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aa:lab:05 [2020/11/05 23:55] 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)] $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. | + | **2.1** 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)] ? | **2.5** $math[O(n) = Θ(n) \cup o(n)] ? | ||
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| Plot the function |$math[sin(n)*n]|, using the command: ''plot abs(sin(x)*x)''. ''abs'' stands for absolute value. | 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), 2*x'' to compare the two functions. | + | Use the command ''plot abs(sin(x)*x), x'' to compare the two functions. |
| **3.3** Write the following script file: | **3.3** Write the following script file: | ||
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| f(x) = abs(sin(x)*x) | f(x) = abs(sin(x)*x) | ||
| - | g(x) = 2*x | + | g(x) = x |
| plot f(x),g(x) | plot f(x),g(x) | ||
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| The output of the script should be a list of values: | The output of the script should be a list of values: | ||
| + | <code> | ||
| a b | a b | ||
| + | </code> | ||
| - | where a is the **size of the input**, expressed as a natural number and b is the running time for gcd. | + | 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. | + | 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. | ||
| - | **3.9** Write a gnuplot script to plot the data collected from the python script, and compare it to the logarithmic function. Comment the results. How does the execution time grow? Find the suitable function. | ||