]> mj.ucw.cz Git - ads1.git/commitdiff
Prvni verze zapisu o kostrach.
authorMartin Mares <mj@ucw.cz>
Sun, 20 May 2007 20:22:59 +0000 (22:22 +0200)
committerMartin Mares <mj@ucw.cz>
Sun, 20 May 2007 20:22:59 +0000 (22:22 +0200)
12-kostry/01.eps [new file with mode: 0644]
12-kostry/02.eps [new file with mode: 0644]
12-kostry/03.eps [new file with mode: 0644]
12-kostry/04.eps [new file with mode: 0644]
12-kostry/05.eps [new file with mode: 0644]
12-kostry/12-kostry.tex [new file with mode: 0644]
12-kostry/Makefile [new file with mode: 0644]
all/Makefile

diff --git a/12-kostry/01.eps b/12-kostry/01.eps
new file mode 100644 (file)
index 0000000..00dc34b
--- /dev/null
@@ -0,0 +1,921 @@
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+%%Title: 01.eps
+%%CreationDate: Sun May 20 18:06:43 2007
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+{currentfile /ASCII85Decode filter /RunLengthDecode filter rstr readstring pop}
+{currentfile /ASCII85Decode filter /RunLengthDecode filter gstr readstring pop}
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diff --git a/12-kostry/02.eps b/12-kostry/02.eps
new file mode 100644 (file)
index 0000000..dd0f94b
--- /dev/null
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+%!PS-Adobe-3.0 EPSF-3.0
+%%Creator: GIMP PostScript file plugin V 1,17 by Peter Kirchgessner
+%%Title: 02.eps
+%%CreationDate: Sun May 20 18:07:19 2007
+%%DocumentData: Clean7Bit
+%%LanguageLevel: 2
+%%Pages: 1
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+%%EndComments
+%%BeginProlog
+% Use own dictionary to avoid conflicts
+10 dict begin
+%%EndProlog
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+{currentfile /ASCII85Decode filter /RunLengthDecode filter gstr readstring pop}
+{currentfile /ASCII85Decode filter /RunLengthDecode filter bstr readstring pop}
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diff --git a/12-kostry/04.eps b/12-kostry/04.eps
new file mode 100644 (file)
index 0000000..1d97c2e
--- /dev/null
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+%!PS-Adobe-3.0 EPSF-3.0
+%%Creator: GIMP PostScript file plugin V 1,17 by Peter Kirchgessner
+%%Title: 04.eps
+%%CreationDate: Sun May 20 18:08:12 2007
+%%DocumentData: Clean7Bit
+%%LanguageLevel: 2
+%%Pages: 1
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+%%EndComments
+%%BeginProlog
+% Use own dictionary to avoid conflicts
+10 dict begin
+%%EndProlog
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+{currentfile /ASCII85Decode filter /RunLengthDecode filter rstr readstring pop}
+{currentfile /ASCII85Decode filter /RunLengthDecode filter gstr readstring pop}
+{currentfile /ASCII85Decode filter /RunLengthDecode filter bstr readstring pop}
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diff --git a/12-kostry/05.eps b/12-kostry/05.eps
new file mode 100644 (file)
index 0000000..0ad4a68
--- /dev/null
@@ -0,0 +1,4601 @@
+%!PS-Adobe-3.0 EPSF-3.0
+%%Creator: GIMP PostScript file plugin V 1,17 by Peter Kirchgessner
+%%Title: 05.eps
+%%CreationDate: Sun May 20 18:08:37 2007
+%%DocumentData: Clean7Bit
+%%LanguageLevel: 2
+%%Pages: 1
+%%BoundingBox: 14 14 744 303
+%%EndComments
+%%BeginProlog
+% Use own dictionary to avoid conflicts
+10 dict begin
+%%EndProlog
+%%Page: 1 1
+% Translate for offset
+14.173228346456694 14.173228346456694 translate
+% Translate to begin of first scanline
+0 288.75 translate
+729.00000000000011 -288.75 scale
+% Image geometry
+972 385 8
+% Transformation matrix
+[ 972 0 0 385 0 0 ]
+% Strings to hold RGB-samples per scanline
+/rstr 972 string def
+/gstr 972 string def
+/bstr 972 string def
+{currentfile /ASCII85Decode filter /RunLengthDecode filter rstr readstring pop}
+{currentfile /ASCII85Decode filter /RunLengthDecode filter gstr readstring pop}
+{currentfile /ASCII85Decode filter /RunLengthDecode filter bstr readstring pop}
+true 3
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+showpage
+%%Trailer
+end
+%%EOF
diff --git a/12-kostry/12-kostry.tex b/12-kostry/12-kostry.tex
new file mode 100644 (file)
index 0000000..6b0c726
--- /dev/null
@@ -0,0 +1,195 @@
+\input lecnotes.tex
+
+\prednaska{10}{Problém minimálni kostry}{(zapsali K. Ka¹èák a M. Vachna)}
+
+Pro zaèátek si definujme, co budeme dìlat, v jakých grafech budeme minimálni kostru hledat.
+
+\s{Zadání úlohy:} Pro neorientovaný graf $G$ s ohodnocením hran $w: E(G) \rightarrow R$,
+chceme najít kostru $T$ s minimálnim ohodnocením $w(T)=\sum_{e\in E(T)} w(e)$.
+
+\s{Pøedpoklady úlohy:}
+\itemize\ibull
+\:BÚNO graf $G$ je souvislý
+\:$\forall e,f \in E(G$) : $e\neq f \Rightarrow w(e)\neq w(f)$
+\endlist
+
+Nyní si uká¾eme tøi algoritmy øe¹íci zadanú úlohu, konkrétne se jedná o Jarníkùv, Borùvkùv a Kruskalùv
+algoritmus.
+
+\s{Algoritmus:} (Jarníkùv)
+
+\def\concat{\mathop{\hbox{.}}}
+
+\algo
+\:Zvolíme libovolný vrchol $v_0\in V(G)$, $T=(\left\{v_0\right\},\emptyset)$.
+\:Dokud $\vert V(T) \vert \neq n$ :
+\:             vybereme hranu $uv\in E(G) : u\in V(T), v\notin V(T)$, min $w(uv)$
+\:             $T\leftarrow T+uv$
+\endalgo
+
+\s{Vìta:} Jarníkùv algorimtus se zastaví po maximálne $n$ iteracích a vydá minimálni kostru grafu $G$.
+
+\proof
+Koneènost - pøi ka¾dé iteraci se pøidá jeden vrchol $\Rightarrow$ po maximálne $n$ iteracích se zastaví.
+Vydaný graf je strom, proto¾e se stále pøidáva list k ji¾ existujícimu stromu. Vydaný graf má $n$
+vrcholù $\Rightarrow$ vydaný graf je kostra. Ostáva nám u¾ jen dokázat, ¾e kostra je minimálni. To
+dokazuje nesledujíci Lemma.
+\qed
+
+Proto aby jsme byli schopni sformulovat i dokázat Lemma, je nutná je¹tì vysvìtlit pojem øez v grafu.
+
+\s{Definice:} Øez v grafu $G=(V,E)$ je mno¾ina hran $F\subseteq E$ taková, ¾e $\exists  U\subset V$ :
+$F=\left\{uv\in E \vert u\in U, v\notin U \right\}$.
+
+\s{Lemma:} Pokud $G$ je graf, $w$ jeho prosté ohodnocení, $F$ je øez v grafu $G$ a $f$ je nejlehèí hrana v øezu
+$F$, pak pro ka¾dou minimálni kostru $G$ je $f\in E(T)$.
+
+\proof
+Buï $T$ kostra a $f\notin E(T)$, pak $\exists$ cesta $P\subseteq T$ spojujíci $u$ a $v$ (vrcholy hrany $f$).
+Cesta musí øez alespoò jednou projít. $\exists e\in P \cap F$ taková, ¾e $w(e) > w(f)$. Uva¾me $T'=T-e+f$.
+$T'$ je rovne¾ kostra grafu $G$, proto¾e odebraním hrany $e$ se graf rozpadne na dvì komponenty a pøidáním
+hrany $f$ se opìt spojí. $w(T')=w(T)-w(e)+w(f)<w(T)$
+\qed
+
+\s{Dùsledky:} Graf $G$ s prostým ohodnocením ma pravì 1 minimálni kostru. Minimálni kostra je
+jednoznaène urèená lineárnim uspoøádaním hran.
+
+\s{Implementace:}
+\itemize\ibull
+\:Pøímoèaøá - pamatujeme si, které vrcholy a hrany jsou v kostøe $T$ a které ne. Èasová slo¾itost je $\O(nm)$
+\:Pro $v\notin V(T)$ si pamatujeme $D(v)=$min$\left\{uv, u\in T\right\}$. Èasová slo¾itost je $\O(n^2+m)=\O(n^2)$
+\endlist
+
+\s{Algoritmus:} (Borùvkùv)
+
+\def\concat{\mathop{\hbox{.}}}
+
+\algo
+\:Les $F=(V(G),\emptyset)$.
+\:Dokud $F$ má alespoò dvì komponenty :
+\:             $\forall$ komponentu $T_i$ vybereme nejlehèí incidentní hranu $t_i$.
+\:             V¹echny hrany $t_i$ pøidáme do $F$.
+\endalgo
+
+\s{Vìta:} Borùvkùv algorimtus se zastaví po $\left\lceil \log_2 n\right\rceil$ iteracích a vydá minimálni kostru grafu $G$.
+
+\proof
+Po $k$ iteracích mají v¹echny stromy minimálne $2^k$ vrcholù, $\forall$ i : $\vert T_i\vert \geq 2^k$.
+Indukcí podle $k$: ka¾dá incidentní hrana vede z $T_i$ do $T_j$. Po $k$ iteracích pøidáním incidentní hrany
+spojíme 2 stromy s alespoò $2^k$ vrcholama, tím vznikne strom s $2^{k+1}$ vrcholama.
+Algoritmus vydá kostru, staèí nahlédnout, zda je minimálni. To se lehce uká¾e u¾itím pøedchozí Lemmy.
+V¾dy pøidávame hranu, která je minimálni mezi $T$ a ostatními vrcholy grafu $G$.
+\qed
+
+\s{Implemntace:}
+\itemize\ibull
+\:Inicializace pøímoèaøá.
+\:Pomocí DFS rozlo¾íme les na komponenty. $\forall$ vrchol si pamatujeme èíslo komponenty.
+\:Pro $\forall$ hranu zjistíme, do které komponenty patøí a pro ka¾dou komponentu
+jsi uchováme nejlehèí hranu.
+
+Rozlo¾ení lesa a zji¹tìní, do jaké komponenty patøí ka¾dá hrana má èasovou slo¾itost $\O(m)$. Výslední èasová slo¾itost je $\O(m\log n)$
+\endlist
+
+\s{Algoritmus:} (Kruskalùv èili `hladový`)
+
+\def\concat{\mathop{\hbox{.}}}
+
+\algo
+\:Zetøídime v¹echny hrany z $E$ : $w(e_1)<...<w(e_m)$.
+\:$F\leftarrow (V(G),\emptyset)$.
+\:Pro $i=1$ do $m$ :
+\:             pokud $F+e_i$ je acyklický :
+\:             $F\leftarrow F+e_i$
+\endalgo
+
+\s{Vìta:} Kruskalùv `hladový` algorimtus se zastaví po $m$ iteracích a vydá minimálni kostru grafu $G$.
+
+\proof
+Podle Lemmy strom, který je výstupem algorimtu, je minimálni kostra. Indukcí snadno doká¾eme, ¾e
+stále platí $F\subseteq$ $minimálni$ $kostry$ a v¾dy do $F$ pøidávame hranu $f$, která je minimálni v øezu
+a spojuje dvì komponenty.
+\qed
+
+\s{Implemntace:}
+\itemize\ibull
+\:Zotøídení $\O(m\log n)$.
+\:Kdy¾ si pamatuji vrcholy $v$ poli:
+Find(u,v) zistí zda vrcholy $u$ a $v$ le¾í v stejné komponente. Jedna operace má èasovou slo¾itost $\O(1)$. Find
+se provede $m-$krát, proto je èasová slo¾itost celkem $\O(m)$.
+Union(u,v) pøidá hranu a tím spojí dvì komponenty. Jedna operace má èasovou slo¾itost $\O(n)$. Union se provede
+$n-$krát, proto je èasová slo¾itost celkem $\O(n^2)$.
+Výslední èasová slo¾itost je $\O(m\log n+m+n^2)=$\O$(n^2+m\log n)$
+
+\:Jinak: $\forall$ komponenta je strom orientovaný smìrem ke koøenu. $\forall$ vrchol si pamatuje
+svého otce, $\forall$ koøen si pamatuje velikost komponenty.
+Find(u,v) najde koøeny a porovná je. Èasová slo¾itost je $\O(hloubka$ $komponenty)$.
+Union(u,v) najde koøeny a pøipojí men¹í komponentu pod vìt¹í (logaritmická hloubka). Èasová slo¾itost je $\O(hloubka$ $komponenty)$.
+Find a Union mají teda èasovou slo¾itost $\O(\log n)$. Výslední èasová slo¾itost je $\O(m\log n)$.
+\endlist
+
+\s{Lemma:} Strom hloubky $h$ má alespoò $2^h$ prvkù.
+
+\proof
+Pokud Union spojí stromy jeden s hloubkou $h$ a druhý s hloubkou men¹í ne¾ $h$, pak hloubka výsledního
+stromu zústava $h$. Pokud spojuje dva stromy hloubky $h$, pak vyslední strom má hloubku $h+1$. Jak ji¾
+víme, strom hloubky $h$ má minimálne $2^h$, proto výslednej strom hloubky $h+1$ má $2^{h+1}$ vrcholù.
+\qed
+
+\h{Èerveno-èerné stromy}
+
+\s{Definice:} Èerveno-èerný strom je binárny vyhledávaci strom s barevnými(èervenými a èernými) vrcholy a
+externými vrcholy. Kdy¾ vrchol nemá nìkterého ze synú, na jeho míste je externý vrchol.
+
+\s{Axiomy:}
+\itemize\ibull
+\:Koøen a externé vrcholy sú èerné.
+\:Otec èerveného vrcholu je èerný.
+\:Na v¹ech cestách z koøene do externího vrcholu je stejný poèet èerných vrcholù.
+\endlist
+
+\s{Vìta:} Èerveno-èerný strom na $n$ vrcholech má hloubku $\O(\log n)$.
+
+K této vìtì nabízime dva dùkazy.
+
+\proof
+1. Skomprimujeme hrany èerný-èervený a èerné vrcholy, které májí dva èervené syny, na vrchol.
+Vznikne $2,4-$strom, který ma hloubku $\log n$, a proto je èerveno-èerný strom maximálne
+2x vìt¹í.
+\qed
+
+\proof
+2. Pøedpokladejme, ¾e $T$ je èerveno-èerný strom, který má na cestì z koøene do listu
+$k$ èerných vrcholù. Pak pre poèet vrcholù stromu $T$ platí $2^k-1\leq n \leq 2^{2k}-1$. Nejmen¹í takový strom má v¹echny vrcholy obarvené èernì a je to úplný
+pravidelný binární strom o hloubce $k-1$, co¾ dáva dolní odhad. Nejvìt¹í takový strom má v¹echny vrcholy v sudých hladinách obarveny èervenì a v lichých hladinách èernì, je to ùplný pravidelný binární strom o hloubce $2k-1$ a tím je dán horní odhad. Tedy $k\leq 1+\log n$. Z vlastností èerveno-èerných stromù plyne, ¾e $k\leq hloubka(T) \leq 2k$.
+\qed
+
+Niní si uká¾eme a rozebereme vkládaní do èerveno-èerných stromù.
+
+\s{Insert:} (v èerveno-èernom strome)
+BÚNO nový uzel $t$ bude èervený.
+Vlo¾íme uzel do stromu jako do standartního binárního vyhledávacího stromu.
+Jestli¾e je pøedek èerný, jsme hotovi (viï obr. 1).
+\figure{01.eps}{obr. 1}{2in}
+Jestli¾e nikoliv, mù¾u nastat tøi nasledujíci pøípady:
+\itemize\ibull
+\:Je-li vrchol $t$ èervený a jeho otec je také èervený, pak øekneme, ¾e $t$ je porucha. Je-li $t$ porucha,
+pak ji musíme nìjak opravit. Situace je na obrázku 2 - nejprve zále¾í na tom, jakou barvu má $s$, strýc $t$:
+\figure{02.eps}{obr. 2}{1.5in}
+\algo
+\:$s$ je èervený. Pak pouze pøebarvíme $o$, $d$ a $s$ podle obrázku 3.
+Nyní $d$ mù¾e být porucha, ov¹em posunutá o 2 hladiny vý¹e.
+\figure{03.eps}{obr. 3}{3.5in}
+Pro splnìní poslední podmínky je je je¹tì nutné pøebarvit koøen $d$ na èerno.
+
+\:$s$ je èerný. Pak zále¾í na tom, zda hodnota $t$ le¾í mezi hodnotami $o$ a $d$ nebo ne. Jinými slovy, zda cesta $t-o-d$ obsahuje zatáèku.
+(a) Bez zatáèky: Provedeme rotaci a pøebarvíme podle obrázku 4. Splnìny budou podmínky 1, 2 i 3, tedy máme èervenoèerný strom:
+\figure{04.eps}{obr. 4}{4in}
+(b) Se zatáèkou. Provedeme dvojitou (LR) rotaci a pøebarvíme podle obrázku 5. Splnìny budou podmínky 1, 2 i 3, opìt máme rovnou èervenoèerný strom.
+\figure{05.eps}{obr. 5}{4in}
+\endalgo
+\endlist
+
+
+
+\bye
diff --git a/12-kostry/Makefile b/12-kostry/Makefile
new file mode 100644 (file)
index 0000000..f79ef54
--- /dev/null
@@ -0,0 +1,3 @@
+P=12-kostry
+
+include ../Makerules
index ca012464afb47f49d41f199067457144ebb144c2..610ec08c688f329787526b5ce97f8f84b0aade8e 100644 (file)
@@ -1,5 +1,5 @@
 P=ads
-X:=$(shell for a in 1 2 3 4 5 6 8 ; do echo ../$$a-*/$$a-*.tex ; done)
+X:=$(shell for a in 1 2 3 4 5 6 8 12 ; do echo ../$$a-*/$$a-*.tex ; done)
 
 %universe: all ChangeLog