From ba8385df7e883beadd6dcd1e7b05e42f660376a5 Mon Sep 17 00:00:00 2001
From: Martin Mares <mj@ucw.cz>
Date: Fri, 14 Dec 2007 21:23:29 +0100
Subject: [PATCH] Nulta verze kapitoly o tridicich sitich.

---
 5-sortnet/5-sortnet.tex | 194 ++++++++++++
 5-sortnet/Makefile      |   3 +
 5-sortnet/sortnet.0     |  65 ++++
 5-sortnet/sortnet.1     | 111 +++++++
 5-sortnet/sortnet.2     | 103 +++++++
 5-sortnet/sortnet.3     | 107 +++++++
 5-sortnet/sortnet.4     | 182 +++++++++++
 5-sortnet/sortnet.5     | 169 +++++++++++
 5-sortnet/sortnet.6     | 212 +++++++++++++
 5-sortnet/sortnet.7     | 106 +++++++
 5-sortnet/sortnet.mp    | 657 ++++++++++++++++++++++++++++++++++++++++
 all/Makefile            |   2 +-
 12 files changed, 1910 insertions(+), 1 deletion(-)
 create mode 100644 5-sortnet/5-sortnet.tex
 create mode 100644 5-sortnet/Makefile
 create mode 100644 5-sortnet/sortnet.0
 create mode 100644 5-sortnet/sortnet.1
 create mode 100644 5-sortnet/sortnet.2
 create mode 100644 5-sortnet/sortnet.3
 create mode 100644 5-sortnet/sortnet.4
 create mode 100644 5-sortnet/sortnet.5
 create mode 100644 5-sortnet/sortnet.6
 create mode 100644 5-sortnet/sortnet.7
 create mode 100644 5-sortnet/sortnet.mp

diff --git a/5-sortnet/5-sortnet.tex b/5-sortnet/5-sortnet.tex
new file mode 100644
index 0000000..fb047a1
--- /dev/null
+++ b/5-sortnet/5-sortnet.tex
@@ -0,0 +1,194 @@
+\input ../lecnotes.tex
+\input epsf
+\def\itm{\item{$\bullet$}}
+
+\prednaska{5}{Goldbergùv algoritmus}{(zapsaly T.~Klimo¹ová 
+a~K.~B\"ohmová)}
+
+\h{Goldbergùv algoritmus -- pokraèování}
+
+Algoritmus upøesníme tak, ¾e místo libovolného vrcholu s~pøebytkem
+budeme v¾dy pracovat s~tím, který je na nejvy¹¹í hladinì. Jak doká¾eme
+v~následujícím lemmatu, sní¾í se tím poèet potøebných nenasycených
+pøevedení. Takto modifikovaný algoritmus budeme znaèit~$G'$.
+
+\s{Lemma $N'$:} V~algoritmu $G'$ je poèet nenasycených pøevedení
+$\O(N^3)$.
+
+\proof
+Definujme $H$~jako maximální vý¹ku vrcholù s~pøebytkem.
+ $$H:=\max\{h(v);v \not= z,s, f^\triangle (v)\geq 0\}$$
+Bìh algoritmu $G'$ rozdìlíme na fáze tak, ¾e fáze skonèí po ka¾dé zmìnì
+$H$. Odhadneme poèet nenasycených pøevedení v~jedné fázi: bude jich
+nanejvý¹ stejnì, jako vrcholù, které se na zaèátku fáze nacházely na
+nejvy¹¹í hladinì -- z~jiných vrcholù v~prùbìhu fáze nic nepøevádíme,
+a~nenasycené pøevedení mù¾eme provést z~ka¾dého vrcholu nejvý¹e
+jednou. Poèet nenasycených pøevedení za fázi je $\leq N$.
+
+Odhadneme poèet fází: rozdìlíme fáze podle toho, jestli konèí sní¾ením
+nebo zvý¹ením~$H$ a~odhadneme jejich poèty. Maximálnì $2N^2$ fází mù¾e
+konèit zvý¹ením~$H$, proto¾e dle lemmatu Z~provedeme nanejvý¹ $2N^2$
+zvednutí. Sní¾ením~$H$ mù¾e konèit nanejvý¹ $2N^2$ fází, proto¾e~$H$
+klesne v¾dy alespoò o~1, pùvodnì bylo 0 a~nikdy nemù¾e být záporné --
+klesnout tedy nemù¾e víckrát ne¾ vzrùst. 
+
+Máme maximálnì $4N^2$ fází o~nejvý¹e~$N$ nenasycených pøevedení,
+celkem tedy algoritmus $G'$ provede $\O(N^3)$ nenasycených pøevedení.
+\qed
+
+\medskip
+Ve skuteènosti je algoritmus je¹tì lep¹í (alespoò pro øídké grafy):
+
+\s{Lemma $N''$:} V~algoritmu $G'$ je poèet nenasycených pøevedení $\O(N^2
+\sqrt M)$.
+
+\proof
+(Nezkou¹í se.) Algoritmus rozdìlíme na fáze stejnì jako v~dùkazu
+pøedchozího lemmatu a~zvolíme pøirozené èíslo~$K$ (jak velké ho
+zvolíme, se rozhodneme a¾ na konci dùkazu).
+Fáze tentokrát budeme dìlit na drahé, v~nich¾ se provede více ne¾~$K$
+nenasycených pøevedení, a~laciné, v~nich¾ se nenasycených pøevedení
+provede maximálnì~$K$.
+
+Nyní budeme zkoumat poèty pøevedení v~obou typech fází.
+Jak víme z~minulého lemmatu, v¹ech fází je nanejvý¹ $4N^2$, tak¾e
+v~laciných se provede nanejvý¹ $N^2K$ nenasycených pøevedení.
+Za úèelem zkoumání drahých fází definujeme potenciál
+$$\psi:=\sum_{v\not=z,s; f^\triangle(x)>0}{ p(v)\over K},$$ kde
+$p(v):= \vert\{u : h(u) \leq h(v)\}\vert$, èili poèet vrcholù ve stejné
+nebo men¹í vý¹ce ne¾~$v$. Víme, ¾e na zaèátku bude $\psi \leq
+{N^2/K}$ a~po celou dobu bude $\psi \geq 0$.
+
+Zvednutím vrcholu~$v$ se hodnota $p(v)$ zvý¹í maximálnì o~$N$, u~libovolného
+jiného vrcholu~$w$ $p(w)$ klesne nebo se nezmìní, potenciál tedy
+vzroste maximálnì o~$N/K$. Pøedchozích èástí
+Pøi sytém pøevedení po hranì z~$u$ do~$v$ (z~minula víme, ¾e se v¾dy
+provádí po hranì spádu nejvý¹e 1) se mohl potenciál zmen¹it o~$p(u)/K$
+a~mohl se zvìt¹it o~$p(v)/K$. Zvìt¹í se tedy nanejvý¹ o~$N/K$
+Pøi nenasyceném pøevedení z~potenciálu urèitì ubyde o~$p(u)/K$
+a~mo¾ná pøibyde $p(v)/K$. Celkovì se tedy sní¾í nanejvý¹
+o~${p(u)/K} - {p(v)/K}$, co¾ je $1/K$~poètu prvkù na nejvy¹¹í
+hladinì. Z~minulého lemmatu víme, ¾e poèet nenasycených pøevedení
+v~jedné fázi je men¹í nebo roven poètu vrcholù na nejvy¹¹í hladinì na
+zaèátku fáze. Vzhledem k~tomu, ¾e zkoumáme drahé fáze v~nich¾ probìhne
+více ne¾~$K$ nenasycených pøevedení, vrcholù na nejvy¹¹í hladinì musí
+být na zaèátku fáze také více ne¾~$K$, potenciál se tedy sní¾í o~více
+ne¾ ${K/K}=1$.
+
+Potenciál $\psi$ tedy pøijme na zaèátku  nanejvý¹ ${N^2/K}$, pøi
+zvedání nevzroste o~víc ne¾ $(N/K)2N^2$, pøi sytém pøevádìní
+nevzroste o~víc ne¾ $(N/K)NM$, celkem dostáváme $\psi\leq
+(N/2)(N+2N^2+NM)=\O({N^2}M/2)$, v~drahých fázích tak mù¾e probìhnout
+nanejvý¹ $\O({N^2}M/2)$ nenasycených pøevedení.
+Celkem algoritmus vykoná $\O(N^2K+{N^2}M/K)$ nenasycených
+pøevedení. Kdy¾ za~$K$ dosadíme $\sqrt M$ dostaneme slíbený poèet
+$\O(N^2\sqrt M)$.
+\qed
+
+\medskip
+\h{Tøídìní}
+
+\s{Definice:} Komparátorová sí» je kombinaèní obvod jeho¾ hradla jsou
+komparátory: 
+
+\centerline{\epsfbox{sortnet.0}}
+
+Výstupy komparátorù se nevìtví.
+
+\medskip
+{\>\sl Bubble sort}
+
+\twofigures{sortnet.1}{Bubble.1}{143pt}{sortnet.2}{Bubble.2}{143pt}
+
+Sna¾íme se výpoèet co nejvíce paralelizovat (viz obrázek Bubble.2). 
+Takto se nám podaøilo výpoèet provést pomocí $\O(n^2)$ komparátorù
+rozmístìných na $\O(n)$ hladinách. Tøídíme v~èase~$N$ a~prostoru
+$N^2$.
+
+\medskip
+{\>\sl Merge sort}
+
+\centerline{\epsfbox{sortnet.4}}
+
+\medskip
+\s{Definice:} Øekneme, ¾e posloupnost $x_0,\dots,x_{n-1} $ je èistì bitonická,
+pokud pro nìjaké $x_j\in\{1\dots n-1\} $ platí, ¾e 
+$$x_0\leq x_1\leq \dots \leq x_j \geq x_{j+1}\geq\dots \geq x_{n-1}$$
+Posloupnost je bitonická, pokud existuje $k\in \{1\dots n-1\}$, pro
+které platí, ¾e posloupnost
+$x_k,x_{k+1 \bmod n},\dots, x_{k+n-1 \bmod n}$ je èistì bitonická.
+
+\s{Definice: Separátor $S_n$}
+Je sí», ve které jsou v¾dy~$i$-tý a~$i+1$-ní (pro $i=0,\dots,
+{n/2}-1$) prvek vstupu propojeny komparátorem, minimum bude~$i$-tým,
+maximum  $i+1$-ním prvkem výstupu.
+\figure{sortnet.3} 
+{$(y_i, y_{i+{n/2}}) = CMP(x_i, x_{i+{n/2}})$} {300pt}
+
+\s{Lemma:} Pokud vstup $S_n$ je bitonická posloupnost, pak výstup
+$y_0,\dots, y_{n-1}$ je posloupnost,  která splòuje:
+
+(i) $y_0,\dots, y_{n/2 -1}$ a~$y_{n/2},\dots, y_{n-1}$ jsou 
+bitonické posloupnosti.
+
+(ii) Pro v¹echna $i,j< {n/2}$ platí $y_i < y_j + {n/2}$
+
+\proof
+(i) Nejprve nahlédneme, ¾e Lemma platí je-li vstupem èistì bitonická
+posloupnost -- najdeme nejmen¹í~$k$ takové, ¾e $x_k$ a~$x_{k+{n/2}}$
+se prohodí. Pokud ho nenajdeme, separátor neudìlá vùbec nic a~obì
+tvrzení lemmatu zøejmì tedy platí. Øeknìme, ¾e $x_m$ je maximum
+vstupní posloupnosti. Pak~$k$ bude jistì men¹í ne¾~$m$
+a~$x_{k+{n/2}}$ bude vìt¹í ne¾~$m$, mezi~$k$ a~$m$ je tedy vstupní
+posloupnost neklesající, mezi $k+{n/2}$ a~$n-1$ nerostoucí.
+Uvìdomíme si, ¾e pro ka¾dé~$i$, $k\leq i\leq {n/2}-1$ se prvky
+$x_i$ a~$x_{i+{n/2}}$ prohodí. Úsek mezi~$k$ a~${n/2}-1$ tedy
+nahradíme nerostoucí posloupností, první polovina výstupu tedy bude
+(dokonce èistì) bitonická. Úsek $k+{n/2}$ a~$n-1$ nahradíme èistì
+bitonickou posloupností, která bude neklesající, je-li $m\geq {n/2}$, 
+v~opaèném pøípadì je úsek ${n/2}-1$ a¾ $k+{n/2}$
+nerostoucí. V~obou pøípadech budou v~posloupnosti maximálnì dva zlomy
+a~mezi $x_{n/2}$ a~$x_{n-1}$ bude správná nerovnost na to, aby
+posloupnost byla bitonická.
+
+Dostaneme-li na vstup obecnou bitonickou posloupnost, pøedstavíme si,
+¾e je to èistì bitonická posloupnost zrotovaná o~$r$ prvkù (BÚNO
+doprava), a~zjistíme, ¾e v~komparátorech se porovnávají tyté¾ prvky
+jako kdyby zrotovaná nebyla. Výstup se od výstupu èistì bitonické
+posloupnosti zrotovaného o~$r$ bude li¹it prohozením úsekù $x_0$ a¾
+$x_{r-1}$ a~$x_{n/2}$ a¾ $x_{{n/2}+r-1}$. Obì polovièní
+posloupnosti tedy budou zrotované o~$r$ prvkù, ale na jejich
+bitoniènosti se nic nezmìní.
+
+(ii) Plyne z~definice separátoru.
+
+\medskip
+\centerline{\epsfbox{sortnet.7}}
+
+\medskip
+\s{Definice: Bitonická tøídièka $B_n$}  
+
+\centerline{\epsfbox{sortnet.5}}
+
+Separátor má jednu hladinu s~${\O}(N)$ hradly, tøídièka tedy bude
+mít 
+$\log N$ hladin s~${\O}(N\log N)$ hradly.
+
+Tøídièka se dá pou¾ít ke slévání, mù¾eme tak s~její pomocí sestavit
+souèástky $M_n$ mergesortové sítì. Setøídìné posloupnosti 
+$x_0,\dots, x_{n-1}$ a~$y_0,\dots, y_{n-1}$ spojíme do jedné bitonické 
+$x_0,\dots, x_{n-1},y_{n-1},\dots, y_0$. Z~takové posloupmosti pomocí
+$B_{2n}$ vytvoøíme setøídìnou posloupnost.
+
+
+\centerline{\epsfbox{sortnet.6}}
+
+Z~bitonických tøídièek tedy mù¾eme postavit mergesortovou sí», která
+bude mít
+${\O}(\log^2 N)$ hladin a~${\O}(N\log^2 N)$ hradel.
+
+Existuje tøídící algoritmus, kterému staèí ${\O}(\log N)$ hladin,
+ale jeho multiplikaèní konstanta je pøíli¹ veliká, tak¾e je v~praxi 
+nepou¾itelný.
+
+\bye
diff --git a/5-sortnet/Makefile b/5-sortnet/Makefile
new file mode 100644
index 0000000..99c4a29
--- /dev/null
+++ b/5-sortnet/Makefile
@@ -0,0 +1,3 @@
+P=5-sortnet
+
+include ../Makerules
diff --git a/5-sortnet/sortnet.0 b/5-sortnet/sortnet.0
new file mode 100644
index 0000000..3dabc76
--- /dev/null
+++ b/5-sortnet/sortnet.0
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diff --git a/5-sortnet/sortnet.7 b/5-sortnet/sortnet.7
new file mode 100644
index 0000000..9b52ee9
--- /dev/null
+++ b/5-sortnet/sortnet.7
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diff --git a/5-sortnet/sortnet.mp b/5-sortnet/sortnet.mp
new file mode 100644
index 0000000..f28a753
--- /dev/null
+++ b/5-sortnet/sortnet.mp
@@ -0,0 +1,657 @@
+%defaultfont:="csr12";
+%verbatimtex \input twelvecs etex
+u:=1 ;
+ahlength:=u*3;
+
+beginfig(0);
+v:=u*7mm;
+
+z12=(1v,1v);
+z13=(2v,1v);
+z21=(0v,2v);
+z22=(1v,2v);
+z23=(2v,2v);
+z24=(3v,2v);
+z31=(0v,4v);
+z32=(1v,4v);
+z33=(2v,4v);
+z34=(3v,4v);
+z42=(1v,5v);
+z43=(2v,5v);
+
+pickup pencircle scaled 0.4pt;
+draw(z21--z22--z23--z24--z34--z33--z32--z31--z21);
+drawarrow(z22--z12);
+drawarrow(z23--z13);
+drawarrow(z42--z32);
+drawarrow(z43--z33);
+
+label.bot(btex \strut a etex,z32);
+label.bot(btex \strut b etex,z33);
+label.top(btex min etex,z22);
+label.top(btex max etex,z23);
+
+
+endfig;
+
+beginfig(1);
+v:=u*5mm;
+z10=(0v,0v);
+z20=(2v,0v);
+z30=(4v,0v);
+z40=(6v,0v);
+z50=(8v,0v);
+
+z11=(0v,1v);
+z12=(0v,2v);
+z13=(0v,3v);
+z14=(0v,4v);
+z15=(0v,5v);
+z16=(0v,6v);
+z17=(0v,7v);
+z18=(0v,8v);
+z19=(0v,9v);
+z110=(0v,10v);
+
+z21=(2v,1v);
+z22=(2v,2v);
+z23=(2v,3v);
+z24=(2v,4v);
+z25=(2v,5v);
+z26=(2v,6v);
+z27=(2v,7v);
+z28=(2v,8v);
+z29=(2v,9v);
+z210=(2v,10v);
+
+z31=(4v,1v);
+z32=(4v,2v);
+z33=(4v,3v);
+z34=(4v,4v);
+z35=(4v,5v);
+z36=(4v,6v);
+z37=(4v,7v);
+z38=(4v,8v);
+z39=(4v,9v);
+z310=(4v,10v);
+
+z41=(6v,1v);
+z42=(6v,2v);
+z43=(6v,3v);
+z44=(6v,4v);
+z45=(6v,5v);
+z46=(6v,6v);
+z47=(6v,7v);
+z48=(6v,8v);
+z49=(6v,9v);
+z410=(6v,10v);
+
+z51=(8v,1v);
+z52=(8v,2v);
+z53=(8v,3v);
+z54=(8v,4v);
+z55=(8v,5v);
+z56=(8v,6v);
+z57=(8v,7v);
+z58=(8v,8v);
+z59=(8v,9v);
+z510=(8v,10v);
+
+z111=(0v,11v);
+z211=(2v,11v);
+z311=(4v,11v);
+z411=(6v,11v);
+z511=(8v,11v);
+
+z1015=(-1v,1.5v);
+z1615=(9v,1.5v);
+z1035=(-1v,3.5v);
+z1635=(9v,3.5v);
+z1065=(-1v,6.5v);
+z1665=(9v,6.5v);
+
+pickup pencircle scaled 0.4pt;
+draw(z10--z111);
+draw(z20--z211);
+draw(z30--z311);
+draw(z40--z411);
+draw(z50--z511);
+drawarrow(z11--z21);
+drawarrow(z13--z23);
+drawarrow(z16--z26);
+drawarrow(z110--z210);
+drawarrow(z22--z32);
+drawarrow(z25--z35);
+drawarrow(z29--z39);
+drawarrow(z34--z44);
+drawarrow(z38--z48);
+drawarrow(z47--z57);
+
+pickup pencircle scaled 0.7pt;
+draw(z1015--z1615) dashed withdots scaled 0.7;
+draw(z1035--z1635) dashed withdots scaled 0.7;
+draw(z1065--z1665) dashed withdots scaled 0.7;
+
+label.top(btex x1 etex,z111);
+label.top(btex x2 etex,z211);
+label.top(btex x3 etex,z311);
+label.top(btex x4 etex,z411);
+label.top(btex x5 etex,z511);
+
+endfig;
+
+beginfig(2);
+v:=u*5mm;
+z13=(0v,3v);
+z23=(2v,3v);
+z33=(4v,3v);
+z43=(6v,3v);
+z53=(8v,3v);
+
+z14=(0v,4v);
+z15=(0v,5v);
+z16=(0v,6v);
+z17=(0v,7v);
+z18=(0v,8v);
+z19=(0v,9v);
+z110=(0v,10v);
+
+z24=(2v,4v);
+z25=(2v,5v);
+z26=(2v,6v);
+z27=(2v,7v);
+z28=(2v,8v);
+z29=(2v,9v);
+z210=(2v,10v);
+
+z34=(4v,4v);
+z35=(4v,5v);
+z36=(4v,6v);
+z37=(4v,7v);
+z38=(4v,8v);
+z39=(4v,9v);
+z310=(4v,10v);
+
+z44=(6v,4v);
+z45=(6v,5v);
+z46=(6v,6v);
+z47=(6v,7v);
+z48=(6v,8v);
+z49=(6v,9v);
+z410=(6v,10v);
+
+z54=(8v,4v);
+z55=(8v,5v);
+z56=(8v,6v);
+z57=(8v,7v);
+z58=(8v,8v);
+z59=(8v,9v);
+z510=(8v,10v);
+
+z111=(0v,11v);
+z211=(2v,11v);
+z311=(4v,11v);
+z411=(6v,11v);
+z511=(8v,11v);
+
+pickup pencircle scaled 0.4pt;
+draw(z13--z111);
+draw(z23--z211);
+draw(z33--z311);
+draw(z43--z411);
+draw(z53--z511);
+drawarrow(z14--z24);
+drawarrow(z16--z26);
+drawarrow(z18--z28);
+drawarrow(z110--z210);
+drawarrow(z25--z35);
+drawarrow(z27--z37);
+drawarrow(z29--z39);
+drawarrow(z36--z46);
+drawarrow(z38--z48);
+drawarrow(z47--z57);
+
+label.top(btex x1 etex,z111);
+label.top(btex x2 etex,z211);
+label.top(btex x3 etex,z311);
+label.top(btex x4 etex,z411);
+label.top(btex x5 etex,z511);
+
+endfig;
+
+
+beginfig(3);
+v:=u*5mm;
+
+z10=(0v,0v);
+z15=(0v,5v);
+z16=(0v,6v);
+
+z20=(2v,0v);
+z24=(2v,4v);
+z26=(2v,6v);
+
+z30=(4v,0v);
+z33=(4v,3v);
+z36=(4v,6v);
+
+z40=(6v,0v);
+z42=(6v,2v);
+z46=(6v,6v);
+
+z50=(8v,0v);
+z51=(8v,1v);
+z56=(8v,6v);
+
+z60=(10v,0v);
+z65=(10v,5v);
+z66=(10v,6v);
+
+z70=(12v,0v);
+z74=(12v,4v);
+z76=(12v,6v);
+
+z80=(14v,0v);
+z83=(14v,3v);
+z86=(14v,6v);
+
+z90=(16v,0v);
+z92=(16v,2v);
+z96=(16v,6v);
+
+z100=(18v,0v);
+z101=(18v,1v);
+z106=(18v,6v);
+
+z110=(20v,0v);
+z116=(20v,6v);
+
+
+pickup pencircle scaled 0.4pt;
+draw(z10--z16);
+draw(z20--z26);
+draw(z30--z36);
+draw(z40--z46);
+draw(z50--z56);
+draw(z60--z66);
+draw(z70--z76);
+draw(z80--z86);
+draw(z90--z96);
+draw(z100--z106);
+draw(z110--z116);
+drawarrow(z15--z65);
+drawarrow(z24--z74);
+drawarrow(z33--z83);
+drawarrow(z42--z92);
+drawarrow(z51--z101);
+
+label.top(btex $x_0$ etex,z16);
+label.top(btex $x_1$ etex,z26);
+label.top(btex $x_2$ etex,z36);
+label.top(btex \dots etex,z66);
+label.top(btex $x_{n-2}$ etex,z106);
+label.top(btex $x_{n-1}$ etex,z116);
+
+endfig;
+
+
+
+beginfig(4);
+v:=u*5mm;
+
+z1075=(7.5v,0v);
+
+z10=(0v,1v);
+z175=(7.5v,1v);
+z115=(15v,1v);
+
+z20=(0v,2v);
+z235=(3.5v,2v);
+z211=(11.5v,2v);
+z215=(15v,2v);
+
+z30=(0v,3v);
+z335=(3.5v,3v);
+z37=(7v,3v);
+z38=(8v,3v);
+z311=(11.5v,3v);
+z315=(15v,3v);
+
+z40=(0v,4v);
+z415=(1.5v,4v);
+z455=(5.5v,4v);
+z47=(7v,4v);
+z48=(8v,4v);
+z495=(9.5v,4v);
+z413=(13.5v,4v);
+z416=(15v,4v);
+
+z50=(0v,5v);
+z515=(1.5v,5v);
+z53=(3v,5v);
+z54=(4v,5v);
+z555=(5.5v,5v);
+z57=(7v,5v);
+z58=(8v,5v);
+z595=(9.5v,5v);
+z511=(11v,5v);
+z512=(12v,5v);
+z513=(13.5v,5v);
+z516=(15v,5v);
+
+z60=(0v,6v);
+z61=(1v,6v);
+z62=(2v,6v);
+z63=(3v,6v);
+z64=(4v,6v);
+z65=(5v,6v);
+z66=(6v,6v);
+z67=(7v,6v);
+z68=(8v,6v);
+z69=(9v,6v);
+z610=(10v,6v);
+z611=(11v,6v);
+z612=(12v,6v);
+z613=(13v,6v);
+z614=(14v,6v);
+z615=(15v,6v);
+
+z71=(1v,7v);
+z72=(2v,7v);
+z75=(5v,7v);
+z76=(6v,7v);
+z79=(9v,7v);
+z710=(10v,7v);
+z713=(13v,7v);
+z714=(14v,7v);
+
+
+
+
+pickup pencircle scaled 0.4pt;
+draw(z10--z115--z215--z20--cycle);
+draw(z30--z37--z47--z40--cycle);
+draw(z38--z315--z416--z48--cycle);
+draw(z50--z53--z63--z60--cycle);
+draw(z54--z57--z67--z64--cycle);
+draw(z58--z511--z611--z68--cycle);
+draw(z512--z516--z615--z612--cycle);
+drawarrow(z71--z61);
+drawarrow(z72--z62);
+drawarrow(z75--z65);
+drawarrow(z76--z66);
+drawarrow(z79--z69);
+drawarrow(z710--z610);
+drawarrow(z713--z613);
+drawarrow(z714--z614);
+drawarrow(z515--z415);
+drawarrow(z555--z455);
+drawarrow(z595--z495);
+drawarrow(z513--z413);
+drawarrow(z335--z235);
+drawarrow(z311--z211);
+drawarrow(z175--z1075);
+
+endfig;
+
+
+beginfig(5);
+v:=u*5mm;
+
+z075=(7.5v,8v);
+
+z10=(0v,7v);
+z175=(7.5v,7v);
+z115=(15v,7v);
+
+z20=(0v,6v);
+z235=(3.5v,6v);
+z211=(11.5v,6v);
+z215=(15v,6v);
+
+z30=(0v,5v);
+z335=(3.5v,5v);
+z37=(7v,5v);
+z38=(8v,5v);
+z311=(11.5v,5v);
+z315=(15v,5v);
+
+z40=(0v,4v);
+z415=(1.5v,4v);
+z455=(5.5v,4v);
+z47=(7v,4v);
+z48=(8v,4v);
+z495=(9.5v,4v);
+z413=(13.5v,4v);
+z416=(15v,4v);
+
+z50=(0v,3v);
+z515=(1.5v,3v);
+z53=(3v,3v);
+z54=(4v,3v);
+z555=(5.5v,3v);
+z57=(7v,3v);
+z58=(8v,3v);
+z595=(9.5v,3v);
+z511=(11v,3v);
+z512=(12v,3v);
+z513=(13.5v,3v);
+z516=(15v,3v);
+
+z60=(0v,2v);
+z61=(1v,2v);
+z62=(2v,2v);
+z63=(3v,2v);
+z64=(4v,2v);
+z65=(5v,2v);
+z66=(6v,2v);
+z67=(7v,2v);
+z68=(8v,2v);
+z69=(9v,2v);
+z610=(10v,2v);
+z611=(11v,2v);
+z612=(12v,2v);
+z613=(13v,2v);
+z614=(14v,2v);
+z615=(15v,2v);
+
+
+pickup pencircle scaled 0.4pt;
+draw(z10--z115--z215--z20--cycle);
+draw(z30--z37--z47--z40--cycle);
+draw(z38--z315--z416--z48--cycle);
+draw(z50--z53--z63--z60--cycle);
+draw(z54--z57--z67--z64--cycle);
+draw(z58--z511--z611--z68--cycle);
+draw(z512--z516--z615--z612--cycle);
+drawarrow(z075--z175);
+drawarrow(z415--z515);
+drawarrow(z455--z555);
+drawarrow(z495--z595);
+drawarrow(z413--z513);
+drawarrow(z235--z335);
+drawarrow(z211--z311);
+
+label.llft(btex $n$ etex,z075);
+label.bot(btex $S_n$ etex,z175);
+label.bot(btex $S_{n\over 2}$ etex,z335);
+label.bot(btex $S_{n\over 2}$ etex,z311);
+label.bot(btex $S_{n\over 4}$ etex,z515);
+label.bot(btex $S_{n\over 4}$ etex,z555);
+label.bot(btex $S_{n\over 4}$ etex,z595);
+label.bot(btex $S_{n\over 4}$ etex,z513);
+
+endfig;
+
+
+
+beginfig(6);
+v:=u*5mm;
+
+z1075=(7.5v,0v);
+
+z10=(0v,1v);
+z175=(7.5v,1v);
+z115=(15v,1v);
+
+z20=(0v,2v);
+z235=(3.5v,2v);
+z211=(11.5v,2v);
+z215=(15v,2v);
+
+z30=(0v,3v);
+z335=(3.5v,3v);
+z37=(7v,3v);
+z38=(8v,3v);
+z311=(11.5v,3v);
+z315=(15v,3v);
+
+z40=(0v,4v);
+z415=(1.5v,4v);
+z455=(5.5v,4v);
+z47=(7v,4v);
+z48=(8v,4v);
+z495=(9.5v,4v);
+z413=(13.5v,4v);
+z416=(15v,4v);
+
+z50=(0v,5v);
+z515=(1.5v,5v);
+z53=(3v,5v);
+z54=(4v,5v);
+z555=(5.5v,5v);
+z57=(7v,5v);
+z58=(8v,5v);
+z595=(9.5v,5v);
+z511=(11v,5v);
+z512=(12v,5v);
+z513=(13.5v,5v);
+z516=(15v,5v);
+
+z60=(0v,6v);
+z61=(1v,6v);
+z62=(2v,6v);
+z63=(3v,6v);
+z64=(4v,6v);
+z65=(5v,6v);
+z66=(6v,6v);
+z67=(7v,6v);
+z68=(8v,6v);
+z69=(9v,6v);
+z610=(10v,6v);
+z611=(11v,6v);
+z612=(12v,6v);
+z613=(13v,6v);
+z614=(14v,6v);
+z615=(15v,6v);
+
+z71=(1v,7v);
+z72=(2v,7v);
+z75=(5v,7v);
+z76=(6v,7v);
+z79=(9v,7v);
+z710=(10v,7v);
+z713=(13v,7v);
+z714=(14v,7v);
+
+
+
+
+pickup pencircle scaled 0.4pt;
+draw(z10--z115--z215--z20--cycle);
+draw(z30--z37--z47--z40--cycle);
+draw(z38--z315--z416--z48--cycle);
+draw(z50--z53--z63--z60--cycle);
+draw(z54--z57--z67--z64--cycle);
+draw(z58--z511--z611--z68--cycle);
+draw(z512--z516--z615--z612--cycle);
+drawarrow(z71--z61);
+drawarrow(z72--z62);
+drawarrow(z75--z65);
+drawarrow(z76--z66);
+drawarrow(z79--z69);
+drawarrow(z710--z610);
+drawarrow(z713--z613);
+drawarrow(z714--z614);
+drawarrow(z515--z415);
+drawarrow(z555--z455);
+drawarrow(z595--z495);
+drawarrow(z513--z413);
+drawarrow(z335--z235);
+drawarrow(z311--z211);
+drawarrow(z175--z1075);
+
+label.top(btex $M_8$ etex,z175);
+label.top(btex $M_4$ etex,z335);
+label.top(btex $M_4$ etex,z311);
+label.top(btex $M_2$ etex,z515);
+label.top(btex $M_2$ etex,z555);
+label.top(btex $M_2$ etex,z595);
+label.top(btex $M_2$ etex,z513);
+
+endfig;
+
+
+beginfig(7);
+v:=u*7mm;
+
+z12=(1v,2v);
+z13=(1v,3v);
+z16=(1v,6v);
+z356=(3.5v,6v);
+z40=(5v,1v);
+z42=(5v,2v);
+z47=(5v,6.5v);
+z72=(9v,2v);
+z74=(9v,4v);
+z76=(9v,6v);
+
+z100=(0.5v,0.5v);
+z101=(1.5v,0.5v);
+
+z0=whatever[z13,z356];
+z1=whatever[z356,z74];
+z1=z0+4v*right;
+z2=whatever[z12,z72];
+z2=z0+whatever*down;
+z3=whatever[z12,z72];
+z3=z1+whatever*down;
+z4=z356+4v*right;
+z5=whatever[z72,z76];
+z5=whatever[z4,z1+4v*right];
+z6=whatever[z40,z47];
+z6=z74+4v*left;
+
+
+pickup pencircle scaled 0.4pt;
+draw(z16--z12--z72--z76);
+draw(z13--z356--z74);
+draw(z40--z47) dashed evenly;
+draw(z1--z4) dashed withdots scaled 0.7;
+draw(z4--z5) dashed withdots scaled 0.7;
+draw(z0--z6) dashed withdots scaled 0.7;
+draw(z0--z2) dashed evenly;
+draw(z1--z3) dashed evenly;
+
+draw(z100--z101) dashed withdots scaled 0.7;
+
+pickup pencircle scaled 3pt;
+drawdot(z0);
+drawdot(z1);
+
+label.bot(btex \strut 0 etex,z12);
+label.bot(btex $k$ etex,z2);
+label.llft(btex \strut ${n\over 2} - 1$ etex,z42);
+label.bot(btex \strut $k+{n\over 2}$ etex,z3);
+label.bot(btex \strut $n-1$ etex,z72);
+label.rt(btex posloupnost prohozen\'a separ\'atorem etex,z101);
+
+endfig;
+
+
+
+
+
+
+
+end;
diff --git a/all/Makefile b/all/Makefile
index 7b16831..90bc7ce 100644
--- a/all/Makefile
+++ b/all/Makefile
@@ -1,5 +1,5 @@
 P=ads2
-X:=$(shell for a in 1 2 3 4 6 7 ; do echo ../$$a-*/$$a-*.tex ; done)
+X:=$(shell for a in 1 2 3 4 5 6 7 ; do echo ../$$a-*/$$a-*.tex ; done)
 
 %universe: all ChangeLog
 
-- 
2.39.2