From f6275f44e00d766f297a4bfa1e0f8f8995054e59 Mon Sep 17 00:00:00 2001 From: Filip Znachor Date: Fri, 3 Mar 2023 00:51:46 +0100 Subject: [PATCH] =?UTF-8?q?P=C5=99id=C3=A1n=C3=AD=203.=20p=C5=99edn=C3=A1?= =?UTF-8?q?=C5=A1ky=20z=20PPA2?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- KIV PPA2/Prednaska03.md | 89 +++++++++++++++++++++++ KIV PPA2/_assets/zanorovani-o-mnozin.png | Bin 0 -> 187694 bytes 2 files changed, 89 insertions(+) create mode 100644 KIV PPA2/Prednaska03.md create mode 100644 KIV PPA2/_assets/zanorovani-o-mnozin.png diff --git a/KIV PPA2/Prednaska03.md b/KIV PPA2/Prednaska03.md new file mode 100644 index 0000000..25b535c --- /dev/null +++ b/KIV PPA2/Prednaska03.md @@ -0,0 +1,89 @@ +# Výpočetní složitost + +- doba výpočtu T(n) v závislosti na vstupu n + - je důležité určit, co je důležité + - zajímá nás, co se děje, když n roste + - různé funkce rostou různě rychle +- druhy algoritmů + - rychle rostoucí funkce = neefektivní algoritmus + - pomalu rostoucí funkce = efektivní algoritmus +- záleží na tom, jestli se funkce vejde pod jinou + - například lineární pod kubickou + - ne hned, ale od nějakého $n_0$ ano + - nezáleží na násobku, jen to $n_{0}$ bude dál + +## O-notace + +$O(f(n))$ je množina všech funkcí, pro které platí, že $g(n) < c \cdot f(n)$ pro všechna $n > n_0 > 0$ a nějaké $c > 0$. + +- je-li $g(n) \in O(f(n))$, pak + - $g(n)$ se vejde pod $f(n)$ + - $g(n)$ neroste rychleji než $f(n)$ + +**Význam O-notace:** +- omezuje funkci jen shora +- je záruka (**nebude to horší než...**) +- příslušnost do $O(f(n))$ implikuje **efektivitu**, pokud $f(n)$ roste dost pomalu + +U funkcí nezáleží na přenásobení konstantou, přičtení konstanty ani na základu logaritmu: + +| pokud | platí také | +| ---------------------- | ------------------ | +| $g(n) \in O(kf(n))$ | $g(n) \in O(f(n))$ | +| $g(n) \in O(k + f(n))$ | $g(n) \in O(f(n))$ | +| $O(\log_{a}(n)) =$ | $O(\log_{b}(n))$ | + +Mezi některými množinami $O(f(n))$ platí tyto vztahy: + +$$ +O(1) \subset O(\log(n)) \subset O(\sqrt{n}) \subset 0(n) \subset O(n \cdot \log(n)) \subset O(n^2) \subset O(n^3) \subset O(2^n) \subset 0(e^n) +$$ + +U 0-notace nezáleží na násobku ani na přičtení konstanty či menší funkce, proto se např. $g(n) = 2 \cdot n^2 + 5 \cdot n + 4$ vejde pod $0(n^2)$. + +![Zanořování O množin](_assets/zanorovani-o-mnozin.png) + +## Omega notace + +$\Omega(f(n))$ je množina všech funkcí, pro které platí, že $g(n) > c \cdot f(n)$ pro všechna $n > n_0 > 0$ a nějaké $c > 0$. (jediná změna je ve znaménku větší než) + +- Omega notace je opakem O-notace +- funkce $g(n)$ patří do $\Omega(f(n))$ když $g(n)$ roste **stejně rychle nebo rychleji** než $f(n)$ + +Význam Omega notace: +- omezuje funkci jen zdola +- říká, že **to nebude lepší než...** +- příslušnost do $\Omega(f(n))$ implikuje **neefektivitu**, pokud $f(n)$ roste rychle + +Mezi Omega množinami existují **obrácené vztahy** oproti O-množinám. + +## Theta notace + +Pokud $g(n) \in O(f(n))$ a zároveň $g(n) \in \Omega(f(n))$, pak $g(n) \in \Theta(f(n))$. +- neroste rychleji ani pomaleji - roste stejně rychle +- graf $g(n)$ je od jistého $n_{0}$ možné uzavřít mezi grafy $c_{1} \cdot f(n)$ a $c_{2} \cdot f(n)$ + - patrně $c_{1} < c_{2}$ + +Theta množiny jsou disjunktní - funkce **nemůže patřit do více množin zároveň**. + +Výjimka: Pokud $g_{1}(n) = 1000n, g_{2}(n) = n\log(n)$, pak pro všechna realistická n platí $g_{2}(n) < g_{1}(n)$, protože funkce $g_{1}$ bude efektivnější až ve **velmi velkých číslech**. + +## Důležitá poznámka + +Význam funkcí může být: +- čas výpočtu pro jakýkoli vstup velikosti n +- čas výpočtu pro nejlepší možný vstup velikosti n +- čas výpočtu pro nejhorší možný vstup velikosti n +- průměrný čas výpočtu pro vstup velikosti n +- počet instrukcí pro ... +- množství paměti nutné pro zpracování ... +- ... + +Je ale potřeba nesrovnávat hrušky s pomeranči. +- Jeden program může mít **v očekávaném případě lepší efektivitu** než jiný **v tom nejhorším případě**, přesto však ten první nemusí být lepší (v nejhorším případě může být mnohem méně efektivní). + +## Určení výpočetní složitosti programu + +- je potřeba určit funkci $g(n)$ +- při určování můžeme zanedbat časy jednotlivých instrukcí, **pokud jsou opravdu konstatní** (nezávislá na parametru n) +- výpočetní složitost neřeší rozdílné výkony počítačů \ No newline at end of file diff --git a/KIV PPA2/_assets/zanorovani-o-mnozin.png b/KIV PPA2/_assets/zanorovani-o-mnozin.png new file mode 100644 index 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