## Description

1) (6 pts) For each of the following pairs of functions, either f(n) is O(g(n)), f(n) is Ω(g(n)), or f(n) is

Θ(g(n)) best describes the relationship. Select one and explain.

a. f(n) = n0.75

; g(n) = n0.5

b. f(n) = log n; g(n) = ln n

c. f(n) = nlog n; g(n) =n√𝑛

d. f(n) = en

; g(n) = 3

n

e. f(n) = 2n

; g(n) = 2n-1

f. f(n) = 4

n

; g(n) = n!

2) (4 pts) Let f1 and f2 be asymptotically positive non-decreasing functions. Prove or disprove each

of the following conjectures. To disprove give a counter example.

a. If f1(n) = (g(n)) and f2(n) = O(g(n)) then f1(n)= (f2(n) ).

b. If f1(n) = O(g1(n)) and f2(n) = O(g2(n)) then f1(n)+ f2(n)= O(g1(n) + g2(n) )

3) (10 pts) Merge Sort and Insertion Sort Programs

Implement merge sort and insertion sort to sort an array/vector of integers. Implement the

algorithms in C++, name the programs “mergesort.cpp” and “insertsort.cpp”. Your programs

should compile with the commands g++ mergesort.cpp and g++ insertsort.cpp. Both should read

inputs from a file called “data.txt” where the first value of each line is the number of integers

that need to be sorted, followed by the integers.

Example values for data.txt:

4 19 2 5 11

8 1 2 3 4 5 6 1 2

The output will be written to files called “merge.txt” and “insert.txt”.

For the above example the output would be:

2 5 11 19

1 1 2 2 3 4 5 6

Submit a copy of all your code files and a README file that explains how to compile and run

your code in a ZIP file to TEACH. We will test execution with an input file named data.txt.

HW 1 – 30 points

4) (10 pts) Merge Sort vs Insertion Sort Running time analysis

a) Modify code- Now that you have verified that your code runs correctly using the data.txt

input file, you can modify the code to collect running time data. Instead of reading arrays from

the file data.txt and sorting, you will now generate arrays of size n containing random integer

values from 0 to 10,000 to sort. Use the system clock to record the running times of each

algorithm for ten different values of n for example: n = 5000, 10000, 15000, 20,000, …, 50,000.

You may need to modify the values of n if an algorithm runs too fast or too slow to collect the

running time data (do not collect times over a minute). Output the array size n and time to the

terminal. Name these new programs insertTime.cpp and mergeTime.cpp.

Submit a copy of the timing programs to TEACH in the Zip file from problem 3, also include a

“text” copy of the modified timing code in the written HW submitted in Canvas.

b) Collect running times – Collect your timing data on the engineering server. You will need at

least eight values of t (time) greater than 0. If there is variability in the times between runs of

the same algorithm you may want to take the average time of several runs for each value of n.

Create a table of running times for each algorithm.

c) Plot data and fit a curve – For each algorithm plot the running time data you collected on an

individual graph with n on the x-axis and time on the y-axis. You may use Excel, Matlab, R or any

other software. What type of curve best fits each data set? Give the equation of the curves that

best “fits” the data and draw that curves on the graphs.

d) Combine – Plot the data from both algorithms together on a combined graph. If the scales

are different you may want to use a log-log plot.

e) Comparison – Compare your experimental running times to the theoretical running times of

the algorithms? Remember, the experimental running times were the “average case” since the

input arrays contained random integers.