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| Both sides previous revision Previous revision Next revision | Previous revision | ||
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aa:lab:notations [2016/11/02 18:02] dalex |
aa:lab:notations [2016/11/02 18:18] (current) dalex |
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| Let $math[f(n) = \log(n)] and $math[g(n) = n]\\ | Let $math[f(n) = \log(n)] and $math[g(n) = n]\\ | ||
| We see that, for $math[c = 1, n_0 = 1 \Rightarrow 0 \le \log(n) \le n]\\ | We see that, for $math[c = 1, n_0 = 1 \Rightarrow 0 \le \log(n) \le n]\\ | ||
| - | for $math[c = 10, n_0 = 64 \Rightarrow 0 \le 10\log(n) \le n]\\ | + | for $math[c = 10, n_0 = 64 \Rightarrow 0 \le 10 \log(n) \le n]\\ |
| - | for $math[c = 100, n_0 = 1024 \Rightarrow 0 \le 100\log(n) \le n]\\ | + | for $math[c = 100, n_0 = 1024 \Rightarrow 0 \le 100 \log(n) \le n]\\ |
| etc.\\ | etc.\\ | ||
| Thus, $math[g(n) \in \omega(f(n))] | Thus, $math[g(n) \in \omega(f(n))] | ||
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| Note that equations are not symmetric and should only be read from left to right. Consider: | Note that equations are not symmetric and should only be read from left to right. Consider: | ||
| $$\Theta(n) = O(n)$$ | $$\Theta(n) = O(n)$$ | ||
| - | While it is true that, for any function in $math[\Theta(n)] there is a function equal to it in $math[O(n)], we can clearly see that there are functions in $math[O(n)] for which there is no correspondent in $math[\Theta(n)] (e.g. $math[f(n) = 1, $f(n) = \log(n)] etc.) | + | While it is true that, for any function in $math[\Theta(n)] there is a function equal to it in $math[O(n)], we can clearly see that there are functions in $math[O(n)] for which there is no correspondent in $math[\Theta(n)] (e.g. $math[f(n) = 1, f(n) = \log(n)] etc.) |
| As a rule, each asymptotic notation on the left side of the equal sign should be read as an **universally quantified** function ($math[\forall f]) from that class and each asymptotic notation on the right should be read as an **existentially quantified** function ($math[\exists g]) from that class. | As a rule, each asymptotic notation on the left side of the equal sign should be read as an **universally quantified** function ($math[\forall f]) from that class and each asymptotic notation on the right should be read as an **existentially quantified** function ($math[\exists g]) from that class. | ||
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| $$\left(\frac{\omega(n^2)}{\Theta(n)}\right) = \Omega(n) + o(n)$$ | $$\left(\frac{\omega(n^2)}{\Theta(n)}\right) = \Omega(n) + o(n)$$ | ||
| - | $math[\forall f \in \omega(n^2)\ and\ \forall g \in \Theta(n),\ \exists h \in \Omega(n)] and $math[\exists j \in o(n)] such that | + | $math[\forall f \in \omega(n^2)\ and\ |
| + | \forall g \in \Theta(n),\ \exists h \in \Omega(n)] and $math[\exists j \in o(n)] such that | ||
| $math[\left(\frac{f(n)}{g(n)}\right) = h(n) + j(n),\ \forall n \in \mathbb{R}^{+}] | $math[\left(\frac{f(n)}{g(n)}\right) = h(n) + j(n),\ \forall n \in \mathbb{R}^{+}] | ||
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| * $math[\log(n\cdot \log(n))\in\Theta(\log(n))] | * $math[\log(n\cdot \log(n))\in\Theta(\log(n))] | ||
| * $math[\sqrt{n}\in\omega(\log(n))] | * $math[\sqrt{n}\in\omega(\log(n))] | ||
| - | * $math[f(n) + g(n) \in O(n\cdot\log(n))$ for $f(n)\in\Theta(n)$ and $g(n)\in O(n\cdot\log n)] | + | * $math[f(n) + g(n) \in O(n\cdot\log(n))] for $math[f(n)\in\Theta(n)] and $math[g(n)\in O(n\cdot\log n)] |
| ===== Exercises (syntactic sugars) ===== | ===== Exercises (syntactic sugars) ===== | ||
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| ===== Practice ===== | ===== Practice ===== | ||
| Prove/disprove the following: | Prove/disprove the following: | ||
| - | * $math[f(n) = \Omega(\log(n))$ and $g(n)=O(n) \implies f(n)=\Omega(\log(g(n))] | + | * $math[f(n) = \Omega(\log(n))] and $math[g(n)=O(n) \implies f(n)=\Omega(\log(g(n))] |
| - | * $math[f(n) = \Omega(\log(n))$ and $g(n)=O(n) \implies f(n)=\Theta(\log(g(n))] | + | * $math[f(n) = \Omega(\log(n))] and $math[g(n)=O(n) \implies f(n)=\Theta(\log(g(n))] |
| - | * $math[f(n) = \Omega(g(n))$ and $g(n)=O(n^2) \implies \frac{g(n)}{f(n)}=O(n)] | + | * $math[f(n) = \Omega(g(n))] and $math[g(n)=O(n^2) \implies \frac{g(n)}{f(n)}=O(n)] |
| Find two functions $math[f] and $math[g] such that: | Find two functions $math[f] and $math[g] such that: | ||