Lab 06 - Asymptotic notations

1.1 If $ f \in O(n \sqrt n)$ and $ g \in Θ(n)$ then $ f \over g$ in ?

1.2 If $ f \in Θ(n)$ and $ g \in O(\sqrt n)$ then $ f \over g$ in ?

1.3 |$ Θ(n) - Θ(n)$ | in ?

2.1 Prove that if $ lim_{n\rightarrow \infty}$ $ g(n) \over f(n)$ $ = 0$ implies that $ g(n) \in o(f(n))$ , for n reaching infinity. Hint: use the “epsilon” or “Cauchy” limit definition for sequences.

2.2 Prove that $ f(n) \in Ω(log(n))$ and $ g(n) \in O(n)$ implies $ f(n) \in Ω(log(g(n)))$ .

2.3 Prove that $ f(n) \in Ω(g(n))$ and $ g(n) \in O(n ^ 2)$ , then $ g(n) \over f(n)$ $ \in O(n)$ .

2.4 $ O(n) \cap Ω(n) = Θ(n)$ ?

2.5 $ O(n) = Θ(n) \cup o(n)$ ?

3.1 Install gnuplot.

3.2 Use the command set xrange [0:] to set the plotting interval to [0,infty).

Plot the function |$ sin(n)*n$ |, using the command: plot abs(sin(x)*x). abs stands for absolute value.

Use the command plot abs(sin(x)*x), x to compare the two functions.

3.3 Write the following script file:

   ###########################
   # plotting script example
   ###########################
   
   reset
   set xrange [0:]
   
   f(x) = abs(sin(x)*x)
   g(x) = x
   
   plot f(x),g(x)

and save it as “example.plot”.

Run gnuplot, and use the command load “example.plot” to run the script.

3.4 Add the following data file “data.dat” to the directory:

   1 2
   2 3
   3 4
   4 5
   5 6
   6 7
   7 8
   8 9

And modify the last command to:

plot “data.dat”,f(x),g(x)

3.5 Write a function which computes the greatest common divisor of two numbers.

3.6 What is the worst-case for the gcd algorithm?

3.7 Write a function which computes fibonacci numbers.

3.8 Start from the following code:

   from time import time
   
   
   l = [80000, 73201, 66003, 60000, 50000, 40000, 30000, 20000, 10000, 5000] 
   
   
   begin_time = time()
   #[test code]
   duration = time() - begin_time

and write a procedure which determines the running times for gcd, on the x-th and (x-1)th fibonacci numbers, from the list l.

The output of the script should be a list of values:

a b

where a is the size of the input, expressed as a natural number and b is the running time for gcd.

For instance, for x = 10000, b will be the running time of gcd, on the 10000th and 9999th fibonacci number. a will be the size of the input, i.e. the number of bits required to store the value $ fibo(x) + fibo(x+1)$ , rounded to a natural number.

3.9 Write a gnuplot script to plot the data collected from the python script, and compare it to the linear function. Comment the results. How does the execution time grow? Find the suitable function.