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aa:lab:notations [2016/11/02 11:52] pdmatei created |
aa:lab:notations [2016/11/02 18:18] (current) dalex |
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| ===== Outline ===== | ===== Outline ===== | ||
| - | An **execution time** is a function $T:\mathbb{N}\rightarrow\mathbb{N}$, which maps a value $n$ representing **the size of the input** of an algorithm to the **number of instructions** (/**execution steps**) performed by the algorithm. | + | An **execution time** is a function $math[T:\mathbb{N}\rightarrow\mathbb{N}], which maps a value $math[n] representing **the size of the input** of an algorithm to the **number of instructions** (**execution steps**) performed by the algorithm. |
| - | Suppose $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ is a non-necessarily monotone function over positive reals. We define the sets $X(f(n)) \subseteq \mathbb{H}om(\mathbb{R}^{+},\mathbb{R}^{+})$, with $X\in\{O,o,\Theta,\omega,\Omega\}$, which we call **asymptotic notations** (or simply notations). | + | Suppose $math[f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}] is a non-necessarily monotone function over positive reals. We define the sets $math[X(f(n)) \subseteq \mathbb{H}om(\mathbb{R}^{+},\mathbb{R}^{+})], with $math[X\in\{O,o,\Theta,\omega,\Omega\}], which we call **asymptotic notations** (or simply notations). |
| - | **Question**: Why are asymptotic notations defined for positive real functions, instead of positive integers? (as in execution times) | + | **Question**: //Why are asymptotic notations defined for positive real functions, instead of positive integers? (as in execution times)// |
| ===== Notations ===== | ===== Notations ===== | ||
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| such\ that\ \forall\ n \ge n_0,\ \ c_1f(n) \le g(n) \le c_2f(n) \}$$ | such\ that\ \forall\ n \ge n_0,\ \ c_1f(n) \le g(n) \le c_2f(n) \}$$ | ||
| - | {{ :labs:theta_fn.png |}} | + | {{ :aa:lab:theta_fn.png |}} |
| - | We say that $\Theta(f(n))$ is the class (set) of functions with //the same asymptotic growth as// f(n). | + | We say that $math[\Theta(f(n))] is the class (set) of functions with **the same asymptotic growth as** $math[f(n)]. |
| === Example === | === Example === | ||
| - | Let $f(n) = n^2 + 8$ and $g(n) = 3n^2 + 4n + 2$\\ | + | Let $math[f(n) = n^2 + 8] and $math[g(n) = 3n^2 + 4n + 2]\\ |
| - | We see that, for $n_0 = 1, c_1 = 1, c_2 = 4$:\\ | + | We see that, for $math[n_0 = 1, c_1 = 1, c_2 = 4 \Rightarrow \ n^2 + 8 \le 3n^2 + 4n + 2 \le 4n^2 + 32]\\ |
| - | $n^2 + 8 \le 3n^2 + 4n + 2 \le 4n^2 + 32$\\ | + | Thus, $math[g(n) \in \Theta(f(n))] |
| - | thus $g(n) \in \Theta(f(n))$ | + | |
| - | ==== $O$ (Big O) Notation ==== | + | ==== $math[O] (Big O) Notation ==== |
| $$ O(f(n)) = \{ g: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\ |\ | $$ O(f(n)) = \{ g: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\ |\ | ||
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| 0 \le g(n) \le cf(n) \}$$ | 0 \le g(n) \le cf(n) \}$$ | ||
| - | {{ :labs:o_fn.png |}} | + | {{ :aa:lab:o_fn.png |}} |
| - | We say that $O(f(n))$ is the class (set) of functions that grow asymptotically //at most as much as// $f(n)$. | + | We say that $math[O(f(n))] is the class (set) of functions that grow asymptotically **at most as much as** $math[f(n)]. |
| === Example === | === Example === | ||
| - | Let $f(n) = nlog(n) + n$ and $g(n) = nlog(n) + n + 10000$\\ | + | Let $math[f(n) = n\log(n) + n] and $math[g(n) = n\log(n) + n + 10000]\\ |
| - | We see that, for $n_0 = 10000, c = 2$:\\ | + | We see that, for $math[n_0 = 10000, c = 2 \Rightarrow 0 \le n\log(n) + n + 10000 \le 2n\log(n) + 2n]\\ |
| - | $0 \le nlog(n) + n + 10000 \le 2nlog(n) + 2n$\\ | + | Thus, $math[g(n) \in O(f(n))] |
| - | thus $g(n) \in O(f(n))$ | + | |
| - | ==== $\Omega$ (Big Omega) Notation ==== | + | ==== $math[\Omega] (Big Omega) Notation ==== |
| $$ \Omega(f(n)) = \{ g: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\ |\ | $$ \Omega(f(n)) = \{ g: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\ |\ | ||
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| 0 \le cf(n) \le g(n) \}$$ | 0 \le cf(n) \le g(n) \}$$ | ||
| - | {{ :labs:omega_fn.png |}} | + | {{ :aa:lab:omega_fn.png |}} |
| - | We say that $\Omega(f(n))$ is the class (set) of functions that grow asymptotically //at least as much as// $f(n)$. | + | We say that $math[\Omega(f(n))] is the class (set) of functions that grow asymptotically **at least as much as** $math[f(n)]. |
| === Example === | === Example === | ||
| - | Let $f(n) = n^4$ and $g(n) = 2^n$\\ | + | Let $math[f(n) = n^4] and $math[g(n) = 2^n]\\ |
| - | We see that, for $n_0 = 16, c = 1$:\\ | + | We see that, for $math[n_0 = 16, c = 1 \Rightarrow 0 \le n^4 \le 2^n]\\ |
| - | $0 \le n^4 \le 2^n$\\ | + | Thus, $math[g(n) \in \Omega(f(n))] |
| - | thus $g(n) \in \Omega(f(n))$ | + | |
| - | ==== $o$ (Little O) Notation ==== | + | ==== $math[o] (Little O) Notation ==== |
| $$ o(f(n)) = \{ g: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\ |\ | $$ o(f(n)) = \{ g: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\ |\ | ||
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| 0 \le g(n) \le cf(n) \}$$ | 0 \le g(n) \le cf(n) \}$$ | ||
| - | We say that $o(f(n))$ is the class (set) of functions that grows asymptotically //strictly less than// $f(n)$. | + | We say that $math[o(f(n))] is the class (set) of functions that grows asymptotically **strictly less than** $math[f(n)]. |
| === Example === | === Example === | ||
| - | Let $f(n) = n^2 + 3n + 6$ and $g(n) = 8n + 3$\\ | + | Let $math[f(n) = n^2 + 3n + 6] and $math[g(n) = 8n + 3]\\ |
| - | We see that, for $c = 1, n_0 = 5$:\\ | + | We see that, for $math[c = 1, n_0 = 5 \Rightarrow 0 \le 8n + 3 \le n^2 + 3n + 6]\\ |
| - | $0 \le 8n + 3 \le n^2 + 3n + 6$\\ | + | for $math[c = 0.5, n_0 = 7 \Rightarrow 0 \le 8n + 3 \le 0.5n^2 + 1.5n + 3]\\ |
| - | \\ | + | for $math[c = 0.1, n_0 = 77 \Rightarrow 0 \le 8n + 3 \le 0.1n^2 + 0.3n + 0.6]\\ |
| - | for $c = 0.5, n_0 = 7$:\\ | + | etc.\\ |
| - | $0 \le 8n + 3 \le 0.5n^2 + 1.5n + 3$\\ | + | Thus, $math[g(n) \in o(f(n))] |
| - | \\ | + | |
| - | for $c = 0.1, n_0 = 77$:\\ | + | |
| - | $0 \le 8n + 3 \le 0.1n^2 + 0.3n + 0.6$\\ | + | |
| - | etc. | + | |
| - | thus $g(n) \in o(f(n))$ | + | |
| ==== $\omega$ (Little Omega) Notation ==== | ==== $\omega$ (Little Omega) Notation ==== | ||
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| 0 \le cf(n) \le g(n) \}$$ | 0 \le cf(n) \le g(n) \}$$ | ||
| - | We say that $\omega(f(n))$ is the class (set) of functions that grows asymptotically //strictly more than// $f(n)$. | + | We say that $math[\omega(f(n))] is the class (set) of functions that grows asymptotically **strictly more than** $math[f(n)]. |
| === Example === | === Example === | ||
| - | Let $f(n) = log(n)$ and $g(n) = n$\\ | + | Let $math[f(n) = \log(n)] and $math[g(n) = n]\\ |
| - | We see that, for $c = 1, n_0 = 1$:\\ | + | We see that, for $math[c = 1, n_0 = 1 \Rightarrow 0 \le \log(n) \le n]\\ |
| - | $0 \le log(n) \le n$\\ | + | 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 $c = 10, n_0 = 64$:\\ | + | |
| - | $0 \le 10log(n) \le n$\\ | + | |
| - | \\ | + | |
| - | for $c = 100, n_0 = 1024$:\\ | + | |
| - | $0 \le 100log(n) \le n$\\ | + | |
| etc.\\ | etc.\\ | ||
| - | thus $g(n) \in \omega(f(n))$ | + | Thus, $math[g(n) \in \omega(f(n))] |
| ===== Syntactic sugars ====== | ===== Syntactic sugars ====== | ||
| Asymptotic notations are often used to refer to arbitrary functions that have a certain growth. To simplify things, we can write //arithmetic expressions// as follows: | Asymptotic notations are often used to refer to arbitrary functions that have a certain growth. To simplify things, we can write //arithmetic expressions// as follows: | ||
| - | $$f(n) = \Theta(n) + O(log(n))$$\\ | + | $$ f(n) = \Theta(n) + O(\log(n)) $$\\ |
| - | which should be read as: $ \exists g \in \Theta(n)$ and $ \exists h \in O(log(n))$ such that $f(n) = g(n) + h(n),\ \forall n \in \mathbb{R}^{+}$. | + | which should be read as: $math[\exists g \in \Theta(n)] and $math[\exists h \in O(\log(n))] such that $math[f(n) = g(n) + h(n),\ \forall n \in \mathbb{R}^{+}]. |
| We can also write //equations// like the following: | We can also write //equations// like the following: | ||
| $$\Theta(n^2) = O(n^2) + o(n)$$\\ | $$\Theta(n^2) = O(n^2) + o(n)$$\\ | ||
| - | which should be read as:\\ | + | which should be read as: |
| - | $ \forall f \in \Theta(n^2),\ \exists g \in O(n^2)$ and $h \in o(n)$ such that $f(n) = g(n) + h(n),\ \forall n \in \mathbb{R}^{+}$. | + | $math[\forall f \in \Theta(n^2),\ \exists g \in O(n^2)] and $math[h \in o(n)] such that $math[f(n) = g(n) + h(n),\ \forall n \in \mathbb{R}^{+}]. |
| 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 $\Theta(n)$ there is a function equal to it in $O(n)$, we can clearly see that there are functions in $O(n)$ for which there is no correspondent in $\Theta(n)$ (e.g. $f(n) = 1$, $f(n) = log(n)$ 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. $math[f(n) = 1, f(n) = \log(n)] etc.) |
| - | As a rule, each asymptotic notation on the left side of the equal sign should be read as an **universally quantified** function ($ \forall f$) from that class and each asymptotic notation on the right should be read as an **existentially quantified** function ($ \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. |
| $$\left(\frac{\omega(n^2)}{\Theta(n)}\right) = \Omega(n) + o(n)$$ | $$\left(\frac{\omega(n^2)}{\Theta(n)}\right) = \Omega(n) + o(n)$$ | ||
| - | $ \forall f \in \omega(n^2)\ and\ \forall g \in \Theta(n),\ \exists h \in \Omega(n)\ and\ \exists j \in o(n)\ such\ that$ | + | $math[\forall f \in \omega(n^2)\ and\ |
| - | $\left(\frac{f(n)}{g(n)}\right) = h(n) + j(n),\ \forall n \in \mathbb{R}^{+}$ | + | \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}^{+}] | ||
| ===== (Counter)intuitive examples ===== | ===== (Counter)intuitive examples ===== | ||
| Which of the following is true: | Which of the following is true: | ||
| - | * $o(f(n)) \cap \omega(f(n)) = \emptyset$ | + | * $math[o(f(n)) \cap \omega(f(n)) = \emptyset] |
| - | * $o(f(n)) \subsetneq O(f(n))$ | + | * $math[o(f(n)) \subsetneq O(f(n))] |
| - | * $O(f(n)) \setminus o(f(n)) = \Theta(f(n))$ | + | * $math[O(f(n)) \setminus o(f(n)) = \Theta(f(n))] |
| ===== Exercises (asymptotic notations) ===== | ===== Exercises (asymptotic notations) ===== | ||
| Check the truth of the following: | Check the truth of the following: | ||
| - | * $\sqrt{n} \in O(\log(n))$ | + | * $math[\sqrt{n} \in O(\log(n))] |
| - | * $log(n) \in O(\log(\log(n)))$ | + | * $math[\log(n) \in O(\log(\log(n)))] |
| - | * $n \in O(\log(n)\cdot\sqrt{n})$ | + | * $math[n \in O(\log(n)\cdot\sqrt{n})] |
| - | * $n + \log(n) \in \Theta(n)$ | + | * $math[n + \log(n) \in \Theta(n)] |
| Prove that: | Prove that: | ||
| - | * $\log(n\cdot \log(n))\in\Theta(\log(n))$ | + | * $math[\log(n\cdot \log(n))\in\Theta(\log(n))] |
| - | * $\sqrt{n}\in\omega(\log(n))$ | + | * $math[\sqrt{n}\in\omega(\log(n))] |
| - | * $f(n) + g(n) \in O(n\cdot\log(n))$ 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) ===== | ||
| - | * $\frac{O(n\sqrt{n})}{\Theta(n)} = \ldots$ | + | * $math[\frac{O(n\sqrt{n})}{\Theta(n)} = \ldots] |
| - | * $\frac{\Theta(n)}{O(\log(n))} = \ldots$ | + | * $math[\frac{\Theta(n)}{O(\log(n))} = \ldots] |
| - | * $|\Theta(f(n)) - \Theta(f(n))| = \ldots$ | + | * $math[\lvert{\Theta(f(n)) - \Theta(f(n))\rvert} = \ldots] |
| ===== Practice ===== | ===== Practice ===== | ||
| Prove/disprove the following: | Prove/disprove the following: | ||
| - | * $f(n) = \Omega(\log(n))$ 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))] |
| - | * $f(n) = \Omega(\log(n))$ 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))] |
| - | * $f(n) = \Omega(g(n))$ 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 $f$ and $g$ such that: | + | Find two functions $math[f] and $math[g] such that: |
| - | * $f(n) = O(g^2(n))$ | + | * $math[f(n) = O(g^2(n))] |
| - | * $f(n) = \omega(\log(g(n)))$ | + | * $math[f(n) = \omega(\log(g(n)))] |
| - | * $f(n) = \Omega(f(n)\cdot g(n))$ | + | * $math[f(n) = \Omega(f(n)\cdot g(n))] |
| - | * $f(n) = \Theta(g(n)) + \Omega(g^2(n))$ | + | * $math[f(n) = \Theta(g(n)) + \Omega(g^2(n))] |