# simple sorting algorithm and two advanced sorting algorithms.

Orçamento $30-250 USD

In this project you will perform an empirical analysis of a simple sorting algorithm and two advanced sorting algorithms. You will implement the algorithms, measure their performance, compare your experimental results with asymptotic efficiency classes, and draw conclusions. This project is to be worked in groups of two students.

The Algorithms

1. Insertion sort.

2. Merge sort.

3. Quick sort.

-Asymptotic Analysis

As stated in the text book, merge sort and quick sort both run in O(nlogn) time. However, Quick sort runs in O(n2) time; if it is the worst case. (You may want to verify this claim by proving it yourself). I would expect your empirical analysis to agree with these asymptotic bounds, but sometimes experiments surprise us.

-Empirical Analysis

To analyze the three algorithms empirically you will need to implement them in a programming language, and then run each of them to sort an array of integer items for various values of n, where n is the number of elements in the array.

For each (algorithm, input data) pair, the program should do the following:

1. Compute the number of comparisons performed.

2. Measure the elapsed running time (in seconds),

3. Graph the results for (input data size, elapsed time) for each sorting algorithm on a scatter plot,

4. Try to infer which complexity class the plot corresponds to (e.g. linear, quadratic curves, etc.).

To conduct an empirical analysis, you should create a test harness program that runs your code and measures the elapsed time of the code corresponding to the algorithm in question.

Your test program should perform the following steps:

1. Input the value of n (input size). Your code should treat n as a variable.

2. Create an array or vector of n random integers to serve as a problem input (you may need to use rand() method to fill your array).

3. Use a clock function to get the current time t1.

4. Execute the algorithms (Insertion Sort) for the same array from step 2.

5. Use a clock function to get the current time t2.

6. Output the elapsed time, t2 - t1.

7. Repeat Steps 3-6 but now you need to execute the Merge sort on the same input.

8. Repeat Steps 3-6 but now you need to execute the Quick sort on the same input.

9. Run your program for different 5 input sizes. E.g. you may ask the user to input the size of the array for the first round as 10 elements. Then for the second round should be 500, and so on.

-Clock Functions

Java has several ways of getting the time, including

System . currentTimeMillis( )

The exact method you use isn’t important, but you do need your results to be accurate and precise enough to measure fractions of seconds.

- What to Measure

The goal is to draw a scatter plot graph for each algorithm’s running times (a total of three plots). Each plot needs to have enough data points to interpolate a fitting curve; 5 is the smallest number that might be reasonable.

So run each algorithm for at least 5 different values of n. You should include at least one very small value of n (less than 10), and one big value that’s large enough to make your code run for at least 5 minutes.

You may not have enough physical memory to get each algorithm to run for a full 5 minutes. If this is the case, use the largest value of n that you can.

--Deliverables--

Produce a written project report (Word format – Moodle submission). Your report should include the following:

1. Three scatter plots, corresponding to the Insertion sort, Merge sort, and Quick sort algorithms. Each plot should have at least 5 points, one with n < 10 and one that corresponds to at least 5 minutes of running time.

2. The scatter plots should have a best-fit line. You can generate your curves in software such as Excel or Mathematica, or draw them by hand.

3. Your source code.

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