Snad jiz finalni Voroneho diagramy.

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 \prednaska{9}{Geometrické algoritmy}{(B. Maslowski, J. Návrat, M. Sta¹a)}
 
-\>Budeme se teï bavit o geometrických problémech v~rovinì. Vìt¹ina algoritmù, které zde uvedeme, má sice své obdoby i pro prostory vy¹¹í nebo ni¾¹í dimenze, ale jednorozmìrné pøípady bývají triviální a vícerozmìrné jsou zase vìt¹inou moc slo¾ité.
-Budeme se tedy zabývat tím, jak tyto problémy øe¹it v~dimenzi dva (v~Euklidovské rovinì).
+\>Budeme se teï bavit o geometrických problémech v~rovinì. Vìt¹ina algoritmù,
+které zde uvedeme, má sice své obdoby i pro prostory vy¹¹í nebo ni¾¹í dimenze,
+ale jednorozmìrné pøípady bývají triviální a vícerozmìrné jsou zase vìt¹inou
+moc slo¾ité.
 
+Budeme se tedy zabývat tím, jak tyto problémy øe¹it v~dimenzi dva
+(v~Euklidovské rovinì).
 
 \h{Hledání konvexního obalu}
-Ptáte se o co pùjde? Zkusme si to pøiblí¾it na problému ledních medvìdù :)
-{\I Lední medvìdi si po dlouhé dobì zmapovali vody severního moøe a zjistili pøesnì místa, kde se nacházejí jejich oblíbené ryby. No a proto¾e to jsou medvìdi chytøí, rozhodli se v¹echny tyto rybky pochytat najednou do jedné velké sítì. A problém, který tady mají, je následující: jaký nejmen¹í obvod mù¾e mít taková sí», aby se dovnitø ve¹ly v¹echny rybky?!}
 
-Neboli budeme øe¹it, jak nìjakou zadanou mno¾inu bodù v~rovinì obalit co nejkrat¹í uzavøenou køivkou, do které se je¹tì v¹echny body vejdou.
+Ptáte se o co pùjde? Zkusme si to pøiblí¾it na problému ledních medvìdù :) {\I
+Lední medvìdi si po dlouhé dobì zmapovali vody severního moøe a zjistili
+pøesnì místa, kde se nacházejí jejich oblíbené ryby. No a proto¾e to jsou
+medvìdi chytøí, rozhodli se v¹echny tyto rybky pochytat najednou do jedné
+velké sítì. A problém, který tady mají, je následující: jaký nejmen¹í obvod
+mù¾e mít taková sí», aby se dovnitø ve¹ly v¹echny rybky?!}
 
-Intuice nám napovídá ¾e výsledek bude nìjaký konvexní\foot{Mno¾ina bodù v~rovinì je konvexní, pokud platí, ¾e pro ka¾dé dva body této mno¾iny le¾í úseèka spojující tyto dva body také celá v~této mno¾inì.} mnohoúhelník, který bude mít vrcholy v~nìkterých uvedených bodech. Ostatní vrcholy pak budou buï nìkde na hranách mnohoúhelníku, nebo uvnitø. Tomuto mnohoulehníku se øíká {\I konvexní obal} dané mno¾iny.
+Neboli budeme øe¹it, jak nìjakou zadanou mno¾inu bodù v~rovinì obalit co
+nejkrat¹í uzavøenou køivkou, do které se je¹tì v¹echny body vejdou.
 
-\>Mo¾ná by se teï hodilo pøedvést názornì, jak vypadají nejmen¹í konvexní obaly:
+Intuice nám napovídá ¾e výsledek bude nìjaký konvexní\foot{Mno¾ina bodù
+v~rovinì je konvexní, pokud platí, ¾e pro ka¾dé dva body této mno¾iny le¾í
+úseèka spojující tyto dva body také celá v~této mno¾inì.} mnohoúhelník, který
+bude mít vrcholy v~nìkterých uvedených bodech. Ostatní vrcholy pak budou buï
+nìkde na hranách mnohoúhelníku, nebo uvnitø. Tomuto mnohoulehníku se øíká {\I
+konvexní obal} dané mno¾iny.
+
+\>Mo¾ná by se teï hodilo pøedvést názornì, jak vypadají nejmen¹í konvexní
+obaly:
 
 \figure{ZakladniObaly.eps}{Základní obaly.}{3in}
 
 \itemize\ibull
+
 \:Konvexní obal prázdné mno¾iny je prázdná mno¾ina.
+
 \:Konvexní obal 1 bodu je bod samotný.
+
 \:Konvexní obal 2 bodù je úseèka spojující tyto body.
+
 \:Konvexní obal 3 bodù je trojúhleník s vrcholy v~tìchto bodech.
+
 \:Konvexní obal 4 bodù \dots to u¾ je slo¾itìj¹í\dots
+
 \endlist
-Konvexní obaly 4 a více bodù, jak si mù¾eme v¹imnout, u¾ nejsou jednoznaèné. Pro $N$-prvkovou mno¾inu bude konvexní obal mnohoúhelník se tøemi a¾ $N$ vrcholy.
 
-Jeden dobrý zpùsob, jak tento konvexní obal sestrojit se nazývá {\I Zametání roviny.}
+Konvexní obaly 4 a více bodù, jak si mù¾eme v¹imnout, u¾ nejsou jednoznaèné.
+Pro $N$-prvkovou mno¾inu bude konvexní obal mnohoúhelník se tøemi a¾ $N$
+vrcholy.
 
-Algoritmus funguje tak, ¾e si v~rovinì zvolíme nìjaký smìr, a v~tomto smìru zaèneme posouvat pøímku. Budeme takto potkávat body le¾ící v~na¹í mno¾inì.
-V~ka¾dém okam¾iku budeme chtít, aby body v~èásti, kterou jsme ji¾ zametli, u¾ mìli spoèítaný konvexní obal. V¾dy kdy¾ pak zametací pøímkou narazíme na nový bod, u¾ si jen rozmyslíme, jak ho do konvexního obalu pøidat.
+Jeden dobrý zpùsob, jak tento konvexní obal sestrojit se nazývá {\I Zametání
+roviny.}
 
-BÚNO pøedpokládáme body v~obecné poloze, tedy takové, ¾e ¾ádné tøi nele¾í na~jedné pøímce. Dále také budeme pøedpokládat, ¾e budeme zametat ve smìru $x$-ové osy a ¾e v¹echny body mají rùznou $x$-ovou souøadnici.
+Algoritmus funguje tak, ¾e si v~rovinì zvolíme nìjaký smìr, a v~tomto smìru
+zaèneme posouvat pøímku. Budeme takto potkávat body le¾ící v~na¹í mno¾inì.
+V~ka¾dém okam¾iku budeme chtít, aby body v~èásti, kterou jsme ji¾ zametli, u¾
+mìli spoèítaný konvexní obal. V¾dy kdy¾ pak zametací pøímkou narazíme na nový
+bod, u¾ si jen rozmyslíme, jak ho do konvexního obalu pøidat.
 
-Je také vidìt, ¾e bod s nejmen¹í a nejvìt¹í $x$-ovou souøadnicí bude le¾et na~konvexním obalu.
+BÚNO pøedpokládáme body v~obecné poloze, tedy takové, ¾e ¾ádné tøi nele¾í
+na~jedné pøímce. Dále také budeme pøedpokládat, ¾e budeme zametat ve smìru
+$x$-ové osy a ¾e v¹echny body mají rùznou $x$-ovou souøadnici.
+
+Je také vidìt, ¾e bod s nejmen¹í a nejvìt¹í $x$-ovou souøadnicí bude le¾et
+na~konvexním obalu.
 
 \s{My¹lenka algoritmu:}
+
 \algo
+
 \:Setøídíme body podle jejich $x$-ové souøadnice.
+
 \:Vezmeme první tøi body a sestrojíme jejich konvexní obal.
-\:Opakuj: Najdeme dal¹í bod a podíváme se, jestli ho mù¾eme do konvexního obalu rovnou pøidat:
-\::Pokud jej mù¾eme rovnou pøidat, tak jej pøidáme.
-\::Pokud jej pøidat rovnou nemù¾eme, pak je potøeba nejdøíve nìjaké body odzadu odebrat a pak teprve pøipojit ná¹ nový bod.
-\endalgo
 
-Ka¾dá iterace tedy bude probíhat tak, ¾e nìjaké body z pùvodního konvexního obalu pozapomínáme a pøidáme nový bod. Aby se to lépe popisovalo, tak celý konvexní obal rozdìlíme na {\I horní obálku} a {\I dolní obálku.}
+\:Opakuj: Najdeme dal¹í bod a podíváme se, jestli ho mù¾eme do konvexního
+obalu rovnou pøidat: \::Pokud jej mù¾eme rovnou pøidat, tak jej pøidáme.
+
+\::Pokud jej pøidat rovnou nemù¾eme, pak je potøeba nejdøíve nìjaké body
+odzadu odebrat a pak teprve pøipojit ná¹ nový bod. \endalgo
+
+Ka¾dá iterace tedy bude probíhat tak, ¾e nìjaké body z pùvodního konvexního
+obalu pozapomínáme a pøidáme nový bod. Aby se to lépe popisovalo, tak celý
+konvexní obal rozdìlíme na {\I horní obálku} a {\I dolní obálku.}
 
 \figure{HD-obalka.eps}{Obrázek obálek.}{3in}
 
-Vidíme teï, ¾e dolní obálka je nìjaká lomená èára, která zatáèí doleva. Horní obálka zatáèí doprava. V~na¹em algoritmu si budeme obálky pamatovat jako dva seznamy vrcholù. Kdy¾ pak v~algoritmu narazíme na nový bod, budeme zvlá¹» øe¹it jak to ovlivní horní obálku a jak ovlivní dolní.
-Je vidìt ¾e bod nejvíce vlevo a bod nejvíce vpravo le¾í v obou obálkách. Ostatní body buï le¾í jen v~jedné z obálek, nebo nele¾í v~¾ádné z nich (tedy nejsou souèástí konvexního obalu).
+Vidíme teï, ¾e dolní obálka je nìjaká lomená èára, která zatáèí doleva. Horní
+obálka zatáèí doprava. V~na¹em algoritmu si budeme obálky pamatovat jako dva
+seznamy vrcholù. Kdy¾ pak v~algoritmu narazíme na nový bod, budeme zvlá¹»
+øe¹it jak to ovlivní horní obálku a jak ovlivní dolní. Je vidìt ¾e bod nejvíce
+vlevo a bod nejvíce vpravo le¾í v obou obálkách. Ostatní body buï le¾í jen
+v~jedné z obálek, nebo nele¾í v~¾ádné z nich (tedy nejsou souèástí konvexního
+obalu).
 
 \s{Algoritmus:}
+
 \algo
+
 \:Setøídíme body podle souøadnice $x$, dostaneme mno¾inu bodù $b_1~-~b_n$.
-\:Spoèítáme konvexní obal ${b_1, b_2, b_3}$, z toho získáme horní a dolní obálku\foot{Body $b_1$ a $b_3$ budou v obou obálkách. Bod $b_2$ bude v~horní obálce pokud le¾í nad pøímkou spojující $b_1$ a $b_3$, v~dolní obálce bude pokud le¾í pod pøímkou.}.
-\:Pro $b$ postupnì zpracováváme $b_3 - b_n$:
+\:Spoèítáme konvexní obal ${b_1, b_2, b_3}$, z toho získáme horní a dolní
+obálku\foot{Body $b_1$ a $b_3$ budou v obou obálkách. Bod $b_2$ bude v~horní
+obálce pokud le¾í nad pøímkou spojující $b_1$ a $b_3$, v~dolní obálce bude
+pokud le¾í pod pøímkou.}. \:Pro $b$ postupnì zpracováváme $b_3 - b_n$:
+
 \::Pøepoèítáme Horní obálku:
-\:::Dokud $(\vert H\vert \geq 2)$ a úhel $(H[-2], H[-1], b)$ je orientovaný doleva:
-\::::Odebereme poslední prvek z obálky.
+
+\:::Dokud $(\vert H\vert \geq 2)$ a úhel $(H[-2], H[-1], b)$ je orientovaný
+doleva: \::::Odebereme poslední prvek z obálky.
+
 \:::Pøidáme do obálky nový vrchol.
+
 \::Pøepoèítáme dolní obálku:
-\:::Dokud $(\vert D\vert \geq 2)$ a úhel $(D[-2], D[-1], b)$ je orientovaný doprava:
-\::::Odebereme poslední prvek z obálky.
+
+\:::Dokud $(\vert D\vert \geq 2)$ a úhel $(D[-2], D[-1], b)$ je orientovaný
+doprava: \::::Odebereme poslední prvek z obálky.
+
 \:::Pøidáme do obálky nový vrchol.
+
 \endalgo
 
-Setøídit body podle $x$-ové souøadnice a sestrojit konvexní obal prvních tøech bodù stihneme v~èase $\O(n \log n)$. Zbytek pak u¾ udìláme dokonce v~èase lineárním $\O(n)$\foot{V této èásti u¾ jen do obálek pøidáváme a odebíráme body.Pøidáváme jich $N$. A odebrat jich mù¾eme maximálnì tolik kolik jsme jich pøidali. Tedy zase maximálnì $N$.}. Platí tedy:
+Setøídit body podle $x$-ové souøadnice a sestrojit konvexní obal prvních tøech
+bodù stihneme v~èase $\O(n \log n)$. Zbytek pak u¾ udìláme dokonce v~èase
+lineárním $\O(n)$\foot{V této èásti u¾ jen do obálek pøidáváme a odebíráme
+body.Pøidáváme jich $N$. A odebrat jich mù¾eme maximálnì tolik kolik jsme jich
+pøidali. Tedy zase maximálnì~$N$.}. Platí tedy:
 
 \s{Vìta:} Konvexní obal doká¾eme sestrojit v~èase $\O(n \log n)$.
 
-\>Na¹i lední medvìdi se tedy ji¾ nauèili, jak si efektivnì obstarat potravu a mohly se pustit do øe¹ení dal¹ího velmi dùle¾itého problému. Pojïme se na nìj podívat s nimi.
-\>A o co ¾e to pùjde?
-{\I Lední medvìdi nejsou na antarktidì sami, kromì nich tam taky bydlí kamarádi eskymáci ve svých iglù. Medvìdi by si teï rádi udìlali mapu, podle které by hned poznali, ke kterému ekymákovi to mají nejblí¾e na náv¹»evu.} 
+\>Na¹i lední medvìdi se tedy ji¾ nauèili, jak si efektivnì obstarat potravu a
+mohly se pustit do øe¹ení dal¹ího velmi dùle¾itého problému. Pojïme se na nìj
+podívat s nimi. \>A o co ¾e to pùjde?
+
+{\I Lední medvìdi nejsou na antarktidì sami, kromì nich tam taky bydlí
+kamarádi eskymáci ve svých iglù. Medvìdi by si teï rádi udìlali mapu, podle
+které by hned poznali, ke kterému ekymákovi to mají nejblí¾e na náv¹»evu.}
 
 My tuhle medvìdí mapu od teï budeme nazývat {\I Voroneho diagramem}.
 
 \h{Voroného diagramy}
 
-Pøed tím, ne¾ vás vystra¹ím nìjakou definicí, si øekneme, co jsi pod tímto, na 
-první pohled ne zøejmým pojmem, pøedstavit. Mìjme mno¾inu teèek T rozmístìných 
-náhodnì po papíru. Ke ka¾dému bodu nakreslíme okraje tak, aby vniklá plo¹ka 
-obsahovala body, které jsou nejblí¾e právì té na¹í vybrané teèce. Samozøejmì 
-\uv{sousední} teèky budou mít tyto hranice spoleèné. Výsledkem na¹eho dlouhého 
-sna¾ení pak bude právì Voroného diagram. V dal¹ích odstavcích se budeme zajímat 
-o to, jak takový útvar správnì popsat, jak ho sestrojit a jaké datové struktury 
-k tomu pou¾ít.
-                                            
-\s{Definice:} {\I Voroného} diagram pro koneènou mno¾inu $M = \{m_1, \dots, m_n\} \in 
-{\bb R}^2$ míst je systém mno¾in $O_1,\dots,O_n$ takových, ¾e pro v¹echna $i$ a $j$ a 
-pro v¹echna $x \in M_i$ je vzdálenost $x$ od $m_i$ men¹í nebo rovna vzdálenosti 
-$x$ od $m_j$ a zároveò sjednocení $O_i$ pøes v¹echna $i$ je celý prostor ${\bb R}^2$, 
-neboli:
+Pøed tím, ne¾ vás vystra¹ím nìjakou definicí, si øekneme, co jsi pod tímto, na
+první pohled ne zøejmým pojmem, pøedstavit. Mìjme mno¾inu teèek $T$
+rozmístìných náhodnì po papíru. Ke ka¾dému bodu nakreslíme okraje tak, aby
+vniklá plo¹ka obsahovala body, které jsou nejblí¾e právì té na¹í vybrané
+teèce. Samozøejmì \uv{sousední} teèky budou mít tyto hranice spoleèné.
+Výsledkem na¹eho dlouhého sna¾ení pak bude právì Voroného diagram. V dal¹ích
+odstavcích se budeme zajímat o to, jak takový útvar správnì popsat, jak ho
+sestrojit a jaké datové struktury k tomu pou¾ít.
+
+\s{Definice:} {\I Voroného diagram} pro koneènou mno¾inu $M = \{m_1, \dots,
+m_n\} \in {\bb R}^2$ míst je systém mno¾in $O_1,\dots,O_n$ takových, ¾e pro
+v¹echna $i$ a $j$ a pro v¹echna $x \in M_i$ je vzdálenost $x$ od $m_i$ men¹í
+nebo rovna vzdálenosti $x$ od $m_j$ a zároveò sjednocení $O_i$ pøes v¹echna
+$i$ je celý prostor ${\bb R}^2$, neboli:
 
 $$d(x,m_i) \leq d(x,m_j) \wedge {\bigcup}_i O_i = {\bb R}^2.$$
 
-Voroného diagram se tedy skládá z nìjakých míst, oblastí a hran, které ty oblasti oddìlují.
+Jednoduchý Voroného diagram:
+
+\figure{voroneho2.eps}{Voroneho diagramu pro dvì místa.}{3in}
 
-\s{Definice:} Øekneme, ¾e {\I hrana} $H$ je taková mno¾ina bodù, ¾e pro ka¾dý bod $x \in H$
-platí.
-\s{Definice:} Øekneme, ¾e {\I vrchol} je takový bod, kde se potkávají alespoò dvì hrany.
+Voroného diagram se tedy skládá z nìjakých míst, oblastí a hran, které ty
+oblasti oddìlují.
+
+\figure{voronoi.eps}{Èásti Voroneho diagramu.}{2in}
+
+\s{Definice:} Øekneme, ¾e {\I hrana} $H$ je taková mno¾ina bodù, ¾e pro ka¾dý
+bod $x \in H$ platí, ¾e existují dvì místa $m_i$ a $m_j$, od kterých má bod
+$x$ stejnou vzdálenost. Tyto dvì místa jsou pro v¹echny body $x$ stejná a
+platí, ¾e v¹echny ostatní místa mají od ka¾dého bodu $x$ del¹í vzdálenost.
+
+\s{Definice:} Øekneme, ¾e {\I vrchol} je takový bod, kde se potkávají alespoò
+dvì hrany.
+
+\s{Pozorování:}
 
-\s{Pozorování:} 
 \itemize\ibull
-\:Ka¾dá mno¾ina $M_i$ je ohranièena konvexní lomenou èarou, tak¾e oblasti mají tvar konvexních mnohoúhelníkù, ale je mo¾né, ¾e jsou oteveøené do nekneèna.
-\:Pro ka¾dou hranu $h$ ve Voroného diagramu existuje $i$ a $j$ takové, ¾e kdy¾ $x \in H$, pak vzdálenost $d(x,m_i) = d(x,m_j)$.
-\:Pro ka¾dý vrchol $v$ Voroného diagramu existují alespoò tøi místa le¾ící
-na kru¾nici se støedem $v$.
+
+\:Voroneho diagramem pro dvì místa jsou dvì poloroviny odìlené takovou pøímkou,
+ ¾e ka¾dý bod pøímky je stejnì vzdálený od obou míst. \:Ka¾dá mno¾ina $M_i$ je
+ohranièena konvexní lomenou èarou, tak¾e oblasti mají tvar konvexních
+mnohoúhelníkù, ale je mo¾né, ¾e jsou oteveøené do nekneèna. \:Pro ka¾dou hranu
+$h$ ve Voroného diagramu existuje $i$ a $j$ takové, ¾e kdy¾ $x \in H$, pak
+vzdálenost $d(x,m_i) = d(x,m_j)$. \:Pro ka¾dý vrchol $v$ Voroného diagramu
+existují alespoò tøi místa le¾ící na kru¾nici se støedem $v$.
+
 \figure{body.eps}{Body na kru¾nici.}{3in}
-\:Poèet vrcholù a hran je lineární.
-\:Poèet krajních oblastí je tak velký, jak velký je konvexní obal té zadané mno¾iny. 
-(Je to dobré vìdìt, ale asi to nebudeme potøebovat.)
+
+\:Poèet vrcholù a hran je lineární k poètu míst\foot{Voroneho diagram si lze
+pøedstavit jako graf, kde místa Voroneho diagramu odpovídají stìnám, vrcholy
+diagramu vrcholùm a hrany odpovídají hranám grafu. Pokud si teï nìkam mimo
+graf pøidáme je¹tì jeden vrchol a v¹echny pøímky vedoucí do nekoneèna navedeme
+do toho bodu, vidíme, ¾e ná¹ graf je roviný. Pro roviný graf platí Eulerova
+formule a z ní u¾ plyne ¾e na¹e linearita.}. \:Poèet krajních oblastí je tak
+velký, jak velký je konvexní obal té zadané mno¾iny. (Je to dobré vìdìt, ale
+asi to nebudeme potøebovat.)
+
 \endlist
 
-Pojïme teï vymyslet, jak takový diagram vyrobit. Mohli bychom zkonstruovat v¹echny 
-dìlící pøímky a poslepovat je, ale vznikl by nám kvadratický algoritmus a to nám 
-nemù¾e staèit. 
-
-Mluvili jsme o zametání roviny, a tak bychom tento trik mohli vyu¾ít právì pøi 
-øe¹ení na¹eho problému. Ov¹em tentokrát má zametání jeden podstatný háèek. Kdy¾
-si vezmeme nìjakou zametací pøímku a pojedeme s ní shora dolù, tak nad ní
-máme nìjakou u¾ zkonstruovanou èást diagramu a kdy¾ narazíme na dal¹í bod, tak 
-se nám mù¾e právì tato èást diagramu pomìrnì slo¾itì zmìnit. Pomù¾eme si malým trikem. 
-Nebudeme pova¾ovat za hotovou celou oblast nad zametací pøímkou, ale jen takové
-body, které mají blí¾e k místùm ($m_i$) ne¾ k zametací pøímce.
-Tak dostaneme nìjakou posloupnost (mno¾inu) parabol. V¹echno, co jsme spoèítali 
-uvnitø této oblasti nám u¾ nikdo nepokazí (ani nevylep¹í), je tam bezpeèno. 
-Vezmeme si tedy dolní obálku tìchto parabol, budeme jí øíkat {\I pobøe¾í}.   
+Pojïme teï vymyslet, jak takový diagram vyrobit. Mohli bychom zkonstruovat
+v¹echny dìlící pøímky a poslepovat je, ale vznikl by nám kvadratický
+algoritmus a to nám nemù¾e staèit.
+
+Mluvili jsme o zametání roviny, a tak bychom tento trik mohli vyu¾ít právì pøi
+øe¹ení na¹eho problému. Ov¹em tentokrát má zametání jeden podstatný háèek.
+Kdy¾ si vezmeme nìjakou zametací pøímku a pojedeme s ní shora dolù, tak nad ní
+máme nìjakou u¾ zkonstruovanou èást diagramu a kdy¾ narazíme na dal¹í bod, tak
+se nám mù¾e právì tato èást diagramu pomìrnì slo¾itì zmìnit. Pomù¾eme si malým
+trikem. Nebudeme pova¾ovat za hotovou celou oblast nad zametací pøímkou, ale
+jen takové body, které mají blí¾e k místùm ($m_i$) ne¾ k zametací pøímce. Tak
+dostaneme nìjakou posloupnost (mno¾inu) parabol. V¹echno, co jsme spoèítali
+uvnitø této oblasti nám u¾ nikdo nepokazí (ani nevylep¹í), je tam bezpeèno.
+Vezmeme si tedy dolní obálku tìchto parabol, budeme jí øíkat {\I pobøe¾í}.
+
 \figure{pobrezi.eps}{Pobøe¾ní linie.}{3in}
-Pobøe¾í je tedy nìjaká posloupnost parabolických obloukù s tím, ¾e nejlevìj¹í a 
-nejpravìj¹í jdou do nekoneèna.
-
-Kdykoliv v prùbìhu zametání narazíme na nìjaký bod, mù¾e nám ovlivnit u¾ jen èást diagramu
-pod pobøe¾ím. Dostáváme se tedy k tomu, co se dìje, kdy¾ hýbeme zametací pøímkou.
-Pakli¾e nenará¾íme na ¾ádné body, tak se v podstatì nedìje nic zajímavého.
-Zajímavá situace nastává a¾ tehdy, narazíme-li na dal¹í bod. V~tom okam¾iku vzniká
-nová parabola. V tuto chvíli je znaènì degenerovaná. Je to toti¾ zatím jen
-polopøímka kolmá na zametací pøímku. S dal¹ím pohybem se zaène parabola rozevírat.
-V¹imnìme si, ¾e prùnik oné pøímky a pobøe¾í je vlastnì vrchol Voroného diagramu.
-
-Mù¾e nastat je¹tì jeden problém. Nìjaká parabola se mù¾e natolik rozevøít, ¾e pohltí
-jiné a ty zmizí z pobøe¾ní linie. V takovém pøípadì se nám ale netriviálnì zmìní
-vzhled pobøe¾í, a proto se této situaci budeme muset více vìnovat.
-
-Shrneme-li na¹e úvahy, mohou se dít celkem tøi vìci. Jedna z nich je posun pøímky.
-To se vlastnì dìje poøád. Pobøe¾í se témìø nemìní a prùseèíky parabol nám kreslí
-hrany. To mù¾eme poèítat najednou. Navíc nejen ¾e bychom mohli s pøímkou skoèit
-o nìjaké epsilon, ale my dokonce mù¾eme skoèit o poøádný kus a prostì pouze dopoèítat,
-jak se pobøe¾í zmìnilo a co se vykreslilo. Dùle¾itým místùm, kde se budeme zastavovat,
-budeme øíkat {\I události}.
+
+Pobøe¾í je tedy nìjaká posloupnost parabolických obloukù s tím, ¾e nejlevìj¹í
+a nejpravìj¹í jdou do nekoneèna. Prùseèíky tìchto obloukù vykreslují hrany
+diagramu. Proè? Odpovìï na tuto otázku není tì¾ká, staèí vyjít z definice
+paraboly tak, jak jí zde pou¾íváme. Nebo-li je to mno¾ina bodù, která je od
+ohniska (pro nás místa) stejnì vzdálená jako od pøímky (øídící). A tedy prùnik
+dvou parabol je místo, které je stejnì vzdálené od obou ohnisek, co¾ je
+vlastnì definice bodu le¾ícího na nìjaké hranì.
+
+Kdykoliv v prùbìhu zametání narazíme na nìjaký bod, mù¾e nám ovlivnit u¾ jen
+èást diagramu pod pobøe¾ím. Dostáváme se tedy k tomu, co se dìje, kdy¾ hýbeme
+zametací pøímkou. Pakli¾e nenará¾íme na ¾ádné body, tak se v podstatì nedìje
+nic zajímavého. Zajímavá situace nastává a¾ tehdy, narazíme-li na dal¹í bod.
+V~tom okam¾iku vzniká nová parabola. V tuto chvíli je znaènì degenerovaná. Je
+to toti¾ zatím jen polopøímka kolmá na zametací pøímku. S dal¹ím pohybem se
+zaène parabola rozevírat. V¹imnìme si, ¾e prùnik oné pøímky a pobøe¾í je
+vlastnì vrchol Voroného diagramu.
+
+Mù¾e nastat je¹tì jeden problém. Nìjaká parabola se mù¾e natolik rozevøít, ¾e
+pohltí jiné a ty zmizí z pobøe¾ní linie. V takovém pøípadì, se nám ale
+netriviálnì zmìní vzhled pobøe¾í, a proto se této situaci budeme muset více
+vìnovat.
+
+Shrneme-li na¹e úvahy, mohou se dít celkem tøi vìci. Jedna z nich je posun
+pøímky. To se vlastnì dìje poøád. Pobøe¾í se témìø nemìní a prùseèíky parabol
+nám kreslí hrany. To mù¾eme poèítat najednou. Navíc nejen ¾e bychom mohli s
+pøímkou skoèit o nìjaké epsilon, ale my dokonce mù¾eme skoèit o poøádný kus a
+prostì pouze dopoèítat, jak se pobøe¾í zmìnilo a co se vykreslilo. Dùle¾itým
+místùm, kde se budeme zastavovat, budeme øíkat {\I události}.
 
 \>{\I Místní událost}
 
-Pokud narazíme na bod, musíme najít místo, kde pobøe¾í rozetnou
-a kam vklínit dal¹í výbì¾ek (parabolu). Takovéto události budeme øíkat místní událost.
-Pokud se pohybujeme v obecné poloze, nestane se nám, ¾e bychom narazili na prùseèík. 
-\figure{mistni.eps}{\vbox{
-\hsize=0.6\hsize\leftskip=0pt plus 0.3\hsize\rightskip=\leftskip\parfillskip=0pt
-\>Místní událost -- èervená kolmice je novì vznikající parabola,
-pøi postupu zametací pøímky dále se bude rozevírat a vytvoøí dal¹í parabolu.}}{3in}
+Pokud narazíme na bod, musíme najít místo, kde pobøe¾í rozetnou a kam vklínit
+dal¹í výbì¾ek (parabolu). Takovéto události budeme øíkat místní událost.
+
+\figure{mistni.eps}{\vbox{\hsize=0.6\hsize\leftskip=0pt plus
+0.3\hsize\rightskip=\leftskip\parfillskip=0pt \>Místní událost -- èervená
+kolmice je novì vznikající parabola, pøi postupu zametací pøímky dále se bude
+rozevírat a vytvoøí dal¹í parabolu.}}{3in}
 
 \>{\I Kru¾nicová událost}
 
 Poslední situace, která mù¾e nastat, je, ¾e se nìjaká parabola schová za jiné.
-Kouknìme se na první obrázek ní¾e, fialový bod le¾í na v¹ech tøech parabolách. 
-Speciálnì tedy ta tøi ohniska parabol jsou vzdálena stejnì a le¾í na kru¾nici. 
-Stane se nám to pravì tehdy, kdy¾ se nám zametací pøímka dotkne kru¾nice zespodu. 
-Tato kru¾nice je urèena právì tìmito tøemi ohnisky. Vzniklé události budeme øíkat kru¾nicová.
-Z obrázku by tedy mìlo být patrné (alespoò èásteènì), ¾e kru¾nicové události
-jsou urèeny trojicemi sousedních obloukù v pobøe¾í.  
-Pojïme si to ukázat lépe na následujících dvou obrázcích. První pøedstavuje 
-situaci pøed kru¾nicovou událostí a druhý právì kru¾nicovou událost.
+Kouknìme se na první obrázek ní¾e, fialový bod je støed kru¾nice opsané
+trojùhelníku tvoøenému tøemi místy. Jak víme, ten le¾í na osách stran takového
+trojùhelníku. Po tìchto osách se v¹ak budou i pohybovat prùseèíky parabol a s
+posunováním øídící pøímky se pak dostanou v¹echny tøi do støedu této kru¾nice.
+Stane se to pravì tehdy, kdy¾ se zametací pøímka dotkne kru¾nice zespodu. Je
+mo¾né nahlédnout, ¾e pøi postupu dále se pak dvì krajní paraboly roz¹íøí
+natolik, ¾e prostøední pohltí a ta zanikne. Takto vzniklé události budeme
+øíkat kru¾nicová.
+
+Pojïme si to ukázat lépe na následujících dvou obrázcích. První pøedstavuje
+situaci pøed kru¾nicovou událostí a druhý právì kru¾nicovou událost. Mimo jiné
+by tedy z obrázkù mìlo být patrné, ¾e kru¾nicové události jsou urèeny
+trojicemi sousedních obloukù v pobøe¾í.
+
 \figure{kruznicova.eps}{Pøed kru¾nicovou událostí.}{3in}
-\figure{kruznicovakonec.eps}{Kru¾nicová událost.}{3in}
 
+\figure{kruznicovakonec.eps}{Kru¾nicová událost.}{3in}
 
 \s{Datové struktury}
 
-Budeme potøebovat haldu událostí (místních i kru¾nicových dohromady), ta nám 
-zabere $\O(\log n)$ pamìti.
+Budeme potøebovat haldu událostí (místních i kru¾nicových dohromady).
 
-Dále bude zapotøebí udr¾ovat si pobøe¾ní linii, neboli posloupnost míst 
-v~ohniscích parabolických obloukù. Zde je potøeba si definovat operace 
-{\I Insert, Delete} a {\I FindX.} Navíc budeme potøebovat vyhledávací strom nad 
-prùseèíky s implicitní reprezentací. Zdá se, ¾e je toho hodnì, ale v¹echno to 
-zvládneme s pamìtí $\O(\log n).$
+Dále bude zapotøebí udr¾ovat si pobøe¾ní linii, neboli posloupnost míst
+v~ohniscích parabolických obloukù. Zde je potøeba si definovat operace {\I
+Insert, Delete} a {\I FindX}, jinak øeèeno pøidat a odebrat oblouk a najít
+oblouk podle x-sové souøadnice nebo-li oblouk do kterého jsme se trefili pøi
+místní události. Navíc budeme potøebovat vyhledávací strom nad prùseèíky s
+implicitní reprezentací, co¾ znamená, ¾e si ve vrcholech nebudeme pamatovat
+pøímo souøadnice prùseèíkù, ale jen instrukci, jak je spoèítat. Tak¾e jakkoli
+se mìní poloha prùseèíkù, tak struktura stromu zùstává stejná.
 
-Poslední datovou strukturou bude samotný diagram, reprezentovaný grafem se 
+S haldou událostí lze pracovat s logaritmickou èasovou slo¾itostí na operaci
+a ve stejném èase doká¾eme pracovat s pobøe¾ní linií. Kdy¾ si pobøe¾í je¹tì
+reprezentujeme zvlá¹» jako seznam tak Insert a Delete budou v konstatním èase
+a operace se stromem pak $\O(\log n)$.
+
+Poslední datovou strukturou bude samotný diagram, reprezentovaný grafem se
 souøadnicemi a vazbami hran na prùseèík.
 
+\s{Fortunùv~algoritmus}
 
-\s{Fortunùv~algoritmus}   
 \algo
+
 \:Pøipravíme si haldu $H$ a vlo¾íme do ní v¹echny místní události.
-\:Pøipravíme pobøe¾ní linii $P \leftarrow 0$. 
+
+\:Pøipravíme pobøe¾ní linii $P \leftarrow 0$.
+
 \:Dokud existuje $h \leftarrow DeleteMin(H)$:
+
 \::Je-li na øadì místní událost:
+
 \:::Najdeme prùseèík s $P(FindX(P))$.
+
 \:::Vlo¾íme do $P$ novou parabolu.
+
 \:::Poznamenáme do $D$ vznik hran.
-\:::Pøepoèítáme kru¾nicové události.   
+
+\:::Pøepoèítáme kru¾nicové události.
+
 \::Je-li na øadì kru¾nicová událost:
+
 \:::Sma¾eme oblouk z $P$.
+
 \:::Poznamenáme vznik a zánik do $D$.
-\:::Pøepoèítáme kru¾nicové události.
-\endalgo
 
-        
-\s{Slo¾itost:} 
-Poèet místních událostí je roven $n$ a poèet kru¾nicových událostí není vìt¹í ne¾ 
- $n$, proto¾e kru¾nicové události odpovídají vrcholùm, a proto tìch, kterých se
- doopravdy provedou je pouze $\O(n)$, neboli velikost $P$ není vìt¹í 
-ne¾ $2n$ a tedy je v¾dy lineární. Zároveò velikost $H$ není vìt¹í ne¾ $2n$, 
-proto¾e aèkoliv~poèet místních událostí je $n$, tak v~$H$ je záznam pro ka¾dou 
-trojici sousedních a tedy v~$H$ je maximálnì $2n$ událostí najednou. Zbývá nám tedy zjistit 
-velikost diagramu $D$, ale ten se urèitì vejde do $\O(n)$ pamìti.
+\:::Pøepoèítáme kru¾nicové události.
 
-Pokud tedy shrneme v¹echny na¹e odhady, pak èasová slo¾itost algoritmu je 
-$\O(n \log n)$ a prostorová $\O(n)$. 
+\endalgo
 
+\s{Slo¾itost:}
+
+Poèet místních událostí je roven $n$ (na ka¾dé místo narazíme právì jednou).
+Poèet kru¾nicových událostí není vìt¹í ne¾ $n$, proto¾e kru¾nicová událost sma¾e
+parabolu a ty vznikají jen pøi místních událostech, tak¾e kru¾nicových událostí
+není více ne¾ místních. Speciálnì z toho plyne, ¾e velikost pobøe¾ní linie je
+v¾dy lineární, proto¾e s ka¾dou místní událostí pøibudou dva úseky do pobøe¾ní
+linie, tak¾e velikost pobøe¾ní linie je maximálnì $2n$. Velikost haldy je pak
+také $2n$, tak¾e pak operace urèitì zvládneme v èase $\O(\log n)$. Jeliko¾
+diagram je lineárnì velký tak i jeho struktura je lineárnì velká. Operace se
+strukturami nás stojí nejvíce $\O(\log n)$. Tak¾e místní i kru¾nicové události
+zvládneme v èase $\O(\log n)$ na jednu na konstantním poètu struktur. Halda má
+velikost $2n$, tak¾e maximálnì provedeme $\O(2n\log n)$ operací.
+Celý algoritmus potøebuje na~inicializaci maximálnì $\O(n \log n)$ (i kdybychom
+ji dìlali neefektivnì) a $\O(2n\log n)$ výpoèet.
+
+Pokud tedy shrneme v¹echny na¹e odhady, pak èasová slo¾itost algoritmu je
+$\O(n \log n)$ a prostorová $\O(n)$.
 
 \bye
-  
 
 -------------------
 
 1) Pøipravíme si haldu $H$ a vlo¾íme do ní v¹echny místní události.
-2) pøipravíme pobøe¾ní linii P <- 0                               } O(n*log n)
-3) pøipraváme reprezentaci diagramu D                            /                              
+
+2) pøipravíme pobøe¾ní linii P <- 0 } O(n\log n)
+
+3) pøipraváme reprezentaci diagramu D /
+
 4) dokud existuje h <- DeleteMin(H)
-5)    je-li na øadì místní událost:                         ----
-        a) najdeme prùseèík s P(FindX(P))   \
-        b) vlo¾íme do P novou parabolu       \
-        c) poznamenáme do D vznik hran        } O(log n)
-        d) pøepoèítáme kru¾nicové události   /                   } <= 2n
-6)    je-li na øadì kru¾nicová událost:
-        a) sma¾eme oblouk z P              \
-        b) poznamenáme vznik a zánik do D   } O(log n)
-        c) pøepoèítáme kru¾nicové události /                ----
 
+5) je-li na øadì místní událost: ----
+
+a) najdeme prùseèík s P(FindX(P)) \
+
+b) vlo¾íme do P novou parabolu \
+
+c) poznamenáme do D vznik hran } O(\log n)
+
+d) pøepoèítáme kru¾nicové události / } <= 2n
+
+6) je-li na øadì kru¾nicová událost:
+
+a) sma¾eme oblouk z P \
 
+b) poznamenáme vznik a zánik do D } O(\log n)
 
+c) pøepoèítáme kru¾nicové události / ----
 
 \bye

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diff --git a/9-geom/body.svg b/9-geom/body.svg
index 73a7a1bbf835b38653591b978a30e3ee92bf6e5f..bd794f01e4274303de5077c2d8480ac1ad619ca7 100644 (file)
--- a/9-geom/body.svg
+++ b/9-geom/body.svg
@@ -15,7 +15,7 @@
    sodipodi:version="0.32"
    inkscape:version="0.45.1"
    version="1.0"
-   sodipodi:docbase="C:\Documents and Settings\Santhos\Plocha\Zapis"
+   sodipodi:docbase="D:\Download\ads\ads2\9-geom"
    sodipodi:docname="body.svg"
    inkscape:output_extension="org.inkscape.output.svg.inkscape">
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@@ -27,15 +27,15 @@
      borderopacity="1.0"
      inkscape:pageopacity="0.0"
      inkscape:pageshadow="2"
-     inkscape:zoom="1"
-     inkscape:cx="215.5"
-     inkscape:cy="104"
+     inkscape:zoom="6.4669697"
+     inkscape:cx="101.8418"
+     inkscape:cy="51.987305"
      inkscape:document-units="px"
      inkscape:current-layer="layer1"
      width="200px"
      height="100px"
-     inkscape:window-width="1024"
-     inkscape:window-height="712"
+     inkscape:window-width="1280"
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