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aa:lab:05 [2020/11/05 23:30]
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 ======
  
  
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 ==== 2. Properties of asymptotic notations ==== ==== 2. Properties of asymptotic notations ====
  
-**2.1** Prove that if $math[lim] $math[g(n) \over f(n) = 0] implies that $math[g(n) \in o(f(n))], for n reaching infinity. Hint: use the "​epsilon"​ or "​Cauchy"​ limit definition for sequences.+**2.1** 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)))]. **2.2** 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)))].
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 **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)]. **2.3** 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)] ?+**2.4** $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:​ 
 + 
 +     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: 
 +<​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.