the priorities. Your implementations should be as follows: â€¢ A class BinaryHeap that implements a binary min-heap as we discussed in class, using an array to store the conceptual complete tree. â€¢ A class ThreeHeap that implements a min-heap where each non-leaf node has 3 children. You should still use a contiguous portion of an array to store the conceptual complete tree. We suggest you make a copy of your BinaryHeap class and make changes as necessary. Put your two implementations in two separate Java files, BinaryHeap.java, ThreeHeap.java, andMyPQ.java. Your priority queues should allow duplicates. That is, two or more copies of the same value should be allowed to exist in the heap at the same time. For example, if you call deleteMin and you have {3.0, 3.0, 6.0, 7.0} in the heap, it would just return one of the 3.0 values, then on the next deleteMin it would return the other 3.0. It does not matter “which” 3.0 is returned first. According to our definition of priority queue, what must be guaranteed is that both 3.0 values will be returned before a 6.0 or 7.0 is returned, and that the 6.0 would be returned before the 7.0. Your implementations should automatically grow as necessary. (If interested, you may also have them shrink when appropriate; this is optional.) For any arrays, you should start with a small array (say, 10 elements) and resize to use an array twice as large whenever the array becomes full, copying over the elements in the smaller array. Do the copying with a for loop rather than any Java library methods (even though using the library is how one would normally do it). You may use the length field of an array as needed. Be sure to test your solutions thoroughly and to turn in your testing code. Part of the grading will involve thorough testing including any difficult cases. For this assignment, we will be grading more strictly for things like style and efficiency than we did on Homework 1. However, your your MyPQ implementation does not need to be more efficient than a good array or linked-list implementation if that is your approach. 2.2. Questions (30 points) The questions include comparing the actual run-time of your implementations. We would expect the reports to be at least a couple of pages long, quite possibly longer to have room for relevant graphs or tables. Submit a report.pdf file, answering the questions in this template report.docx file: 1. What is the worst case asymptotic running time of isEmpty, size, insert, findMin, and deleteMin operations on all your heap implementations? For this analysis you should ignore the cost of growing the array. That is, assume that you have enough space when you are inserting a value. 2. Timing your code: Perform several timing experiments (similar to what you did in Homework 2, where you timed pieces of code), to examine the running time of both of your heap implementations. An experiment will include running the same client code (that uses a Priority Queue) for your two different heap implementations for at least four different values of N and timing this. It is up to you to write and to determine what this client code should be. Just be sure that it exercises your insert and deleteMin operations in a reasonable manner, including eventually deleting everything that has been inserted into the heap. You are not required to explicitly measure calls to findMin, size, and isEmpty but feel free to do so if interested. Similar to Homework 2, graphing your results is recommended, but a table of results will work also. Please note that similar to Homework 2, you are required to turn in the code you used to do your timing experiments. 3. Compare what you see in your experiments, to what you expected to see based on a big-O analysis. (This is also similar to what you did in Homework 2.) In your discussion, answer these questions: a. How useful was the asymptotic analysis for predicting the measured run time of insert and deleteMin for your two implementations? b. If your predictions differed substantially from your measured times, gives reasons why this might have occurred. c. Which of your two implementations would you recommend to someone who needs to use a heap? Why is that one preferred? Are there any conditions under which you might suggest using your other implementations? 4. Briefly discuss how you went about testing your two heap implementations. Feel free to refer to your testing files, which you should submit. 5. You implemented a binary-heap and a three-heap. Now think if you can implement a four-heap, a five-heap, etc. a. In a short table, indicate for a binary heap, a three-heap, a four-heap and a five-heap, where the children for the node at array index i are. For example, the first row of the table would indicate that for a binary heap, the two children would be at i*2 and i*2+1. b. For a d-heap where d is a variable representing the number of children (like two, three, four, five, …), give an arithmetic formula for calculating where the left-most child for the node at array index i are. For example, a wrong answer in the right format would be i*d+14. Naturally, your formula should produce the right answer for all the rows in your table from part (a). 2.3. Bonus Components (10 points) The following suggestion is meant for you to try if you finish the requirements early. 1. (5pts) Implement a d-heap where d is the number of children for non-leaf nodes. Your class should implement the same priority queue interface and it should use a contiguous array portion as in your first two implementations. It should include an empty constructor and additional constructor that takes d as an argument, work correctly for any d greater than or equal to 2, and use d as the number of children for nodes. 2. (5pts) Implement a binary heap that works for any type (not just doubles). It should use Java generic types to allow any priority type that implements an appropriate interface for comparing two priorities and your heap should allow items of a second generic type that are “attached” to each priority. That is, each node contains a key-value pair, where key is the priority. Note this implementation will not implement the provided interface, so provide any additional comments necessary to explain how your class should be used. 3. Grading notes If your program does not compile, you receive zero points for that program. Additional deductions: 1. (5 points) Your code does not follow the style guide discussed in class/textbook. 2. (30 points) Your code does not have author name, date, purpose of this program, comments on the variables and methods, etc. 4. Turn in You should ZIP the following files and Submit the ZIP â€¢ BinaryHeap.java â€¢ ThreeHeap.java â€¢ Any additional Java files needed, if any. â€¢ The Java files you used to test your three implementations. â€¢ The Java files you used to time your three implementations. â€¢ report.pdf, containing answers to Questions in 2.2. â€¢ Any additional files for the bonus credits in a zip file named extracredit.zip. Please make sure that this zip file decompresses its contents into a folder called extracredit and not into a bunch of individual files. Do not turn in PriorityQueue.java and EmptyPQException.java. You must not change these files. Your implementations must work with the code as provided to you.