From f1bd5ecf34086b304c06414d39f5dbd48dd2187d Mon Sep 17 00:00:00 2001 From: Martin Mares Date: Thu, 25 Jan 2007 13:17:04 +0100 Subject: [PATCH] Brouseni, lesteni a zametani kapitoly o dekompozicich. --- 9-decomp/9-decomp.tex | 43 +++++------ 9-decomp/mima.eps | 12 +-- 9-decomp/mima.vrr | 168 +++++++++++++++++++++--------------------- ga.bib | 2 +- 4 files changed, 113 insertions(+), 112 deletions(-) diff --git a/9-decomp/9-decomp.tex b/9-decomp/9-decomp.tex index 81fbf31..f3f2bed 100644 --- a/9-decomp/9-decomp.tex +++ b/9-decomp/9-decomp.tex @@ -18,8 +18,9 @@ komponent souvislosti neorientovan zda dva vrcholy le¾í v~té¾e komponentì. To se hodí v~mnoha algoritmech, kupøíkladu v~Kruskalovì algoritmu pro hledání minimální kostry. -\s{Triviální øe¹ení:} Ka¾dé tøídì pøiøadíme unikátní barvu, kterou obarvíme prvky tøídy. Operace \ -porovnává barvy, operace \ prvky jedné tøídy pøebarvuje. +\s{Triviální øe¹ení:} Prvky ka¾dé tøídy obarvíme unikátní barvou (identifikátorem +tøídy). Operace \ porovnává barvy, \ prvky jedné ze~sjednocovaných +tøíd pøebarvuje. Operace \ tak pracuje v~konstantním èase, \ mù¾e zabrat a¾ lineární èas. Mù¾eme si pomoci tím, ¾e v¾dy pøebarvíme {\I men¹í} ze~sluèovaných ekvivalenèních tøíd (budeme @@ -32,7 +33,7 @@ slo \s{Chytøej¹í øe¹ení:} Ka¾dou tøídu budeme reprezentovat zakoøenìným stromem s~hranami orientovanými smìrem ke~koøeni (jinými slovy pro ka¾dý prvek si pamatujeme jeho otce nebo ¾e je to koøen). \ nalezne koøeny stromù a porovná je, \ pøipojí koøen -jedné tøídy pod koøen druhé. Aby stromy nedegenerovaly, pøidáme dvì podmínky: +jedné tøídy pod koøen druhé. Aby stromy nedegenerovaly, pøidáme dvì pravidla: \itemize\ibull \:{\I Union by rank:} ka¾dý koøen $v$ si pamatuje svùj rank $r(v)$. Pokud spojujeme @@ -48,10 +49,10 @@ rovnou pod ko \s{Pozorování:} Samotné pravidlo Union by rank zajistí, ¾e strom ranku $r$ bude mít hloubku nejvý¹e $r$ a minimálnì $2^r$ vrcholù, tak¾e èasová slo¾itost operací -bude omezena $\O(\log n)$.% +bude omezena $\O(\log n)$ v~nejhor¹ím pøípadì.% \foot{Mimochodem, Path compression samotná by také na~slo¾itost $\O(\log n)$ amortizovanì staèila.} -Ve~skuteènosti se popsaná struktura chová daleko lépe: +Amortizovanì se ale popsaná struktura chová daleko lépe: \s{Vìta:} (Tarjan, van Leeuwen \cite{tarjan84setunion}) Kombinace Union by rank a Path compression vede k~amortizované slo¾itosti obou operací $\O(\alpha(n))$, kde $\alpha$ je inverzní Ackermannova funkce.% @@ -63,10 +64,10 @@ p Dále nás bude zajímat speciální varianta Union-Find problemu, v~ní¾ dopøedu známe posloupnost Unionù, èili strom, který spojováním komponent vznikne.\foot{Kdy se to hodí? Tøeba v~Thorupovì lineárním algoritmu \cite{thorup:usssp} na~nejkrat¹í cesty nebo -v~Weiheho takté¾ lineárním algoritmu \cite{weihe:paths} na~hledání hranovì disjunktních +ve~Weiheho takté¾ lineárním algoritmu \cite{weihe:paths} na~hledání hranovì disjunktních cest v~rovinných grafech.} Jiná interpretace tého¾ (jen pozpátku) je dekrementální udr¾ování komponent -souvislosti lesa: na~poèátku je dán les a umíme smazat hranu a otestovat, zda jsou +souvislosti lesa: na~poèátku je dán les, umíme smazat hranu a otestovat, zda jsou dva vrcholy v~tém¾e stromu. Popí¹eme algoritmus, @@ -77,7 +78,7 @@ a \cite{alstrup98marked}. \s{Definice:} {\I (Microtree/Macrotree dekompozice)} Pro zakoøenìný strom $T$ o~$n$ vrcholech definujeme: \itemize\ibull -\:{\I Koøeny mikrostromù} $R$ budou nejvy¹¹í vrcholy, pod~nimi¾ je nejvý¹e $\log n$ listù +\:{\I Koøeny mikrostromù} budou nejvy¹¹í vrcholy v~$T$, pod~nimi¾ je nejvý¹e $\log n$ listù a které nejsou koøenem celého~$T$. \:{\I Mikrostromy} le¾í v~$T$ od~tìchto koøenù ní¾e. \:{\I Spojovací hrany} vedou z~koøenù mikrostromù do~jejich otcù. @@ -143,7 +144,7 @@ p \s{Algoritmus pro mikrostromy:} Po~kompresi cest má ka¾dý mikrostrom nejvý¹e $2\log n$ vrcholù, èili také nejvý¹e tolik hran. Hrany si oèíslujeme pøirozenými èísly, ka¾dou -mno¾inu hran pak mù¾eme reprezentovat $2\log n$-bitovým èíslem a mno¾inové operace +mno¾inu hran pak mù¾eme reprezentovat $(2\log n)$-bitovým èíslem a mno¾inové operace provádìt pomocí bitových v~konstantním èase. Pro ka¾dý mikrostrom si pøedpoèítáme pro v¹echny jeho vrcholy~$v$ mno¾iny~$P_v$ hran le¾ících @@ -160,12 +161,12 @@ p \>$\(x,y):$ \algo -\:$P \leftarrow P_x \mathop{\Delta} P_v$ (mno¾ina hran le¾ících na~cestì z~$x$ do~$y$). +\:$P \leftarrow P_x \mathop{\Delta} P_y$ (mno¾ina hran le¾ících na~cestì z~$x$ do~$y$). \:Pokud $P\setminus F=\emptyset$, le¾í $x$ a $y$ ve~stejnì komponentì, jinak ne. \endalgo -\s{Algoritmus pro celý problém:} Strom rozlo¾íme na~mikrostromy, makrostromy a spojovací -hrany. V~mikrostromech i makrostromech zkomprimujeme cesty. Pro cesty a mikrostromy pou¾ijeme +\s{Algoritmus pro celý problém:} Strom rozlo¾íme na~mikrostromy, makrostrom a spojovací +hrany. V~mikrostromech i makrostromu zkomprimujeme cesty. Pro cesty a mikrostromy pou¾ijeme vý¹e popsané struktury, pro ka¾dou spojovací hranu si budeme pamatovat jen znaèku, zda je pøítomna, a pro makrostrom pøebarvovací strukturu. @@ -190,7 +191,7 @@ zda je p \s{Analýza:} Operace \ trvá konstantní èas, proto¾e se rozlo¾í na~$\O(1)$ \ù v~dílèích strukturách a ka¾dý z~nich trvá konstantnì dlouho. V¹ech $n$ operací \ trvá $\O(n)$, jeliko¾ zpùsobí $\O(n)$ amortizovanì konstantních operací s~mikrostromy, spojovacími -hranami a cestami a $\O(n/\log n)$ operací s~makrostromy, které trvají $\O(\log n)$ amortizovanì +hranami a cestami a $\O(n/\log n)$ operací s~makrostromem, které trvají $\O(\log n)$ amortizovanì ka¾dá.% \foot{To je v~prùmìru $\O(1)$ na~operaci a dokonce i amortizovanì, pokud necháme inicializaci struktury, která je lineární, naspoøit potenciál $\O(n)$, ze~kterého budeme prùbì¾nì platit @@ -204,20 +205,20 @@ nejbli \h{Fredericksonova clusterizace} Mikro/makro-stromová dekompozice není jediný zpùsob, jak stromy rozkládat. Nìkdy -se hodí napøíklad následující my¹lenka: \cite{frederickson91ambivalent} +se hodí napøíklad následující my¹lenka: -\s{Definice:} (Fredericksonova clusterizace) Nech» $G$ je graf s~vrcholy stupòù nejvý¹e~3 +\s{Definice:} {\I (Fredericksonova clusterizace)} Nech» $G$ je graf s~vrcholy stupòù nejvý¹e~3 a $c\ge 1$. Pak $c$-clusterizací grafu $G$ nazveme libovolný rozklad -$G$ na~souvislé podgrafy (clustery) $C_1, C_2, \ldots, C_k$ takový, ¾e platí: +$G$ na~souvislé podgrafy {\I (clustery)} $C_1, C_2, \ldots, C_k$ takový, ¾e platí: \itemize\ibull -\:$\forall v \in V \exists ! i: v \in C_i$. -\:$\forall i: \vert C_i\vert \le c$. -\:$\forall i$ je vnìj¹í stupeò $C_i$ (tj. poèet hran, které vedou mezi $C_i$ a zbytkem grafu) -nejvý¹e~3. Navíc pokud je právì~3, je cluster triviální, èili $\vert C_i \vert = 1$. +\:Ka¾dý vrchol se nachází v~právì jednom clusteru (hrany mohou vést i mezi clustery). +\:Ka¾dý cluster má nejvý¹e~$c$ vrcholù. +\:Vnìj¹í stupeò ka¾dého clusteru (tj. poèet hran, které vedou mezi $C_i$ a zbytkem grafu) +je nejvý¹e~3. Navíc pokud je právì~3, je cluster triviální, èili $\vert C_i \vert = 1$. \:®ádné dva sousední clustery nelze spojit. \endlist -\s{Vìta:} (Frederickson) Ka¾dá $c$-clusterizace grafu $G$ má $\O(V(G)/c)$ clusterù. Existuje +\s{Vìta:} (Frederickson \cite{frederickson91ambivalent}) Ka¾dá $c$-clusterizace grafu $G$ má $\O(V(G)/c)$ clusterù. Existuje algoritmus, který jednu takovou najde v~lineárním èase. \proof První èást rozborem pøípadù, druhá hladovì pomocí DFS. \qed diff --git a/9-decomp/mima.eps b/9-decomp/mima.eps index fade593..a8296e8 100644 --- a/9-decomp/mima.eps +++ b/9-decomp/mima.eps @@ -1,7 +1,7 @@ %!PS-Adobe-3.0 EPSF-3.0 %%Title: mima %%Creator: VRR (http://vrr.ucw.cz/) -%%CreationDate: Wed Jan 24 22:35:30 2007 +%%CreationDate: Thu Jan 25 13:08:39 2007 %%Pages: 1 %%BoundingBox: 0 0 334 77 %%DocumentData: Clean7Bit @@ -1007,7 +1007,7 @@ cp st gr gs -0.000000 slw +0.100000 slw 0 slc 0 slj [] sdh @@ -1059,7 +1059,7 @@ cp st gr gs -0.000000 slw +0.100000 slw 0 slc 0 slj [] sdh @@ -1111,7 +1111,7 @@ cp st gr gs -0.000000 slw +0.100000 slw 0 slc 0 slj [] sdh @@ -1293,7 +1293,7 @@ cp st gr gs -0.000000 slw +0.100000 slw 0 slc 0 slj [] sdh @@ -1449,7 +1449,7 @@ cp st gr gs -0.000000 slw +0.100000 slw 0 slc 0 slj [] sdh diff --git a/9-decomp/mima.vrr b/9-decomp/mima.vrr index 82bded2..fc8b8a4 100644 --- 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137807744 137808376 137817384 137809704)))))) \ No newline at end of file diff --git a/ga.bib b/ga.bib index 31c389a..792e03a 100644 --- a/ga.bib +++ b/ga.bib @@ -451,7 +451,7 @@ } @article{ weihe:paths, - title={{Edge-Disjoint (s, t)-Paths in Undirected Planar Graphs in Linear Time}}, + title={{Edge-Disjoint $(s,t)$-Paths in Undirected Planar Graphs in Linear Time}}, author={Weihe, K.}, journal={Journal of Algorithms}, volume={23}, -- 2.39.5