Nove obrazky do kapitoly o kostrach a spousta kosmetickych vylepseni tamtez.

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diff --git a/5-mst/5-mst.tex b/5-mst/5-mst.tex
index 06fdb617fe47c1d1d37a8609b251a8e30adde7ac..b93fa4c682d870307f5ac59ddc3b8817c0f2abb2 100644 (file)
--- a/5-mst/5-mst.tex
+++ b/5-mst/5-mst.tex
@@ -1,7 +1,7 @@
 \input ../sgr.tex
 \prednaska{5}{Minimální kostry}{}
 
-\def\symdiff{\mathop{\Delta}}
+\def\Tmin{T_{min}}
 
 \>Tato kapitola uvede problém minimální kostry, základní vìty o~kostrách a klasické
 algoritmy na~hledání minimálních koster. Budeme se inspirovat Tarjanovým pøístupem
@@ -25,7 +25,7 @@ ka
 \endlist
 
 Toto je sice standardní definice MST, ale jinak je dosti ne¹ikovná, proto¾e vy¾aduje,
-aby bylo váhy mo¾né sèítat. Za~chvíli si uká¾eme, ¾e to není potøeba.
+aby bylo váhy mo¾né sèítat. Pojïme ukázat, ¾e to není potøeba.
 
 \s{Definice:}
 Buï $T \subseteq G$ nìjaká kostra grafu~$G$. Pak:
@@ -41,11 +41,9 @@ Bu
 \s{Vìta:} Kostra~$T$ je minimální $\Leftrightarrow$ neexistuje hrana lehká vzhledem k~$T$.
 
 Tato vìta nám dává pìknou alternativní definici MST, která místo sèítání vah váhy
-pouze porovnává, èili jí místo èísel staèí (kvazi)uspoøádání na~hranách. Ne¾ se dostaneme
+pouze porovnává, èili jí místo èísel staèí lineární (kvazi)uspoøádání na~hranách. Ne¾ se dostaneme
 k~jejímu dùkazu, prozkoumejme nejdøíve, jak se dá mezi jednotlivými kostrami pøecházet.
 
-%\centerline{\epsfysize=3cm\epsfbox{01.eps}}
-
 \s{Definice:} Pro kostru~$T$ a hrany $e, e'$
 zaveïme $\<swap>(T,e,e^\prime) := T-e+e'$.
 
@@ -58,7 +56,7 @@ a vynech
 Máme-li libovolné kostry $T$ a $T'$, pak lze z~$T$ dostat $T'$ koneèným poètem operací \<swap>.
 
 \proof
-Jeliko¾ $\vert T \vert = \vert T' \vert$, musí existovat $e' \in T'\setminus T$.
+Pokud $T \ne T'$, musí existovat hrana $e' \in T'\setminus T$, proto¾e $\vert T \vert = \vert T' \vert$.
 Kru¾nice $T[e']+e'$ nemù¾e být celá obsa¾ena v~$T$, tak¾e existuje hrana
 $e\in T[e']\setminus T'$ a $\check{T} := \<swap>(T,e,e')$ je kostra,
 pro kterou $\vert \check{T} \symdiff T' \vert = \vert T \symdiff T' \vert -2$.
@@ -88,17 +86,17 @@ ne
 
 \s{Dùkaz vìty:}
 \itemize\ibull
-\:$\Rightarrow$ Chceme dokázat, ¾e $\exists$ lehká hrana $\Rightarrow$ $T$ není minimální.
+\:$\exists$ lehká hrana $\Rightarrow$ $T$ není minimální.
 
-Nech» $\exists e$ lehká. Najdeme $e' \in T[e] : w(e) < w(e')$ (ta musí existovat z def. lehké hrany).
+Nech» $\exists e$ lehká. Najdeme $e' \in T[e] : w(e) < w(e')$ (ta musí existovat z~definice lehké hrany).
 Kostra $T' := \<swap>(T,e,e')$ je lehèí ne¾~$T$.
 
 \medskip
 
-\:$\Leftarrow$ Pokud k~$T$ neexistuje lehká hrana, je $T$ minimální.
+\:K~$T$ neexistuje lehká hrana $\Rightarrow$ $T$ je minimální.
 
-Uva¾me nìjakou minimální kostru $T_{min}$ a pou¾ijme monotónní swapovací lemma na~$T$ a $T_{min}$. Z~nìj plyne $w(T)\le w(T_{min})$,
-a~tedy $w(T)=w(T_{min})$.
+Uva¾me nìjakou minimální kostru $\Tmin$ a pou¾ijme monotónní swapovací lemma na~$T$ a $\Tmin$. Z~nìj plyne $w(T)\le w(\Tmin)$,
+a~tedy $w(T)=w(\Tmin)$.
 \qeditem
 \endlist
 
@@ -141,29 +139,31 @@ podle vah plat
 \:Modøe obarvené hrany tvoøí minimální kostru.
 \endalgo
 
-\s{Modré lemma:} Je-li hrana~$e$ kdykoliv algoritmem obarvena na~modro, pak $e\in T_{min}$.
+\proof
+Nejdøíve si doká¾eme nìkolik lemmat. Jeliko¾ hrany mají navzájem rùzné váhy,
+mù¾eme pøedpokládat, ¾e algoritmus má sestrojit jednu konkrétní minimální kostru~$\Tmin$.
+
+\s{Modré lemma:} Je-li libovolná hrana~$e$ algoritmem kdykoliv obarvena na~modro, pak $e\in \Tmin$.
 
-\proof Minimální kostra $T_{min}$ je urèena jednoznaènì (váhy jsou rùzné).
-Hrana~$e$ byla omodøena jako nejlehèí hrana nìjakého øezu~$C$.
-Pokud by existovala nìjaká jiná $e' \in C \cap T_{min}[e]$, mù¾eme provést
-$\<swap>(T_{min},e',e)$ a tím z~$T_{min}$ vytvoøit je¹tì lehèí kostru,
-co¾ je spor.
+\proof Sporem: Hrana~$e$ byla omodøena jako nejlehèí hrana nìjakého øezu~$C$.
+Pokud $e\not\in \Tmin$, musí cesta $\Tmin[e]$ obsahovat nìjakou jinou
+hranu~$e'$ øezu~$C$. Jen¾e $e'$ je tì¾¹í ne¾~$e$, tak¾e operací $\<swap>(\Tmin,e',e)$
+získáme je¹tì lehèí kostru, co¾ není mo¾né.
 \qed
 
-\fig{02.eps}{3cm}
+\figure{mst-rb.eps}{Situace v~dùkazu Modrého a Èerveného lemmatu}{\epsfxsize}
 
-\s{Èervené lemma:} Je-li hrana~$e$ kdykoliv algoritmem obarvena na~èerveno, pak $e\not\in T_{min}$.
+\s{Èervené lemma:} Je-li libovolná hrana~$e$ algoritmem kdykoliv obarvena na~èerveno,
+pak $e\not\in \Tmin$.
 
 \proof Opìt sporem: Pøedpokládejme, ¾e~$e$ byla obarvena èervenì jako nejtì¾¹í na~nìjaké kru¾nici~$C$
-a ¾e $e\in T_{min}$. Odebráním~$e$ se nam $T_min$ rozpadne na dvì komponenty $T_x$ a $T_y$.
-Nìkteré vrcholy kru¾nice pøipadnou do komponenty $T_x$, ostatní do $T_y$.
-Lze jednodu¹e nahlédnout, ¾e musí existovat hrana $e'\ne e$ taková, ¾e $x' \in T_x$ a $y' \in T_y$.
-Hrana~$e$ byla nejtì¾¹í na~kru¾nici, tak¾e $w(e') < w(e)$ a $\<swap>(T_{min},e,e')$ nám dá~lehèí kostru,
-co¾ je spor.
+a ¾e $e\in \Tmin$. Odebráním~$e$ se nám $\Tmin$ rozpadne na~dvì komponenty $T_x$ a $T_y$.
+Nìkteré vrcholy kru¾nice pøipadnou do komponenty $T_x$, ostatní do~$T_y$. Na~$C$ ale musí
+existovat nìjaká hrana $e'\ne e$, její¾ krajní vrcholy le¾í v~rùzných komponentách,
+a~jeliko¾ hrana~$e$ byla na~kru¾nici nejtì¾¹í, je $w(e') < w(e)$. Pomocí $\<swap>(\Tmin,e,e')$
+proto získáme lehèí kostru, a~to je spor.
 \qed
 
-\fig{03.eps}{4cm}
-
 \s{Bezbarvé lemma:} Pokud existuje nìjaká neobarvená hrana, lze je¹tì pou¾ít nìkteré
 z~pravidel.
 
@@ -175,22 +175,19 @@ do~nich
 neexistují ¾ádné lehké hrany, tak¾e hrana $e$ je nejdra¾¹í na~cyklu tvoøeném modrou cestou a~touto hranou
 a mohu na ni pou¾ít èervené pravidlo.
 
-\fig{04.eps}{3cm}
+\figure{mst-bez.eps}{Situace v~dùkazu Bezbarvého lemmatu}{\epsfxsize}
 
 \:$y \notin M$: Tehdy øez $\delta(M)$ neobsahuje ¾ádné modré hrany a alespoò jednu, která není
 èervená (konkrétnì hranu~$e$), tak¾e na~tento øez mù¾eme pou¾ít modré pravidlo.
-
-\fig{05.eps}{3cm}
-
 \qeditem
 \endlist
 
 \s{Dùkaz vìty:}
 \itemize\ibull
-\:{\I Zastaví se:} Z~èerveného a modrého lemmatu plyne, ¾e ¾ádnou hranu nikdy nepøebarvíme, pøibude ka¾dým krokem
-  alespoò jedna obarvená hrana, tak¾e se algoritmus zastaví.
+\:{\I Zastaví se:} Z~èerveného a modrého lemmatu plyne, ¾e ¾ádnou hranu nikdy nepøebarvíme. Ka¾dým krokem pøibude
+  alespoò jedna obarvená hrana, tak¾e se algoritmus po~nejvý¹e $m$~krocích zastaví.
 \:{\I Obarví v¹e:} Pokud existuje alespoò jedna neobarvená hrana, pak podle bezbarvého lemmatu algoritmus pokraèuje.
-\:{\I Najde modrou MST:} Podle èerveného a modrého lemmatu le¾í v~$T_{min}$ právì modré hrany.
+\:{\I Najde modrou MST:} Podle èerveného a modrého lemmatu le¾í v~$\Tmin$ právì modré hrany.
 \qeditem
 \endlist
 
@@ -223,7 +220,7 @@ dob
 
 \s{Borùvkùv:}
 
-Opìt si budeme pìstovat modrý les, av¹ak tentokrát jej budeme roz¹iøovat ve fázích. V~jedné fázi nalezneme ke ka¾dému stromeèku nejlevnìj¹í incidentní hranu
+Opìt si budeme pìstovat modrý les, av¹ak tentokrát jej budeme roz¹iøovat ve~fázích. V~jedné fázi nalezneme ke ka¾dému stromeèku nejlevnìj¹í incidentní hranu
 a v¹echny tyto nalezené hrany naráz pøidáme (aplikujeme nìkolik modrých pravidel najednou). Pokud jsou v¹echny váhy rùzné, cyklus
 tím nevznikne.
 
@@ -236,7 +233,7 @@ Mimo to lze ka
 Jarníkùv algoritmus je podobný Borùvkovi, ale s tím rozdílem, ¾e nenecháme rùst celý les, ale jen jeden modrý strom. V~ka¾dém
 okam¾iku nalezneme nejlevnìj¹í hranu vedoucí mezi stromem a zbytkem grafu a pøidáme ji ke~stromu (modré pravidlo);
 hrany vedoucí uvnitø stromu prùbì¾nì zahazujeme (èervené pravidlo). Kroky opakujeme, dokud se strom nerozroste pøes v¹echny vrcholy.
-Pøi ¹ikovné implementaci pomocí haldy má èasovou slo¾itost $\O(m\log n)$, v~pøí¹tí kapitole uká¾eme implementaci je¹tì ¹ikovnìj¹í.
+Pøi ¹ikovné implementaci pomocí haldy dosáhneme èasové slo¾itosti $\O(m\log n)$, v~pøí¹tí kapitole uká¾eme implementaci je¹tì ¹ikovnìj¹í.
 
 \s{Cvièení:}
 Naleznìte algoritmus pro výpoèet MST v~grafech ohodnocených vahami $\{1,\ldots k\}$ se slo¾itostí $\O(mk)$.

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+%%BeginResource: font CMSY7
+%!PS-AdobeFont-1.1: CMSY7 1.0
+%%CreationDate: 1991 Aug 15 07:21:52
+
+% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.
+
+11 dict begin
+/FontInfo 7 dict dup begin
+/version (1.0) readonly def
+/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
+/FullName (CMSY7) readonly def
+/FamilyName (Computer Modern) readonly def
+/Weight (Medium) readonly def
+/ItalicAngle -14.035 def
+/isFixedPitch false def
+end readonly def
+/FontName /CMSY7 def
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+/FontMatrix [0.001 0 0 0.001 0 0] readonly def
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+% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.
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diff --git a/ga.bib b/ga.bib
index d24d0521a78005605c2d6c0210ebc44d39ca65dc..cd16e4d77247466b10a1e89e6b5a03686b684fe5 100644 (file)
--- a/ga.bib
+++ b/ga.bib
@@ -383,3 +383,14 @@
  publisher = {Society for Industrial and Applied Mathematics},
  address = {Philadelphia, PA, USA},
 }
+
+@article{ kruskal:mst,
+  title={{On the Shortest Spanning Subtree of a Graph and the Traveling Salesman Problem}},
+  author={Kruskal Jr, J.B.},
+  journal={Proceedings of the American Mathematical Society},
+  volume={7},
+  number={1},
+  pages={48--50},
+  year={1956},
+  publisher={JSTOR}
+}

diff --git a/sgr.tex b/sgr.tex
index 3347f16a301071d8857de3025374deef428f2e4d..f0e9b7e7c086fa7d053ae8a34113dd4bff215349 100644 (file)
--- a/sgr.tex
+++ b/sgr.tex
@@ -120,3 +120,6 @@
 % Predefinujeme odsazovani odrazek z mjmacu
 \def\itemspacingrules{\ifitem\advance\leftskip by \parindent\parindent=0pt\itemfalse\else
 \advance\leftskip by 2.5em\fi\rightskip=\leftskip}
+
+% Matematicke symboly
+\def\symdiff{\mathop{\Delta}}