--- /dev/null
+\input lecnotes.tex
+
+\prednaska{7}{Geometrické algoritmy}{}
+
+\>Uká¾eme si nìkolik základních algoritmù na øe¹ení geometrických problémù v~rovinì. Proè zrovna v~rovinì? Inu, jednorozmìrné problémy bývají triviální
+a naopak pro vy¹¹í dimenze jsou velice komplikované. Rovina je proto rozumným kompromisem mezi obtí¾ností a zajímavostí.
+
+Celou kapitolou nás bude provázet pohádka ze ¾ivota ledních medvìdù. Pokusíme se vyøe¹it jejich \uv{ka¾dodenní} problémy~\dots
+
+\h{Hledání konvexního obalu}
+
+{\I Daleko na severu ¾ili lední medvìdi. Ve vodách tamního moøe byla hojnost ryb a jak je známo, ryby jsou oblíbenou pochoutkou ledních medvìdù.
+Proto¾e medvìdi z~na¹í pohádky rozhodnì nejsou ledajací a ani chytrost jim neschází, rozhodli se v¹echny ryby pochytat. Znají pøesná místa výskytu
+ryb a rádi by vyrobili obrovskou sí», do které by je v¹echny chytili. Pomozte medvìdùm zjistit, jaký nejmen¹í obvod taková sí» mù¾e mít.}
+
+\figure{7-geom5_rybi_motivace.eps}{Problém ledních mìdvìdù: Jaký je nejmen¹í obvod sítì?}{3in}
+
+Neboli v~øeèi matematické, chceme pro zadanou mno¾inu bodù v~rovinì nalézt její konvexní obal. Co je to konvexní obal? Mno¾ina bodù je {\I konvexní},
+pokud pro ka¾dé dva body obsahuje i celou úseèku mezi nimi. {\I Konvexní obal} je nejmen¹í konvexní podmno¾ina roviny, která obsahuje v¹echny zadané
+body.\foot{Pamatujete si na lineární obaly ve vektorových prostorech? Lineární obal mno¾iny vektorù je nejmen¹í vektorový podprostor, který tyto
+vektory obsahuje. Není náhoda, ¾e tato definice pøipomíná definici konvexního obalu. Na druhou stranu ka¾dý vektor z~lineárního obalu lze vyjádøit
+jako lineární kombinaci daných vektorù. Podobnì platí i pro konvexní obaly, ¾e ka¾dý bod z~obalu je konvexní kombinací daných bodù. Ta se li¹í od
+lineární v~tom, ¾e v¹echny koeficienty jsou v~intervalu $[0,1]$ a navíc souèet v¹ech koeficientù je $1$. Tento algebraický pohled mù¾e mnohé vìci
+zjednodu¹it. Zkuste si dokázat, ¾e obì definice konvexního obalu jsou ekvivalentní.} Z~algoritmického hlediska nás v¹ak bude zajímat jenom jeho
+hranice, kterou budeme dále oznaèovat jako konvexní obal.
+
+Na¹ím úkolem je nalézt konvexní obal koneèné mno¾iny bodù. To je v¾dy konvexní mnohoúhelník, navíc s~vrcholy v~zadaných bodech. Øe¹ením problému tedy
+bude posloupnost bodù, které tvoøí konvexní obal. Pro malé mno¾iny je konvexní obal nakreslen na obrázku, pro více bodù je v¹ak situace mnohem
+slo¾itìj¹í.
+
+\figure{7-geom1_male_obaly.eps}{Konvexní obaly malých mno¾in.}{3in}
+
+Pro jednoduchost budeme pøedpokládat, ¾e v¹echny body mají rùzné $x$-ové souøadnice. Tedy utøídìní bodù zleva doprava je urèené jednoznaènì.\foot{To si
+mù¾eme dovolit pøedpokládat, nebo» se v¹emi body staèí nepatrnì pootoèit. Tím konvexní obal urèitì nezmìníme. Av¹ak jednodu¹¹í øe¹ení je naprogramovat
+tøídìní lexikograficky (druhotnì podle souøadnice $y$) a vyøadit identické body.} Tím máme zaji¹tìné, ¾e existují dva body, nejlevìj¹í a
+nejpravìj¹í, pro které platí následující invariant:
+
+\s{Invariant:} Nejlevìj¹í a nejpravìj¹í body jsou v¾dy v~konvexním obalu.
+
+Algoritmus na nalezení konvexního obalu funguje na následujícím jednoduchém principu, kterému se nìkdy øíká {\I zametání roviny}. Procházíme body
+zleva doprava a postupnì roz¹iøujeme doposud nalezený konvexní obal o~dal¹í body. Na zaèátku bude konvexní obal jediného bodu samotný bod. Na konci
+$k$-tého kroku algoritmu známe konvexní obal prvních $k$ bodù. Kdy¾ algoritmus skonèí, známe hledaný konvexní obal. Podle invariantu musíme v~$k$-tém
+kroku pøidat do obalu $k$-tý nejlevìj¹í bod. Zbývá si jen rozmyslet, jak pøesnì tento bod pøidat.
+
+Pøidání dal¹ího bodu do konvexního obalu funguje, jak je naznaèeno na obrázku. Podle invariantu víme, ¾e bod nejvíc vpravo je souèástí konvexního
+obalu. Za nìj napojíme novì pøidávaný bod. Tím jsme získali nìjaký obal, ale zpravidla nebude konvexní. To lze v¹ak snadno napravit, staèí
+odebírat body, v obou smìrech podél konvexního obalu, tak dlouho, dokud nezískáme konvexní obal. Na pøíkladu z obrázku nemusíme po smìru hodinových
+ruèièek odebrat ani jeden bod, obal je v poøádku. Naopak proti smìru hodinových ruèièek musíme odebrat dokonce dva body.
+
+\figure{7-geom2_pridani_bodu.eps}{Pøidání bodu do konvexního obalu.}{4.5in}
+
+Pro pøípadnou implementaci a rozbor slo¾itosti si nyní popí¹eme algoritmus detailnìji. Aby se lépe popisoval, rozdìlíme si konvexní obal na dvì èásti
+spojující nejlevìj¹í a nejpravìj¹í bod obalu. Budeme jim øíkat {\I horní obálka} a {\I dolní obálka}.
+
+\figure{7-geom3_obalky.eps}{Horní a dolní obálka konvexního obalu.}{3.4in}
+
+Obì obálky jsou lomené èáry, navíc horní obálka poøád zatáèí doprava a dolní naopak doleva. Pro udr¾ování bodù v~obálkách staèí dva zásobníky.
+V~$k$-tém kroku algoritmu pøidáme zvlá¹» $k$-tý bod do horní i dolní obálky. Pøidáním $k$-tého bodu se v¹ak mù¾e poru¹it smìr, ve kterém obálka
+zatáèí. Proto budeme nejprve body z~obálky odebírat a $k$-tý bod pøidáme a¾ ve chvíli, kdy jeho pøidání smìr zatáèení neporu¹í.
+
+\s{Algoritmus:}
+
+\algo
+
+\:Setøídíme body podle $x$-ové souøadnice, oznaème body $b_1, \ldots, b_n$.
+\:Vlo¾íme do horní a dolní obálky bod $b_1$: $H = D = (b_1)$.
+\:Pro ka¾dý dal¹í bod $b = b_2,\ldots,b_n$:
+\::Pøepoèítáme horní obálku:
+\:::Dokud $\vert H\vert \ge 2$, $H = (\ldots, h_{k-1}, h_k)$ a úhel $h_{k-1} h_k b$ je orientovaný doleva:
+\::::Odebereme poslední bod $h_k$ z~obálky $H$.
+\:::Pøidáme bod $b$ do obálky $H$.
+\::Symetricky pøepoèteme dolní obálku (s orientací doprava).
+\: Výsledný obal je tvoøen body v~obálkách $H$ a $D$.
+
+\endalgo
+
+Rozebereme si èasovou slo¾itost algoritmu. Setøídit body podle $x$-ové souøadnice doká¾eme v~èase $\O(n \log n)$. Pøidání dal¹ího bodu do obálek
+trvá lineárnì vzhledem k~poètu odebraných bodù. Zde vyu¾ijeme obvyklý postup: Ka¾dý bod je odebrán nejvý¹e jednou, a tedy v¹echna odebrání trvají
+dohromady $\O(n)$. Konvexní obal doká¾eme sestrojit v~èase $\O(n \log n)$, a pokud bychom mìli seznam bodù ji¾ utøídený, doká¾eme to dokonce v
+$\O(n)$.
+
+\s{Algebraický dodatek:} Existuje jednoduchý postup, jak zjistit orientaci úhlu? Uká¾eme si jeden zalo¾ený na lineární algebøe. Budou se hodit
+vlastnosti determinantu. Absolutní hodnota determinantu je objem rovnobì¾nostìnu urèeného øádkovými vektory matice. Dùle¾itìj¹í v¹ak je, ¾e znaménko
+determinantu urèuje \uv{orientaci} vektorù, zda je levotoèivá èi pravotoèivá. Proto¾e ná¹ problém je rovinný, budeme uva¾ovat determinanty matic $2
+\times 2$.
+
+Uva¾me souøadnicový systém v~rovinì, kde $x$-ová souøadnice roste smìrem doprava a~$y$-ová smìrem nahoru. Chceme zjistit orientaci úhlu $h_{k-1} h_k
+b$. Polo¾me $\vec u = (x_1, y_1)$ jako rozdíl souøadnic $h_k$ a~$h_{k-1}$ a podobnì $\vec v = (x_2, y_2)$ je rozdíl souøadnic $b$ a~$h_k$. Matice $M$
+je definována následovnì:
+$$M = \pmatrix{\vec u \cr \vec v} = \pmatrix {x_1&y_1\cr x_2&y_2}.$$
+Úhel $h_{k-1} h_k b$ je orientován doleva, právì kdy¾ $\det M = x_1y_2 - x_2y_1$ je nezáporný,\foot{Neboli vektory $\vec u$ a $\vec v$ odpovídají
+rozta¾ení a zkosení vektorù báze $\vec x = (1,0)$ a $\vec y = (0,1)$, pro nì¾ je determinant nezáporný.} a spoèítat hodnotu determinantu je jednoduché.
+Mo¾né situace jsou nakresleny na obrázku. Poznamenejme, ¾e k~podobnému vzorci se lze také dostat pøes vektorový souèin vektorù $\vec u$ a $\vec v$.
+
+\figure{7-geom4_determinant.eps}{Jak vypadají determinanty rùzných znamének v~rovinì.}{4.6in}
+
+\s{©lo by to vyøe¹it rychleji?} Také vám vrtá hlavou, zda existují rychlej¹í algoritmy? Na závìr si uká¾eme nìco, co na pøedná¹ce nebylo.\foot{A také
+se nebude zkou¹et.} Nejrychlej¹í známý algoritmus, jeho¾ autorem je T.~Chan, funguje v~èase $\O(n \log h)$, kde $h$ je poèet bodù le¾ících na
+konvexním obalu, a pøitom je pøekvapivì jednoduchý. Zde si naznaèíme, jak tento algoritmus funguje.
+
+Algoritmus pøichází s~následující my¹lenkou. Pøedpokládejme, ¾e bychom znali velikost konvexního obalu $h$. Rozdìlíme body libovolnì do $\lceil {n
+\over h} \rceil$ mno¾in $Q_1, \ldots, Q_k$ tak, ¾e $\vert Q_i \vert \le h$. Pro ka¾dou z~tìchto mno¾in nalezneme konvexní obal pomocí vý¹e popsaného
+algoritmu. To doká¾eme pro jednu v~èase $\O(h \log h)$ a pro v¹echny v~èase $\O(n \log h)$. V druhé fázi spustíme hledání konvexního obalu pomocí
+provázkového algoritmu a pro zrychlení pou¾ijeme pøedpoèítané obaly men¹ích mno¾in. Nejprve popí¹eme jeho my¹lenku. Pou¾ijeme následující pozorování:
+
+\s{Pozorování:} Úseèka spojující dva body $a$ a $b$ le¾í na konvexním obalu, právì kdy¾ v¹echny ostatní body le¾í pouze na jedné její
+stranì.\foot{Formálnì je podmínka následující: Pøímka $ab$ urèuje dvì poloroviny. Úseèka le¾í na konvexním obalu, právì kdy¾ v¹echny body le¾í v jedné
+z polorovin.}
+
+Algoritmu se øíká {\I provázkový}, proto¾e svojí èinností pøipomíná namotávání provázku podél konvexního obalu. Zaèneme s bodem, který na konvexním
+obalu urèitì le¾í, to je tøeba ten nejlevìj¹í. V ka¾dém kroku nalezneme následující bod po obvodu konvexního obalu. To udìláme napøíklad tak, ¾e
+projdeme v¹echny body a vybereme ten, který svírá nejmen¹í úhel s poslední stranou konvexního obalu. Novì pøidaná úseèka vyhovuje pozorování a proto
+do konvexního obalu patøí. Po $h$ krocích se dostaneme zpìt k nejlevìj¹ímu bodu a výpoèet ukonèíme. V ka¾dém kroku potøebujeme projít v¹echny body a
+vybrat následníka, co¾ doká¾eme v èase $\O(n)$. Celková slo¾itost algoritmu je tedy $\O(n \cdot h)$.
+
+\twofigures{7-geom6_provazkovy_algoritmus.eps}{Provázkový algoritmus.}{1.25in}{7-geom7_naslednik_pres_konvexni_obal.eps}{Hledání kandidáta v pøedpoèítaném obalu.}{2.5in}
+
+Provázkový algoritmus funguje, ale má jednu obrovskou nevýhodu -- je toti¾ ukrutnì pomalý. Ký¾eného zrychlení dosáhneme, pokud pou¾ijeme pøedpoèítané
+konvexní obaly. Ty umo¾ní rychleji hledat následníka. Pro ka¾dou z mno¾in $Q_i$ najdeme zvlá¹» kandidáta a poté z nich vybereme toho nejlep¹ího.
+Mo¾ný kandidát v¾dy le¾í na konvexním obalu mno¾iny $Q_i$. Vyu¾ijeme toho, ¾e body obalu jsou \uv{uspoøádané}, i kdy¾ trochu netypicky do kruhu.
+Kandidáta mù¾eme hledat metodou pùlení intervalu, i kdy¾ detaily jsou malièko slo¾itìj¹í ne¾ je obvyklé. Jak pùlit zjistíme podle smìru zatáèení
+konvexního obalu. Detaily si rozmyslí ètenáø sám.
+
+Èasová slo¾itost pùlení je $\O(\log h)$ pro jednu mno¾inu. Mno¾in je nejvý¹e $\O({n \over h})$, tedy následující bod konvexního obalu nalezneme v èase
+$\O({n \over h} \log h)$. Celý obal nalezneme ve slibovaném èase $\O(n \log h)$.
+
+Popsanému algoritmu schází jedna dùle¾itá vìc: Ve skuteènosti vìt¹inou neznáme velikost $h$. Budeme proto algoritmus iterovat s~rostoucí hodnotou $h$,
+dokud konvexní obal nesestrojíme. Pokud pøi slepování konvexních obalù zjistíme, ¾e konvexní obal je vìt¹í ne¾ $h$, výpoèet ukonèíme. Zbývá je¹tì
+zvolit, jak rychle má $h$ rùst. Pokud by rostlo moc pomalu, budeme poèítat zbyteènì mnoho fází, naopak pøi rychlém rùstu by nás poslední fáze mohla
+stát pøíli¹ mnoho.
+
+V~$k$-té iteraci polo¾íme $h = 2^{2^k}$. Dostáváme celkovou slo¾itost algoritmu:
+$$\sum_{m=0}^{\O(\log \log h)} \O(n \log 2^{2^m}) = \sum_{m=0}^{\O(\log \log h)} \O(n \cdot 2^m) = \O(n \log h),$$
+kde poslední rovnost dostaneme jako souèet prvních $\O(\log \log h)$ èlenù geometrické øady $\sum 2^m$.
+
+\>Kdy¾ s geometrickými problémy poøádnì nezametete, ony vám to vrátí! Ale kdy¾ u¾ zametat, tak urèitì ne pod koberec a místo smetáku pou¾ijte pøímku.
+V této pøedná¹ce nás spolu s dvìma geometrickými problémy samozøejmì èeká pokraèování pohádky o ledních medvìdech.
+
+{\I Medvìdi vyøe¹ili rybí problém a hlad je ji¾ netrápí. Av¹ak na severu ne¾ijí sami, za sousedy mají Eskymáky. Proto¾e je rozhodnì lep¹í se sousedy
+dobøe vycházet, jsou medvìdi a Eskymáci velcí pøátelé. Skoro ka¾dý se se svými pøáteli rád schází. Av¹ak to je musí nejprve nalézt~\dots}
+
+\h{Hledání prùseèíkù úseèek}
+
+Zkusíme nejprve Eskymákùm vyøe¹it lokalizaci ledních medvìdù.
+
+{\I Kdy¾ takový medvìd nemá co na práci, rád se prochází. Na místech, kde se trasy protínají, je zvý¹ená ¹ance, ¾e se dva medvìdi potkají a zapovídají
+-- ostatnì co byste èekali od medvìdù. To jsou ta správná místa pro Eskymáka, který chce potkat medvìda. Jenom¾e jak tato køí¾ení najít?}
+
+Pro zjednodu¹ení pøedpokládejme, ¾e medvìdi chodí po úseèkách tam a zpìt. Budeme tedy chtít nalézt v¹echny prùseèíky úseèek v rovinì.
+
+\bigskip
+\centerline{\epsfxsize=1.5in\epsfbox{8-geom2_0_bear.eps}\hskip 4em\epsfxsize2in\epsfbox{8-geom2_1_usecky.eps}}
+\smallskip
+\centerline{Problém Eskymákù: Kde v¹ude se køí¾í medvìdí trasy?}
+\bigskip
+
+Pro $n$ úseèek mù¾e existovat a¾ $\Omega(n^2)$ prùseèíkù.\foot{Zkuste takový pøíklad zkonstruovat.} Tedy optimální slo¾itosti by dosáhl i algoritmus,
+který by pro ka¾dou dvojici úseèek testoval, zda se protínají. Èasovou slo¾itost algoritmu v¹ak posuzujeme i vzhledem k velikosti výstupu $p$. Typické
+rozmístìní úseèek mívá toti¾ prùseèíkù spí¹e pomálu. Pro tento pøípad si uká¾eme podstatnì rychlej¹í algoritmus.
+
+Pro jednodu¹¹í popis pøedpokládejme, ¾e úseèky le¾í v obecné poloze. To znamená, ¾e ¾ádné tøi úseèky se neprotínají v jednom bodì a prùnikem ka¾dých
+dvou úseèek je nejvý¹e jeden bod. Navíc pøedpokládejme, ¾e krajní bod ¾ádné úseèky nele¾í na jiné úseèce a také neexistují vodorovné úseèky. Na závìr si
+uká¾eme, jak se s tìmito pøípady vypoøádat.
+
+Algoritmus funguje na principu zametání roviny, popsaném v minulé pøedná¹ce. Budeme posouvat vodorovnou pøímku odshora dolù. V¾dy, kdy¾ narazíme na
+nový prùnik, ohlásíme jeho výskyt. Samozøejmì spojité posouvání nahradíme diskrétním a pøímku v¾dy posuneme do dal¹ího zajímavého bodu.
+
+Zajímavé události jsou {\I zaèátky úseèek}, {\I konce úseèek} a {\I prùseèíky úseèek}. Po utøídìní známe pro první dva typy událostí poøadí, v jakém
+se objeví. Výskyty prùseèíkù budeme poèítat prùbì¾nì, jinak bychom celý problém nemuseli øe¹it.
+
+V ka¾dém kroku si pamatujeme {\I prùøez} $P$ -- posloupnost úseèek aktuálnì protnutých zametací pøímkou. Tyto úseèky máme utøídìné zleva doprava. Navíc si
+udr¾ujeme kalendáø $K$ budoucích událostí. Z hlediska prùseèíkù budeme na úseèky nahlí¾et jako na polopøímky. Pro sousední dvojice úseèek si
+udr¾ujeme, zda se jejich smìry nìkde protnou. Algoritmus pro hledání prùnikù úseèek funguje následovnì:
+
+\s{Algoritmus:}
+
+\algo
+
+\:$P \leftarrow \emptyset$.
+\:Do $K$ vlo¾íme zaèátky a konce v¹ech úseèek.
+\:Dokud $K \ne \emptyset$:
+\::Odebereme nejvy¹¹í událost.
+\::Pokud je to zaèátek úseèky, zatøídíme novou úseèku do $P$.
+\::Pokud je to konec úseèky, odebereme úseèku z $P$.
+\::Pokud je to prùseèík, nahlásíme ho a prohodíme úseèky v $P$.
+\::Navíc v¾dy pøepoèítáme prùseèíkové události, v¾dy maximálnì dvì odebereme a dvì nové pøidáme.
+\endalgo
+
+Zbývá rozmyslet si, jaké datové struktury pou¾ijeme, abychom prùseèíky nalezli dostateènì rychle. Pro kalendáø pou¾ijeme napøíklad haldu. Prùøez si
+budeme udr¾ovat ve vyhledávacím stromì. Poznamenejme, ¾e nemusíme znát souøadnice úseèek, staèí znát jejich poøadí, které se mezi jednotlivými
+událostmi nemìní. Pøi pøidávání úseèek procházíme stromem a porovnáváme souøadnice v prùøezu, které prùbì¾nì dopoèítáváme.
+
+Kalendáø obsahuje v¾dy nejvý¹e $\O(n)$ událostí. Podobnì prùøez obsahuje v ka¾dém okam¾iku nejvý¹e $\O(n)$ úseèek. Jednu událost kalendáøe doká¾eme
+o¹etøit v èase $\O(\log n)$. V¹ech událostí je $\O(n+p)$, a tedy celková slo¾itost algoritmu je $\O((n+p) \log n)$.
+
+Slíbili jsme, ¾e popí¹eme, jak se vypoøádat s vý¹e uvedenými podmínkami na vstup. Události kalendáøe se stejnou $y$-ovou souøadnicí budeme tøídit v
+poøadí zaèátky, prùseèíky a konce úseèek. Tím nahlásíme i prùseèíky krajù úseèek a ani vodorovné úseèky nebudou vadit. Podobnì se není tøeba obávat
+prùseèíkù více úseèek v jednom bodì. Úseèky jdoucí stejným smìrem, jejich¾ prùnik je úseèka, jsou komplikovanìj¹í, ale lze jejich prùseèíky o¹etøit a
+vypsat tøeba souøadnice úseèky tvoøící jejich prùnik.
+
+Na závìr poznamenejme, ¾e Balaban vymyslel efektivnìj¹í algoritmus, který funguje v èase $\O(n \log n + p)$, ale je podstatnì komplikovanìj¹í.
+
+\h{Hledání nejbli¾¹ích bodù a Voroného diagramy}
+
+Nyní se pokusíme vyøe¹it i problém druhé strany -- pomù¾eme medvìdùm nalézt Eskymáky.
+
+{\I Eskymáci tráví vìt¹inu èasu doma, ve svém iglù. Takový medvìd je na své toulce zasnì¾enou krajinou, kdy¾ tu se najednou rozhodne nav¹tívit nìjakého
+Eskymáka. Proto se podívá do své medvìdí mapy a nalezne nejbli¾¹í iglù. Má to ale jeden háèek, iglù jsou spousty a medvìd by dávno usnul, ne¾ by
+nejbli¾¹í objevil.}\foot{Zlí jazykové by øekli, ¾e medvìdi jsou moc líní a nebo v mapách ani èíst neumí!}
+
+Popí¹eme si nejprve, jak vypadá medvìdí mapa. Medvìdí mapa obsahuje celou Arktidu a jsou v ní vyznaèena v¹echna iglù. Navíc obsahuje vyznaèené
+oblasti tvoøené body, které jsou nejblí¾e k jednomu danému iglù. Takovému schématu se øíká {\I Voroného diagram}. Ten pro zadané body $x_1, \ldots, x_n$
+obsahuje rozdìlení roviny na oblasti $B_1, \ldots, B_n$, kde $B_i$ je mno¾ina bodù, které jsou blí¾e k $x_i$ ne¾ k ostatním bodùm $x_j$. Formálnì jsou
+tyto oblasti definovány následovnì:
+$$B_i = \left\{y \in {\bb R}^2\ \vert\ \forall j:\rho(x_i,y) \le \rho(x_j,y)\right\},$$
+kde $\rho(x,y)$ znaèí vzdálenost bodù $x$ a $y$.
+
+Uká¾eme si, ¾e Voroného diagram má pøekvapivì jednoduchou strukturu. Nejprve uva¾me, jak budou vypadat oblasti $B_a$ a $B_b$ pouze pro dva body
+$a$ a $b$, jak je naznaèeno na obrázku. V¹echny body stejnì vzdálené od $a$ i $b$ le¾í na pøímce $p$ -- ose úseèky $ab$. Oblasti $B_a$ a $B_b$
+jsou tedy tvoøeny polorovinami ohranièenými osou $p$. Tedy obecnì tvoøí mno¾ina v¹ech bodù bli¾¹ích k $x_i$ ne¾ k $x_j$ nìjakou polorovinu. Oblast
+$B_i$ obsahuje v¹echny body, které jsou souèasnì bli¾¹í k $x_i$ ne¾ ke v¹em ostatním bodùm $x_j$ -- tedy le¾í ve v¹ech polorovinách souèasnì.
+Ka¾dá z oblastí $B_i$ je tvoøena prùnikem $n-1$ polorovin, tedy je to (mo¾ná neomezený) mnohoúhelník.\foot{Sly¹eli jste u¾ o lineárním programování?
+Jak název vùbec nenapoví, {\I lineární programování} je teorii zabývající se øe¹ením a vlastnostmi soustav lineárních nerovnic. Lineární program je
+popsaný lineární funkcí, kterou chceme maximalizovat za podmínek popsaných soustavou lineárních nerovnic. Ka¾dá nerovnice urèuje poloprostor, ve
+kterém se pøípustná øe¹ení nachází. Proto¾e pøípustné øe¹ení splòuje v¹echny nerovnice zároveò, je mno¾ina v¹ech pøípustných øe¹ení (mo¾ná neomezený)
+mnohostìn, obecnì ve veliké dimenzi ${\bb R}^d$, kde $d$ je poèet promìnných. Mno¾iny $B_i$ lze snadno popsat jako mno¾iny v¹ech pøípustných øe¹ení
+lineárních programù pomocí vý¹e ukázaných polorovin. Na závìr poznamenejme, ¾e dlouho otevøená otázka, zda lze nalézt optimální øe¹ení lineárního
+programu v polynomiálním èase, byla pozitivnì vyøe¹ena -- je znám polynomiální algoritmus, kterému se øíká {\I metoda vnitøního bodu}. Na druhou
+stranu, pokud chceme najít pøípustné celoèíselné øe¹ení, je úloha NP-úplná a je jednoduché na ni pøevést spoustu optimalizaèních problémù. Dokázat
+NP-tì¾kost není pøíli¹ tì¾ké. Na druhou stranu ukázat, ¾e tento problém le¾í v NP, není vùbec jednoduché.}
+Pøíklad Voroného diagramu je naznaèen na obrázku. Zadané body jsou oznaèeny prázdnými krou¾ky a hranice oblastí $B_i$ jsou vyznaèené èernými èárami.
+
+\twofigures{8-geom2_2_polorovina.eps}{Body bli¾¹í k $a$ ne¾ $b$.}{1.25in}{8-geom2_3_voroneho_diagram.eps}{Voroného diagram.}{2.5in}
+
+Není náhoda, pokud vám hranice oblastí pøipomíná rovinný graf. Jeho vrcholy jsou body, které jsou stejnì vzdálené od alespoò tøí zadaných bodù. Jeho
+stìny jsou oblasti $B_i$. Jeho hrany jsou tvoøeny èástí hranice mezi dvìma oblastmi -- body, které mají dvì oblasti spoleèné. Obecnì prùnik dvou
+oblastí mù¾e být, v závislosti na jejich sousedìní, prázdný, bod, úseèka, polopøímka nebo dokonce celá pøímka. V dal¹ím textu si pøedstavme, ¾e celý
+Voroného diagram uzavøeme do dostateènì velkého obdélníka,\foot{Pøeci jenom i celá Arktida je omezenì velká.} èím¾ dostaneme omezený rovinný graf.
+
+Poznamenejme, ¾e pøeru¹ované èáry tvoøí hrany duálního rovinného grafu s vrcholy v zadaných bodech. Hrany spojují sousední body na kru¾nicích, které
+obsahují alespoò tøi ze zadaných bodù. Napøíklad na obrázku dostáváme skoro samé trojúhelníky, proto¾e vìt¹ina kru¾nic obsahuje pøesnì tøi zadané
+body. Av¹ak nalezneme i jeden ètyøúhelník, jeho¾ vrcholy le¾í na jedné kru¾nici.
+
+Zkusíme nyní odhadnout, jak velký je rovinný graf popisující Voroného diagram. Podle slavné Eulerovy formule má ka¾dý rovinný graf nejvý¹e lineárnì
+mnoho vrcholù, hran a stìn -- pro $v$ vrcholù, $e$ hran a $f$ stìn je $e \le 3v-6$ a navíc $v+f = e+2$. Tedy slo¾itost diagramu je lineární vzhledem k
+poètu zadaných bodù $n=f$, $\O(n)$. Navíc Voroného diagram lze zkonstruovat v èase $\O(n \log n)$, napøíklad pomocí zametání roviny nebo metodou
+rozdìl a panuj. Tím se v¹ak zabývat nebudeme,\foot{Pro zvídavé, kteøí nemají zkou¹ku druhý den ráno: Detaily naleznete v zápiscích z pøedloòského
+ADSka.} místo toho si uká¾eme, jak v ji¾ spoèteném Voroného diagramu rychle hledat nejbli¾¹í body.
+
+\h{Lokalizace bodu uvnitø mnohoúhelníkové sítì}
+
+Problém medvìdù je najít v medvìdí mapì co nejrychleji nejbli¾¹í iglù. Máme v rovinì sí» tvoøenou mnohoúhelníky. Chceme pro jednotlivé body rychle
+rozhodovat, do kterého mnohoúhelníku patøí. Na¹e øe¹ení budeme optimalizovat pro jeden pevný rozklad a obrovské mno¾ství rùzných dotazù, které chceme
+co nejrychleji zodpovìdìt.\foot{Pøedstavujme si to tøeba tak, ¾e medvìdùm zprovozníme server. Ten jednou schroustá celou mapu a potom co nejrychleji
+odpovídá na jejich dotazy. Medvìdi tak nemusí v mapách nic hledat, staèí se pøipojit na server a poèkat na odpovìï.} Nejprve pøedzpracujeme zadané
+mnohoúhelníky a vytvoøíme strukturu, která nám umo¾ní rychlé dotazy na jednotlivé body.
+
+Uka¾me si pro zaèátek øe¹ení bez pøedzpracování. Rovinu budeme zametat pøímkou shora dolù. Podobnì jako pøi hledání prùseèíkù úseèek, udr¾ujeme si prùøez
+pøímkou. V¹imnìte si, ¾e tento prùøez se mìní jenom ve vrcholech mnohoúhelníkù. Ve chvíli, kdy narazíme na hledaný bod, podíváme se, do kterého
+intervalu v prùøezu patøí. To nám dá mnohoúhelník, který nahlásíme. Prùøez budeme uchovávat ve vyhledávacím stromì. Takové øe¹ení má slo¾itost $\O(n
+\log n)$ na dotaz, co¾ je hroznì pomalé.
+
+Pøedzpracování bude fungovat následovnì. Jak je naznaèeno na obrázku pøeru¹ovanými èárami, rozøe¾eme si celou rovinu na pásy, bìhem kterých se prùøez
+pøímkou nemìní. Pro ka¾dý z nich si pamatujeme stav stromu popisující, jak vypadal prùøez pøi procházení tímto pásem. Kdy¾ chceme lokalizovat nìjaký bod,
+nejprve pùlením nalezneme pás, ve kterém se nachází. Poté polo¾íme dotaz na pøíslu¹ný strom. Strom procházíme a po cestì si dopoèítáme souøadnice
+prùøezu, a¾ lokalizujeme správný interval v prùøezu. Dotaz doká¾eme zodpovìdìt v èase $\O(\log n)$. Hledaný bod je na obrázku naznaèen prázdným
+koleèkem a nalezený interval v prùøezu je vyta¾ený tuènì.
+
+\figure{8-geom2_4_pasy_mnohouhelniku.eps}{Mnohoúhelníky rozøezané na pásy.}{2.5in}
+
+Jenom¾e na¹e øe¹ení má jeden háèek: Jak zkonstruovat jednotlivé verze stromu dostateènì rychle? K tomu napomohou {\I èásteènì perzistentní} datové
+struktury. Pod perzistencí se myslí, ¾e struktura umo¾òuje uchovávat svoji historii. Èásteènì perzistentní struktury nemohou svoji historii
+modifikovat.
+
+Popí¹eme si, jak vytvoøit perzistentní strom s pamìtí $\O(\log n)$ na zmìnu. Pokud provádíme operaci na stromì, mìní se jenom malá èást stromu.
+Napøíklad pøi vkládání do stromu se mìní jenom prvky na jedné cestièce z koøene do listu (a pøípadnì rotací i na jejím nejbli¾¹ím okolí). Proto si
+ulo¾íme upravenou cestièku a zbytek stromu budeme sdílet s pøedchozí verzí. Na obrázku je vyznaèena cesta, její¾ vrcholy jsou upravovány. ©edì
+oznaèené podstromy navì¹ené na tuto cestu se nemìní, a proto na nì staèí zkopírovat ukazatele. Mimochodem zmìny ka¾dé operace se slo¾itostí $\O(k)$
+lze zapsat v pamìti $\O(k)$, prostì operace nemá tolik èasu, aby mohla pozmìnit pøíli¹ velikou èást stromu.
+
+\figure{8-geom2_5_upravy_stromu.eps}{Jedna operace mìní pouze okolí cesty -- navì¹ené podstromy se nemìní.}{2in}
+
+Celková èasová slo¾itost je tedy $\O(n \log n)$ na pøedzpracování Voroného diagramu a vytvoøení persistentního stromu. Kvùli persistenci potøebuje
+toto pøedzpracování pamì» $\O(n \log n)$. Na dotaz spotøebujeme èas $\O(\log n)$, nebo» nejprve vyhledáme pùlením pøíslu¹ný pás a poté polo¾íme dotaz
+na pøíslu¹nou verzi stromu. Rychleji to ani provést nepùjde, nebo» potøebujeme utøídit souøadnice bodù.
+
+\s{Lze to lépe?} Na závìr poznamenejme, ¾e se umí provést vý¹e popsaná persistence vyhledávacího stromu v amortizované pamìti $\O(1)$ na zmìnu. Ve
+struènosti naznaèíme my¹lenku. Pou¾ijeme stromy, které pøi insertu a deletu provádí amortizovanì jenom konstantnì mnoho úprav své struktury. To nám
+napøíklad zaruèí 2-4 stromy z pøedná¹ky a podobnou vlastnost lze dokázat i o èerveno-èerných stromech. Pøi zmìnì potom nebudeme upravovat celou cestu,
+ale upravíme jenom jednotlivé vrcholy, kterých se zmìna týká. Ka¾dý vrchol stromu si v sobì bude pamatovat a¾ dvì své verze. Pokud chceme vytvoøit
+tøetí verzi, vrchol zkopírujeme stranou. To v¹ak mù¾e vyvolat zmìny v jeho rodièích a¾ do koøene. Situace je naznaèena na obrázku. Pøi vytvoøení nové
+verze $3$ pro vrcholu $v$ vytvoøíme jeho kopii $v'$, do které ulo¾íme tuto verzi. Av¹ak musíme také zmìnit rodièe $u$, kterému vytvoøíme novou verzi
+ukazující na $v'$. Abychom dosáhli ký¾ené konstantní pamì»ové slo¾itosti, pomù¾e potenciálový argument -- zmìn se provádí amortizovanì jenom
+konstantnì mnoho. Navíc si pro ka¾dou verzi pamatujeme její koøen, ze kterého máme dotaz spustit.
+
+\figure{8-geom2_6_rychla_perzistence.eps}{Vytvoøení nové verze vrcholu.}{2in}
+
+\bye
--- /dev/null
+input lib
+
+figname("7-geom");
+
+figtag("male_obaly");
+beginfig(1);
+ pickup boldpen;
+ labeloffset:=1cm;
+ pair c,pos; c := (0,0); pos := c;
+ drawemptyvertex(c);
+ label.bot(btex $n=1$ etex, pos);
+ c := (2cm,0);
+ pos := pos + c;
+ pair A[];
+ A[0] := (-0.3cm, -0.2cm)+c; A[1] := (0.2cm, 0.3cm)+c;
+ draw A[0]--A[1];
+ drawemptyvertex(A[0]); drawemptyvertex(A[1]);
+ label.bot(btex $n=2$ etex, pos);
+
+ pos := pos + c;
+ A[2] := (+0.3cm, -0.4cm)+c;
+ for i := 0 upto 2: A[i] := A[i] shifted c; endfor
+ draw A[0]--A[1]--A[2]--cycle;
+ for i := 0 upto 2: drawemptyvertex(A[i]); endfor
+ label.bot(btex $n=3$ etex, pos);
+
+ pos := pos + c;
+ A[3] := (A[0]+A[1]+A[2])/3;
+ for i := 0 upto 3: A[i] := A[i] shifted c; endfor
+ draw A[0]--A[1]--A[2]--cycle;
+ for i := 0 upto 2: drawemptyvertex(A[i]); endfor
+ draw vertex(A[3]);
+
+ c := (1cm,0);
+ pos := pos + c/2;
+ A[3] := A[1]+(0.3cm,-0.2cm);
+ for i := 0 upto 3: A[i] := A[i] shifted c; endfor
+ draw A[0]--A[1]--A[3]--A[2]--cycle;
+ for i := 0 upto 3: drawemptyvertex(A[i]); endfor
+ label.bot(btex $n=4$ etex, pos);
+endfig;
+
+figtag("pridani_bodu");
+beginfig(2);
+ pair A[],B[],C,shift; shift := (4.5cm,0);
+ A[0] := (-1.7cm,1.1cm);
+ A[1] := (-1.2cm,1.2cm);
+ A[2] := (-0.4cm,1cm);
+ A[3] := (0.2cm,0.2cm);
+ A[4] := (0.4cm,-0.7cm);
+ A[5] := (-0.8cm,-1.3cm);
+ A[6] := (-1.4cm,-1.4cm);
+ B[0] := (-1.1cm, 0.7cm);
+ B[1] := (-0.6cm, 0.1cm);
+ B[2] := (-1.3cm, -0.6cm);
+ C := (1cm, 0.1cm);
+
+ % krok 1
+ pickup boldpen;
+ draw A[0] for i := 1 upto 6: --A[i] endfor;
+ for i := 1 upto 5: drawemptyvertex(A[i]); endfor
+ for i := 0 upto 2: draw vertex(B[i]); endfor
+ draw vertex(C);
+ drawarrow (C+(0.5cm,0)--C+(1.5cm,0)) shifted (0, -0.1cm) withpen normalpen;
+ for i:=0 upto 6: A[i] := A[i] shifted shift; endfor
+ for i:=0 upto 2: B[i] := B[i] shifted shift; endfor
+ C := C shifted shift;
+
+ % krok 2
+ draw A[0] for i := 1 upto 6: --A[i] endfor;
+ draw A[4]{dir 70}..C;
+ draw A[4]{dir 45}..C;
+ draw C--A[2] dashed evenly withpen normalpen;
+ draw C--A[3] dashed evenly withpen normalpen;
+ for i := 1 upto 5: drawemptyvertex(A[i]); endfor
+ for i := 0 upto 2: draw vertex(B[i]); endfor
+ drawemptyvertex(C);
+ drawarrow (C+(0.5cm,0)--C+(1.5cm,0)) shifted (0,-0.1cm) withpen normalpen;
+ for i:=0 upto 6: A[i] := A[i] shifted shift; endfor
+ for i:=0 upto 2: B[i] := B[i] shifted shift; endfor
+ C := C shifted shift;
+
+ % krok 3
+ draw for i := 0 upto 2: A[i]-- endfor C for i := 4 upto 6: --A[i] endfor;
+ for i := 1 upto 2: drawemptyvertex(A[i]); endfor
+ for i := 4 upto 5: drawemptyvertex(A[i]); endfor
+ for i := 0 upto 2: draw vertex(B[i]); endfor
+ draw vertex(A[3]);
+ drawemptyvertex(C);
+endfig;
+
+figtag("obalky");
+beginfig(3);
+ labeloffset := 0.2cm;
+ pickup boldpen;
+ pair A[],B[];
+ A[0] := (-7cm, 0cm);
+ A[1] := (-6.2cm, 0.9cm);
+ A[2] := (-4.6cm,1.5cm);
+ A[3] := (-2.4cm,1.8cm);
+ A[4] := (-0.8cm,1.5cm);
+ A[5] := (0.4cm,0.6cm);
+ A[6] := (0.8cm,-0.10cm);
+ A[7] := (-1.6cm,-1.9cm);
+ A[8] := (-4cm,-2.1cm);
+ A[9] := (-6cm, -1.5cm);
+ A[10] := (-7cm, 0cm);
+
+ B[0] := (-2.2cm, 0.7cm);
+ B[1] := (-1.2cm, 0.1cm);
+ B[2] := (-2.6cm, -0.6cm);
+ B[3] := (-3.6cm, -0.4cm);
+ B[4] := (-3cm, 0.6cm);
+ B[5] := (-2.6cm, 1cm);
+ B[6] := (-1cm, -1.2cm);
+ B[7] := (-6.5cm, 0.2cm);
+ B[8] := (-5cm, 0.8cm);
+ B[9] := (-6cm, -0.6cm);
+ B[10] := (-5cm, -1.2cm);
+
+ draw createpath(for i := 0 upto 5: A[i]-- endfor A[6]);
+ draw (for i := 6 upto 9: A[i]-- endfor A[10]) dashed evenly;
+ for i := 0 upto 9: drawemptyvertex(A[i]); endfor
+ for i := 0 upto 10: draw vertex(B[i]); endfor
+
+ label(btex \font\myfont=csr10 \myfont horní obálka etex, ((-7cm+0.8cm)/2,2.2cm));
+ label(btex \font\myfont=csr10 \myfont dolní obálka etex, ((-7cm+0.8cm)/2,-2.5cm));
+ label.lft(btex $L$ etex, A[0]);
+ label.rt(btex $P$ etex, A[6]);
+endfig;
+
+figtag("determinant");
+beginfig(4);
+ labeloffset := 0.1cm;
+ pair A[], shift; shift := (4cm,1cm);
+
+ % det(M) > 0
+ A[0] := (-2cm, 0);
+ A[1] := (0,-1cm);
+ A[2] := (1.5cm, 0cm);
+ A[3] := A[0] + A[2] - A[1];
+
+ fill A[0]--A[1]--A[2]--A[3]--cycle withcolor 0.8white;
+ draw A[0]--A[1]--A[2]--A[3]--cycle dashed evenly;
+ drawarrow A[0]--0.96[A[0],A[1]] withpen boldpen;
+ drawarrow A[1]--0.96[A[1],A[2]] withpen boldpen;
+ for i:=0 upto 2: draw vertex(A[i]); endfor
+ label.lft(btex $h_{k-1}$ etex, A[0]);
+ label.bot(btex $h_k$ etex, A[1]);
+ label.llft(btex $\vec u$ etex, 0.5[A[0],A[1]]);
+ label.lrt(btex $\vec v$ etex, 0.5[A[1],A[2]]);
+ label.rt(btex $b$ etex, A[2]);
+ label(btex $\det(M) > 0$ etex, 0.5[A[0],A[2]]);
+
+ % det(M) = 0
+ A[0] := (-1cm, -0.5cm) + shift;
+ A[1] := (0, -1cm) + shift;
+ A[2] := (1cm, -1.5cm) + shift;
+ drawarrow A[0]--0.96[A[0],A[1]] withpen boldpen;
+ drawarrow A[1]--0.96[A[1],A[2]] withpen boldpen;
+ for i:=0 upto 2: draw vertex(A[i]); endfor
+ label.lft(btex $h_{k-1}$ etex, A[0]);
+ label.llft(btex $h_k$ etex, A[1]);
+ label.llft(btex $\vec u$ etex, 0.5[A[0],A[1]]);
+ label.llft(btex $\vec v$ etex, 0.5[A[1],A[2]]);
+ label.bot(btex $b$ etex, A[2]);
+ label(btex $\det(M) = 0$ etex, origin) shifted (0,0.3cm) rotated -28 shifted 0.5[A[0], A[2]];
+
+ % det(M) < 0
+ shift := (7.5cm, 1.25cm);
+ A[0] := (-1cm, -0.5cm) + shift;
+ A[1] := (1.5cm, -1cm) + shift;
+ A[2] := (2cm, -2.5cm) + shift;
+ A[3] := A[0] + A[2] - A[1];
+ fill A[0]--A[1]--A[2]--A[3]--cycle withcolor 0.8white;
+ draw A[0]--A[1]--A[2]--A[3]--cycle dashed evenly;
+ drawarrow A[0]--0.96[A[0],A[1]] withpen boldpen;
+ drawarrow A[1]--0.96[A[1],A[2]] withpen boldpen;
+ for i:=0 upto 2: draw vertex(A[i]); endfor
+ label.lft(btex $h_{k-1}$ etex, A[0]);
+ label.urt(btex $h_k$ etex, A[1]);
+ label.top(btex $\vec u$ etex, 0.5[A[0],A[1]]);
+ label.rt(btex $\vec v$ etex, 0.5[A[1],A[2]]);
+ label.rt(btex $b$ etex, A[2]);
+ label(btex $\det(M) < 0$ etex, 0.5[A[0],A[2]]);
+endfig;
+
+figtag("rybi_motivace");
+beginfig(5);
+ u := 0.3cm;
+ def draw_fish(expr pos,size,rot) =
+ draw ((-1.3u*size,0){dir 60}..{right}(u*size,-u*size/4)) rotated rot shifted pos;
+ draw ((-1.3u*size,0){dir -50}..{right}(u*size,u*size/4)) rotated rot shifted pos;
+ draw ((u*size,-u*size/4)--(u*size,u*size/4)) rotated rot shifted pos;
+ draw (-1u*size,u*size/15) rotated rot shifted pos withpen pencircle scaled (u/8);
+ for i:=1 upto 3: draw (dirs((u*size,-u*size/4+i*u*size/8), 180, u*size/6)) rotated rot shifted pos; endfor
+ enddef;
+
+ pair A[],B[];
+ A[0] := (-7cm, 0cm);
+ A[1] := (-6.2cm, 0.9cm);
+ A[2] := (-4.6cm,1.5cm);
+ A[3] := (-2.4cm,1.8cm);
+ A[4] := (-0.8cm,1.5cm);
+ A[5] := (0.4cm,0.6cm);
+ A[6] := (0.8cm,-0.10cm);
+ A[7] := (-1.6cm,-1.9cm);
+ A[8] := (-4cm,-2.1cm);
+ A[9] := (-6cm, -1.5cm);
+ A[10] := (-7cm, 0cm);
+
+ B[0] := (-2.2cm, 0.7cm);
+ B[1] := (-1.2cm, 0.1cm);
+ B[2] := (-2.6cm, -0.6cm);
+ B[3] := (-3.6cm, -0.4cm);
+ B[4] := (-3cm, 0.6cm);
+ B[5] := (-2.6cm, 1cm);
+ B[6] := (-1cm, -1.2cm);
+ B[7] := (-6.5cm, 0.2cm);
+ B[8] := (-5cm, 0.8cm);
+ B[9] := (-6cm, -0.6cm);
+ B[10] := (-5cm, -1.2cm);
+
+ for i:=0 upto 9: draw_fish(A[i], 1, 0); endfor;
+ for i:=0 upto 10: draw_fish(B[i], 1, 0); endfor;
+ draw createpath(for i:=0 upto 9: A[i]-- endfor cycle) scaled 1.13 shifted (0.4cm,0) withpen boldpen;
+endfig;
+
+figtag("provazkovy_algoritmus");
+beginfig(6);
+ pickup boldpen;
+ pair A[],B[],u; u := (-3cm, 0);
+ for i := 0 upto 3: A[i] := u rotated (-30*i) yscaled 0.7; endfor;
+ A[2] := A[2] + (0,0.1cm);
+ draw for i:=0 upto 2: A[i]-- endfor A[3];
+ drawarrow ((u/2) for i:=1 upto 3: ..u/2 rotated (-30*i) endfor) yscaled 0.7 withpen normalpen;
+ B[0] := (-2cm,0.5cm);
+ B[1] := (-1cm,1.5cm);
+ B[2] := (-0.5cm,0.2cm);
+ for i:=0 upto 2: draw vertex(B[i]); endfor
+
+ path ub; ub := (-20cm,3cm)--(20cm,3cm);
+
+ numeric ang[]; ang[0] = 90; ang[1] = angle(A[1]-A[0]); ang[2] = angle(A[2]-A[1]); ang[3] = angle(A[3]-A[2]);
+ for i:=0 upto 2:
+ draw reverse(dirs(A[i],ang[i],6cm) cutafter ub) withpen normalpen dashed evenly;
+ l := 1cm + (i-1)*0.2cm;
+ drawarrow from(A[i],ang[i],l)..from(A[i],(ang[i]+ang[i+1])/2,l)..from(A[i],ang[i+1],l) withpen normalpen;
+ endfor
+
+ for i:=0 upto 3: drawemptyvertex(A[i]); endfor;
+endfig;
+
+figtag("naslednik_pres_konvexni_obal");
+beginfig(7);
+ pair A[], C;
+ label.lrt(btex $Q_i$ etex, (1.5cm,-0.8cm));
+ pickup boldpen;
+
+ C := (-4cm,-0.3cm);
+ for i:=0 upto 6:
+ A[i] := (2cm,0) rotated (360*i/7+5) yscaled 0.7;
+ draw vertex(A[i]);
+ endfor;
+ draw for i:=0 upto 6: A[i]-- endfor cycle withpen normalpen;
+ draw C--A[2] dashed evenly;
+
+ draw dirs(C, -140, 0.5cm);
+ drawemptyvertex(A[2]);
+ drawemptyvertex(C);
+ drawdblarrow (fullcircle scaled 2cm rotated (360*2/7-5) yscaled 0.7) cutbefore (origin--(3cm,0) rotated (360*2/7+25)) withpen normalpen;
+ %drawarrow C+(0,0.5cm){dir 60}..A[2]+(0,0.5cm) withpen normalpen;
+ %drawarrow 0.6A[5]{dir 170}..(0.6A[3] rotated -15) withpen normalpen;
+ %drawarrow 0.7A[6]{dir 60}..(0.5A[1] rotated 30) withpen normalpen;
+endfig;
+end
--- /dev/null
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--- /dev/null
+input lib
+
+figname("8-geom2_");
+figtag("usecky");
+beginfig(1);
+ def drawusecka(expr p,q) = draw vertex(p); draw vertex(q); draw p--q; enddef;
+ pair A[],B[];
+ A0 := origin; B0 := (4cm,1.5cm);
+ A1 := (0.5cm,1.5cm); B1 := (1cm,0);
+ A2 := (2cm,1.3cm); B2 := (3cm, -0.9cm);
+ A3 := (1cm,-1.3cm); B3 := (3.5cm,0);
+ z0 = whatever[A0,B0]; z0 = whatever[A1,B1];
+ z1 = whatever[A0,B0]; z1 = whatever[A2,B2];
+ z2 = whatever[A2,B2]; z2 = whatever[A3,B3];
+
+ for i:=0 upto 2: fill fullcircle scaled 7pt shifted z[i]; drawemptyvertex(z[i]); endfor
+ for i:=0 upto 3: drawusecka(A[i], B[i]); endfor
+endfig;
+
+figtag("polorovina");
+beginfig(2);
+ pair A,B;
+ A := origin;
+ B := (1.7cm,0.5cm);
+ an := angle(B-A);
+ path p; p := from(.5[A,B],an+90,3cm)--from(.5[A,B],an-90,3cm);
+ fill p--reverse(p shifted (A-0.8[A,B]))--cycle withcolor 0.8white;
+ %fill p{dir (an+180)}..-0.35[A,B]..{dir an}cycle withcolor 0.8white;
+ draw p withpen boldpen;
+
+ pair C; C := point(0.9) of p;
+ drawarrow from(C,an+180,0.1cm)--from(C,an+180,1cm) withpen boldpen;
+ C := point(0.1) of p;
+ drawarrow from(C,an+180,0.1cm)--from(C,an+180,1cm) withpen boldpen;
+ label.rt(btex $p$ etex, point(0.15) of p);
+
+ draw A--B dashed evenly;
+ drawemptyvertex(A); drawemptyvertex(B);
+ label.llft(btex $a$ etex, A);
+ label.urt(btex $b$ etex, B);
+ label(btex $B_a$ etex, A+2(-0.1cm,0.5cm));
+ label(btex $B_b$ etex, B+2(-0.1cm,0.5cm));
+endfig;
+
+figtag("voroneho_diagram");
+beginfig(3);
+ u := 1.35cm;
+ pair A[],B[];
+ A0 := origin; A1 := (-u,-u); A2 := (1.3u,-u); A3 := (1.50821u,0.7u); A4 := (0.6u,0.8u); A5 := (0.2u,1.9u); A6 := (-1.3u,1u);
+ def osa(expr a, b, an,l) = from(.5[a,b], angle(b-a)+an, l) enddef;
+ vardef prusecik_os(expr p,q,r) =
+ save b; pair b;
+ b = whatever[osa(p,q, 90,1cm), osa(p,q,-90,1cm)];
+ b = whatever[osa(q,r, 90,1cm), osa(q,r,-90,1cm)];
+ b
+ enddef;
+ B[0] := prusecik_os(A0,A1,A2); B[1] := prusecik_os(A0,A1,A6); B[2] := prusecik_os(A0,A4,A6); B[3] := prusecik_os(A4,A5,A6);
+ B[4] := prusecik_os(A3,A4,A5); B[5] := prusecik_os(A0,A3,A4); B[6] := prusecik_os(A0,A2,A4);
+ draw osa(A1,A2,-90,2cm)--B[0]--B[1]--osa(A1,A6,90,1.3cm) withpen boldpen;
+ draw B[1]--B[2]--B[3]--osa(A6,A5,90,2cm) withpen boldpen;
+ draw B[3]--B[4]--osa(A3,A5,-90,2.7cm) withpen boldpen;
+ draw B[4]--B[5]--B[2] withpen boldpen;
+ draw B[5]--B[6]--osa(A2,A3,-90,1.3cm) withpen boldpen;
+ draw B[6]--B[0] withpen boldpen;
+
+ draw A0--A1--A2--A0--A6--A1 dashed evenly;
+ draw A6--A5--A4--A0 dashed evenly; draw A3--A2 dashed evenly;
+ draw A3--A5 dashed evenly; draw A6--A4--A3 dashed evenly;
+
+ for i:=0 upto 6: draw vertex(B[i]); endfor
+ for i:=0 upto 6: drawemptyvertex(A[i]); endfor
+endfig;
+
+figtag("pasy_mnohouhelniku");
+beginfig(4);
+ u := 1.35cm;
+ pair A[],B[];
+ A0 := origin; A1 := (-u,-u); A2 := (1.3u,-u); A3 := (1.50821u,0.7u); A4 := (0.6u,0.8u); A5 := (0.2u,1.9u); A6 := (-1.3u,1u);
+ def osa(expr a, b, an,l) = from(.5[a,b], angle(b-a)+an, l) enddef;
+ vardef prusecik_os(expr p,q,r) =
+ save b; pair b;
+ b = whatever[osa(p,q, 90,1cm), osa(p,q,-90,1cm)];
+ b = whatever[osa(q,r, 90,1cm), osa(q,r,-90,1cm)];
+ b
+ enddef;
+ def drawline(expr p) = draw ((-2.3u,p)--(2.5u,p)) cutbefore (D0--D1) cutafter (D2--D3) enddef;
+ B[0] := prusecik_os(A0,A1,A2); B[1] := prusecik_os(A0,A1,A6); B[2] := prusecik_os(A0,A4,A6); B[3] := prusecik_os(A4,A5,A6);
+ B[4] := prusecik_os(A3,A4,A5); B[5] := prusecik_os(A0,A3,A4); B[6] := prusecik_os(A0,A2,A4);
+
+ pair C; C := origin; for i:=0 upto 6: C := C + B[i]; endfor C := C/6;
+ pair D[];
+ D0 := C+(-2.25,2.25)*u; D1 := C+(-2.25,-2.25)*u; D2 := C+(2.25,-2.25)*u; D3 := C+(2.25,2.25)*u;
+
+ draw B[0]--B[1]--B[2]--B[3]--B[4]--B[5]--B[6]--B[0] withpen boldpen;
+ draw B[5]--B[2] withpen boldpen;
+ pair E; E := 0.6[B0,B4];
+ drawline(ypart(E));
+ drawline(ypart(E)) cutbefore (B2--B5) cutafter (B4--B5) dashed evenly withpen bolderpen;
+ drawemptyvertex(E);
+
+ for i:=0 upto 6: drawline(ypart(B[i])) dashed evenly; endfor
+ for i:=0 upto 6: draw vertex(B[i]); endfor
+endfig;
+
+figtag("upravy_stromu");
+beginfig(5);
+ u := 1cm;
+ draw (0,0.1u)--(2.05u,-2.05u)--(-2.05u,-2.05u)--cycle;
+ pair A[]; A0 := from(origin,-90,0.1u); A1 := from(A0, -70, 0.5u); A2 := from(A1, -110, 0.5u); A3 := from(A2, -80, 0.5u); A4 := from(A3, -120, 0.45u);
+ path p; p := createpath(A0--A1--A2--A3--A4);
+ path q; q := (p scaled 0.93 shifted (-0.15u,-0.15u))--(-1.85u,-1.95u)--cycle; fill q withcolor 0.8white; draw q;
+ q := (p scaled 0.93 shifted (0.15u,-0.15u))--(1.85u,-1.95u)--cycle; fill q withcolor 0.8white; draw q;
+ p := createpath(A0--from(A0,-110,0.1u)--A1--A2--A3--A4); draw p withpen boldpen;
+endfig;
+
+figtag("rychla_perzistence");
+beginfig(6);
+ u := 1cm;
+ def drawtable(expr p, lab, sa, sb) =
+ draw centersquare xscaled 2u yscaled (2u/3) shifted (p+(0,u/3));
+ draw centersquare xscaled 2u yscaled (2u/3) shifted (p-(0,u/3));
+ label(sa, p+(0,u/3));
+ label(sb, p-(0,u/3));
+ label.bot(lab, p-(0,2u/3));
+ enddef;
+
+ pair A[];
+ for i:=0 upto 2: A[i] := (0,2u) rotated 120i; endfor
+ drawtable(A1, btex $v$ etex, btex verze 2 etex, btex verze 1 etex);
+ drawtable(A2, btex $v'$ etex, btex verze 3 etex, "");
+ drawtable(A0, btex $u$ etex, btex verze 2 etex, btex verze 1 etex);
+
+ drawarrow from(A[2]+(0,u/3), 180, 7u/6)--from(A[1]+(0,u/3), 0, 7u/6);
+ drawarrow from(A[0]+(0,-u/3), 180, 7u/6)..{dir -90}from(A[1]+(-3u/4,2u/3), 90, u/6);
+ drawarrow from(A[0]+(0,u/3), 0, 7u/6)..{dir -90}from(A[2]+(3u/4,2u/3), 90, u/6);
+endfig;
+end
--- /dev/null
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--- /dev/null
+P=6-geom
+
+include ../Makerules
--- /dev/null
+% implementation of figure naming and figure transparency
+string name,tag; name := ""; tag := "";
+def updatefigname =
+ if tag="": filenametemplate (name & "%c.eps");
+ else: filenametemplate (name & "%c_" & tag & ".eps");
+ fi;
+enddef;
+
+def figname(expr n) = name := n; updatefigname; enddef;
+def figtag(expr t) = tag := t; updatefigname; enddef;
+
+picture transparent_picture;
+color transparent_color; transparent_color := 0.9white;
+def drawtransparent(expr num) =
+ transparent_picture := currentpicture;
+endfig;
+
+if tag="": filenametemplate (name & "%c_transparent.eps");
+else: filenametemplate (name & "%c_" & tag & "_transparent.eps");
+fi;
+beginfig(num);
+ draw transparent_picture withcolor transparent_color;
+endfig;
+
+updatefigname;
+enddef;
+
+def from(expr p,d,len) =
+ p+dir(d)*len
+enddef;
+
+def dirs(expr p,d,len) =
+ p--from(p,d,len)
+enddef;
+
+def drawvertices(expr s,n) =
+ for i:=s upto n:
+ draw vertex(PQ[i]);
+ endfor
+enddef;
+
+def drawfvertices(expr s,n,flags) =
+ for i:=s upto n:
+ draw vertex(PQ[i]) flags;
+ endfor
+enddef;
+
+path centersquare; centersquare := (-0.5,-0.5)--(0.5,-0.5)--(0.5,0.5)--(-0.5,0.5)--cycle;
+
+def vertex(expr p) = p withpen pencircle scaled 4pt enddef;
+def drawemptyvertex(expr p) = unfill fullcircle scaled 4pt shifted p; draw fullcircle scaled 4pt shifted p; enddef;
+def drawendpointvertex(expr p) = draw vertex(p) withcolor red; draw fullcircle scaled 6pt shifted p; enddef;
+def createpath(expr p) = shakepath(p, 0.015cm,0.1cm) enddef;
+vardef shakepath(expr p,d,l) =
+ save r,b;
+ path r; r := point(arctime 0 of p) of p;
+ b := -1;
+ for i:=l step l until arclength(p):
+ r := r--(point(arctime i of p) of p)+dir(angle(direction(arctime i of p) of p rotated 90))*d*b;
+ b := -b;
+ endfor
+ r--point(arctime arclength(p) of p) of p
+enddef;
+
+pen normalpen; normalpen := pencircle scaled 0.6pt;
+pen boldpen; boldpen := pencircle scaled 1.5pt;
+pen bolderpen; bolderpen := pencircle scaled 2pt;
+def dotline = withdots scaled 0.82 withpen boldpen enddef;
+
+vardef unclosedbubblec(expr p,c) =
+ bubblec((p..reverse p..cycle),c)
+enddef;
+
+vardef bubblec(expr p,c) =
+ save r;
+ path r; r := (point(arctime 0 of p) of p)+dir(angle(direction(arctime 0 of p) of p rotated 90))*c;
+ for i:=0.01cm step 0.025cm until arclength(p):
+ r := r..(point(arctime i of p) of p)+dir(angle(direction(arctime i of p) of p rotated 90))*c;
+ endfor
+ r..(point(arctime arclength(p) of p) of p)+dir(angle(direction(arctime arclength(p) of p) of p rotated 90))*c..cycle
+enddef;
+
+vardef bubble(expr p) = bubblec(p,0.12cm) enddef;
+vardef unclosedbubble(expr p) = unclosedbubblec(p,0.12cm) enddef;
+++ /dev/null
-input lib
-
-figname("7-geom");
-
-figtag("male_obaly");
-beginfig(1);
- pickup boldpen;
- labeloffset:=1cm;
- pair c,pos; c := (0,0); pos := c;
- drawemptyvertex(c);
- label.bot(btex $n=1$ etex, pos);
- c := (2cm,0);
- pos := pos + c;
- pair A[];
- A[0] := (-0.3cm, -0.2cm)+c; A[1] := (0.2cm, 0.3cm)+c;
- draw A[0]--A[1];
- drawemptyvertex(A[0]); drawemptyvertex(A[1]);
- label.bot(btex $n=2$ etex, pos);
-
- pos := pos + c;
- A[2] := (+0.3cm, -0.4cm)+c;
- for i := 0 upto 2: A[i] := A[i] shifted c; endfor
- draw A[0]--A[1]--A[2]--cycle;
- for i := 0 upto 2: drawemptyvertex(A[i]); endfor
- label.bot(btex $n=3$ etex, pos);
-
- pos := pos + c;
- A[3] := (A[0]+A[1]+A[2])/3;
- for i := 0 upto 3: A[i] := A[i] shifted c; endfor
- draw A[0]--A[1]--A[2]--cycle;
- for i := 0 upto 2: drawemptyvertex(A[i]); endfor
- draw vertex(A[3]);
-
- c := (1cm,0);
- pos := pos + c/2;
- A[3] := A[1]+(0.3cm,-0.2cm);
- for i := 0 upto 3: A[i] := A[i] shifted c; endfor
- draw A[0]--A[1]--A[3]--A[2]--cycle;
- for i := 0 upto 3: drawemptyvertex(A[i]); endfor
- label.bot(btex $n=4$ etex, pos);
-endfig;
-
-figtag("pridani_bodu");
-beginfig(2);
- pair A[],B[],C,shift; shift := (4.5cm,0);
- A[0] := (-1.7cm,1.1cm);
- A[1] := (-1.2cm,1.2cm);
- A[2] := (-0.4cm,1cm);
- A[3] := (0.2cm,0.2cm);
- A[4] := (0.4cm,-0.7cm);
- A[5] := (-0.8cm,-1.3cm);
- A[6] := (-1.4cm,-1.4cm);
- B[0] := (-1.1cm, 0.7cm);
- B[1] := (-0.6cm, 0.1cm);
- B[2] := (-1.3cm, -0.6cm);
- C := (1cm, 0.1cm);
-
- % krok 1
- pickup boldpen;
- draw A[0] for i := 1 upto 6: --A[i] endfor;
- for i := 1 upto 5: drawemptyvertex(A[i]); endfor
- for i := 0 upto 2: draw vertex(B[i]); endfor
- draw vertex(C);
- drawarrow (C+(0.5cm,0)--C+(1.5cm,0)) shifted (0, -0.1cm) withpen normalpen;
- for i:=0 upto 6: A[i] := A[i] shifted shift; endfor
- for i:=0 upto 2: B[i] := B[i] shifted shift; endfor
- C := C shifted shift;
-
- % krok 2
- draw A[0] for i := 1 upto 6: --A[i] endfor;
- draw A[4]{dir 70}..C;
- draw A[4]{dir 45}..C;
- draw C--A[2] dashed evenly withpen normalpen;
- draw C--A[3] dashed evenly withpen normalpen;
- for i := 1 upto 5: drawemptyvertex(A[i]); endfor
- for i := 0 upto 2: draw vertex(B[i]); endfor
- drawemptyvertex(C);
- drawarrow (C+(0.5cm,0)--C+(1.5cm,0)) shifted (0,-0.1cm) withpen normalpen;
- for i:=0 upto 6: A[i] := A[i] shifted shift; endfor
- for i:=0 upto 2: B[i] := B[i] shifted shift; endfor
- C := C shifted shift;
-
- % krok 3
- draw for i := 0 upto 2: A[i]-- endfor C for i := 4 upto 6: --A[i] endfor;
- for i := 1 upto 2: drawemptyvertex(A[i]); endfor
- for i := 4 upto 5: drawemptyvertex(A[i]); endfor
- for i := 0 upto 2: draw vertex(B[i]); endfor
- draw vertex(A[3]);
- drawemptyvertex(C);
-endfig;
-
-figtag("obalky");
-beginfig(3);
- labeloffset := 0.2cm;
- pickup boldpen;
- pair A[],B[];
- A[0] := (-7cm, 0cm);
- A[1] := (-6.2cm, 0.9cm);
- A[2] := (-4.6cm,1.5cm);
- A[3] := (-2.4cm,1.8cm);
- A[4] := (-0.8cm,1.5cm);
- A[5] := (0.4cm,0.6cm);
- A[6] := (0.8cm,-0.10cm);
- A[7] := (-1.6cm,-1.9cm);
- A[8] := (-4cm,-2.1cm);
- A[9] := (-6cm, -1.5cm);
- A[10] := (-7cm, 0cm);
-
- B[0] := (-2.2cm, 0.7cm);
- B[1] := (-1.2cm, 0.1cm);
- B[2] := (-2.6cm, -0.6cm);
- B[3] := (-3.6cm, -0.4cm);
- B[4] := (-3cm, 0.6cm);
- B[5] := (-2.6cm, 1cm);
- B[6] := (-1cm, -1.2cm);
- B[7] := (-6.5cm, 0.2cm);
- B[8] := (-5cm, 0.8cm);
- B[9] := (-6cm, -0.6cm);
- B[10] := (-5cm, -1.2cm);
-
- draw createpath(for i := 0 upto 5: A[i]-- endfor A[6]);
- draw (for i := 6 upto 9: A[i]-- endfor A[10]) dashed evenly;
- for i := 0 upto 9: drawemptyvertex(A[i]); endfor
- for i := 0 upto 10: draw vertex(B[i]); endfor
-
- label(btex \font\myfont=csr10 \myfont horní obálka etex, ((-7cm+0.8cm)/2,2.2cm));
- label(btex \font\myfont=csr10 \myfont dolní obálka etex, ((-7cm+0.8cm)/2,-2.5cm));
- label.lft(btex $L$ etex, A[0]);
- label.rt(btex $P$ etex, A[6]);
-endfig;
-
-figtag("determinant");
-beginfig(4);
- labeloffset := 0.1cm;
- pair A[], shift; shift := (4cm,1cm);
-
- % det(M) > 0
- A[0] := (-2cm, 0);
- A[1] := (0,-1cm);
- A[2] := (1.5cm, 0cm);
- A[3] := A[0] + A[2] - A[1];
-
- fill A[0]--A[1]--A[2]--A[3]--cycle withcolor 0.8white;
- draw A[0]--A[1]--A[2]--A[3]--cycle dashed evenly;
- drawarrow A[0]--0.96[A[0],A[1]] withpen boldpen;
- drawarrow A[1]--0.96[A[1],A[2]] withpen boldpen;
- for i:=0 upto 2: draw vertex(A[i]); endfor
- label.lft(btex $h_{k-1}$ etex, A[0]);
- label.bot(btex $h_k$ etex, A[1]);
- label.llft(btex $\vec u$ etex, 0.5[A[0],A[1]]);
- label.lrt(btex $\vec v$ etex, 0.5[A[1],A[2]]);
- label.rt(btex $b$ etex, A[2]);
- label(btex $\det(M) > 0$ etex, 0.5[A[0],A[2]]);
-
- % det(M) = 0
- A[0] := (-1cm, -0.5cm) + shift;
- A[1] := (0, -1cm) + shift;
- A[2] := (1cm, -1.5cm) + shift;
- drawarrow A[0]--0.96[A[0],A[1]] withpen boldpen;
- drawarrow A[1]--0.96[A[1],A[2]] withpen boldpen;
- for i:=0 upto 2: draw vertex(A[i]); endfor
- label.lft(btex $h_{k-1}$ etex, A[0]);
- label.llft(btex $h_k$ etex, A[1]);
- label.llft(btex $\vec u$ etex, 0.5[A[0],A[1]]);
- label.llft(btex $\vec v$ etex, 0.5[A[1],A[2]]);
- label.bot(btex $b$ etex, A[2]);
- label(btex $\det(M) = 0$ etex, origin) shifted (0,0.3cm) rotated -28 shifted 0.5[A[0], A[2]];
-
- % det(M) < 0
- shift := (7.5cm, 1.25cm);
- A[0] := (-1cm, -0.5cm) + shift;
- A[1] := (1.5cm, -1cm) + shift;
- A[2] := (2cm, -2.5cm) + shift;
- A[3] := A[0] + A[2] - A[1];
- fill A[0]--A[1]--A[2]--A[3]--cycle withcolor 0.8white;
- draw A[0]--A[1]--A[2]--A[3]--cycle dashed evenly;
- drawarrow A[0]--0.96[A[0],A[1]] withpen boldpen;
- drawarrow A[1]--0.96[A[1],A[2]] withpen boldpen;
- for i:=0 upto 2: draw vertex(A[i]); endfor
- label.lft(btex $h_{k-1}$ etex, A[0]);
- label.urt(btex $h_k$ etex, A[1]);
- label.top(btex $\vec u$ etex, 0.5[A[0],A[1]]);
- label.rt(btex $\vec v$ etex, 0.5[A[1],A[2]]);
- label.rt(btex $b$ etex, A[2]);
- label(btex $\det(M) < 0$ etex, 0.5[A[0],A[2]]);
-endfig;
-
-figtag("rybi_motivace");
-beginfig(5);
- u := 0.3cm;
- def draw_fish(expr pos,size,rot) =
- draw ((-1.3u*size,0){dir 60}..{right}(u*size,-u*size/4)) rotated rot shifted pos;
- draw ((-1.3u*size,0){dir -50}..{right}(u*size,u*size/4)) rotated rot shifted pos;
- draw ((u*size,-u*size/4)--(u*size,u*size/4)) rotated rot shifted pos;
- draw (-1u*size,u*size/15) rotated rot shifted pos withpen pencircle scaled (u/8);
- for i:=1 upto 3: draw (dirs((u*size,-u*size/4+i*u*size/8), 180, u*size/6)) rotated rot shifted pos; endfor
- enddef;
-
- pair A[],B[];
- A[0] := (-7cm, 0cm);
- A[1] := (-6.2cm, 0.9cm);
- A[2] := (-4.6cm,1.5cm);
- A[3] := (-2.4cm,1.8cm);
- A[4] := (-0.8cm,1.5cm);
- A[5] := (0.4cm,0.6cm);
- A[6] := (0.8cm,-0.10cm);
- A[7] := (-1.6cm,-1.9cm);
- A[8] := (-4cm,-2.1cm);
- A[9] := (-6cm, -1.5cm);
- A[10] := (-7cm, 0cm);
-
- B[0] := (-2.2cm, 0.7cm);
- B[1] := (-1.2cm, 0.1cm);
- B[2] := (-2.6cm, -0.6cm);
- B[3] := (-3.6cm, -0.4cm);
- B[4] := (-3cm, 0.6cm);
- B[5] := (-2.6cm, 1cm);
- B[6] := (-1cm, -1.2cm);
- B[7] := (-6.5cm, 0.2cm);
- B[8] := (-5cm, 0.8cm);
- B[9] := (-6cm, -0.6cm);
- B[10] := (-5cm, -1.2cm);
-
- for i:=0 upto 9: draw_fish(A[i], 1, 0); endfor;
- for i:=0 upto 10: draw_fish(B[i], 1, 0); endfor;
- draw createpath(for i:=0 upto 9: A[i]-- endfor cycle) scaled 1.13 shifted (0.4cm,0) withpen boldpen;
-endfig;
-
-figtag("provazkovy_algoritmus");
-beginfig(6);
- pickup boldpen;
- pair A[],B[],u; u := (-3cm, 0);
- for i := 0 upto 3: A[i] := u rotated (-30*i) yscaled 0.7; endfor;
- A[2] := A[2] + (0,0.1cm);
- draw for i:=0 upto 2: A[i]-- endfor A[3];
- drawarrow ((u/2) for i:=1 upto 3: ..u/2 rotated (-30*i) endfor) yscaled 0.7 withpen normalpen;
- B[0] := (-2cm,0.5cm);
- B[1] := (-1cm,1.5cm);
- B[2] := (-0.5cm,0.2cm);
- for i:=0 upto 2: draw vertex(B[i]); endfor
-
- path ub; ub := (-20cm,3cm)--(20cm,3cm);
-
- numeric ang[]; ang[0] = 90; ang[1] = angle(A[1]-A[0]); ang[2] = angle(A[2]-A[1]); ang[3] = angle(A[3]-A[2]);
- for i:=0 upto 2:
- draw reverse(dirs(A[i],ang[i],6cm) cutafter ub) withpen normalpen dashed evenly;
- l := 1cm + (i-1)*0.2cm;
- drawarrow from(A[i],ang[i],l)..from(A[i],(ang[i]+ang[i+1])/2,l)..from(A[i],ang[i+1],l) withpen normalpen;
- endfor
-
- for i:=0 upto 3: drawemptyvertex(A[i]); endfor;
-endfig;
-
-figtag("naslednik_pres_konvexni_obal");
-beginfig(7);
- pair A[], C;
- label.lrt(btex $Q_i$ etex, (1.5cm,-0.8cm));
- pickup boldpen;
-
- C := (-4cm,-0.3cm);
- for i:=0 upto 6:
- A[i] := (2cm,0) rotated (360*i/7+5) yscaled 0.7;
- draw vertex(A[i]);
- endfor;
- draw for i:=0 upto 6: A[i]-- endfor cycle withpen normalpen;
- draw C--A[2] dashed evenly;
-
- draw dirs(C, -140, 0.5cm);
- drawemptyvertex(A[2]);
- drawemptyvertex(C);
- drawdblarrow (fullcircle scaled 2cm rotated (360*2/7-5) yscaled 0.7) cutbefore (origin--(3cm,0) rotated (360*2/7+25)) withpen normalpen;
- %drawarrow C+(0,0.5cm){dir 60}..A[2]+(0,0.5cm) withpen normalpen;
- %drawarrow 0.6A[5]{dir 170}..(0.6A[3] rotated -15) withpen normalpen;
- %drawarrow 0.7A[6]{dir 60}..(0.5A[1] rotated 30) withpen normalpen;
-endfig;
-end
+++ /dev/null
-\input lecnotes.tex
-
-\prednaska{7}{Geometrické algoritmy}{(sepsal Pavel Klavík)}
-
-\>Uká¾eme si nìkolik základních algoritmù na øe¹ení geometrických problémù v~rovinì. Proè zrovna v~rovinì? Inu, jednorozmìrné problémy bývají triviální
-a naopak pro vy¹¹í dimenze jsou velice komplikované. Rovina je proto rozumným kompromisem mezi obtí¾ností a zajímavostí.
-
-Celou kapitolou nás bude provázet pohádka ze ¾ivota ledních medvìdù. Pokusíme se vyøe¹it jejich \uv{ka¾dodenní} problémy~\dots
-
-\h{Hledání konvexního obalu}
-
-{\I Daleko na severu ¾ili lední medvìdi. Ve vodách tamního moøe byla hojnost ryb a jak je známo, ryby jsou oblíbenou pochoutkou ledních medvìdù.
-Proto¾e medvìdi z~na¹í pohádky rozhodnì nejsou ledajací a ani chytrost jim neschází, rozhodli se v¹echny ryby pochytat. Znají pøesná místa výskytu
-ryb a rádi by vyrobili obrovskou sí», do které by je v¹echny chytili. Pomozte medvìdùm zjistit, jaký nejmen¹í obvod taková sí» mù¾e mít.}
-
-\figure{7-geom5_rybi_motivace.eps}{Problém ledních mìdvìdù: Jaký je nejmen¹í obvod sítì?}{3in}
-
-Neboli v~øeèi matematické, chceme pro zadanou mno¾inu bodù v~rovinì nalézt její konvexní obal. Co je to konvexní obal? Mno¾ina bodù je {\I konvexní},
-pokud pro ka¾dé dva body obsahuje i celou úseèku mezi nimi. {\I Konvexní obal} je nejmen¹í konvexní podmno¾ina roviny, která obsahuje v¹echny zadané
-body.\foot{Pamatujete si na lineární obaly ve vektorových prostorech? Lineární obal mno¾iny vektorù je nejmen¹í vektorový podprostor, který tyto
-vektory obsahuje. Není náhoda, ¾e tato definice pøipomíná definici konvexního obalu. Na druhou stranu ka¾dý vektor z~lineárního obalu lze vyjádøit
-jako lineární kombinaci daných vektorù. Podobnì platí i pro konvexní obaly, ¾e ka¾dý bod z~obalu je konvexní kombinací daných bodù. Ta se li¹í od
-lineární v~tom, ¾e v¹echny koeficienty jsou v~intervalu $[0,1]$ a navíc souèet v¹ech koeficientù je $1$. Tento algebraický pohled mù¾e mnohé vìci
-zjednodu¹it. Zkuste si dokázat, ¾e obì definice konvexního obalu jsou ekvivalentní.} Z~algoritmického hlediska nás v¹ak bude zajímat jenom jeho
-hranice, kterou budeme dále oznaèovat jako konvexní obal.
-
-Na¹ím úkolem je nalézt konvexní obal koneèné mno¾iny bodù. To je v¾dy konvexní mnohoúhelník, navíc s~vrcholy v~zadaných bodech. Øe¹ením problému tedy
-bude posloupnost bodù, které tvoøí konvexní obal. Pro malé mno¾iny je konvexní obal nakreslen na obrázku, pro více bodù je v¹ak situace mnohem
-slo¾itìj¹í.
-
-\figure{7-geom1_male_obaly.eps}{Konvexní obaly malých mno¾in.}{3in}
-
-Pro jednoduchost budeme pøedpokládat, ¾e v¹echny body mají rùzné $x$-ové souøadnice. Tedy utøídìní bodù zleva doprava je urèené jednoznaènì.\foot{To si
-mù¾eme dovolit pøedpokládat, nebo» se v¹emi body staèí nepatrnì pootoèit. Tím konvexní obal urèitì nezmìníme. Av¹ak jednodu¹¹í øe¹ení je naprogramovat
-tøídìní lexikograficky (druhotnì podle souøadnice $y$) a vyøadit identické body.} Tím máme zaji¹tìné, ¾e existují dva body, nejlevìj¹í a
-nejpravìj¹í, pro které platí následující invariant:
-
-\s{Invariant:} Nejlevìj¹í a nejpravìj¹í body jsou v¾dy v~konvexním obalu.
-
-Algoritmus na nalezení konvexního obalu funguje na následujícím jednoduchém principu, kterému se nìkdy øíká {\I zametání roviny}. Procházíme body
-zleva doprava a postupnì roz¹iøujeme doposud nalezený konvexní obal o~dal¹í body. Na zaèátku bude konvexní obal jediného bodu samotný bod. Na konci
-$k$-tého kroku algoritmu známe konvexní obal prvních $k$ bodù. Kdy¾ algoritmus skonèí, známe hledaný konvexní obal. Podle invariantu musíme v~$k$-tém
-kroku pøidat do obalu $k$-tý nejlevìj¹í bod. Zbývá si jen rozmyslet, jak pøesnì tento bod pøidat.
-
-Pøidání dal¹ího bodu do konvexního obalu funguje, jak je naznaèeno na obrázku. Podle invariantu víme, ¾e bod nejvíc vpravo je souèástí konvexního
-obalu. Za nìj napojíme novì pøidávaný bod. Tím jsme získali nìjaký obal, ale zpravidla nebude konvexní. To lze v¹ak snadno napravit, staèí
-odebírat body, v obou smìrech podél konvexního obalu, tak dlouho, dokud nezískáme konvexní obal. Na pøíkladu z obrázku nemusíme po smìru hodinových
-ruèièek odebrat ani jeden bod, obal je v poøádku. Naopak proti smìru hodinových ruèièek musíme odebrat dokonce dva body.
-
-\figure{7-geom2_pridani_bodu.eps}{Pøidání bodu do konvexního obalu.}{4.5in}
-
-Pro pøípadnou implementaci a rozbor slo¾itosti si nyní popí¹eme algoritmus detailnìji. Aby se lépe popisoval, rozdìlíme si konvexní obal na dvì èásti
-spojující nejlevìj¹í a nejpravìj¹í bod obalu. Budeme jim øíkat {\I horní obálka} a {\I dolní obálka}.
-
-\figure{7-geom3_obalky.eps}{Horní a dolní obálka konvexního obalu.}{3.4in}
-
-Obì obálky jsou lomené èáry, navíc horní obálka poøád zatáèí doprava a dolní naopak doleva. Pro udr¾ování bodù v~obálkách staèí dva zásobníky.
-V~$k$-tém kroku algoritmu pøidáme zvlá¹» $k$-tý bod do horní i dolní obálky. Pøidáním $k$-tého bodu se v¹ak mù¾e poru¹it smìr, ve kterém obálka
-zatáèí. Proto budeme nejprve body z~obálky odebírat a $k$-tý bod pøidáme a¾ ve chvíli, kdy jeho pøidání smìr zatáèení neporu¹í.
-
-\s{Algoritmus:}
-
-\algo
-
-\:Setøídíme body podle $x$-ové souøadnice, oznaème body $b_1, \ldots, b_n$.
-\:Vlo¾íme do horní a dolní obálky bod $b_1$: $H = D = (b_1)$.
-\:Pro ka¾dý dal¹í bod $b = b_2,\ldots,b_n$:
-\::Pøepoèítáme horní obálku:
-\:::Dokud $\vert H\vert \ge 2$, $H = (\ldots, h_{k-1}, h_k)$ a úhel $h_{k-1} h_k b$ je orientovaný doleva:
-\::::Odebereme poslední bod $h_k$ z~obálky $H$.
-\:::Pøidáme bod $b$ do obálky $H$.
-\::Symetricky pøepoèteme dolní obálku (s orientací doprava).
-\: Výsledný obal je tvoøen body v~obálkách $H$ a $D$.
-
-\endalgo
-
-Rozebereme si èasovou slo¾itost algoritmu. Setøídit body podle $x$-ové souøadnice doká¾eme v~èase $\O(n \log n)$. Pøidání dal¹ího bodu do obálek
-trvá lineárnì vzhledem k~poètu odebraných bodù. Zde vyu¾ijeme obvyklý postup: Ka¾dý bod je odebrán nejvý¹e jednou, a tedy v¹echna odebrání trvají
-dohromady $\O(n)$. Konvexní obal doká¾eme sestrojit v~èase $\O(n \log n)$, a pokud bychom mìli seznam bodù ji¾ utøídený, doká¾eme to dokonce v
-$\O(n)$.
-
-\s{Algebraický dodatek:} Existuje jednoduchý postup, jak zjistit orientaci úhlu? Uká¾eme si jeden zalo¾ený na lineární algebøe. Budou se hodit
-vlastnosti determinantu. Absolutní hodnota determinantu je objem rovnobì¾nostìnu urèeného øádkovými vektory matice. Dùle¾itìj¹í v¹ak je, ¾e znaménko
-determinantu urèuje \uv{orientaci} vektorù, zda je levotoèivá èi pravotoèivá. Proto¾e ná¹ problém je rovinný, budeme uva¾ovat determinanty matic $2
-\times 2$.
-
-Uva¾me souøadnicový systém v~rovinì, kde $x$-ová souøadnice roste smìrem doprava a~$y$-ová smìrem nahoru. Chceme zjistit orientaci úhlu $h_{k-1} h_k
-b$. Polo¾me $\vec u = (x_1, y_1)$ jako rozdíl souøadnic $h_k$ a~$h_{k-1}$ a podobnì $\vec v = (x_2, y_2)$ je rozdíl souøadnic $b$ a~$h_k$. Matice $M$
-je definována následovnì:
-$$M = \pmatrix{\vec u \cr \vec v} = \pmatrix {x_1&y_1\cr x_2&y_2}.$$
-Úhel $h_{k-1} h_k b$ je orientován doleva, právì kdy¾ $\det M = x_1y_2 - x_2y_1$ je nezáporný,\foot{Neboli vektory $\vec u$ a $\vec v$ odpovídají
-rozta¾ení a zkosení vektorù báze $\vec x = (1,0)$ a $\vec y = (0,1)$, pro nì¾ je determinant nezáporný.} a spoèítat hodnotu determinantu je jednoduché.
-Mo¾né situace jsou nakresleny na obrázku. Poznamenejme, ¾e k~podobnému vzorci se lze také dostat pøes vektorový souèin vektorù $\vec u$ a $\vec v$.
-
-\figure{7-geom4_determinant.eps}{Jak vypadají determinanty rùzných znamének v~rovinì.}{4.6in}
-
-\s{©lo by to vyøe¹it rychleji?} Také vám vrtá hlavou, zda existují rychlej¹í algoritmy? Na závìr si uká¾eme nìco, co na pøedná¹ce nebylo.\foot{A také
-se nebude zkou¹et.} Nejrychlej¹í známý algoritmus, jeho¾ autorem je T.~Chan, funguje v~èase $\O(n \log h)$, kde $h$ je poèet bodù le¾ících na
-konvexním obalu, a pøitom je pøekvapivì jednoduchý. Zde si naznaèíme, jak tento algoritmus funguje.
-
-Algoritmus pøichází s~následující my¹lenkou. Pøedpokládejme, ¾e bychom znali velikost konvexního obalu $h$. Rozdìlíme body libovolnì do $\lceil {n
-\over h} \rceil$ mno¾in $Q_1, \ldots, Q_k$ tak, ¾e $\vert Q_i \vert \le h$. Pro ka¾dou z~tìchto mno¾in nalezneme konvexní obal pomocí vý¹e popsaného
-algoritmu. To doká¾eme pro jednu v~èase $\O(h \log h)$ a pro v¹echny v~èase $\O(n \log h)$. V druhé fázi spustíme hledání konvexního obalu pomocí
-provázkového algoritmu a pro zrychlení pou¾ijeme pøedpoèítané obaly men¹ích mno¾in. Nejprve popí¹eme jeho my¹lenku. Pou¾ijeme následující pozorování:
-
-\s{Pozorování:} Úseèka spojující dva body $a$ a $b$ le¾í na konvexním obalu, právì kdy¾ v¹echny ostatní body le¾í pouze na jedné její
-stranì.\foot{Formálnì je podmínka následující: Pøímka $ab$ urèuje dvì poloroviny. Úseèka le¾í na konvexním obalu, právì kdy¾ v¹echny body le¾í v jedné
-z polorovin.}
-
-Algoritmu se øíká {\I provázkový}, proto¾e svojí èinností pøipomíná namotávání provázku podél konvexního obalu. Zaèneme s bodem, který na konvexním
-obalu urèitì le¾í, to je tøeba ten nejlevìj¹í. V ka¾dém kroku nalezneme následující bod po obvodu konvexního obalu. To udìláme napøíklad tak, ¾e
-projdeme v¹echny body a vybereme ten, který svírá nejmen¹í úhel s poslední stranou konvexního obalu. Novì pøidaná úseèka vyhovuje pozorování a proto
-do konvexního obalu patøí. Po $h$ krocích se dostaneme zpìt k nejlevìj¹ímu bodu a výpoèet ukonèíme. V ka¾dém kroku potøebujeme projít v¹echny body a
-vybrat následníka, co¾ doká¾eme v èase $\O(n)$. Celková slo¾itost algoritmu je tedy $\O(n \cdot h)$.
-
-\twofigures{7-geom6_provazkovy_algoritmus.eps}{Provázkový algoritmus.}{1.25in}{7-geom7_naslednik_pres_konvexni_obal.eps}{Hledání kandidáta v pøedpoèítaném obalu.}{2.5in}
-
-Provázkový algoritmus funguje, ale má jednu obrovskou nevýhodu -- je toti¾ ukrutnì pomalý. Ký¾eného zrychlení dosáhneme, pokud pou¾ijeme pøedpoèítané
-konvexní obaly. Ty umo¾ní rychleji hledat následníka. Pro ka¾dou z mno¾in $Q_i$ najdeme zvlá¹» kandidáta a poté z nich vybereme toho nejlep¹ího.
-Mo¾ný kandidát v¾dy le¾í na konvexním obalu mno¾iny $Q_i$. Vyu¾ijeme toho, ¾e body obalu jsou \uv{uspoøádané}, i kdy¾ trochu netypicky do kruhu.
-Kandidáta mù¾eme hledat metodou pùlení intervalu, i kdy¾ detaily jsou malièko slo¾itìj¹í ne¾ je obvyklé. Jak pùlit zjistíme podle smìru zatáèení
-konvexního obalu. Detaily si rozmyslí ètenáø sám.
-
-Èasová slo¾itost pùlení je $\O(\log h)$ pro jednu mno¾inu. Mno¾in je nejvý¹e $\O({n \over h})$, tedy následující bod konvexního obalu nalezneme v èase
-$\O({n \over h} \log h)$. Celý obal nalezneme ve slibovaném èase $\O(n \log h)$.
-
-Popsanému algoritmu schází jedna dùle¾itá vìc: Ve skuteènosti vìt¹inou neznáme velikost $h$. Budeme proto algoritmus iterovat s~rostoucí hodnotou $h$,
-dokud konvexní obal nesestrojíme. Pokud pøi slepování konvexních obalù zjistíme, ¾e konvexní obal je vìt¹í ne¾ $h$, výpoèet ukonèíme. Zbývá je¹tì
-zvolit, jak rychle má $h$ rùst. Pokud by rostlo moc pomalu, budeme poèítat zbyteènì mnoho fází, naopak pøi rychlém rùstu by nás poslední fáze mohla
-stát pøíli¹ mnoho.
-
-V~$k$-té iteraci polo¾íme $h = 2^{2^k}$. Dostáváme celkovou slo¾itost algoritmu:
-$$\sum_{m=0}^{\O(\log \log h)} \O(n \log 2^{2^m}) = \sum_{m=0}^{\O(\log \log h)} \O(n \cdot 2^m) = \O(n \log h),$$
-kde poslední rovnost dostaneme jako souèet prvních $\O(\log \log h)$ èlenù geometrické øady $\sum 2^m$.
-
-\bye
+++ /dev/null
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+++ /dev/null
-P=7-geom
-
-include ../Makerules
+++ /dev/null
-% implementation of figure naming and figure transparency
-string name,tag; name := ""; tag := "";
-def updatefigname =
- if tag="": filenametemplate (name & "%c.eps");
- else: filenametemplate (name & "%c_" & tag & ".eps");
- fi;
-enddef;
-
-def figname(expr n) = name := n; updatefigname; enddef;
-def figtag(expr t) = tag := t; updatefigname; enddef;
-
-picture transparent_picture;
-color transparent_color; transparent_color := 0.9white;
-def drawtransparent(expr num) =
- transparent_picture := currentpicture;
-endfig;
-
-if tag="": filenametemplate (name & "%c_transparent.eps");
-else: filenametemplate (name & "%c_" & tag & "_transparent.eps");
-fi;
-beginfig(num);
- draw transparent_picture withcolor transparent_color;
-endfig;
-
-updatefigname;
-enddef;
-
-def from(expr p,d,len) =
- p+dir(d)*len
-enddef;
-
-def dirs(expr p,d,len) =
- p--from(p,d,len)
-enddef;
-
-def drawvertices(expr s,n) =
- for i:=s upto n:
- draw vertex(PQ[i]);
- endfor
-enddef;
-
-def drawfvertices(expr s,n,flags) =
- for i:=s upto n:
- draw vertex(PQ[i]) flags;
- endfor
-enddef;
-
-def vertex(expr p) = p withpen pencircle scaled 4pt enddef;
-def drawemptyvertex(expr p) = unfill fullcircle scaled 4pt shifted p; draw fullcircle scaled 4pt shifted p; enddef;
-def drawendpointvertex(expr p) = draw vertex(p) withcolor red; draw fullcircle scaled 6pt shifted p; enddef;
-def createpath(expr p) = shakepath(p, 0.015cm,0.1cm) enddef;
-vardef shakepath(expr p,d,l) =
- save r,b;
- path r; r := point(arctime 0 of p) of p;
- b := -1;
- for i:=l step l until arclength(p):
- r := r--(point(arctime i of p) of p)+dir(angle(direction(arctime i of p) of p rotated 90))*d*b;
- b := -b;
- endfor
- r--point(arctime arclength(p) of p) of p
-enddef;
-
-pen normalpen; normalpen := pencircle scaled 0.6pt;
-pen boldpen; boldpen := pencircle scaled 1.5pt;
-pen bolderpen; bolderpen := pencircle scaled 2pt;
-def dotline = withdots scaled 0.82 withpen boldpen enddef;
-
-vardef unclosedbubblec(expr p,c) =
- bubblec((p..reverse p..cycle),c)
-enddef;
-
-vardef bubblec(expr p,c) =
- save r;
- path r; r := (point(arctime 0 of p) of p)+dir(angle(direction(arctime 0 of p) of p rotated 90))*c;
- for i:=0.01cm step 0.025cm until arclength(p):
- r := r..(point(arctime i of p) of p)+dir(angle(direction(arctime i of p) of p rotated 90))*c;
- endfor
- r..(point(arctime arclength(p) of p) of p)+dir(angle(direction(arctime arclength(p) of p) of p rotated 90))*c..cycle
-enddef;
-
-vardef bubble(expr p) = bubblec(p,0.12cm) enddef;
-vardef unclosedbubble(expr p) = unclosedbubblec(p,0.12cm) enddef;
+++ /dev/null
-input lib
-
-figname("8-geom2_");
-figtag("usecky");
-beginfig(1);
- def drawusecka(expr p,q) = draw vertex(p); draw vertex(q); draw p--q; enddef;
- pair A[],B[];
- A0 := origin; B0 := (4cm,1.5cm);
- A1 := (0.5cm,1.5cm); B1 := (1cm,0);
- A2 := (2cm,1.3cm); B2 := (3cm, -0.9cm);
- A3 := (1cm,-1.3cm); B3 := (3.5cm,0);
- z0 = whatever[A0,B0]; z0 = whatever[A1,B1];
- z1 = whatever[A0,B0]; z1 = whatever[A2,B2];
- z2 = whatever[A2,B2]; z2 = whatever[A3,B3];
-
- for i:=0 upto 2: fill fullcircle scaled 7pt shifted z[i]; drawemptyvertex(z[i]); endfor
- for i:=0 upto 3: drawusecka(A[i], B[i]); endfor
-endfig;
-
-figtag("polorovina");
-beginfig(2);
- pair A,B;
- A := origin;
- B := (1.7cm,0.5cm);
- an := angle(B-A);
- path p; p := from(.5[A,B],an+90,3cm)--from(.5[A,B],an-90,3cm);
- fill p--reverse(p shifted (A-0.8[A,B]))--cycle withcolor 0.8white;
- %fill p{dir (an+180)}..-0.35[A,B]..{dir an}cycle withcolor 0.8white;
- draw p withpen boldpen;
-
- pair C; C := point(0.9) of p;
- drawarrow from(C,an+180,0.1cm)--from(C,an+180,1cm) withpen boldpen;
- C := point(0.1) of p;
- drawarrow from(C,an+180,0.1cm)--from(C,an+180,1cm) withpen boldpen;
- label.rt(btex $p$ etex, point(0.15) of p);
-
- draw A--B dashed evenly;
- drawemptyvertex(A); drawemptyvertex(B);
- label.llft(btex $a$ etex, A);
- label.urt(btex $b$ etex, B);
- label(btex $B_a$ etex, A+2(-0.1cm,0.5cm));
- label(btex $B_b$ etex, B+2(-0.1cm,0.5cm));
-endfig;
-
-figtag("voroneho_diagram");
-beginfig(3);
- u := 1.35cm;
- pair A[],B[];
- A0 := origin; A1 := (-u,-u); A2 := (1.3u,-u); A3 := (1.50821u,0.7u); A4 := (0.6u,0.8u); A5 := (0.2u,1.9u); A6 := (-1.3u,1u);
- def osa(expr a, b, an,l) = from(.5[a,b], angle(b-a)+an, l) enddef;
- vardef prusecik_os(expr p,q,r) =
- save b; pair b;
- b = whatever[osa(p,q, 90,1cm), osa(p,q,-90,1cm)];
- b = whatever[osa(q,r, 90,1cm), osa(q,r,-90,1cm)];
- b
- enddef;
- B[0] := prusecik_os(A0,A1,A2); B[1] := prusecik_os(A0,A1,A6); B[2] := prusecik_os(A0,A4,A6); B[3] := prusecik_os(A4,A5,A6);
- B[4] := prusecik_os(A3,A4,A5); B[5] := prusecik_os(A0,A3,A4); B[6] := prusecik_os(A0,A2,A4);
- draw osa(A1,A2,-90,2cm)--B[0]--B[1]--osa(A1,A6,90,1.3cm) withpen boldpen;
- draw B[1]--B[2]--B[3]--osa(A6,A5,90,2cm) withpen boldpen;
- draw B[3]--B[4]--osa(A3,A5,-90,2.7cm) withpen boldpen;
- draw B[4]--B[5]--B[2] withpen boldpen;
- draw B[5]--B[6]--osa(A2,A3,-90,1.3cm) withpen boldpen;
- draw B[6]--B[0] withpen boldpen;
-
- draw A0--A1--A2--A0--A6--A1 dashed evenly;
- draw A6--A5--A4--A0 dashed evenly; draw A3--A2 dashed evenly;
- draw A3--A5 dashed evenly; draw A6--A4--A3 dashed evenly;
-
- for i:=0 upto 6: draw vertex(B[i]); endfor
- for i:=0 upto 6: drawemptyvertex(A[i]); endfor
-endfig;
-
-figtag("pasy_mnohouhelniku");
-beginfig(4);
- u := 1.35cm;
- pair A[],B[];
- A0 := origin; A1 := (-u,-u); A2 := (1.3u,-u); A3 := (1.50821u,0.7u); A4 := (0.6u,0.8u); A5 := (0.2u,1.9u); A6 := (-1.3u,1u);
- def osa(expr a, b, an,l) = from(.5[a,b], angle(b-a)+an, l) enddef;
- vardef prusecik_os(expr p,q,r) =
- save b; pair b;
- b = whatever[osa(p,q, 90,1cm), osa(p,q,-90,1cm)];
- b = whatever[osa(q,r, 90,1cm), osa(q,r,-90,1cm)];
- b
- enddef;
- def drawline(expr p) = draw ((-2.3u,p)--(2.5u,p)) cutbefore (D0--D1) cutafter (D2--D3) enddef;
- B[0] := prusecik_os(A0,A1,A2); B[1] := prusecik_os(A0,A1,A6); B[2] := prusecik_os(A0,A4,A6); B[3] := prusecik_os(A4,A5,A6);
- B[4] := prusecik_os(A3,A4,A5); B[5] := prusecik_os(A0,A3,A4); B[6] := prusecik_os(A0,A2,A4);
-
- pair C; C := origin; for i:=0 upto 6: C := C + B[i]; endfor C := C/6;
- pair D[];
- D0 := C+(-2.25,2.25)*u; D1 := C+(-2.25,-2.25)*u; D2 := C+(2.25,-2.25)*u; D3 := C+(2.25,2.25)*u;
-
- draw B[0]--B[1]--B[2]--B[3]--B[4]--B[5]--B[6]--B[0] withpen boldpen;
- draw B[5]--B[2] withpen boldpen;
- pair E; E := 0.6[B0,B4];
- drawline(ypart(E));
- drawline(ypart(E)) cutbefore (B2--B5) cutafter (B4--B5) dashed evenly withpen bolderpen;
- drawemptyvertex(E);
-
- for i:=0 upto 6: drawline(ypart(B[i])) dashed evenly; endfor
- for i:=0 upto 6: draw vertex(B[i]); endfor
-endfig;
-
-figtag("upravy_stromu");
-beginfig(5);
- u := 1cm;
- draw (0,0.1u)--(2.05u,-2.05u)--(-2.05u,-2.05u)--cycle;
- pair A[]; A0 := from(origin,-90,0.1u); A1 := from(A0, -70, 0.5u); A2 := from(A1, -110, 0.5u); A3 := from(A2, -80, 0.5u); A4 := from(A3, -120, 0.45u);
- path p; p := createpath(A0--A1--A2--A3--A4);
- path q; q := (p scaled 0.93 shifted (-0.15u,-0.15u))--(-1.85u,-1.95u)--cycle; fill q withcolor 0.8white; draw q;
- q := (p scaled 0.93 shifted (0.15u,-0.15u))--(1.85u,-1.95u)--cycle; fill q withcolor 0.8white; draw q;
- p := createpath(A0--from(A0,-110,0.1u)--A1--A2--A3--A4); draw p withpen boldpen;
-endfig;
-
-figtag("rychla_perzistence");
-beginfig(6);
- u := 1cm;
- def drawtable(expr p, lab, sa, sb) =
- draw centersquare xscaled 2u yscaled (2u/3) shifted (p+(0,u/3));
- draw centersquare xscaled 2u yscaled (2u/3) shifted (p-(0,u/3));
- label(sa, p+(0,u/3));
- label(sb, p-(0,u/3));
- label.bot(lab, p-(0,2u/3));
- enddef;
-
- pair A[];
- for i:=0 upto 2: A[i] := (0,2u) rotated 120i; endfor
- drawtable(A1, btex $v$ etex, btex verze 2 etex, btex verze 1 etex);
- drawtable(A2, btex $v'$ etex, btex verze 3 etex, "");
- drawtable(A0, btex $u$ etex, btex verze 2 etex, btex verze 1 etex);
-
- drawarrow from(A[2]+(0,u/3), 180, 7u/6)--from(A[1]+(0,u/3), 0, 7u/6);
- drawarrow from(A[0]+(0,-u/3), 180, 7u/6)..{dir -90}from(A[1]+(-3u/4,2u/3), 90, u/6);
- drawarrow from(A[0]+(0,u/3), 0, 7u/6)..{dir -90}from(A[2]+(3u/4,2u/3), 90, u/6);
-endfig;
-end
+++ /dev/null
-\input lecnotes.tex
-
-\prednaska{8}{Geometrie vrací úder}{(sepsal Pavel Klavík)}
-
-\>Kdy¾ s geometrickými problémy poøádnì nezametete, ony vám to vrátí! Ale kdy¾ u¾ zametat, tak urèitì ne pod koberec a místo smetáku pou¾ijte pøímku.
-V této pøedná¹ce nás spolu s dvìma geometrickými problémy samozøejmì èeká pokraèování pohádky o ledních medvìdech.
-
-{\I Medvìdi vyøe¹ili rybí problém a hlad je ji¾ netrápí. Av¹ak na severu ne¾ijí sami, za sousedy mají Eskymáky. Proto¾e je rozhodnì lep¹í se sousedy
-dobøe vycházet, jsou medvìdi a Eskymáci velcí pøátelé. Skoro ka¾dý se se svými pøáteli rád schází. Av¹ak to je musí nejprve nalézt~\dots}
-
-\h{Hledání prùseèíkù úseèek}
-
-Zkusíme nejprve Eskymákùm vyøe¹it lokalizaci ledních medvìdù.
-
-{\I Kdy¾ takový medvìd nemá co na práci, rád se prochází. Na místech, kde se trasy protínají, je zvý¹ená ¹ance, ¾e se dva medvìdi potkají a zapovídají
--- ostatnì co byste èekali od medvìdù. To jsou ta správná místa pro Eskymáka, který chce potkat medvìda. Jenom¾e jak tato køí¾ení najít?}
-
-Pro zjednodu¹ení pøedpokládejme, ¾e medvìdi chodí po úseèkách tam a zpìt. Budeme tedy chtít nalézt v¹echny prùseèíky úseèek v rovinì.
-
-\bigskip
-\centerline{\epsfxsize=1.5in\epsfbox{8-geom2_0_bear.eps}\hskip 4em\epsfxsize2in\epsfbox{8-geom2_1_usecky.eps}}
-\smallskip
-\centerline{Problém Eskymákù: Kde v¹ude se køí¾í medvìdí trasy?}
-\bigskip
-
-Pro $n$ úseèek mù¾e existovat a¾ $\Omega(n^2)$ prùseèíkù.\foot{Zkuste takový pøíklad zkonstruovat.} Tedy optimální slo¾itosti by dosáhl i algoritmus,
-který by pro ka¾dou dvojici úseèek testoval, zda se protínají. Èasovou slo¾itost algoritmu v¹ak posuzujeme i vzhledem k velikosti výstupu $p$. Typické
-rozmístìní úseèek mívá toti¾ prùseèíkù spí¹e pomálu. Pro tento pøípad si uká¾eme podstatnì rychlej¹í algoritmus.
-
-Pro jednodu¹¹í popis pøedpokládejme, ¾e úseèky le¾í v obecné poloze. To znamená, ¾e ¾ádné tøi úseèky se neprotínají v jednom bodì a prùnikem ka¾dých
-dvou úseèek je nejvý¹e jeden bod. Navíc pøedpokládejme, ¾e krajní bod ¾ádné úseèky nele¾í na jiné úseèce a také neexistují vodorovné úseèky. Na závìr si
-uká¾eme, jak se s tìmito pøípady vypoøádat.
-
-Algoritmus funguje na principu zametání roviny, popsaném v minulé pøedná¹ce. Budeme posouvat vodorovnou pøímku odshora dolù. V¾dy, kdy¾ narazíme na
-nový prùnik, ohlásíme jeho výskyt. Samozøejmì spojité posouvání nahradíme diskrétním a pøímku v¾dy posuneme do dal¹ího zajímavého bodu.
-
-Zajímavé události jsou {\I zaèátky úseèek}, {\I konce úseèek} a {\I prùseèíky úseèek}. Po utøídìní známe pro první dva typy událostí poøadí, v jakém
-se objeví. Výskyty prùseèíkù budeme poèítat prùbì¾nì, jinak bychom celý problém nemuseli øe¹it.
-
-V ka¾dém kroku si pamatujeme {\I prùøez} $P$ -- posloupnost úseèek aktuálnì protnutých zametací pøímkou. Tyto úseèky máme utøídìné zleva doprava. Navíc si
-udr¾ujeme kalendáø $K$ budoucích událostí. Z hlediska prùseèíkù budeme na úseèky nahlí¾et jako na polopøímky. Pro sousední dvojice úseèek si
-udr¾ujeme, zda se jejich smìry nìkde protnou. Algoritmus pro hledání prùnikù úseèek funguje následovnì:
-
-\s{Algoritmus:}
-
-\algo
-
-\:$P \leftarrow \emptyset$.
-\:Do $K$ vlo¾íme zaèátky a konce v¹ech úseèek.
-\:Dokud $K \ne \emptyset$:
-\::Odebereme nejvy¹¹í událost.
-\::Pokud je to zaèátek úseèky, zatøídíme novou úseèku do $P$.
-\::Pokud je to konec úseèky, odebereme úseèku z $P$.
-\::Pokud je to prùseèík, nahlásíme ho a prohodíme úseèky v $P$.
-\::Navíc v¾dy pøepoèítáme prùseèíkové události, v¾dy maximálnì dvì odebereme a dvì nové pøidáme.
-\endalgo
-
-Zbývá rozmyslet si, jaké datové struktury pou¾ijeme, abychom prùseèíky nalezli dostateènì rychle. Pro kalendáø pou¾ijeme napøíklad haldu. Prùøez si
-budeme udr¾ovat ve vyhledávacím stromì. Poznamenejme, ¾e nemusíme znát souøadnice úseèek, staèí znát jejich poøadí, které se mezi jednotlivými
-událostmi nemìní. Pøi pøidávání úseèek procházíme stromem a porovnáváme souøadnice v prùøezu, které prùbì¾nì dopoèítáváme.
-
-Kalendáø obsahuje v¾dy nejvý¹e $\O(n)$ událostí. Podobnì prùøez obsahuje v ka¾dém okam¾iku nejvý¹e $\O(n)$ úseèek. Jednu událost kalendáøe doká¾eme
-o¹etøit v èase $\O(\log n)$. V¹ech událostí je $\O(n+p)$, a tedy celková slo¾itost algoritmu je $\O((n+p) \log n)$.
-
-Slíbili jsme, ¾e popí¹eme, jak se vypoøádat s vý¹e uvedenými podmínkami na vstup. Události kalendáøe se stejnou $y$-ovou souøadnicí budeme tøídit v
-poøadí zaèátky, prùseèíky a konce úseèek. Tím nahlásíme i prùseèíky krajù úseèek a ani vodorovné úseèky nebudou vadit. Podobnì se není tøeba obávat
-prùseèíkù více úseèek v jednom bodì. Úseèky jdoucí stejným smìrem, jejich¾ prùnik je úseèka, jsou komplikovanìj¹í, ale lze jejich prùseèíky o¹etøit a
-vypsat tøeba souøadnice úseèky tvoøící jejich prùnik.
-
-Na závìr poznamenejme, ¾e Balaban vymyslel efektivnìj¹í algoritmus, který funguje v èase $\O(n \log n + p)$, ale je podstatnì komplikovanìj¹í.
-
-\h{Hledání nejbli¾¹ích bodù a Voroného diagramy}
-
-Nyní se pokusíme vyøe¹it i problém druhé strany -- pomù¾eme medvìdùm nalézt Eskymáky.
-
-{\I Eskymáci tráví vìt¹inu èasu doma, ve svém iglù. Takový medvìd je na své toulce zasnì¾enou krajinou, kdy¾ tu se najednou rozhodne nav¹tívit nìjakého
-Eskymáka. Proto se podívá do své medvìdí mapy a nalezne nejbli¾¹í iglù. Má to ale jeden háèek, iglù jsou spousty a medvìd by dávno usnul, ne¾ by
-nejbli¾¹í objevil.}\foot{Zlí jazykové by øekli, ¾e medvìdi jsou moc líní a nebo v mapách ani èíst neumí!}
-
-Popí¹eme si nejprve, jak vypadá medvìdí mapa. Medvìdí mapa obsahuje celou Arktidu a jsou v ní vyznaèena v¹echna iglù. Navíc obsahuje vyznaèené
-oblasti tvoøené body, které jsou nejblí¾e k jednomu danému iglù. Takovému schématu se øíká {\I Voroného diagram}. Ten pro zadané body $x_1, \ldots, x_n$
-obsahuje rozdìlení roviny na oblasti $B_1, \ldots, B_n$, kde $B_i$ je mno¾ina bodù, které jsou blí¾e k $x_i$ ne¾ k ostatním bodùm $x_j$. Formálnì jsou
-tyto oblasti definovány následovnì:
-$$B_i = \left\{y \in {\bb R}^2\ \vert\ \forall j:\rho(x_i,y) \le \rho(x_j,y)\right\},$$
-kde $\rho(x,y)$ znaèí vzdálenost bodù $x$ a $y$.
-
-Uká¾eme si, ¾e Voroného diagram má pøekvapivì jednoduchou strukturu. Nejprve uva¾me, jak budou vypadat oblasti $B_a$ a $B_b$ pouze pro dva body
-$a$ a $b$, jak je naznaèeno na obrázku. V¹echny body stejnì vzdálené od $a$ i $b$ le¾í na pøímce $p$ -- ose úseèky $ab$. Oblasti $B_a$ a $B_b$
-jsou tedy tvoøeny polorovinami ohranièenými osou $p$. Tedy obecnì tvoøí mno¾ina v¹ech bodù bli¾¹ích k $x_i$ ne¾ k $x_j$ nìjakou polorovinu. Oblast
-$B_i$ obsahuje v¹echny body, které jsou souèasnì bli¾¹í k $x_i$ ne¾ ke v¹em ostatním bodùm $x_j$ -- tedy le¾í ve v¹ech polorovinách souèasnì.
-Ka¾dá z oblastí $B_i$ je tvoøena prùnikem $n-1$ polorovin, tedy je to (mo¾ná neomezený) mnohoúhelník.\foot{Sly¹eli jste u¾ o lineárním programování?
-Jak název vùbec nenapoví, {\I lineární programování} je teorii zabývající se øe¹ením a vlastnostmi soustav lineárních nerovnic. Lineární program je
-popsaný lineární funkcí, kterou chceme maximalizovat za podmínek popsaných soustavou lineárních nerovnic. Ka¾dá nerovnice urèuje poloprostor, ve
-kterém se pøípustná øe¹ení nachází. Proto¾e pøípustné øe¹ení splòuje v¹echny nerovnice zároveò, je mno¾ina v¹ech pøípustných øe¹ení (mo¾ná neomezený)
-mnohostìn, obecnì ve veliké dimenzi ${\bb R}^d$, kde $d$ je poèet promìnných. Mno¾iny $B_i$ lze snadno popsat jako mno¾iny v¹ech pøípustných øe¹ení
-lineárních programù pomocí vý¹e ukázaných polorovin. Na závìr poznamenejme, ¾e dlouho otevøená otázka, zda lze nalézt optimální øe¹ení lineárního
-programu v polynomiálním èase, byla pozitivnì vyøe¹ena -- je znám polynomiální algoritmus, kterému se øíká {\I metoda vnitøního bodu}. Na druhou
-stranu, pokud chceme najít pøípustné celoèíselné øe¹ení, je úloha NP-úplná a je jednoduché na ni pøevést spoustu optimalizaèních problémù. Dokázat
-NP-tì¾kost není pøíli¹ tì¾ké. Na druhou stranu ukázat, ¾e tento problém le¾í v NP, není vùbec jednoduché.}
-Pøíklad Voroného diagramu je naznaèen na obrázku. Zadané body jsou oznaèeny prázdnými krou¾ky a hranice oblastí $B_i$ jsou vyznaèené èernými èárami.
-
-\twofigures{8-geom2_2_polorovina.eps}{Body bli¾¹í k $a$ ne¾ $b$.}{1.25in}{8-geom2_3_voroneho_diagram.eps}{Voroného diagram.}{2.5in}
-
-Není náhoda, pokud vám hranice oblastí pøipomíná rovinný graf. Jeho vrcholy jsou body, které jsou stejnì vzdálené od alespoò tøí zadaných bodù. Jeho
-stìny jsou oblasti $B_i$. Jeho hrany jsou tvoøeny èástí hranice mezi dvìma oblastmi -- body, které mají dvì oblasti spoleèné. Obecnì prùnik dvou
-oblastí mù¾e být, v závislosti na jejich sousedìní, prázdný, bod, úseèka, polopøímka nebo dokonce celá pøímka. V dal¹ím textu si pøedstavme, ¾e celý
-Voroného diagram uzavøeme do dostateènì velkého obdélníka,\foot{Pøeci jenom i celá Arktida je omezenì velká.} èím¾ dostaneme omezený rovinný graf.
-
-Poznamenejme, ¾e pøeru¹ované èáry tvoøí hrany duálního rovinného grafu s vrcholy v zadaných bodech. Hrany spojují sousední body na kru¾nicích, které
-obsahují alespoò tøi ze zadaných bodù. Napøíklad na obrázku dostáváme skoro samé trojúhelníky, proto¾e vìt¹ina kru¾nic obsahuje pøesnì tøi zadané
-body. Av¹ak nalezneme i jeden ètyøúhelník, jeho¾ vrcholy le¾í na jedné kru¾nici.
-
-Zkusíme nyní odhadnout, jak velký je rovinný graf popisující Voroného diagram. Podle slavné Eulerovy formule má ka¾dý rovinný graf nejvý¹e lineárnì
-mnoho vrcholù, hran a stìn -- pro $v$ vrcholù, $e$ hran a $f$ stìn je $e \le 3v-6$ a navíc $v+f = e+2$. Tedy slo¾itost diagramu je lineární vzhledem k
-poètu zadaných bodù $n=f$, $\O(n)$. Navíc Voroného diagram lze zkonstruovat v èase $\O(n \log n)$, napøíklad pomocí zametání roviny nebo metodou
-rozdìl a panuj. Tím se v¹ak zabývat nebudeme,\foot{Pro zvídavé, kteøí nemají zkou¹ku druhý den ráno: Detaily naleznete v zápiscích z pøedloòského
-ADSka.} místo toho si uká¾eme, jak v ji¾ spoèteném Voroného diagramu rychle hledat nejbli¾¹í body.
-
-\h{Lokalizace bodu uvnitø mnohoúhelníkové sítì}
-
-Problém medvìdù je najít v medvìdí mapì co nejrychleji nejbli¾¹í iglù. Máme v rovinì sí» tvoøenou mnohoúhelníky. Chceme pro jednotlivé body rychle
-rozhodovat, do kterého mnohoúhelníku patøí. Na¹e øe¹ení budeme optimalizovat pro jeden pevný rozklad a obrovské mno¾ství rùzných dotazù, které chceme
-co nejrychleji zodpovìdìt.\foot{Pøedstavujme si to tøeba tak, ¾e medvìdùm zprovozníme server. Ten jednou schroustá celou mapu a potom co nejrychleji
-odpovídá na jejich dotazy. Medvìdi tak nemusí v mapách nic hledat, staèí se pøipojit na server a poèkat na odpovìï.} Nejprve pøedzpracujeme zadané
-mnohoúhelníky a vytvoøíme strukturu, která nám umo¾ní rychlé dotazy na jednotlivé body.
-
-Uka¾me si pro zaèátek øe¹ení bez pøedzpracování. Rovinu budeme zametat pøímkou shora dolù. Podobnì jako pøi hledání prùseèíkù úseèek, udr¾ujeme si prùøez
-pøímkou. V¹imnìte si, ¾e tento prùøez se mìní jenom ve vrcholech mnohoúhelníkù. Ve chvíli, kdy narazíme na hledaný bod, podíváme se, do kterého
-intervalu v prùøezu patøí. To nám dá mnohoúhelník, který nahlásíme. Prùøez budeme uchovávat ve vyhledávacím stromì. Takové øe¹ení má slo¾itost $\O(n
-\log n)$ na dotaz, co¾ je hroznì pomalé.
-
-Pøedzpracování bude fungovat následovnì. Jak je naznaèeno na obrázku pøeru¹ovanými èárami, rozøe¾eme si celou rovinu na pásy, bìhem kterých se prùøez
-pøímkou nemìní. Pro ka¾dý z nich si pamatujeme stav stromu popisující, jak vypadal prùøez pøi procházení tímto pásem. Kdy¾ chceme lokalizovat nìjaký bod,
-nejprve pùlením nalezneme pás, ve kterém se nachází. Poté polo¾íme dotaz na pøíslu¹ný strom. Strom procházíme a po cestì si dopoèítáme souøadnice
-prùøezu, a¾ lokalizujeme správný interval v prùøezu. Dotaz doká¾eme zodpovìdìt v èase $\O(\log n)$. Hledaný bod je na obrázku naznaèen prázdným
-koleèkem a nalezený interval v prùøezu je vyta¾ený tuènì.
-
-\figure{8-geom2_4_pasy_mnohouhelniku.eps}{Mnohoúhelníky rozøezané na pásy.}{2.5in}
-
-Jenom¾e na¹e øe¹ení má jeden háèek: Jak zkonstruovat jednotlivé verze stromu dostateènì rychle? K tomu napomohou {\I èásteènì perzistentní} datové
-struktury. Pod perzistencí se myslí, ¾e struktura umo¾òuje uchovávat svoji historii. Èásteènì perzistentní struktury nemohou svoji historii
-modifikovat.
-
-Popí¹eme si, jak vytvoøit perzistentní strom s pamìtí $\O(\log n)$ na zmìnu. Pokud provádíme operaci na stromì, mìní se jenom malá èást stromu.
-Napøíklad pøi vkládání do stromu se mìní jenom prvky na jedné cestièce z koøene do listu (a pøípadnì rotací i na jejím nejbli¾¹ím okolí). Proto si
-ulo¾íme upravenou cestièku a zbytek stromu budeme sdílet s pøedchozí verzí. Na obrázku je vyznaèena cesta, její¾ vrcholy jsou upravovány. ©edì
-oznaèené podstromy navì¹ené na tuto cestu se nemìní, a proto na nì staèí zkopírovat ukazatele. Mimochodem zmìny ka¾dé operace se slo¾itostí $\O(k)$
-lze zapsat v pamìti $\O(k)$, prostì operace nemá tolik èasu, aby mohla pozmìnit pøíli¹ velikou èást stromu.
-
-\figure{8-geom2_5_upravy_stromu.eps}{Jedna operace mìní pouze okolí cesty -- navì¹ené podstromy se nemìní.}{2in}
-
-Celková èasová slo¾itost je tedy $\O(n \log n)$ na pøedzpracování Voroného diagramu a vytvoøení persistentního stromu. Kvùli persistenci potøebuje
-toto pøedzpracování pamì» $\O(n \log n)$. Na dotaz spotøebujeme èas $\O(\log n)$, nebo» nejprve vyhledáme pùlením pøíslu¹ný pás a poté polo¾íme dotaz
-na pøíslu¹nou verzi stromu. Rychleji to ani provést nepùjde, nebo» potøebujeme utøídit souøadnice bodù.
-
-\s{Lze to lépe?} Na závìr poznamenejme, ¾e se umí provést vý¹e popsaná persistence vyhledávacího stromu v amortizované pamìti $\O(1)$ na zmìnu. Ve
-struènosti naznaèíme my¹lenku. Pou¾ijeme stromy, které pøi insertu a deletu provádí amortizovanì jenom konstantnì mnoho úprav své struktury. To nám
-napøíklad zaruèí 2-4 stromy z pøedná¹ky a podobnou vlastnost lze dokázat i o èerveno-èerných stromech. Pøi zmìnì potom nebudeme upravovat celou cestu,
-ale upravíme jenom jednotlivé vrcholy, kterých se zmìna týká. Ka¾dý vrchol stromu si v sobì bude pamatovat a¾ dvì své verze. Pokud chceme vytvoøit
-tøetí verzi, vrchol zkopírujeme stranou. To v¹ak mù¾e vyvolat zmìny v jeho rodièích a¾ do koøene. Situace je naznaèena na obrázku. Pøi vytvoøení nové
-verze $3$ pro vrcholu $v$ vytvoøíme jeho kopii $v'$, do které ulo¾íme tuto verzi. Av¹ak musíme také zmìnit rodièe $u$, kterému vytvoøíme novou verzi
-ukazující na $v'$. Abychom dosáhli ký¾ené konstantní pamì»ové slo¾itosti, pomù¾e potenciálový argument -- zmìn se provádí amortizovanì jenom
-konstantnì mnoho. Navíc si pro ka¾dou verzi pamatujeme její koøen, ze kterého máme dotaz spustit.
-
-\figure{8-geom2_6_rychla_perzistence.eps}{Vytvoøení nové verze vrcholu.}{2in}
-
-\bye
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+++ /dev/null
-%!PS
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-%%BeginProlog
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+++ /dev/null
-P=8-geom2
-
-include ../Makerules
+++ /dev/null
-% implementation of figure naming and figure transparency
-string name,tag; name := ""; tag := "";
-def updatefigname =
- if tag="": filenametemplate (name & "%c.eps");
- else: filenametemplate (name & "%c_" & tag & ".eps");
- fi;
-enddef;
-
-def figname(expr n) = name := n; updatefigname; enddef;
-def figtag(expr t) = tag := t; updatefigname; enddef;
-
-picture transparent_picture;
-color transparent_color; transparent_color := 0.9white;
-def drawtransparent(expr num) =
- transparent_picture := currentpicture;
-endfig;
-
-if tag="": filenametemplate (name & "%c_transparent.eps");
-else: filenametemplate (name & "%c_" & tag & "_transparent.eps");
-fi;
-beginfig(num);
- draw transparent_picture withcolor transparent_color;
-endfig;
-
-updatefigname;
-enddef;
-
-def from(expr p,d,len) =
- p+dir(d)*len
-enddef;
-
-def dirs(expr p,d,len) =
- p--from(p,d,len)
-enddef;
-
-def drawvertices(expr s,n) =
- for i:=s upto n:
- draw vertex(PQ[i]);
- endfor
-enddef;
-
-def drawfvertices(expr s,n,flags) =
- for i:=s upto n:
- draw vertex(PQ[i]) flags;
- endfor
-enddef;
-
-path centersquare; centersquare := (-0.5,-0.5)--(0.5,-0.5)--(0.5,0.5)--(-0.5,0.5)--cycle;
-
-def vertex(expr p) = p withpen pencircle scaled 4pt enddef;
-def drawemptyvertex(expr p) = unfill fullcircle scaled 4pt shifted p; draw fullcircle scaled 4pt shifted p; enddef;
-def drawendpointvertex(expr p) = draw vertex(p) withcolor red; draw fullcircle scaled 6pt shifted p; enddef;
-def createpath(expr p) = shakepath(p, 0.015cm,0.1cm) enddef;
-vardef shakepath(expr p,d,l) =
- save r,b;
- path r; r := point(arctime 0 of p) of p;
- b := -1;
- for i:=l step l until arclength(p):
- r := r--(point(arctime i of p) of p)+dir(angle(direction(arctime i of p) of p rotated 90))*d*b;
- b := -b;
- endfor
- r--point(arctime arclength(p) of p) of p
-enddef;
-
-pen normalpen; normalpen := pencircle scaled 0.6pt;
-pen boldpen; boldpen := pencircle scaled 1.5pt;
-pen bolderpen; bolderpen := pencircle scaled 2pt;
-def dotline = withdots scaled 0.82 withpen boldpen enddef;
-
-vardef unclosedbubblec(expr p,c) =
- bubblec((p..reverse p..cycle),c)
-enddef;
-
-vardef bubblec(expr p,c) =
- save r;
- path r; r := (point(arctime 0 of p) of p)+dir(angle(direction(arctime 0 of p) of p rotated 90))*c;
- for i:=0.01cm step 0.025cm until arclength(p):
- r := r..(point(arctime i of p) of p)+dir(angle(direction(arctime i of p) of p rotated 90))*c;
- endfor
- r..(point(arctime arclength(p) of p) of p)+dir(angle(direction(arctime arclength(p) of p) of p rotated 90))*c..cycle
-enddef;
-
-vardef bubble(expr p) = bubblec(p,0.12cm) enddef;
-vardef unclosedbubble(expr p) = unclosedbubblec(p,0.12cm) enddef;
projects / ads2.git / commitdiff