algoritmù pracuje s~konkrétním vnoøením grafu do~roviny (rovinným nakreslením).
Proto se zamìøíme na~algoritmus, který zadaný graf buïto vnoøí do~roviny, nebo se
-zastaví s~tím, ¾e graf není rovinný. Tarjan ji¾ v~roce 1974 \cite{tarjan:planarity}
-ukázal, ¾e je to mo¾né provést v~lineárním èase, ale jeho algoritmus je ponìkud
+zastaví s~tím, ¾e graf není rovinný. Tarjan ji¾ v~roce 1974 ukázal \cite{tarjan:planarity},
+¾e je to mo¾né provést v~lineárním èase, ale jeho algoritmus je ponìkud
komplikovaný. Od~té doby se objevilo mnoho zjednodu¹ení, prozatím vrcholících
algoritmem Boyera a Myrvoldové \cite{boyer:cutting}, který zde uká¾eme.
V¹imnìme si, ¾e pokud vede z~nìjakého u¾ nakresleného vrcholu~$v$ je¹tì nenakreslená hrana,
lze pokraèovat po~nenakreslených hranách a¾ do~koøene. V¹echny vrcholy, ke~kterým
je¹tì bude potøeba nìco pøipojit (takovým budeme øíkat {\I externì aktivní} a za~chvíli
-to nadefinujeme preciznì), tedy musí le¾et v~té¾e stìnì dosud nakresleného
+to nadefinujeme formálnì), tedy musí le¾et v~té¾e stìnì dosud nakresleného
podgrafu a bez újmy na~obecnosti si vybereme, ¾e to bude vnìj¹í stìna.
Základním krokem algoritmu tedy bude roz¹íøit nakreslení o~nový vrchol~$v$,
touto hranou (tím vytvoøí novou stìnu) a také mù¾e slouèit nìkolik
blokù do~jednoho:
-\todo{Obrázek}
+\twofigures{planar1.eps}{Pøed nakreslením zpìtných hran \dots}{\epsfxsize}{planar2.eps}{\dots\ po nìm}{\epsfxsize}
Bude se nám hodit, ¾e èas potøebný na~tuto operaci je pøímo úmìrný poètu
hran, které ubyly z~vnìj¹í stìny, co¾ je amortizovanì konstanta.
\s{Definice:} Vrchol~$w$ je {\I externì aktivní} pokud buïto z~$w$ vede zpìtná
hrana do~je¹tì nenakresleného vrcholu, nebo je k~$w$ pøipojen externì aktivní
-blok, èili blok obsahující alespoò jeden externì aktivní vrchol.
+blok, èili blok obsahující alespoò jeden externì aktivní vrchol. Externì
+aktivní vrcholy budeme kreslit jako ètvereèky.
Jinými slovy $w$ je externì aktivní pøi zpracování vrcholu~$v$, pokud je $\<Ancestor>(w) < \<Enter>(v)$,
nebo pokud pro nìkterého ze~synù $x$ le¾ícího v~jiném bloku je $\<LowPoint>(x) < \<Enter>(x)$.
bity pøenese ve~smìru od~koøene k~potomkùm a urèí tak absolutní orientaci
v¹ech seznamù.
-\h{Pøidávání vrcholu}
+\h{®ivý podgraf}
+
+Kdy¾ nakreslíme nový vrchol~$v$ a z~nìj vedoucí stromové hrany, musíme obejít
+ka¾dý podstrom a ve~vhodném poøadí nakreslit zpìtné hrany do~$v$ a podle
+potøeby pøeklopit bloky. V~podstromu ov¹em mù¾e být mnoho blokù, které,
+aè jsou externì aktivní, ¾ádnou pozornost nevy¾adují a bìh algoritmu by
+zbyteènì brzdily. Proto podobnì jako externí aktivitu nadefinujeme je¹tì
+¾ivost vrcholu, které bude odpovídat zpìtným hranám vedoucím do~$v$:
+
+\s{Definice:}
+Vrchol~$w$ je {\I ¾ivý,} pokud z~nìj buïto vede zpìtná hrana do~právì
+zpracovávaného vrcholu~$v$, nebo pokud k~nìmu je pøipojen ¾ivý blok,
+tj. blok obsahující ¾ivý vrchol. Není-li ¾ivý vrchol èi blok externì aktivní,
+budeme mu øíkat {\I internì aktivní.} Pakli¾e není vrchol/blok ani ¾ivý, ani externì aktivní,
+budeme ho nazývat {\I neaktivní.}
+
+Pøed procházením podstromù tedy nejprve probereme v¹echny zpìtné hrany vedoucí do~$v$
+a oznaèíme ¾ivé vrcholy. Pro ka¾dou zpìtnou hranu potøebujeme o¾ivit vrchol, z~nìj¾
+hrana vede, dále artikulaci, pod~ní¾ je tento blok pøipojen a dal¹í artikulace
+na~cestì do~$v$. Poka¾dé, kdy¾ vstoupíme do~bloku (nìjakým vrcholem na~vnìj¹í stìnì),
+tedy potøebujeme nalézt koøen bloku. To udìláme tak, ¾e zaèneme obcházet vnìj¹í
+stìnu obìma smìry souèasnì, ne¾ dojdeme v~jednom smìru do~koøene. Navíc si v¹echny
+vrcholy, pøes nì¾ jsme pro¹li, oznaèkujeme a pøiøadíme k~nim rovnou ukazatel na~koøen,
+tak¾e po~¾ádné èásti hranice neprojdeme vícekrát.\foot{Znaèky ani nebude potøeba
+mazat, kdy¾ si u nich poznamenáme, který vrchol byl koøenem v~okam¾iku, kdy jsme
+znaèku vytvoøili, a znaèky patøící ke~starým koøenùm budeme ignorovat, resp. pøepisovat.}
+
+\s{Lemma:} Pro ka¾dý koøen trvá znaèení ¾ivých vrcholù èas $\O(k+l)$, kde $k$ je poèet
+kreslených zpìtných hran a $l$ poèet vrcholù, které zmizely z~vnìj¹í stìny, èili
+amortizovaná konstanta.
+
+\s{Dùkaz:} Alespoò polovina vrcholù, po~nich¾ jsme v~libovolném bloku pro¹li,
+zmizí z~vnìj¹í stìny, tak¾e hledání koøenù blokù trvá $\O(l)$. Pro ka¾dou zpìtnou
+hranu oznaèíme jeden vrchol jako ¾ivý a pak pokraèujeme hledáním koøenù.
+\qed
+
+\h{Kreslení zpìtných hran}
+
+Nyní ji¾ máme v¹e pøipraveno -- datové struktury, detekci externích vrcholù
+a oznaèování ¾ivého podgrafu -- a zbývá doplnit, jak algoritmus kreslí zpìtné
+hrany. Jeliko¾ zpìtné hrany vedoucí do~$v$ nemohou zpùsobit slouèení blokù
+le¾ících pod~$v$ (na~to jsou potøeba zpìtné hrany vedoucí nìkam nad~$v$ a ty
+je¹tì nekreslíme), zpracováváme ka¾dý podstrom zvlá¹». Pøidáme 2-cyklus
+pro stromovou hranu, pod nìj pøipojíme blokovou strukturu zatím nakreslené
+èásti podstromu a vydáme se po~hranici této struktury nejdøíve jedním
+a pak druhým smìrem.
+
+Oba prùchody vypadají následovnì: Procházíme seznam vrcholù na~hranici a neaktivní
+vrcholy pøeskakujeme. Pokud objevíme ¾ivý vrchol, nakreslíme v¹e, co z~nìj vede,
+pøípadnì se zanoøíme do~¾ivých blokù, které jsou pøipojeny pod tímto vrcholem.
+Pokud objevíme externì aktivní vrchol (pøípadnì poté, co jsme ho o¹etøili jako ¾ivý),
+procházení zastavíme, proto¾e za externì aktivní vrchol ji¾ nemù¾eme po~této stranì
+hranice nic pøipojit, ani¾ by se externì aktivní vrchol dostal dovnitø nakreslení.
+
+Pøitom se øídíme dvìma jednoduchými pravidly:
+
+\s{Pravidlo \#1:} V~ka¾dém ¾ivém vrcholu zpracováváme nejdøíve zpìtné hrany do~$v$,
+pak internì aktivní bloky a koneènì externì aktivní bloky pøipojené pod vrcholem.
+
+\s{Pravidlo \#2:} Pokud vstoupíme do~dal¹ího bloku, vybereme si smìr, ve~kterém
+budeme pokraèovat (pokud se li¹í od~smìru, ve~kterém zatím hranici
+obcházíme, blok pøeklopíme) následovnì: preferujeme smìr k~internì
+aktivnímu vrcholu, pokud takový neexistuje, pak k~¾ivému externì aktivnímu
+vrcholu.
+
+Èasová slo¾itost této èásti algoritmu je lineární ve~velikosti ¾ivého podgrafu
+a¾ na~dvì výjimky. Jednou je konec prohledávání od~posledního ¾ivého vrcholu
+k~bodu zastavení, druhou pak vybírání strany hranice pøi vstupu do~bloku.
+V~obou mù¾eme procházet a¾ lineárnì mnoho neaktivních vrcholù. Pomù¾eme si
+ov¹em snadno: kdykoliv projdeme souvislý úsek hranice tvoøený neaktivními
+vrcholy, pøidáme pomocnou hranu, která tento úsek pøeklene. Mù¾eme ji dokonce
+pøidat do~nakreslení a podrozdìlit si tak vnìj¹í stìnu.
+
+\h{Hotový algoritmus}
+
+Celý algoritmus tedy bude vypadat takto:
+
+\algo
+\:Pokud má graf více ne¾ $3n-6$ vrcholù, odmítneme ho rovnou jako nerovinný.
+\:Prohledáme graf $G$ do~hloubky, spoèteme \<Enter,> \<Ancestor> a \<LowPoint> v¹ech vrcholù.
+\:Inicializujeme \<SepBlockList> v¹ech vrcholù.
+\:Procházíme vrcholy v~poøadí klesajících \<Enter>ù, pro ka¾dý vrchol~$v$:
+\::Nakreslíme v¹echny stromové hrany z~$v$ jako 2-cykly.
+\::Oznaèíme ¾ivý podgraf.
+\::Pro ka¾dého syna vrcholu~$v$ obcházíme ¾ivý podgraf nále¾ící k~tomuto vrcholu
+ a kreslíme zpìtné hrany do~$v$.
+\::Zkontrolujeme, zda v¹echny hrany incidentní s~$v$ byly nakresleny, pokud ne,
+ prohlásíme graf za~nerovinný a zastavíme se.
+\:Projdeme hotové nakreslení do~hloubky a zorientujeme seznamy sousedù.
+\endalgo
+
+\s{Vìta:} Tento algoritmus pro ka¾dý graf dobìhne v~èase $\O(n)$ a pokud byl graf rovinný,
+vydá jeho nakreslení, v~opaèném pøípadì ohlásí nerovinnost.
+
+\s{Dùkaz:} První krok je korektní, jeliko¾ pro v¹echny rovinné grafy je $m\le 3n-6$; nadále
+tedy mù¾eme pøedpokládat, ¾e $m=\O(n)$. Lineární èasovou slo¾itost krokù 4--6 jsme ji¾
+diskutovali, kroky~7--8 jsou lineární ve~velikosti ¾ivého podgrafu, a tedy také $\O(n)$.
+Nakreslení vydané algoritmem je v¾dy rovinné a v¹echny stromové hrany jsou v¾dy
+nakresleny, zbývá tedy ukázat, ¾e zpìtnou hranu mù¾eme nenakreslit jen pokud
+graf nebyl rovinný. Tomu vìnujeme zbytek kapitoly.
+\qed
-\todo{Kostra algoritmu. ®ivé vrcholy a jejich nalezení.}
+\h{Dùkaz korektnosti}
-\h{Pøidávání hran}
+\s{Lemma:} Pokud existuje zpìtná hrana, kterou algoritmus nenakreslil, graf na~vstupu
+není rovinný.
-\todo{Jak se obchází hranice a v~jakém poøadí se co zpracovává.}
+\s{Dùkaz:} Rozborem pøípadù uká¾eme, ¾e kdykoliv existuje nenakreslená zpìtná hrana,
+pak algoritmus buïto poru¹il Pravidla \#1 a \#2 nebo v~zadaném grafu nalezneme
+minor $K_5$ èi $K_{3,3}$.
-\h{Dùkaz korektnosti}
+\todo{Rozebrat hou¹» pøípadù.}
-\todo{Èasová slo¾itost se doká¾e snadno, ale na~korektnost je potøeba rozebírat hou¹» pøípadù.}
+\qed
-\todo{Izolace Kuratowského podgrafù.}
+\s{Poznámka:} Podle tohoto dùkazu bychom také mohli v~lineárním èase v~ka¾dém nerovinném
+grafu nalézt Kuratowského podgraf, dokonce také v~$O(n)$, jeliko¾ kdy¾ je $m>3n-6$,
+mù¾eme se omezit na~libovolných $3n-5$ hran, které urèitì tvoøí nerovinný podgraf.
\references
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projects / ga.git / commitdiff