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--- aa:lab:05 [2020/11/05 22:38]
calin_andrei.bucur
+++ aa:lab:05 [2020/11/15 13:11] (current)
fabianpatras
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-====== Lab 05 - Asymptotic notations ======
+====== Lab 06 - Asymptotic notations ======
 ==== 1. Asymptotic notations ====
-**1.1** If f in $math[O(n \sqrt n)]
+**1.1** If $math[f \in O(n \sqrt n)] and $math[g \in Θ(n)] then $math[f \over g] in ?
+**1.2** If $math[f \in Θ(n)] and $math[g \in O(\sqrt n)] then $math[f \over g] in ?
+**1.3** |$math[Θ(n) - Θ(n)]| in ?
+==== 2. Properties of asymptotic notations ====
+**2.1** Prove that if $math[lim_{n\rightarrow \infty}] $math[g(n) \over f(n)] $math[= 0] implies that $math[g(n) \in o(f(n))], for n reaching infinity. Hint: use the "epsilon" or "Cauchy" limit definition for sequences.
+**2.2** Prove that $math[f(n) \in Ω(log(n))] and $math[g(n) \in O(n)] implies $math[f(n) \in Ω(log(g(n)))].
+**2.3** Prove that $math[f(n) \in Ω(g(n))] and $math[g(n) \in O(n ^ 2)], then $math[g(n) \over f(n)] $math[\in O(n)].
+**2.4** $math[O(n) \cap
+ Ω(n) = Θ(n)] ?
+**2.5** $math[O(n) = Θ(n) \cup o(n)] ?
+==== 3. Execution time vs. Asymptotic growth ====
+**3.1** Install gnuplot.
+**3.2** Use the command ''set xrange [0:]'' to set the plotting interval to [0,infty).
+Plot the function |$math[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:
+2
+3
+4
+5
+6
+7
+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:
+<code>
+a b
+</code>
+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 $math[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.