From: Martin Mares Date: Thu, 24 Nov 2011 22:45:30 +0000 (+0100) Subject: Geometrie: Spojeni dvojich starych zapisku X-Git-Url: http://mj.ucw.cz/gitweb/?a=commitdiff_plain;h=523bcad0f9c7ec0b6579bc0ce9ba392eb136670c;p=ads2.git Geometrie: Spojeni dvojich starych zapisku --- diff --git a/6-geom/6-geom.tex b/6-geom/6-geom.tex new file mode 100644 index 0000000..d2856e4 --- /dev/null +++ b/6-geom/6-geom.tex @@ -0,0 +1,298 @@ +\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 diff --git a/6-geom/7-geom.mp b/6-geom/7-geom.mp new file mode 100644 index 0000000..7375c63 --- /dev/null +++ b/6-geom/7-geom.mp @@ -0,0 +1,276 @@ +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 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+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 diff --git a/6-geom/8-geom2_0_bear.eps b/6-geom/8-geom2_0_bear.eps new file mode 100644 index 0000000..9e6ec74 --- /dev/null +++ b/6-geom/8-geom2_0_bear.eps @@ -0,0 +1,391 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%Creator: Adobe Illustrator by AutoTrace version 0.31.1 +%%Title: /tmp/potraceguiTmp-r4ayx1 +%%CreationDate: Tue Dec 15 14:58:41 2009 +%%BoundingBox: 0 0 437 436 +%%DocumentData: Clean7Bit +%%EndComments +%%BeginProlog +/bd { bind def } bind def +/incompound false def +/m { moveto } bd +/l { lineto } bd +/c { curveto } bd 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+++ b/6-geom/Makefile @@ -0,0 +1,3 @@ +P=6-geom + +include ../Makerules diff --git a/6-geom/lib.mp b/6-geom/lib.mp new file mode 100644 index 0000000..cbeedf5 --- /dev/null +++ b/6-geom/lib.mp @@ -0,0 +1,84 @@ +% 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; diff --git a/old/7-geom/7-geom.mp b/old/7-geom/7-geom.mp deleted file mode 100644 index 7375c63..0000000 --- a/old/7-geom/7-geom.mp +++ /dev/null @@ -1,276 +0,0 @@ -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 diff --git a/old/7-geom/7-geom.tex b/old/7-geom/7-geom.tex deleted file mode 100644 index d600d11..0000000 --- a/old/7-geom/7-geom.tex +++ /dev/null @@ -1,136 +0,0 @@ -\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 diff --git a/old/7-geom/7-geom1_male_obaly.eps b/old/7-geom/7-geom1_male_obaly.eps deleted file mode 100644 index 94a501d..0000000 --- a/old/7-geom/7-geom1_male_obaly.eps +++ /dev/null @@ -1,324 +0,0 @@ -%!PS -%%BoundingBox: -13 -35 216 12 -%%HiResBoundingBox: -12.12236 -34.76685 215.33812 11.24376 -%%Creator: MetaPost 0.993 -%%CreationDate: 2009.11.17:1821 -%%Pages: 1 -%*Font: cmmi10 9.96265 9.96265 6e:8 -%*Font: cmr10 9.96265 9.96265 31:f008 -%%BeginProlog -%%EndProlog -%%Page: 1 1 - 1 1 1 setrgbcolor -newpath 1.99252 0 moveto -1.99252 0.52847 1.78256 1.03523 1.4089 1.4089 curveto -1.03523 1.78256 0.52847 1.99252 0 1.99252 curveto --0.52847 1.99252 -1.03523 1.78256 -1.4089 1.4089 curveto --1.78256 1.03523 -1.99252 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b/old/7-geom/Makefile deleted file mode 100644 index b06fe95..0000000 --- a/old/7-geom/Makefile +++ /dev/null @@ -1,3 +0,0 @@ -P=7-geom - -include ../Makerules diff --git a/old/7-geom/lib.mp b/old/7-geom/lib.mp deleted file mode 100644 index 2536305..0000000 --- a/old/7-geom/lib.mp +++ /dev/null @@ -1,82 +0,0 @@ -% 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; diff --git a/old/8-geom2/8-geom2.mp b/old/8-geom2/8-geom2.mp deleted file mode 100644 index 5a44396..0000000 --- a/old/8-geom2/8-geom2.mp +++ /dev/null @@ -1,137 +0,0 @@ -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 diff --git a/old/8-geom2/8-geom2.tex b/old/8-geom2/8-geom2.tex deleted file mode 100644 index 034b395..0000000 --- a/old/8-geom2/8-geom2.tex +++ /dev/null @@ -1,167 +0,0 @@ -\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. 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b/old/8-geom2/lib.mp deleted file mode 100644 index cbeedf5..0000000 --- a/old/8-geom2/lib.mp +++ /dev/null @@ -1,84 +0,0 @@ -% 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;