From c76ffcadd60454e286eb77169d51293a5071ba31 Mon Sep 17 00:00:00 2001
From: Martin Mares <mj@ucw.cz>
Date: Wed, 24 Jan 2007 22:37:34 +0100
Subject: [PATCH] Opravena chyba v mikro/makro-dekompozici stromu: cestovou
 kompresi musime provadet na kazdem podstromu zvlast, nestaci ji provest na
 vstupni strom. Take uz se objevil obrazek s priklady dekompozic, ze ktereho
 je videt, proc je to potreba.

---
 10-decomp/10-decomp.tex |   62 +-
 10-decomp/mima.eps      | 1517 +++++++++++++++++++++++++++++++++++++++
 10-decomp/mima.vrr      |   90 +++
 3 files changed, 1649 insertions(+), 20 deletions(-)
 create mode 100644 10-decomp/mima.eps
 create mode 100644 10-decomp/mima.vrr

diff --git a/10-decomp/10-decomp.tex b/10-decomp/10-decomp.tex
index b5509b6..b76b00b 100644
--- a/10-decomp/10-decomp.tex
+++ b/10-decomp/10-decomp.tex
@@ -77,27 +77,33 @@ 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} $R$ budou nejvy¹¹í vrcholy, 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.
-\:{\I Makrostrom} je tvoøen ostatními vrcholy a hranami stromu~$T$.
+\:{\I Makrostrom} je tvoøen zbývajícími vrcholy a hranami stromu~$T$.
 \endlist
 
 \s{Pozorování:} Ka¾dý mikrostrom má nejvý¹e $\log n$ listù. Pod ka¾dým listem makrostromu le¾í
-alespoò jeden mikrostrom (mù¾e jich být i více, pøedstavte si dekompozici hvìzdy), tak¾e
+alespoò jeden mikrostrom (mù¾e jich být i více, viz dekompozice hvìzdy na~obrázku), tak¾e
 listù makrostromu je nejvý¹e $n/\log n$.
 
 Vnitøních vrcholù makro- i mikrostromù ale mù¾e být ne¹ikovnì mnoho, proto¾e se ve~stromech mohou
 vyskytovat dlouhé cesty. Pomù¾eme si snadno: ka¾dou cestu si budeme pamatovat zvlá¹» a ve~stromu
 ji nahradíme hranou, která bude vlo¾ena právì tehdy, kdy¾ budou pøítomny v¹echny hrany cesty.
 
+\s{Pøíklad:} Následující obrázek ukazuje dekompozici nìkolika stromù za~pøepokladu,
+¾e $\log n=4$. Vrcholy mikrostromù jsou èerné, makrostromu bílé. Spojovací hrany kreslíme teèkovanì,
+hrany komprimovaných cest tuènì.
+
+\fig{mima.eps}{\epsfxsize}
+
 \s{Algoritmus pro cesty:} Cestu délky~$l$ rozdìlíme na~úseky délky $\log n$, pro nì¾ si ulo¾íme
 mno¾iny ji¾ pøítomných hran (po~bitech jako èísla). Pak si je¹tì pamatujeme zkomprimovanou cestu (hrany
 odpovídají úsekùm a jsou pøítomny právì tehdy, jsou-li pøítomny v¹echny hrany pøíslu¹ného úseku)
 délky $l/\log n$ a pro ni \uv{pøebarvovací} strukturu pro Union-Find.
 
 \>$\<Union>(x,y)$ (pøidání hrany $e=xy$ do~cesty):
-
 \algo
 \:Pøidáme $e$ do mno¾iny hran pøítomných v~pøíslu¹ném úseku.
 \:Pokud se tím úsek naplnil, pøidáme odpovídající hranu do~zkomprimované cesty.
@@ -114,6 +120,27 @@ d
 Operace uvnitø úsekù pracují v~èase $\O(1)$, operace na~zkomprimované cestì v~$\O(\log l)$
 amortizovanì, ale za~dobu ¾ivota struktury je jich $\O(l/\log n)=\O(l/\log l)$, tak¾e celkovì zaberou lineární èas.
 
+\s{Cestová komprese:} Operace na~mikro/makro-stromech budeme následujícím zpùsobem
+pøevádìt na~operace s~jejich cestovì komprimovanými podobami a na~operace s~cestovými strukturami:
+
+\>$\<Union>(x,y)$:
+\algo
+\:Pokud $e=xy$ le¾í uvnitø nìjaké cesty, pøidáme ji do~cesty, co¾ buïto zpùsobí
+  pøidávání jiné hrany, a~nebo u¾ jsme hotovi.
+\:Provedeme \<Union> v~komprimovaném stromu.
+\endalgo
+
+\>$\<Find>(x,y)$:
+\algo
+\:Pokud $x$ a $y$ le¾í uvnitø jedné cesty, zeptáme se cestové struktury a konèíme.
+\:Pokud $x$ le¾í uvnitø nìjaké cesty, zjistíme dotazem na~cestovou strukturu,
+  ke~kterému krajnímu vrcholu cesty je pøipojen, a~$x$ nahradíme tímto vrcholem.
+  Není-li pøipojen k~¾ádnému, je~evidentnì odpovìï na~celý \<Find> negativní;
+  pokud k~obìma, vybereme si libovolný, proto¾e jsou stejnì v~cestovì komprimovaném
+  stromu spojeny hranou. Analogicky pro~$y$.
+\:Zeptáme se struktury pro komprimovaný strom.
+\endalgo
+
 \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
@@ -137,36 +164,26 @@ p
 \:Pokud $P\setminus F=\emptyset$, le¾í $x$ a $y$ ve~stejnì komponentì, jinak ne.
 \endalgo
 
-\s{Algoritmus pro celý problém:} Zkomprimujeme cesty a výsledný strom rozlo¾íme
-na~mikrostromy, makrostromy a spojovací hrany. Pro cesty a mikrostromy pou¾ijeme
+\s{Algoritmus pro celý problém:} Strom rozlo¾íme na~mikrostromy, makrostromy a spojovací
+hrany. V~mikrostromech i makrostromech 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.
 
-\>Operace $\<Union>(x,y)$:
+\>$\<Union>(x,y)$:
 
 \algo
-\:Pokud $e=xy$ le¾í uvnitø cesty, pøidáme ji do~cesty, co¾ buïto zpùsobí
-   pøidávání jiné hrany do~stromu, a~nebo u¾ jsme hotovi.
-\:Pokud $e$ je spojovací, poznamenáme si, ¾e je pøítomna, a~konèíme.
-\:Pokud $e$ le¾í uvnitø mikrostromu nebo makrostromu, provedeme \<Union>
+\:Pokud $e=xy$ je spojovací, poznamenáme si, ¾e je pøítomna, a~konèíme.
+\:Nyní víme, ¾e $e$ le¾í uvnitø mikrostromu nebo makrostromu, a~tak provedeme \<Union>
    na~pøíslu¹né struktuøe.
 \endlist
 
-\>Operace $\<Find>(x,y)$:
+\>$\<Find>(x,y)$:
 
 \algo
-\:Pokud $x$ a $y$ le¾í uvnitø jedné cesty, zeptáme se cestové struktury a konèíme.
-\:Pokud $x$ le¾í uvnitø nìjaké cesty, zjistíme dotazem na~cestovou strukturu,
-   ke~kterému krajnímu vrcholu cesty je pøipojen, a~$x$ nahradíme tímto vrcholem.
-   Není-li pøipojen k~¾ádnému, je~evidentnì odpovìï na~celý \<Find> negativní;
-   pokud k~obìma, vybereme si libovolný, proto¾e jsou stejnì v~cestovì komprimovaném
-   stromu spojeny hranou. Analogicky pro~$y$.
-   \hfil\break [Nyní jsou $x$ a $y$ vrcholy cestovì zkomprimovaného stromu.]
 \:Le¾í-li $x$ a $y$ v~jednom mikrostromu, zeptáme se struktury pro~mikrostrom.
 \:Je-li $x$ uvnitø mikrostromu, zeptáme se mikrostruktury na~spojení s~koøenem mikrostromu.
   Není-li, odpovíme {\sc ne}, stejnì jako kdy¾ není pøítomna pøíslu¹ná spojovací hrana.
   Jinak $x$ nahradíme listem makrostromu, do~kterého spojovací hrana vede. Podobnì pro~$y$.
-  \hfil\break [Nyní jsou $x$ a $y$ vrcholy makrostromu.]
 \:Odpovíme podle struktury pro makrostrom.
 \endalgo
 
@@ -179,6 +196,11 @@ ka
 struktury, která je lineární, naspoøit potenciál $\O(n)$, ze~kterého budeme prùbì¾nì platit
 sluèování v~makrostromu.}
 
+\s{Cvièení:} Zkuste pomocí dekompozice vyøe¹it následující problém: je dán strom,
+jeho¾ ka¾dý vrchol mù¾e být oznaèený. Navrhnìte datovou strukturu, která bude umìt
+v~èase $\O(\log\log n)$ oznaèit nebo odznaèit vrchol a v~èase $\O(\log n/\log\log n)$ najít
+nejbli¾¹ího oznaèeného pøedchùdce.
+
 \h{Fredericksonova clusterizace}
 
 Mikro/makro-stromová dekompozice není jediný zpùsob, jak stromy rozkládat. Nìkdy
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2.39.2