Description

• to explore time complexity and “real time” of a well-known algorithm

• USE THIS FILE AS THE STARTING DOCUMENT YOU WILL TURN IN. DO NOT DELETE ANYTHING FROM THIS FILE: JUST INSERT YOUR ANSWERS.
• IF USING HAND WRITING (STRONGLY DISCOURAGED), USE THIS FILE BY CREATING SUFFICIENT SPACE AND WRITE IN YOUR ANSWERS.
• FAILING TO FOLLOW TURN IN DIRECTIONS /GUIDELINES WILL COST A 30% PENALTY.

What you need to do: (Use this file to INSERT your answers as indicated below)
1. Implement the Merge-Sort algorithm to sort an array. (See Appendix for the Merge-Sort algorithm)
2. Collect the execution time T(n) as a function of n

3. Plot the functions T(n)/𝑛, T(n)/n.log2(n), and T(n)/𝑛 𝑛 as a function of n on three separate graphs.
4. In Module 4, we establish that the running time T(n) of Merge-Sort is (n.log(n)). Discuss T(n) in light of the graph you plotted above. Use the prediction techniques learned in M1: Programming Assignment (See Early questions trying to infer the shape)

Objective: The objective of this programming assignment is to design and implement in Java the Merge-Sort algorithm presented in the lecture to sort a list of numbers. We are interested in exploring the relationship between the time complexity and the “real time”. For this exploration, you will collect the execution time T(n) as a function of n and plot

the functions T(n)/𝑛, T(n)/n.log2(n), and T(n)/𝑛 𝑛 on the same graph (If you cannot see clearly the shape of the plots, feel free

to separate plots.). Try to predict ahead the shapes of T(n)/𝑛, T(n)/n.log2(n), and T(n)/𝑛 𝑛 to check whether your plots are correct. Finally, discuss your results.

Program to implement
collectData()
Generate an array G of HUGE length L (as huge as your language allows) with random values capped at some max value (as supported by your chosen language).
for n = 1,000 to L (with step 1,000)
copy in Array A n first values from Array G // (declare Array A only ONCE out of the loop)

Take current time Start // We time the sorting of Array A of length n
Merge-Sort(A,0,n-1)
Take current time End // T(n) = End – Start

Store the value n and the values T(n)/𝑛, T(n)/n.log2(n), and T(n)/𝑛 𝑛 in a file F where T(n) is the execution time

Advice:
1) The pseudocode assumes arrays that start with index 1. So, an array A with n elements is an array A[1], A[2]…, A[n-1], A[n]. With most programming languages, an array A with n elements is an array A[0], A[2]…, A[n-1], A[n-1]. When implementing pseudocode that uses some array A with 𝒏 elements, I advise you to declare an array with 𝒏+𝟏 elements and just ignore (not use) A[0]. This way, you can directly implement the algorithm without worrying about indices changes.

Data Analysis

Use any plotting software (e.g., Excel) to plot the values T(n)/𝑛, T(n)/n.log2(n), and T(n)/𝑛 𝑛 in File F as a function of n. File F is the file produced by the program you implemented. Discuss your results based on the plots. (Hint: is T(n) closer to 𝐾.𝑛, 𝐾.𝑛.𝑙𝑜𝑔2(𝑛), or 𝐾.𝑛 𝑛 where K is a constant? See M1: Programming Assignment).

Answer where indicated below. Recall that answers must be well written, documented, justified, and presented to get full credit.

1. (25 points) Implement the Merge-Sort algorithm to sort an array. (See Appendix for the Merge-Sort algorithm) a) State here whether your algorithm works.
b) Insert here a screenshot showing that your implementation sorts correctly an array that contains 10 numbers.

2. (10 points) Collect the execution time T(n) as a function of n. Record the values n, T(n), T(n)/𝑛, T(n)/n.log2(n), and T(n)/𝑛 𝑛 in csv (comma-separated-values) file.
Turn in this csv file with your submission

3. (3×15 points) Plot the functions T(n)/𝑛, T(n)/n.log2(n), and T(n)/𝑛 𝑛 as a function of n on three separate graphs (15 points per graph).
Insert here the three graphs/plots

4. (20 points) In Module 4, we establish that the running time T(n) of Merge-Sort is (n.log(n)).
Discuss here T(n) in light of the graphs you plotted above. Use the prediction techniques learned in M1: Programming Assignment (See Early questions trying to infer the shape of T(n) and determine the asymptotic growth). Discuss whether your plots confirm what we learned in Module M4.
Answer/elaborate/Justify.

What you need to turn in:
• Electronic copy of your source program of collectData program

• Electronic copy of the csv file recording the values n, T(n), T(n)/𝑛, T(n)/n.log2(n), and T(n)/𝑛 𝑛.
• Electronic copy of this file (including your answers) (standalone). Submit the file as a Microsoft Word or PDF file.

Grading
At this stage, you do NOT need to understand Merge-Sort (It will be presented and explained in Module 4)). Implement Merge-Sort exactly the way it is described below. Replace the infinity value () with 0x0fff ffff.