]> mj.ucw.cz Git - ads2.git/commitdiff
Oziveni hradlovych siti a bitonickeho trideni, zatim bez uprav
authorMartin Mares <mj@ucw.cz>
Fri, 18 Nov 2011 18:04:34 +0000 (19:04 +0100)
committerMartin Mares <mj@ucw.cz>
Fri, 18 Nov 2011 18:04:34 +0000 (19:04 +0100)
96 files changed:
5-hradla/1_9_deleni_bloku.eps [new file with mode: 0644]
5-hradla/5-hradla.tex [new file with mode: 0644]
5-hradla/Makefile [new file with mode: 0644]
5-hradla/blok_scitani.eps [new file with mode: 0644]
5-hradla/blok_scitani.svg [new file with mode: 0644]
5-hradla/bloky_1bit.eps [new file with mode: 0644]
5-hradla/bloky_1bit.svg [new file with mode: 0644]
5-hradla/chytry_or.eps [new file with mode: 0644]
5-hradla/chytry_or.svg [new file with mode: 0644]
5-hradla/deleni_bloku.eps [new file with mode: 0644]
5-hradla/hloupe_scitani.eps [new file with mode: 0644]
5-hradla/hloupe_scitani.svg [new file with mode: 0644]
5-hradla/hloupy_or.eps [new file with mode: 0644]
5-hradla/hloupy_or.svg [new file with mode: 0644]
5-hradla/hradlo_and.eps [new file with mode: 0644]
5-hradla/hradlo_and.svg [new file with mode: 0644]
5-hradla/hradlo_ternbior.eps [new file with mode: 0644]
5-hradla/hradlo_ternbior.svg [new file with mode: 0644]
5-hradla/hradlo_ternor.eps [new file with mode: 0644]
5-hradla/hradlo_ternor.svg [new file with mode: 0644]
5-hradla/hradlova_sit.eps [new file with mode: 0644]
5-hradla/hradlova_sit.svg [new file with mode: 0644]
5-hradla/obvod.eps [new file with mode: 0644]
5-hradla/obvod.svg [new file with mode: 0644]
5-hradla/obvod_real.eps [new file with mode: 0644]
5-hradla/obvod_real.svg [new file with mode: 0644]
5-hradla/skolni_scitani.eps [new file with mode: 0644]
5-hradla/skolni_scitani.svg [new file with mode: 0644]
5-hradla/sl_stromecek.eps [new file with mode: 0644]
5-hradla/sl_stromecek.svg [new file with mode: 0644]
5-hradla/stromecek.eps [new file with mode: 0644]
5-hradla/stromecek.svg [new file with mode: 0644]
5-hradla/tabulka_skladani_bloku.eps [new file with mode: 0644]
5-hradla/tabulka_skladani_bloku.svg [new file with mode: 0644]
5-hradla/vypocet_site.eps [new file with mode: 0644]
5-hradla/vypocet_site.svg [new file with mode: 0644]
6-bitonic/6-bitonic.tex [new file with mode: 0644]
6-bitonic/Makefile [new file with mode: 0644]
6-bitonic/sortnet.0 [new file with mode: 0644]
6-bitonic/sortnet.1 [new file with mode: 0644]
6-bitonic/sortnet.2 [new file with mode: 0644]
6-bitonic/sortnet.3 [new file with mode: 0644]
6-bitonic/sortnet.4 [new file with mode: 0644]
6-bitonic/sortnet.5 [new file with mode: 0644]
6-bitonic/sortnet.6 [new file with mode: 0644]
6-bitonic/sortnet.7 [new file with mode: 0644]
6-bitonic/sortnet.mp [new file with mode: 0644]
6-bitonic/sortnet.mpx [new file with mode: 0644]
old/4-hradla/4-hradla.tex [deleted file]
old/4-hradla/Makefile [deleted file]
old/4-hradla/chytry_or.eps [deleted file]
old/4-hradla/chytry_or.svg [deleted file]
old/4-hradla/hloupy_or.eps [deleted file]
old/4-hradla/hloupy_or.svg [deleted file]
old/4-hradla/hradlo_and.eps [deleted file]
old/4-hradla/hradlo_and.svg [deleted file]
old/4-hradla/hradlo_ternbior.eps [deleted file]
old/4-hradla/hradlo_ternbior.svg [deleted file]
old/4-hradla/hradlo_ternor.eps [deleted file]
old/4-hradla/hradlo_ternor.svg [deleted file]
old/4-hradla/hradlova_sit.eps [deleted file]
old/4-hradla/hradlova_sit.svg [deleted file]
old/4-hradla/vypocet_site.eps [deleted file]
old/4-hradla/vypocet_site.svg [deleted file]
old/5-addsort/1_9_deleni_bloku.eps [deleted file]
old/5-addsort/5-addsort.tex [deleted file]
old/5-addsort/Makefile [deleted file]
old/5-addsort/blok_scitani.eps [deleted file]
old/5-addsort/blok_scitani.svg [deleted file]
old/5-addsort/bloky_1bit.eps [deleted file]
old/5-addsort/bloky_1bit.svg [deleted file]
old/5-addsort/deleni_bloku.eps [deleted file]
old/5-addsort/hloupe_scitani.eps [deleted file]
old/5-addsort/hloupe_scitani.svg [deleted file]
old/5-addsort/obvod.eps [deleted file]
old/5-addsort/obvod.svg [deleted file]
old/5-addsort/obvod_real.eps [deleted file]
old/5-addsort/obvod_real.svg [deleted file]
old/5-addsort/skolni_scitani.eps [deleted file]
old/5-addsort/skolni_scitani.svg [deleted file]
old/5-addsort/sl_stromecek.eps [deleted file]
old/5-addsort/sl_stromecek.svg [deleted file]
old/5-addsort/sortnet.0 [deleted file]
old/5-addsort/sortnet.1 [deleted file]
old/5-addsort/sortnet.2 [deleted file]
old/5-addsort/sortnet.3 [deleted file]
old/5-addsort/sortnet.4 [deleted file]
old/5-addsort/sortnet.5 [deleted file]
old/5-addsort/sortnet.6 [deleted file]
old/5-addsort/sortnet.7 [deleted file]
old/5-addsort/sortnet.mp [deleted file]
old/5-addsort/sortnet.mpx [deleted file]
old/5-addsort/stromecek.eps [deleted file]
old/5-addsort/stromecek.svg [deleted file]
old/5-addsort/tabulka_skladani_bloku.eps [deleted file]
old/5-addsort/tabulka_skladani_bloku.svg [deleted file]

diff --git a/5-hradla/1_9_deleni_bloku.eps b/5-hradla/1_9_deleni_bloku.eps
new file mode 100644 (file)
index 0000000..2c96b1c
--- /dev/null
@@ -0,0 +1,3225 @@
+%!PS-Adobe-3.0 EPSF-3.0\r
+%%Creator: Mayura Draw, Version 4.3\r
+%%Title: 1_9_deleni_bloku.ai\r
+%%CreationDate: Mon Oct 29 23:24:27 2007\r
+%%BoundingBox: -9 -9 276 157\r
+%%EndComments\r
+%%BeginProlog\r
+%%BeginResource: procset MayuraDraw_ops\r
+%%Version: 4.3\r
+%%Copyright: (c) 1993-2003 Mayura Software\r
+/PDXDict 100 dict def\r
+PDXDict begin\r
+% width height matrix proc key cache\r
+% definepattern -\> font\r
+/definepattern { %def\r
+  7 dict begin\r
+    /FontDict 9 dict def\r
+    FontDict begin\r
+      /cache exch def\r
+      /key exch def\r
+      /proc exch cvx def\r
+      /mtx exch matrix invertmatrix def\r
+      /height exch def\r
+      /width exch def\r
+      /ctm matrix currentmatrix def\r
+      /ptm matrix identmatrix def\r
+      /str\r
+      (xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx)\r
+      def\r
+    end\r
+    /FontBBox [ %def\r
+      0 0 FontDict /width get\r
+      FontDict /height get\r
+    ] def\r
+    /FontMatrix FontDict /mtx get def\r
+    /Encoding StandardEncoding def\r
+    /FontType 3 def\r
+    /BuildChar { %def\r
+      pop begin\r
+      FontDict begin\r
+        width 0 cache { %ifelse\r
+          0 0 width height setcachedevice\r
+        }{ %else\r
+          setcharwidth\r
+        } ifelse\r
+        0 0 moveto width 0 lineto\r
+        width height lineto 0 height lineto\r
+        closepath clip newpath\r
+        gsave proc grestore\r
+      end end\r
+    } def\r
+    FontDict /key get currentdict definefont\r
+  end\r
+} bind def\r
+\r
+% dict patternpath -\r
+% dict matrix patternpath -\r
+/patternpath { %def\r
+  dup type /dicttype eq { %ifelse\r
+    begin FontDict /ctm get setmatrix\r
+  }{ %else\r
+    exch begin FontDict /ctm get setmatrix\r
+    concat\r
+  } ifelse\r
+  currentdict setfont\r
+  FontDict begin\r
+    FontMatrix concat\r
+    width 0 dtransform\r
+    round width div exch round width div exch\r
+    0 height dtransform\r
+    round height div exch\r
+    round height div exch\r
+    0 0 transform round exch round exch\r
+    ptm astore setmatrix\r
+\r
+    pathbbox\r
+    height div ceiling height mul 4 1 roll\r
+    width div ceiling width mul 4 1 roll\r
+    height div floor height mul 4 1 roll\r
+    width div floor width mul 4 1 roll\r
+\r
+    2 index sub height div ceiling cvi exch\r
+    3 index sub width div ceiling cvi exch\r
+    4 2 roll moveto\r
+\r
+    FontMatrix ptm invertmatrix pop\r
+    { %repeat\r
+      gsave\r
+        ptm concat\r
+        dup str length idiv { %repeat\r
+          str show\r
+        } repeat\r
+        dup str length mod str exch\r
+        0 exch getinterval show\r
+      grestore\r
+      0 height rmoveto\r
+    } repeat\r
+    pop\r
+  end end\r
+} bind def\r
+\r
+% dict patternfill -\r
+% dict matrix patternfill -\r
+/patternfill { %def\r
+  gsave\r
+    eoclip patternpath\r
+  grestore\r
+  newpath\r
+} bind def\r
+\r
+/img { %def\r
+  gsave\r
+  /imgh exch def\r
+  /imgw exch def\r
+  concat\r
+  imgw imgh 8\r
+  [imgw 0 0 imgh neg 0 imgh]\r
+  /colorstr 768 string def\r
+  /colorimage where {\r
+    pop\r
+    { currentfile colorstr readhexstring pop }\r
+    false 3 colorimage\r
+  }{\r
+    /graystr 256 string def\r
+    {\r
+      currentfile colorstr readhexstring pop\r
+      length 3 idiv\r
+      dup 1 sub 0 1 3 -1 roll\r
+      {\r
+        graystr exch\r
+        colorstr 1 index 3 mul get 30 mul\r
+        colorstr 2 index 3 mul 1 add get 59 mul\r
+        colorstr 3 index 3 mul 2 add get 11 mul\r
+        add add 100 idiv\r
+        put\r
+      } for\r
+      graystr 0 3 -1 roll getinterval\r
+    } image\r
+  } ifelse\r
+  grestore\r
+} bind def\r
+\r
+/arrowhead {\r
+  gsave\r
+    [] 0 setdash\r
+    strokeC strokeM strokeY strokeK setcmykcolor\r
+    2 copy moveto\r
+    4 2 roll exch 4 -1 roll exch\r
+    sub 3 1 roll sub\r
+    exch atan rotate dup scale\r
+    arrowtype\r
+    dup 0 eq {\r
+      -1 2 rlineto 7 -2 rlineto -7 -2 rlineto\r
+      closepath fill\r
+    } if\r
+    dup 1 eq {\r
+      0 3 rlineto 9 -3 rlineto -9 -3 rlineto\r
+      closepath fill\r
+    } if\r
+    dup 2 eq {\r
+      -6 -6 rmoveto 6 6 rlineto -6 6 rlineto\r
+      -1.4142 -1.4142 rlineto 4.5858 -4.5858 rlineto\r
+      -4.5858 -4.5858 rlineto closepath fill\r
+    } if\r
+    dup 3 eq {\r
+      -6 0 rmoveto -1 2 rlineto 7 -2 rlineto -7 -2 rlineto\r
+      closepath fill\r
+    } if\r
+    dup 4 eq {\r
+      -9 0 rmoveto 0 3 rlineto 9 -3 rlineto -9 -3 rlineto\r
+      closepath fill\r
+    } if\r
+    dup 5 eq {\r
+      currentpoint newpath 3 0 360 arc\r
+      closepath fill\r
+    } if\r
+    dup 6 eq {\r
+      2.5 2.5 rmoveto 0 -5 rlineto -5 0 rlineto 0 5 rlineto\r
+      closepath fill\r
+    } if\r
+    pop\r
+  grestore\r
+} bind def\r
+\r
+/setcmykcolor where { %ifelse\r
+  pop\r
+}{ %else\r
+  /setcmykcolor {\r
+     /black exch def /yellow exch def\r
+     /magenta exch def /cyan exch def\r
+     cyan black add dup 1 gt { pop 1 } if 1 exch sub\r
+     magenta black add dup 1 gt { pop 1 } if 1 exch sub\r
+     yellow black add dup 1 gt { pop 1 } if 1 exch sub\r
+     setrgbcolor\r
+  } bind def\r
+} ifelse\r
+\r
+/RE { %def\r
+  findfont begin\r
+  currentdict dup length dict begin\r
+    { %forall\r
+      1 index /FID ne { def } { pop pop } ifelse\r
+    } forall\r
+    /FontName exch def dup length 0 ne { %if\r
+      /Encoding Encoding 256 array copy def\r
+      0 exch { %forall\r
+        dup type /nametype eq { %ifelse\r
+          Encoding 2 index 2 index put\r
+          pop 1 add\r
+        }{ %else\r
+          exch pop\r
+        } ifelse\r
+      } forall\r
+    } if pop\r
+  currentdict dup end end\r
+  /FontName get exch definefont pop\r
+} bind def\r
+\r
+/spacecount { %def\r
+  0 exch\r
+  ( ) { %loop\r
+    search { %ifelse\r
+      pop 3 -1 roll 1 add 3 1 roll\r
+    }{ pop exit } ifelse\r
+  } loop\r
+} bind def\r
+\r
+/WinAnsiEncoding [\r
+  39/quotesingle 96/grave 130/quotesinglbase/florin/quotedblbase\r
+  /ellipsis/dagger/daggerdbl/circumflex/perthousand\r
+  /Scaron/guilsinglleft/OE 145/quoteleft/quoteright\r
+  /quotedblleft/quotedblright/bullet/endash/emdash\r
+  /tilde/trademark/scaron/guilsinglright/oe/dotlessi\r
+  159/Ydieresis 164/currency 166/brokenbar 168/dieresis/copyright\r
+  /ordfeminine 172/logicalnot 174/registered/macron/ring\r
+  177/plusminus/twosuperior/threesuperior/acute/mu\r
+  183/periodcentered/cedilla/onesuperior/ordmasculine\r
+  188/onequarter/onehalf/threequarters 192/Agrave/Aacute\r
+  /Acircumflex/Atilde/Adieresis/Aring/AE/Ccedilla\r
+  /Egrave/Eacute/Ecircumflex/Edieresis/Igrave/Iacute\r
+  /Icircumflex/Idieresis/Eth/Ntilde/Ograve/Oacute\r
+  /Ocircumflex/Otilde/Odieresis/multiply/Oslash\r
+  /Ugrave/Uacute/Ucircumflex/Udieresis/Yacute/Thorn\r
+  /germandbls/agrave/aacute/acircumflex/atilde/adieresis\r
+  /aring/ae/ccedilla/egrave/eacute/ecircumflex\r
+  /edieresis/igrave/iacute/icircumflex/idieresis\r
+  /eth/ntilde/ograve/oacute/ocircumflex/otilde\r
+  /odieresis/divide/oslash/ugrave/uacute/ucircumflex\r
+  /udieresis/yacute/thorn/ydieresis\r
+] def\r
+\r
+/SymbolEncoding [\r
+  32/space/exclam/universal/numbersign/existential/percent\r
+  /ampersand/suchthat/parenleft/parenright/asteriskmath/plus\r
+  /comma/minus/period/slash/zero/one/two/three/four/five/six\r
+  /seven/eight/nine/colon/semicolon/less/equal/greater/question\r
+  /congruent/Alpha/Beta/Chi/Delta/Epsilon/Phi/Gamma/Eta/Iota\r
+  /theta1/Kappa/Lambda/Mu/Nu/Omicron/Pi/Theta/Rho/Sigma/Tau\r
+  /Upsilon/sigma1/Omega/Xi/Psi/Zeta/bracketleft/therefore\r
+  /bracketright/perpendicular/underscore/radicalex/alpha\r
+  /beta/chi/delta/epsilon/phi/gamma/eta/iota/phi1/kappa/lambda\r
+  /mu/nu/omicron/pi/theta/rho/sigma/tau/upsilon/omega1/omega\r
+  /xi/psi/zeta/braceleft/bar/braceright/similar\r
+  161/Upsilon1/minute/lessequal/fraction/infinity/florin/club\r
+  /diamond/heart/spade/arrowboth/arrowleft/arrowup/arrowright\r
+  /arrowdown/degree/plusminus/second/greaterequal/multiply\r
+  /proportional/partialdiff/bullet/divide/notequal/equivalence\r
+  /approxequal/ellipsis/arrowvertex/arrowhorizex/carriagereturn\r
+  /aleph/Ifraktur/Rfraktur/weierstrass/circlemultiply\r
+  /circleplus/emptyset/intersection/union/propersuperset\r
+  /reflexsuperset/notsubset/propersubset/reflexsubset/element\r
+  /notelement/angle/gradient/registerserif/copyrightserif\r
+  /trademarkserif/product/radical/dotmath/logicalnot/logicaland\r
+  /logicalor/arrowdblboth/arrowdblleft/arrowdblup/arrowdblright\r
+  /arrowdbldown/lozenge/angleleft/registersans/copyrightsans\r
+  /trademarksans/summation/parenlefttp/parenleftex/parenleftbt\r
+  /bracketlefttp/bracketleftex/bracketleftbt/bracelefttp\r
+  /braceleftmid/braceleftbt/braceex\r
+  241/angleright/integral/integraltp/integralex/integralbt\r
+  /parenrighttp/parenrightex/parenrightbt/bracketrighttp\r
+  /bracketrightex/bracketrightbt/bracerighttp/bracerightmid\r
+  /bracerightbt\r
+] def\r
+\r
+/patarray [\r
+/leftdiagonal /rightdiagonal /crossdiagonal /horizontal\r
+/vertical /crosshatch /fishscale /wave /brick\r
+] def\r
+/arrowtype 0 def\r
+/fillC 0 def /fillM 0 def /fillY 0 def /fillK 0 def\r
+/strokeC 0 def /strokeM 0 def /strokeY 0 def /strokeK 1 def\r
+/pattern -1 def\r
+/mat matrix def\r
+/mat2 matrix def\r
+/nesting 0 def\r
+/deferred /N def\r
+/c /curveto load def\r
+/c2 { pop pop c } bind def\r
+/C /curveto load def\r
+/C2 { pop pop C } bind def\r
+/e { gsave concat 0 0 moveto } bind def\r
+/F {\r
+  nesting 0 eq { %ifelse\r
+    pattern -1 eq { %ifelse\r
+      fillC fillM fillY fillK setcmykcolor eofill\r
+    }{ %else\r
+      gsave fillC fillM fillY fillK setcmykcolor eofill grestore\r
+      0 0 0 1 setcmykcolor\r
+      patarray pattern get findfont patternfill\r
+    } ifelse\r
+  }{ %else\r
+    /deferred /F def\r
+  } ifelse\r
+} bind def\r
+/f { closepath F } bind def\r
+/K { /strokeK exch def /strokeY exch def\r
+     /strokeM exch def /strokeC exch def } bind def\r
+/k { /fillK exch def /fillY exch def\r
+     /fillM exch def /fillC exch def } bind def\r
+/opc { pop } bind def\r
+/Opc { pop } bind def\r
+/L /lineto load def\r
+/L2 { pop pop L } bind def\r
+/m /moveto load def\r
+/m2 { pop pop m } bind def\r
+/n /newpath load def\r
+/N {\r
+  nesting 0 eq { %ifelse\r
+    newpath\r
+  }{ %else\r
+    /deferred /N def\r
+  } ifelse\r
+} def\r
+/S {\r
+  nesting 0 eq { %ifelse\r
+    strokeC strokeM strokeY strokeK setcmykcolor stroke\r
+  }{ %else\r
+    /deferred /S def\r
+  } ifelse\r
+} bind def\r
+/s { closepath S } bind def\r
+/Tx { fillC fillM fillY fillK setcmykcolor show\r
+      0 leading neg translate 0 0 moveto } bind def\r
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+%%PageTrailer\r
+_PDX_savepage restore\r
+%%Trailer\r
+end\r
+% showpage\r
+%%EOF\r
diff --git a/5-hradla/5-hradla.tex b/5-hradla/5-hradla.tex
new file mode 100644 (file)
index 0000000..f995d4f
--- /dev/null
@@ -0,0 +1,318 @@
+\input ../lecnotes.tex
+
+\prednaska{5}{Hradlové sítì}{}
+
+\def\land{\mathbin{\&}}
+
+Výkon poèítaèù nelze zvy¹ovat donekoneèna a pøesto¾e ji¾ pìkných pár let platí,
+¾e se jejich rychlost s~èasem exponenciálnì zvìt¹uje, jednou urèitì narazíme
+pøinejmen¹ím na~fyzikální limity.
+
+Kdy¾ tedy nemù¾eme zvy¹ovat rychlost jednoho procesoru, jak poèítat rychleji?
+Øe¹ením by mohlo být poøídit si procesorù víc. U¾~dnes na~bì¾ném PéCéèku máme
+k~dispozici vícejádrové procesory, díky nim¾ mù¾eme vyu¾ít pararelní poèítání
+a úlohu øe¹it tak, ¾e práci ¹ikovnì rozdìlíme mezi procesory (èi jádra) a
+zamìstnáme je pøi~výpoètu v¹echny.
+
+My se podíváme na~abstraktní výpoèetní model, který je je¹tì paralelnìj¹í.
+Techniky, které si uká¾eme na~tomto modelu, se v¹ak dají pøekvapivì vyu¾ít i~pøi~reálném
+paralelizování na~nìkolika málo procesorech. Konec koncù i proto, ¾e vnitøní
+architektura procesoru se na¹emu modelu velmi podobá. Budeme se zabývat 
+jednoduchým modelem paralelního poèítaèe, toti¾ hradlovou sítí.
+
+\h{Hradlové sítì}
+
+\s{Definice:} {\I Hradlo} je prvek, který umí vyhodnocovat nìjakou funkci
+nad~koneènou abecedou $\Sigma$.
+
+Obecnì se na~hradlo díváme jako na~funkci $f: {\Sigma}^{k} \rightarrow \Sigma$, která dostane $k$ vstupù
+a~vrátí jeden výstup, pøièem¾ hodnoty, nad~kterými pracuje, budou z~nìjaké koneèné
+abecedy -- tedy z~nìjaké koneèné mno¾iny symbolù $\Sigma$. Písmenku $k$ zde øíkáme {\I arita
+hradla}.
+
+\s{Pøíklad:} Èasto studujeme hradla booleovská (pracující nad abecedou $\{0,1\}$), která poèítají logické funkce.
+
+Z~nich nejèastìji potkáme:
+
+\itemize\ibull
+\:nulární: to jsou konstanty (FALSE=0, TRUE=1),
+\:unární: napø. negace (znaèíme~$\lnot$),
+\:binární: logický souèin ({\csc and},~$\land$), souèet ({\csc or},~$\lor$), ...
+\endlist
+
+\>Hradla kreslíme tøeba následovnì:
+
+\figure{hradlo_and.eps}{Binární hradlo provádící logickou operaci {\csc and}.}{1in}
+
+Jednotlivá hradla mù¾eme navzájem urèitým zpùsobem propojovat a vytváøet
+z nich {\I hradlové sítì}. Pokud pou¾íváme pouze booleovská hradla, øíkáme takto vytvoøeným
+sítím {\I booleovské obvody}. Pokud pracujeme s~operacemi nad nìjakou obecnìj¹í (ale koneènou) 
+mno¾inou symbolù (abecedou), nazývají se {\I kombinaèní obvody.}
+
+Ka¾dá hradlová sí» má nìjaké vstupy, nìjaké výstupy a uvnitø jsou propojovaná
+hradla. Ka¾dý vstup hradla je pøipojen buïto na~nìkterý ze~vstupù sítì nebo
+na~výstup jiného hradla. Výstupy hradel mohou být propojeny na~vstupy dal¹ích
+hradel (mohou se vìtvit), nebo na výstupy sítì. Pøitom máme zakázáno vytváøet
+cykly. 
+
+Ne¾ si øekneme formální definici, podívejme se na obrázek.
+
+\figure{hradlova_sit.eps}{Hradlová sí» -- tøívstupová verze funkce {\I majorita}.}{3in}
+
+Obrazek znázoròuje hradlovou sí», která poèítá, zda je alespoò na~dvou ze~vstupù
+jednièka. Pojïme si ale {\I hradlovou sí»} definovat formálnì.
+
+\s{Definice:} {\I Hradlová sí»} je urèena:
+\itemize\ibull
+\:{\I abecedou} $\Sigma$ (to je nìjaká koneèná mno¾ina symbolù, obvykle $\Sigma=\{0,1\}$);
+\:po dvou disjunktními koneènými mno¾inami $I$~({\I vstupy}), $O$~({\I výstupy}) \hfil\break a~$H$~({\I hradla});
+\:acyklickým orientovaným multigrafem~$(V,E)$, kde~$V = I \cup O \cup H$;
+\:zobrazením~$F$, které ka¾dému hradlu $h\in H$ pøiøadí nìjakou funkci~$F(h):
+  \Sigma^{a(h)} \rightarrow \Sigma$, co¾ je funkce, kterou toto hradlo vykonává.
+  Èíslu $a(h)$ øíkáme {\I arita} hradla~$h$;
+\:zobrazením~$z: E \rightarrow {\bb N}$, které ka¾dé hranì vedoucí do~nìjakého
+  hradla pøiøazuje nìkterý ze vstupù tohoto hradla.
+\endlist
+
+\>Pøitom jsou splnìny následující podmínky:
+
+\itemize\ibull
+\:$\forall i \in I: \deg^{in}(i)=0$ (do~vstupù nic nevede);
+\:$\forall o \in O: \deg^{in}(o)=1~\land~\deg^{out}(o)=0$ (z~výstupù nic nevede a do~ka¾dého vede právì jedna hrana);
+\:$\forall h \in H: \deg^{in}(v)=a(v)$ (do~ka¾dého hradla vede tolik hran, kolik je jeho arita);
+\:$\forall h \in H~\forall j: 1\le j\le a(h)$ existuje právì jedena hrana~$e$ taková, ¾e~$e$ konèí v~$h$ a~$z(e)=j$, 
+  (v¹echny vstupy hradel jsou zapojeny).
+\endlist
+
+\s{Pozorování:} Kdybychom pøipustili hradla s~libovolnì vysokým poètem vstupù, mohli
+bychom libovolný problém se vstupem délky~$n$ vyøe¹it jedním hradlem o~$n$~vstupech,
+kterému bychom pøiøadili funkci, která by na¹i úlohu rovnou vyøe¹ila. Tento model
+v¹ak není ani realistický, ani pìkný. Proto pøijmìme omezení, ¾e~arity v¹ech hradel
+budou omezeny nìjakou pevnou konstantou $k$ (uká¾e se, ¾e nám bude staèit dvojka
+a~vystaèíme si tedy pouze s nulárními, unárními a binárními hradly). 
+Následující obrázky ukazují, jak hradla o~více vstupech nahradit dvouvstupovými:
+
+\twofigures{hradlo_ternor.eps}{Trojvstupové hradlo \csc or.}{0.5in}{hradlo_ternbior.eps}{Jeho nahrazení 2-vstupovými hradly.}{0.6in}
+
+
+Nyní bychom je¹tì mìli definovat, co taková hradlová sí» vlastnì poèítá a~jak
+její výpoèet probíhá.
+
+\s{Definice:} {\I Výpoèet sítì} probíhá po~{\I taktech.} V nultém taktu jsou definovány pouze 
+hodnoty na~vstupech sítì a na~výstupech hradel arity 0. Mù¾eme si to pøedstavit
+tak, ¾e na~zaèátku nemá ¾ádné hradlo definovánu výstupní hodnotu (a¾ na ji¾ 
+zmínìná hradla nulární). V~ka¾dém dal¹ím taktu pak vydají výstup hradla, která
+na~konci minulého taktu mìla definovány v¹echny hodnoty na vstupech. Jakmile
+budou po~nìjakém koneèném poètu taktù definované i hodnoty v¹ech výstupù, sí»
+se zastaví a~vydá výsledek.
+
+\s{Pozorování:} Proto¾e je sí» acyklická, je jasné, ¾e jakmile jednou nìjaké hradlo vydá
+výstup, tak se tento výstup bìhem dal¹ího výpoètu sítì ji¾ nezmìní.
+
+\figure{vypocet_site.eps}{Výpoèet hradlové sítì.}{6cm}
+
+\>Podle toho, jak sí» poèítá, si ji mù¾eme rozdìlit na~vrstvy:
+
+\s{Definice:} {\I $i$-tá vrstva $S_i$} obsahuje právì takové vrcholy~$v$, pro~které nejdel¹í cesta ze~vstupù
+sítì do $v$ má délku rovnou~$i$.
+
+\s{Pozorování:} V¹imnìme si, ¾e v $i$-tém taktu vydají hodnoty právì hradla z $S_i$.
+Dává tedy smysl prohlásit za~{\I èasovou slo¾itost} sítì poèet
+jejích vrstev. Podobnì {\I prostorovou slo¾itost} definujeme jako poèet hradel v~síti.
+
+\s{Pøíklad:} Sestrojme sí», která zjistí, zda se mezi jejími~$n$ vstupy
+vyskytuje alespoò jedna jednièka.
+
+\>{\I První øe¹ení:} zapojíme hradla za~sebe (sériovì). Èasová i prostorová
+slo¾itost odpovídají~$\Theta(n)$. Zde ov¹em vùbec nevyu¾íváme toho, ¾e by mohlo poèítat více
+hradel souèasnì.
+
+\figure{hloupy_or.eps}{Hradlová sí», která zjistí, zdali je na vstupu alespoò jedna jednièka.}{0.7in}
+
+\>{\I Druhé øe¹ení:} Hradla budeme spojovat do~dvojic, pak výsledky z~tìchto dvojic opìt
+do~dvojic a tak dále. Díky paralelnímu zapojení dosáhneme èasové slo¾itosti $\Theta(\log n)$,
+prostorová slo¾itost zùstane lineární.
+
+\figure{chytry_or.eps}{Chytøej¹í øe¹ení stejného problému pro vstup velikosti 16.}{3in}
+
+Minule jsme si zavedli paralelní výpoèetní model, ve kterém si nyní nìco naprogramujeme \dots
+
+\h{Sèítání binárních èísel}
+
+Mìjme dvì èísla $x$ a $y$ zapsané ve~dvojkové soustavì. Jejich èíslice oznaème
+$x_{n-1}\ldots x_0$ a $y_{n-1}\ldots y_0$, kde $i$-tý øád má váhu $2^i$. Nyní budeme chtít tato èísla
+seèíst.
+
+K tomuto úèelu se~ihned nabízí pou¾ít starý dobrý \uv{¹kolní algoritmus}, který
+funguje ve~dvojkové soustavì stejnì dobøe jako v~desítkové. Zkrátka sèítáme èísla
+zprava doleva. V¾dy seèteme pøíslu¹né èíslice pod~sebou a~pøièteme pøenos z~ni¾¹ího
+øádu. Tím dostaneme jednu èíslici výsledku a~pøenos do~vy¹¹ího øádu. Formálnì
+bychom to mohli zapsat tøeba takto:
+$$
+z_i=x_i \oplus y_i \oplus c_i,
+$$
+kde $z_i$ je $i$-tá èíslice souètu, $\oplus$ znaèí operaci {\csc xor} (souèet modulo~2) a~$c_i$ je {\I pøenos}
+z~$(i-1)$-ního øádu do~$i$-tého. Pøenos pøitom nastane tehdy, pokud se~nám potkají
+dvì jednièky pod~sebou, nebo kdy¾ se~vyskytne alespoò jedna jednièka a~k~tomu
+pøenos z~ni¾¹ího øádu. Jinými slovy tehdy, kdy¾ ze~tøí xorovaných èíslic jsou alespoò
+dvì jednièky (pomocí obvodu pro majoritu z minulé pøedná¹ky lehce zkonstruujeme):
+$$
+\eqalign{
+c_{0} &= 0, \cr
+c_{i+1} &= (x_i~\&~y_i)\lor(x_i~\&~c_i)\lor(y_i~\&~c_i).\cr
+}
+$$
+
+Takovéto sèítání sice perfektnì funguje, nicménì je bohu¾el pomìrnì pomalé.
+Pokud bychom stavìli hradlovou sí» podle tohoto pøedpisu, byla by slo¾ená z~nìjakých
+podsítí (\uv{krabièek}), které budou mít na~vstupu $x_i$, $y_i$ a~$c_i$ a jejich výstupem
+bude $z_i$ a~$c_{i+1}$. 
+
+\figure{hloupe_scitani.eps}{Sèítání ¹kolním algoritmem.}{1.5in}
+
+V¹imìme si, ¾e~ka¾dá krabièka závisí na~výstupu té pøedcházející. Jednotlivé
+krabièky tedy musí urèitì le¾et na~rùzných hladinách. Celkovì bychom museli pou¾ít
+$\Theta{(n)}$ hladin a~jeliko¾ je ka¾dá krabièka konstantnì velká, také $\Theta{(n)}$ hradel. To dává
+lineární èasovou i~prostorovou slo¾itost, èili oproti sekvenènímu algoritmu jsme si nepomohli.
+
+Zamysleme se nad tím, jak by se proces sèítání mohl zrychlit.
+
+\h{Pøenosy v~blocích}
+Jediné, co nás pøi sèítání brzdí, jsou pøenosy mezi jednotlivými øády. Ka¾dý øád,
+aby~vydal souèet, musí poèkat na~to, a¾~dopoèítají v¹echny pøedcházející øády.
+Teprve pak se~toti¾ dozví pøenos. Kdybychom ov¹em pøenosy dokázali spoèítat
+nìjakým zpùsobem paralelnì, máme vyhráno. Jakmile známe v¹echny pøenosy, souèet
+u¾~zvládneme dopoèítat na~konstantnì mnoho hladin -- tedy v~konstantním èase.
+
+Podívejme se na~libovolný {\I blok} v~na¹em souètu. Tak budeme øíkat èíslùm
+$x_j\ldots x_i$ a $y_j\ldots y_i$ v~nìjakém intervalu indexù $\left<i,j\right>$. Pøenos $c_{j+1}$ vystupující z~tohoto bloku závisí mimo hodnot sèítancù u¾ pouze na~pøenosu $c_{i}$, který do bloku vstupuje.
+
+\figure{blok_scitani.eps}{Blok souètu.}{3in}
+
+Z tohoto pohledu se mù¾eme na blok také dívat jako na nìjakou funkci, která
+dostane jednobitový vstup a vydá jednobitový výstup. To je nám pøíjemné, nebo» 
+takových funkcí existují jenom ètyøi typy:
+\numlist\ndotted
+\:$f(x) = 0$,  ~~~~(0)
+\:$f(x) = 1$,  ~~~~(1)
+\:$f(x) = x$,  ~~~~($<$ -- kopírování)
+\:$f(x) = \neg{x}$.
+\endlist
+Jak se za~chvíli uká¾e, poslední pøípad, kdy by~nìjaký blok pøedával opaèný
+pøenos, ne¾ do~nìj vstupuje, navíc nikdy nemù¾e nastat. Pojïme si to rozmyslet.
+Jednobitové bloky se chovají velice jednodu¹e:
+
+\figure{bloky_1bit.eps}{Tabulka triviálních blokù.}{1.1in}
+
+Z prvního bloku evidentnì v¾dy vyleze 0, a»~do~nìj vstoupí jakýkoli pøenos.
+Poslední blok naopak sám o~sobì pøenos vytváøí, a»~ji¾ do~nìj vleze jakýkoliv.
+Bloky prostøední se chovají stejnì, a~to tak, ¾e~samy o~sobì ¾ádný pøenos nevytvoøí,
+ale~pokud do~nich nìjaký pøijde, tak~také odejde.
+
+Mìjme nyní nìjaký vìt¹í blok~$C$ slo¾ený ze~dvou men¹ích podblokù $A$ a~$B$, jejich¾
+chování u¾ známe. Z~toho mù¾eme odvodit, jak se chová celý blok:
+
+\figure{tabulka_skladani_bloku.eps}{Skládání chování blokù.}{1.3in}
+
+Pokud vy¹¹í blok pøenos pohlcuje, pak a»~se u¾~ni¾¹í blok chová jakkoli, slo¾ení
+tìchto blokù musí v¾dy pohlcovat. V~prvním øádku tabulky jsou tudí¾ nuly. Analogicky,
+pokud vy¹¹í blok generuje pøenos, tak~ten ni¾¹í na~tom nic nezmìní. V~druhém
+øádku tabulky jsou tedy samé jednièky. Zajímavìj¹í pøípad nastává, pokud vy¹¹í blok
+kopíruje -- tehdy zále¾í èistì na~chování ni¾¹ího bloku.
+
+V¹imnìme si, ¾e~skládání chování blokù je vlastnì úplnì obyèejné skládání
+funkcí. Nyní bychom mohli prohlásit, ¾e~budeme poèítat nad~tøíprvkovou abecedou,
+a~¾e~celou tabulku doká¾eme spoèítat jedním jediným hradlem. Pojïmì si v¹ak
+rozmyslet, jak~bychom takovouto vìc popsali èistì binárnì. Jak tedy tyto tøi stavy
+popisovat pouze nìkolika bity?
+
+Evidentnì nám k tomuto binárnímu zakódování tøí stavù budou staèit bity dva.
+Oznaème si je jako $p$ a $q$. Tato dvojice mù¾e nabývat hned ètyø mo¾ných hodnot,
+kterým pøiøadíme tøi mo¾ná chování bloku. Toto kódování mù¾eme zvolit zcela
+libovolnì, ale pokud si ho zvolíme ¹ikovnì, u¹etøíme si dále práci pøi kompozici.
+Zvolme si tedy kódování takto:
+
+\itemize\ibull
+\:$(1,*) = <$,
+\:$(0,0) = 0$,
+\:$(0,1) = 1$
+\endlist
+
+Tomu, ¾e blok kopíruje, odpovídá dvojice $p = 1$; $q =$ \<cokoliv>. V~ostatních
+pøípadech bude~$p$ nulové a~$q$ nám bude øíkat, co je na~výstupu pøíslu¹ného bloku.
+Jinými slovy $p = 0$ znamená, ¾e funkce je konstanta, pøièem¾ $q$ øíká jaká; naproti
+tomu $p = 1$~znamená, ¾e funkce je identita, a»~u¾~je $q$ cokoli.
+
+Pojïme si nyní ukázat, jak bude celé skládání blokù vypadat. Rozmysleme si,
+kdy je~$p$ celého dvojbloku jednièkové, tedy kdy celý dvojblok kopíruje. To nastane
+tehdy, pokud kopírují obì jeho èásti, a tedy  $p = p_a~\&~p_b$. Dále $q$ bude rovno jednièce,
+pokud $q = (\neg{p_a}~\&~q_a) \lor (p_a~\&~q_b)$.
+
+Skládání chování blokù lze tedy popsat buï ternárnì -- tabulkou, ale lze to
+i~binárnì vý¹e uvedeným pøedpisem.
+
+Nyní si tedy mù¾eme dopøedu vypoèítat chování bloku velikosti jedna, poté
+ z~nich skládáním blokù velikosti dva, dál velikosti ètyøi, osm, atd \dots
+
+\h{Paralelní sèítání}
+
+Paralelní algoritmus na~sèítání u¾~zkonstruujeme pomìrnì snadno. Bez újmy
+na~obecnosti budeme pøedpokládat, ¾e~poèet bitù vstupních èísel je mocnina dvojky,
+jinak si vstup doplníme nulami, co¾ výsedný èas bìhu algoritmu zhor¹í maximálnì
+konstanta-krát.
+
+\algo
+\:Spoèteme chování blokù velikosti~1. ($\O(1)$ hladin)
+\:Postupnì poèítáme chování blokù\foot{myslíme \uv{pøirozenì zarovnané} bloky, tedy takové, jejich¾ poloha je násobkem velikosti} velikosti 2, 4, 8, ..., $2^k$. 
+  ($\O(\log n)$ hladin, na~nich¾ se skládají bloky)
+\:$c_0 \leftarrow 0$ (pøenos do nejni¾¹ího øádu je v¾dy 0)
+\:Urèíme $c_n$ podle $c_0$ a chování (jediného) bloku velikosti~$n$.
+\:Postupnì dopoèítáme pøenosy na~hranicích dìlitelných $2^k$ \uv{zahu¹»ováním}:
+  jakmile víme $c_{2^k}$, mù¾eme dopoèítat $c_{2^k+2^{k-1}}$ podle
+  chování bloku $\left< 2^k, 2^k+2^{k-1}\right>$. ($\O(\log n)$ hladin,
+  na~nich¾ se dosazuje)
+\:Seèteme: $\forall i: z_i = x_i \oplus y_i \oplus c_i$.
+\endalgo
+
+\figure{1_9_deleni_bloku.eps}{Výpoèet pøenosu.}{2.5in}
+
+Algoritmus pracuje v~èase $\O(\log n)$. Hradel je pou¾ito lineárnì: na~jednotlivých
+hladinách kroku~2 poèet hradel exponenciálnì klesá od~$n$ k~1, na~hladinách kroku~5
+exponenciálnì stoupá od~1 k~$n$, tak¾e dohromady se seète na~$\Theta(n)$.
+
+\h{Paralelní násobení}
+
+Podobnì jako u~sèítání si vzpomeneme na~¹kolní algoritmus -- tentokráte v¹ak pro násobení.
+To fungovalo tak, ¾e~jsme si jedno ze~dvou binárních èísel na~vstupu (øíkejme mu tøeba $x$) $n$-krát posouvali. Tam kde pak byly v~èísle~$y$ jednièky, pøíslu¹né kopie $x$ jsme seèetli. Jinými slovy tedy násobení umíme pøevést na~nìjaké posuny (ty lze realizovat pouze \uv{pøedrátováním} -- nic nás nestojí), násobení jedním bitem (co¾ je 
+{\csc and} ) a~nakonec potøebujeme výsledných~$n$ èísel seèíst.
+\figure{skolni_scitani.eps}{©kolní sèítání.}{2in}
+
+Jak nyní seèíst $n$ $n$-bitových èísel..? Nabízí se vyu¾ít osvìdèený \uv{stromeèek} -- sèítat dvojice èísel, výsledky pak opìt po~dvojicích seèíst, a¾ na~konci vyjde jediný výsledek.
+\figure{stromecek.eps}{Stromeèek}{1.4in}
+
+Toto øe¹ení by v¹ak vedlo na~èasovou slo¾itost $\Theta (\log^2 n)$, nebo» sèítat umíme v~èase $\Theta (\log n)$. To je sice dle na¹ich mìøítek
+docela efektivní, ale pøekvapivì to jde je¹tì lépe -- toti¾ na~$\Theta (\log n)$ hladin. Této 
+slo¾itosti dosáhneme malým trikem.
+
+Vymyslíme obvod konstantní hloubky, který na~vstupu dostane tøi èísla. Odpoví pak dvìma èísly
+takovými, ¾e budou mít stejný souèet jako pùvodní tøi èísla. Jinými slovy pomocí
+tohoto obvodu budeme umìt seètení tøí èísel pøevést na seètení dvou èísel.
+
+\figure{obvod.eps}{$x+y+z = p+q$}{0.3in}
+
+V¹imnìme si, ¾e~kdy¾ sèítáme tøi bity, mù¾e být pøenos do~vy¹¹ího øádu nula èi~jednièka. Vezmeme si tedy bity $x_i$, $y_i$, $z_i$ a~ty seèteme. To nám dá dvoubitový výsledek, pøièem¾ ni¾¹í bit z~tohoto výsledku po¹leme do~èísla~$p$, vy¹¹í do~èísla~$q$.
+
+\figure{obvod_real.eps}{Redukování sèítání}{1.2in}
+
+Toto zredukování sèítání nám nyní umo¾ní opìt stavit strom, by» o malièko slo¾itìj¹í.
+
+\figure{sl_stromecek.eps}{Slo¾itìj¹í stromeèek}{0.9in}
+
+Pokud jsme mìli na~zaèátku $n$ èísel, po~první redukci nám jich zbývá jen $2/3 \cdot n$ a~obecnì v~$k$-té hladiné $(2/3)^k \cdot n$. Znamená to, ¾e èísel nám ubývá exponenciálnì, tak¾e poèet hladin bude logaritmický. Redukující obvod je pøi tom jen konstantnì hluboký, tak¾e celé redukování zvládneme v~èase $\Theta (\log n)$. Na~konci Slo¾itìj¹ího stromeèku pak máme umístìnou jednu obyèejnou sèítaèku, která zbývající dvì èísla seète v~logaritmickém èase.
+
+Seètení v¹ech $n$ èísel tedy zabere $\Theta (\log n)$ hladin.
+
+Kdy¾ se nyní vrátíme k~násobení, zbývá nám vyøe¹it posouvání a~{\csc and}ování. Uvìdomme si, ¾e to je plnì paralelení a~zvládneme ho za~konstantnì mnoho hladin. Celé násobení tedy zvládneme v~logaritmickém èase.
+
+\bye
diff --git a/5-hradla/Makefile b/5-hradla/Makefile
new file mode 100644 (file)
index 0000000..9d4d631
--- /dev/null
@@ -0,0 +1,3 @@
+P=5-hradla
+
+include ../Makerules
diff --git a/5-hradla/blok_scitani.eps b/5-hradla/blok_scitani.eps
new file mode 100644 (file)
index 0000000..225a323
--- /dev/null
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diff --git a/6-bitonic/6-bitonic.tex b/6-bitonic/6-bitonic.tex
new file mode 100644 (file)
index 0000000..549e2f8
--- /dev/null
@@ -0,0 +1,130 @@
+\input ../lecnotes.tex
+
+\prednaska{6}{Bitonické tøídìní}{}
+
+Nyní se podíváme na~druhý problém, a~to na~problém tøídìní. Ji¾ známe pomìrnì efektivní sekvenèní algoritmy, které dovedou tøídit v~èase $\O(n\log n)$. Byli bychom jistì rádi, kdybychom to zvládli je¹tì rychleji. Pojïme se podívat, zda by nám v tom nepomohlo problém paralelizovat.
+
+Budeme pøi tom pracovat ve~výpoèetním modelu, kterému se øíká komparátorová sí». Ta je postavená z~hradel, kterým se øíká komparátory.
+
+\s{Definice:} {\I Komparátorová sí»} je kombinaèní obvod, jeho¾ hradla jsou
+komparátory.
+
+\figure{sortnet.0}{Komparátor}{0.7in}
+
+Komparátor umí porovnat dvì hodnoty a~rozhodnout, která z~nich je vìt¹í
+a~která men¹í. Nevrací v¹ak booleovský výsledek jako bì¾né hradlo, ale má dva
+výstupy, pøièem¾ na~jednom vrátí men¹í ze~vstupních hodnot a~na~druhém vìt¹í.
+
+Výstupy komparátorù se nevìtví. Nemù¾eme tedy jeden výstup \uv{rozdvojit} a~pøipojit ho na~dva vstupy. (Vìtvení by dokonce ani nemìlo smysl, proto¾e zatímco rozdvojit bychom mohli, slouèit u¾~ne. Pokud tedy chceme, aby sí» mìla $n$~vstupù i~$n$~výstupù, rozdvojení stejnì nesmíme provést, i kdybychom jej mìli povolené.)
+
+\s{Pøíklad:} {\sl Bubble sort}
+
+Obrázek Bubble 1 ilustruje pou¾ití komparátorù pro tøídìní Bubble sortem.
+©ipky pøedstavují jednotlivé komparátory. Výpoèet v¹ak je¹tì mù¾eme vylep¹it.
+
+\twofigures{sortnet.1}{Bubble 1}{143pt}{sortnet.2}{Bubble 2}{143pt}
+
+Sna¾íme se výpoèet co nejvíce paralelizovat (viz obrázek Bubble 2).
+Jak je vidìt, komparátory na sebe nemusejí èekat. Tím mù¾eme výpoèet urychlit a místo èasu $\Theta{(n^2)}$ docílit èasové slo¾itosti $\Theta{(n)}$. V obou pøípadech je zachován kvadratický prostor.
+
+Nyní si uká¾eme je¹tì rychlej¹í tøídící algoritmus. Pùjdeme na nìj v¹ak trochu \uv{od lesa}. Nejdøíve vymyslíme sí», která bude umìt tøídit jenom nìco -- toti¾ bitonické posloupnosti. Bez újmy na obecnosti budeme pøedpokládat, ¾e ka¾dé dva prvky na vstupu jsou navzájem rùzné.
+
+\medskip
+\s{Definice:} Øekneme, ¾e posloupnost $x_0,\dots,x_{n-1} $ je {\I èistì bitonická} právì tehdy, kdy¾
+pro nìjaké $x_k, k\in\{0, \dots, n-1\} $ platí, ¾e~v¹echny prvky pøed ním (vèetnì jeho samotného)
+tvoøí rostoucí poslopnost, kde¾to prvky stojící za~ním tvoøí poslopnost klesající.
+Formálnì zapsáno musí platit, ¾e:
+$$x_0\leq x_1\leq \dots \leq x_k \geq x_{k+1}\geq\dots \geq x_{n-1}.$$
+
+\s{Definice:} Posloupnost $x_0 \dots x_{n-1}$ je {\I bitonická}, právì kdy¾ $\exists~j\in \{0,\dots ,n-1\}$, pro
+které je rotace pùvodní posloupnosti o $j$ prvkù, tedy posloupnost
+$$x_j,x_{(j+1) \bmod n},\dots, x_{(j+n-1) \bmod n},$$ èistì bitonická.
+
+\s{Definice:} {\I Separátor $S_n$} je sí», ve které jsou v¾dy~$i$-tý a~$(i+{n/2})$-tý prvek vstupu
+(pro $i=0,\dots, {n/2}-1$) propojeny komparátorem. Minimum se pak stane~$i$-tým,
+maximum  $(i+{n/2})$-tým prvkem výstupu.
+\figure{sortnet.3}{$(y_i, y_{i+{n/2}}) = \<CMP>(x_i, x_{i+{n/2}})$} {300pt}
+
+\s{Lemma:} Pokud separátor dostane na~vstupu bitonickou posloupnost, pak jeho výstup $y_0, \dots, y_{n-1}$
+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 $y_i < y_{j + {n/2}}$.
+
+Separátor nám tedy jednu bitonickou posloupnost na~vstupu rozdìlí na~dvì
+bitonické posloupnosti, pøièem¾ navíc ka¾dý prvek první posloupnosti ($y_0,\dots, y_{n/2 -1}$)
+je men¹í nebo roven prvkùm druhé posloupnosti ($y_{n/2}, \dots, y_{n-1}$).
+
+\>Ne¾ pøistoupíme k dùkazu lemmatu, uka¾me si, k èemu se nám bude hodit.
+
+
+\s{Definice:} {\I Bitonická tøídièka $B_n$} je obvod sestavený ze separátorù, který dostane-li na vstupu bitonickou posloupnost délky $n$ (BÚNO konstruujeme tøídièku pro $n=2^k$), vydá setøídìnou zadanou posloupnost délky $n$. 
+
+Tøídièka dostane na~vstupu bitonickou posloupnost. Separátor~$S_n$ ji pak dle lemmatu rozdìlí na~dvì bitonické posloupnosti, kdy je ka¾dý prvek z~první posloupnosti men¹í ne¾ libovolný prvek z~druhé. Tyto poloviny pak dal¹í separátory rozdìlí na~ètvrtiny, ..., a¾~na~konci zbudou pouze jednoduché posloupnosti délky jedna (zjevnì setøídìné), které mezi sebou mají po¾adovanou nerovnost -- tedy ka¾dá posloupnost (nebo spí¹e prvek) nalevo je $\leq$ ne¾ prvek napravo od~nìj.
+
+\centerline{\epsfbox{sortnet.5}}
+
+Jak je vidìt, bitonická tøídièka nám libovolnou bitonickou posloupnost délky~$n$
+setøídí na~$\Theta(\log n)$ hladin.
+
+Nyní se dá odvodit, ¾e pokud umíme tøídit bitonické posloupnosti, umíme setøídit v¹echno.
+Vzpomeòme si na~tøídìní sléváním -- Merge sort. To funguje tak, ¾e zaène s~jednoprvkovými posloupnostmi, které jsou evidentnì setøídìné, a~poté v¾dy dvì setøídìné posloupnosti slévá do~jedné. Kdybychom nyní umìli paralelnì slévat, mohli bychom vytvoøit i~paralelní Merge sort. Jinými slovy, potøebujeme dvì rostoucí posloupnosti nìjak efektivnì slít do~jedné setøídìné. Uvìdomme si, ¾e to zvládneme jednodu¹e -- staèí druhou z~posloupností obrátit a~\uv{pøilepit} za~první, èím¾ vznikne bitonická posloupnost, kterou poté mù¾eme setøídit na¹í bitonickou tøídièkou.
+
+\s{Pøíklad:} {\sl Merge sort}
+
+Bitonická tøídièka se tedy dá pou¾ít ke~slévání setøídìných posloupností. 
+Uka¾me si, jak s~její pomocí sestavíme souèástky {\I slévaèky} $M_n$:
+
+Setøídìné posloupnosti $x_0,\dots, x_{n-1}$ a~$y_0,\dots, y_{n-1}$ spojíme do jedné bitonické
+posloupnosti $x_0,\dots, x_{n-1},y_{n-1},\dots, y_0$. Z~této posloupnosti vytvoøíme pomocí bitonické tøídièky
+$B_{2n}$ setøídìnou posloupnost. Vytvoøíme tedy blok $M_{2n}$, který se ov¹em sestává de~facto pouze z~bloku $B_{2n}$, jeho¾ druhá polovina vstupù je zapojena v~obráceném poøadí.
+\figure{sortnet.6}{Paralelní MergeSort.}{3in}
+
+Nyní se pokusme odhadnout èasovou slo¾itost. Ná¹ MergeSort bude mít øádovì hloubku blokù $\log n$.
+V ka¾dém bloku $M_n$ je navíc ukryta bitonická tøídièka s takté¾ logaritmickou hloubkou.
+Celková hloubka tedy bude $\log 2 + \log4 + \dots + \log 2^k + \dots + \log n$.
+Po seètení nakonec dostáváme výslednou èasovou slo¾itost $\Theta (\log^2 n)$.
+
+Dodejme je¹tì, ¾e existuje i~tøídicí algoritmus, kterému staèí jen ${\O}(\log n)$ hladin.
+Jeho multiplikativní konstanta je v¹ak pøíli¹ veliká, tak¾e je v~praxi nepou¾itelný.
+
+Vra»me se nyní k dùkazu lemmatu, který jsme na pøedminulé stránce vynechali.
+
+\h{Dùkaz Lemmatu:}
+(i) Nejprve nahlédneme, ¾e lemma platí, je-li vstupem èistì bitonická
+posloupnost. Dále BÚNO pøedpokládejme, ¾e vrchol posloupnosti je v~první polovinì (kdyby
+byl vrchol za~polovinou, staèilo by zrcadlovì obrátit posloupnost i~komparátory a~øe¹ili
+bychom stejný problém). Nyní si definujme $k := \min j: x_j > x_{j+n/2}$. (Pokud
+by~takové~$k$ neexistovalo, znamenalo by~to, ¾e vstupní posloupnost je monotónní.
+Separátor by tedy nic nedìlal a~pouze zkopíroval vstup na~výstup, co¾ jistì lemma splòuje.) Nyní si v¹imìme,
+¾e jakmile jednou zaène platit, ¾e prvek na~levé stranì je men¹í ne¾ na~pravé, bude
+nám tato relace platit a¾~do~konce. Oznaème vrchol vstupní posloupnosti jako~$x_m$.
+Pak~$k$ bude jistì men¹í ne¾~$m$ a~$k+{n/2}$ bude vìt¹í ne¾~$m$. Mezi~$k$ a~$m$ je tedy
+vstupní posloupnost neklesající, mezi $k+{n/2}$ a~$n-1$ nerostoucí.
+
+Do pozice~$k$ tedy separátor bude pouze kopírovat vstup na~výstup, od~pozice~$k$
+dál u¾~jen prohazuje. 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. Podobnì úsek $k+{n/2}$ a¾~$n-1$ nahradíme èistì
+bitonickou posloupností. Obì poloviny tedy budou bitonické.
+
+Dostaneme-li na vstupu obecnou bitonickou posloupnost, pøedstavíme si,
+¾e je to èistì bitonická posloupnost zrotovaná o~$r$ prvkù (BÚNO
+doprava). 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 výstupní
+posloupnosti tedy budou zrotované o~$r$ prvkù, ale na jejich
+bitoniènosti se nic nezmìní.
+
+(ii) Z dùkazu (i) pro èistì bitonickou posloupnost víme, ¾e $y_0\dots y_{n/2-1}$ je èistì bitonická a bude rovna $x_0\dots x_{k-1},x_{k+n/2}\dots x_{n-1}$ pro vhodné $k$ a navíc bude mít maximum v $x_{k-1}$ nebo $x_k+{n/2}$. Mezi tìmito body ov¹em ve vstupní posloupnosti urèitì nele¾el ¾ádný $x_i$ men¹í ne¾ $x_k-1$ nebo $x_k+{n/2}$ (jak je vidìt z obrázku) a posloupnost  $x_k \dots x_{k-1+{n/2}}$ je rovna $y_{n/2}\dots y_{n-1}$. Pro obecné bitonické posloupnosti uká¾eme stejnì jako v (i).
+\qed
+
+\medskip
+
+\centerline{\epsfbox{sortnet.7}}
+
+\bye
diff --git a/6-bitonic/Makefile b/6-bitonic/Makefile
new file mode 100644 (file)
index 0000000..496cb3d
--- /dev/null
@@ -0,0 +1,3 @@
+P=6-bitonic
+
+include ../Makerules
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+(2) cmr7 6.97385 fshow
+82.16853 29.11678 moveto
+(\000) cmsy10 9.96265 fshow
+92.13113 29.11678 moveto
+(1) cmr10 9.96265 fshow
+117.86421 28.21677 moveto
+(k) cmmi10 9.96265 fshow
+125.57831 28.21677 moveto
+(+) cmr10 9.96265 fshow
+136.7364 32.13937 moveto
+(n) cmmi7 6.97385 fshow
+newpath 136.7364 30.70747 moveto
+141.66132 30.70747 lineto stroke
+137.21321 24.78127 moveto
+(2) cmr7 6.97385 fshow
+167.0135 28.21677 moveto
+(n) cmmi10 9.96265 fshow
+175.2073 28.21677 moveto
+(\000) cmsy10 9.96265 fshow
+185.1699 28.21677 moveto
+(1) cmr10 9.96265 fshow
+32.76373 7.43059 moveto
+(p) cmr10 9.96265 fshow
+38.57533 7.43059 moveto
+(osloupnost) cmr10 9.96265 fshow
+87.94583 7.43059 moveto
+(prohozen\023) cmr10 9.96265 fshow
+127.27063 7.43059 moveto
+(a) cmr10 9.96265 fshow
+135.57283 7.43059 moveto
+(separ\023) cmr10 9.96265 fshow
+158.34863 7.43059 moveto
+(atorem) cmr10 9.96265 fshow
+showpage
+%%EOF
diff --git a/6-bitonic/sortnet.mp b/6-bitonic/sortnet.mp
new file mode 100644 (file)
index 0000000..081f9f4
--- /dev/null
@@ -0,0 +1,675 @@
+%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);
+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,z96);
+label.top(btex $x_{n-1}$ etex,z106);
+
+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);
+
+% ve skutecnosti dle znaceni 
+% by melo byt z6* ale uz 
+% obsazeno
+z815=(1.5v,2v);
+z855=(5.5v,2v);
+z895=(9.5v,2v);
+z813=(13.5v,2v);
+
+z715=(1.5v,1v);
+z755=(5.5v,1v);
+z795=(9.5v,1v);
+z713=(13.5v,1v);
+
+z9=(4v,0v);
+
+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);
+drawarrow(z815--z715);
+drawarrow(z855--z755);
+drawarrow(z895--z795);
+drawarrow(z813--z713);
+
+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);
+label.rt(btex Bitonick\'a t\v r\'\i di\v cka $B_{n}$ etex,z9);
+
+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/6-bitonic/sortnet.mpx b/6-bitonic/sortnet.mpx
new file mode 100644 (file)
index 0000000..9814d6d
--- /dev/null
@@ -0,0 +1,548 @@
+% Written by DVItoMP, Version 0.64/color (Web2C 7.5.4)
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n0="cmr10";
+_s("a",_n0,1.00000,0.0000,0.0000,);
+setbounds _p to (0,-3.4869)--(4.9813,-3.4869)--
+ (4.9813,8.4682)--(0,8.4682)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n0="cmr10";
+_s("b",_n0,1.00000,0.0000,0.0000,);
+setbounds _p to (0,-3.4869)--(5.5348,-3.4869)--
+ (5.5348,8.4682)--(0,8.4682)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n0="cmr10";
+_s("min",_n0,1.00000,0.0000,0.0000,);
+setbounds _p to (0,0.0000)--(16.6045,0.0000)--
+ (16.6045,6.6536)--(0,6.6536)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n0="cmr10";
+_s("max",_n0,1.00000,0.0000,0.0000,);
+setbounds _p to (0,0.0000)--(18.5416,0.0000)--
+ (18.5416,4.2895)--(0,4.2895)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n0="cmr10";
+_s("x1",_n0,1.00000,0.0000,0.0000,);
+setbounds _p to (0,0.0000)--(10.2394,0.0000)--
+ (10.2394,6.4204)--(0,6.4204)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n0="cmr10";
+_s("x2",_n0,1.00000,0.0000,0.0000,);
+setbounds _p to (0,0.0000)--(10.2394,0.0000)--
+ (10.2394,6.4204)--(0,6.4204)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n0="cmr10";
+_s("x3",_n0,1.00000,0.0000,0.0000,);
+setbounds _p to (0,0.0000)--(10.2394,0.0000)--
+ (10.2394,6.4204)--(0,6.4204)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n0="cmr10";
+_s("x4",_n0,1.00000,0.0000,0.0000,);
+setbounds _p to (0,0.0000)--(10.2394,0.0000)--
+ (10.2394,6.4204)--(0,6.4204)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n0="cmr10";
+_s("x5",_n0,1.00000,0.0000,0.0000,);
+setbounds _p to (0,0.0000)--(10.2394,0.0000)--
+ (10.2394,6.4204)--(0,6.4204)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n0="cmr10";
+_s("x1",_n0,1.00000,0.0000,0.0000,);
+setbounds _p to (0,0.0000)--(10.2394,0.0000)--
+ (10.2394,6.4204)--(0,6.4204)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n0="cmr10";
+_s("x2",_n0,1.00000,0.0000,0.0000,);
+setbounds _p to (0,0.0000)--(10.2394,0.0000)--
+ (10.2394,6.4204)--(0,6.4204)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n0="cmr10";
+_s("x3",_n0,1.00000,0.0000,0.0000,);
+setbounds _p to (0,0.0000)--(10.2394,0.0000)--
+ (10.2394,6.4204)--(0,6.4204)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n0="cmr10";
+_s("x4",_n0,1.00000,0.0000,0.0000,);
+setbounds _p to (0,0.0000)--(10.2394,0.0000)--
+ (10.2394,6.4204)--(0,6.4204)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n0="cmr10";
+_s("x5",_n0,1.00000,0.0000,0.0000,);
+setbounds _p to (0,0.0000)--(10.2394,0.0000)--
+ (10.2394,6.4204)--(0,6.4204)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("x",_n1,1.00000,0.0000,0.0000,);
+_n2="cmr7";
+_s("0",_n2,1.00000,5.6939,-1.4944,);
+setbounds _p to (0,-1.4944)--(10.1633,-1.4944)--
+ (10.1633,4.2895)--(0,4.2895)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("x",_n1,1.00000,0.0000,0.0000,);
+_n2="cmr7";
+_s("1",_n2,1.00000,5.6939,-1.4944,);
+setbounds _p to (0,-1.4944)--(10.1633,-1.4944)--
+ (10.1633,4.2895)--(0,4.2895)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("x",_n1,1.00000,0.0000,0.0000,);
+_n2="cmr7";
+_s("2",_n2,1.00000,5.6939,-1.4944,);
+setbounds _p to (0,-1.4944)--(10.1633,-1.4944)--
+ (10.1633,4.2895)--(0,4.2895)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s(":",_n1,1.00000,0.0000,0.0000,);
+_s(":",_n1,1.00000,4.4278,0.0000,);
+_s(":",_n1,1.00000,8.8556,0.0000,);
+setbounds _p to (0,0.0000)--(13.2834,0.0000)--
+ (13.2834,1.0516)--(0,1.0516)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("x",_n1,1.00000,0.0000,0.0000,);
+_n3="cmmi7";
+_s("n",_n3,1.00000,5.6939,-1.4944,);
+_n4="cmsy7";
+_s(char0,_n4,1.00000,10.6188,-1.4944,);
+_n2="cmr7";
+_s("2",_n2,1.00000,16.8454,-1.4944,);
+setbounds _p to (0,-2.3246)--(21.3148,-2.3246)--
+ (21.3148,4.2895)--(0,4.2895)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("x",_n1,1.00000,0.0000,0.0000,);
+_n3="cmmi7";
+_s("n",_n3,1.00000,5.6939,-1.4944,);
+_n4="cmsy7";
+_s(char0,_n4,1.00000,10.6188,-1.4944,);
+_n2="cmr7";
+_s("1",_n2,1.00000,16.8454,-1.4944,);
+setbounds _p to (0,-2.3246)--(21.3148,-2.3246)--
+ (21.3148,4.2895)--(0,4.2895)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("n",_n1,1.00000,0.0000,0.0000,);
+setbounds _p to (0,0.0000)--(5.9799,0.0000)--
+ (5.9799,4.2895)--(0,4.2895)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("S",_n1,1.00000,0.0000,0.0000,);
+_n3="cmmi7";
+_s("n",_n3,1.00000,6.1090,-1.4944,);
+setbounds _p to (0,-1.4944)--(11.5320,-1.4944)--
+ (11.5320,6.8078)--(0,6.8078)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("S",_n1,1.00000,0.0000,0.0000,);
+_n5="cmmi5";
+_s("n",_n5,1.00000,7.3045,1.1829,);
+interim linecap:=0;
+vardef _r(expr _a,_w)(text _t) =
+  addto _p doublepath _a withpen pencircle scaled _w _t enddef;
+_r((7.3045,0.2491)..(11.6921,0.2491), 0.3985,);
+_n6="cmr5";
+_s("2",_n6,1.00000,7.8033,-3.8949,);
+setbounds _p to (0,-3.8949)--(13.3857,-3.8949)--
+ (13.3857,6.8078)--(0,6.8078)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("S",_n1,1.00000,0.0000,0.0000,);
+_n5="cmmi5";
+_s("n",_n5,1.00000,7.3045,1.1829,);
+interim linecap:=0;
+vardef _r(expr _a,_w)(text _t) =
+  addto _p doublepath _a withpen pencircle scaled _w _t enddef;
+_r((7.3045,0.2491)..(11.6921,0.2491), 0.3985,);
+_n6="cmr5";
+_s("2",_n6,1.00000,7.8033,-3.8949,);
+setbounds _p to (0,-3.8949)--(13.3857,-3.8949)--
+ (13.3857,6.8078)--(0,6.8078)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("S",_n1,1.00000,0.0000,0.0000,);
+_n5="cmmi5";
+_s("n",_n5,1.00000,7.3045,1.1829,);
+interim linecap:=0;
+vardef _r(expr _a,_w)(text _t) =
+  addto _p doublepath _a withpen pencircle scaled _w _t enddef;
+_r((7.3045,0.2491)..(11.6921,0.2491), 0.3985,);
+_n6="cmr5";
+_s("4",_n6,1.00000,7.8033,-3.8949,);
+setbounds _p to (0,-3.8949)--(13.3857,-3.8949)--
+ (13.3857,6.8078)--(0,6.8078)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("S",_n1,1.00000,0.0000,0.0000,);
+_n5="cmmi5";
+_s("n",_n5,1.00000,7.3045,1.1829,);
+interim linecap:=0;
+vardef _r(expr _a,_w)(text _t) =
+  addto _p doublepath _a withpen pencircle scaled _w _t enddef;
+_r((7.3045,0.2491)..(11.6921,0.2491), 0.3985,);
+_n6="cmr5";
+_s("4",_n6,1.00000,7.8033,-3.8949,);
+setbounds _p to (0,-3.8949)--(13.3857,-3.8949)--
+ (13.3857,6.8078)--(0,6.8078)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("S",_n1,1.00000,0.0000,0.0000,);
+_n5="cmmi5";
+_s("n",_n5,1.00000,7.3045,1.1829,);
+interim linecap:=0;
+vardef _r(expr _a,_w)(text _t) =
+  addto _p doublepath _a withpen pencircle scaled _w _t enddef;
+_r((7.3045,0.2491)..(11.6921,0.2491), 0.3985,);
+_n6="cmr5";
+_s("4",_n6,1.00000,7.8033,-3.8949,);
+setbounds _p to (0,-3.8949)--(13.3857,-3.8949)--
+ (13.3857,6.8078)--(0,6.8078)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("S",_n1,1.00000,0.0000,0.0000,);
+_n5="cmmi5";
+_s("n",_n5,1.00000,7.3045,1.1829,);
+interim linecap:=0;
+vardef _r(expr _a,_w)(text _t) =
+  addto _p doublepath _a withpen pencircle scaled _w _t enddef;
+_r((7.3045,0.2491)..(11.6921,0.2491), 0.3985,);
+_n6="cmr5";
+_s("4",_n6,1.00000,7.8033,-3.8949,);
+setbounds _p to (0,-3.8949)--(13.3857,-3.8949)--
+ (13.3857,6.8078)--(0,6.8078)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n0="cmr10";
+_s("Bitonic",_n0,1.00000,0.0000,0.0000,);
+_s("k"&char19,_n0,1.00000,31.1333,0.0000,);
+_s("a",_n0,1.00000,36.3914,0.0000,);
+_s("t",_n0,1.00000,44.6936,0.0000,);
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+_s("r",_n0,1.00000,48.5680,0.0000,);
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+_s(char16&"di",_n0,1.00000,52.4700,0.0000,);
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+_s("a",_n0,1.00000,72.6721,0.0000,);
+_n1="cmmi10";
+_s("B",_n1,1.00000,80.9743,0.0000,);
+_n3="cmmi7";
+_s("n",_n3,1.00000,88.5311,-1.4944,);
+setbounds _p to (0,-1.4944)--(93.9541,-1.4944)--
+ (93.9541,6.9185)--(0,6.9185)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("M",_n1,1.00000,0.0000,0.0000,);
+_n2="cmr7";
+_s("8",_n2,1.00000,9.6652,-1.4944,);
+setbounds _p to (0,-1.4944)--(14.1345,-1.4944)--
+ (14.1345,6.8078)--(0,6.8078)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("M",_n1,1.00000,0.0000,0.0000,);
+_n2="cmr7";
+_s("4",_n2,1.00000,9.6652,-1.4944,);
+setbounds _p to (0,-1.4944)--(14.1345,-1.4944)--
+ (14.1345,6.8078)--(0,6.8078)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("M",_n1,1.00000,0.0000,0.0000,);
+_n2="cmr7";
+_s("4",_n2,1.00000,9.6652,-1.4944,);
+setbounds _p to (0,-1.4944)--(14.1345,-1.4944)--
+ (14.1345,6.8078)--(0,6.8078)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("M",_n1,1.00000,0.0000,0.0000,);
+_n2="cmr7";
+_s("2",_n2,1.00000,9.6652,-1.4944,);
+setbounds _p to (0,-1.4944)--(14.1345,-1.4944)--
+ (14.1345,6.8078)--(0,6.8078)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("M",_n1,1.00000,0.0000,0.0000,);
+_n2="cmr7";
+_s("2",_n2,1.00000,9.6652,-1.4944,);
+setbounds _p to (0,-1.4944)--(14.1345,-1.4944)--
+ (14.1345,6.8078)--(0,6.8078)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("M",_n1,1.00000,0.0000,0.0000,);
+_n2="cmr7";
+_s("2",_n2,1.00000,9.6652,-1.4944,);
+setbounds _p to (0,-1.4944)--(14.1345,-1.4944)--
+ (14.1345,6.8078)--(0,6.8078)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("M",_n1,1.00000,0.0000,0.0000,);
+_n2="cmr7";
+_s("2",_n2,1.00000,9.6652,-1.4944,);
+setbounds _p to (0,-1.4944)--(14.1345,-1.4944)--
+ (14.1345,6.8078)--(0,6.8078)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n0="cmr10";
+_s("0",_n0,1.00000,0.0000,0.0000,);
+setbounds _p to (0,-3.4869)--(4.9813,-3.4869)--
+ (4.9813,8.4682)--(0,8.4682)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("k",_n1,1.00000,0.0000,0.0000,);
+setbounds _p to (0,0.0000)--(5.5002,0.0000)--
+ (5.5002,6.9185)--(0,6.9185)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n3="cmmi7";
+_s("n",_n3,1.00000,1.1955,3.9226,);
+interim linecap:=0;
+vardef _r(expr _a,_w)(text _t) =
+  addto _p doublepath _a withpen pencircle scaled _w _t enddef;
+_r((1.1955,2.4907)..(6.1204,2.4907), 0.3985,);
+_n2="cmr7";
+_s("2",_n2,1.00000,1.6723,-3.4355,);
+_n7="cmsy10";
+_s(char0,_n7,1.00000,9.5298,0.0000,);
+_n0="cmr10";
+_s("1",_n0,1.00000,19.4924,0.0000,);
+setbounds _p to (0,-3.4869)--(24.4737,-3.4869)--
+ (24.4737,8.4682)--(0,8.4682)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("k",_n1,1.00000,0.0000,0.0000,);
+_n0="cmr10";
+_s("+",_n0,1.00000,7.7141,0.0000,);
+_n3="cmmi7";
+_s("n",_n3,1.00000,18.8722,3.9226,);
+interim linecap:=0;
+vardef _r(expr _a,_w)(text _t) =
+  addto _p doublepath _a withpen pencircle scaled _w _t enddef;
+_r((18.8722,2.4907)..(23.7971,2.4907), 0.3985,);
+_n2="cmr7";
+_s("2",_n2,1.00000,19.3490,-3.4355,);
+setbounds _p to (0,-3.4869)--(24.9926,-3.4869)--
+ (24.9926,8.4682)--(0,8.4682)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("n",_n1,1.00000,0.0000,0.0000,);
+_n7="cmsy10";
+_s(char0,_n7,1.00000,8.1938,0.0000,);
+_n0="cmr10";
+_s("1",_n0,1.00000,18.1564,0.0000,);
+setbounds _p to (0,-3.4869)--(23.1377,-3.4869)--
+ (23.1377,8.4682)--(0,8.4682)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n0="cmr10";
+_s("p",_n0,1.00000,0.0000,0.0000,);
+_s("osloupnost",_n0,1.00000,5.8116,0.0000,);
+_s("prohozen"&char19,_n0,1.00000,55.1821,0.0000,);
+_s("a",_n0,1.00000,94.5069,0.0000,);
+_s("separ"&char19,_n0,1.00000,102.8091,0.0000,);
+_s("atorem",_n0,1.00000,125.5849,0.0000,);
+setbounds _p to (0,-1.9372)--(156.0540,-1.9372)--
+ (156.0540,6.9185)--(0,6.9185)--cycle;
+_p endgroup
+mpxbreak
diff --git a/old/4-hradla/4-hradla.tex b/old/4-hradla/4-hradla.tex
deleted file mode 100644 (file)
index ccb0c50..0000000
+++ /dev/null
@@ -1,139 +0,0 @@
-\input ../lecnotes.tex
-
-\prednaska{4}{Hradlové sítì}{(zapsal: Petr Jankovský)}
-
-\def\land{\mathbin{\&}}
-
-Výkon poèítaèù nelze zvy¹ovat donekoneèna a pøesto¾e ji¾ pìkných pár let platí,
-¾e se jejich rychlost s~èasem exponenciálnì zvìt¹uje, jednou urèitì narazíme
-pøinejmen¹ím na~fyzikální limity.
-
-Kdy¾ tedy nemù¾eme zvy¹ovat rychlost jednoho procesoru, jak poèítat rychleji?
-Øe¹ením by mohlo být poøídit si procesorù víc. U¾~dnes na~bì¾ném PéCéèku máme
-k~dispozici vícejádrové procesory, díky nim¾ mù¾eme vyu¾ít pararelní poèítání
-a úlohu øe¹it tak, ¾e práci ¹ikovnì rozdìlíme mezi procesory (èi jádra) a
-zamìstnáme je pøi~výpoètu v¹echny.
-
-My se podíváme na~abstraktní výpoèetní model, který je je¹tì paralelnìj¹í.
-Techniky, které si uká¾eme na~tomto modelu, se v¹ak dají pøekvapivì vyu¾ít i~pøi~reálném
-paralelizování na~nìkolika málo procesorech. Konec koncù i proto, ¾e vnitøní
-architektura procesoru se na¹emu modelu velmi podobá. Budeme se zabývat 
-jednoduchým modelem paralelního poèítaèe, toti¾ hradlovou sítí.
-
-\h{Hradlové sítì}
-
-\s{Definice:} {\I Hradlo} je prvek, který umí vyhodnocovat nìjakou funkci
-nad~koneènou abecedou $\Sigma$.
-
-Obecnì se na~hradlo díváme jako na~funkci $f: {\Sigma}^{k} \rightarrow \Sigma$, která dostane $k$ vstupù
-a~vrátí jeden výstup, pøièem¾ hodnoty, nad~kterými pracuje, budou z~nìjaké koneèné
-abecedy -- tedy z~nìjaké koneèné mno¾iny symbolù $\Sigma$. Písmenku $k$ zde øíkáme {\I arita
-hradla}.
-
-\s{Pøíklad:} Èasto studujeme hradla booleovská (pracující nad abecedou $\{0,1\}$), která poèítají logické funkce.
-
-Z~nich nejèastìji potkáme:
-
-\itemize\ibull
-\:nulární: to jsou konstanty (FALSE=0, TRUE=1),
-\:unární: napø. negace (znaèíme~$\lnot$),
-\:binární: logický souèin ({\sc and},~$\land$), souèet ({\sc or},~$\lor$), ...
-\endlist
-
-\>Hradla kreslíme tøeba následovnì:
-
-\figure{hradlo_and.eps}{Binární hradlo provádící logickou operaci {\sc and}.}{1in}
-
-Jednotlivá hradla mù¾eme navzájem urèitým zpùsobem propojovat a vytváøet
-z nich {\I hradlové sítì}. Pokud pou¾íváme pouze booleovská hradla, øíkáme takto vytvoøeným
-sítím {\I booleovské obvody}. Pokud pracujeme s~operacemi nad nìjakou obecnìj¹í (ale koneènou) 
-mno¾inou symbolù (abecedou), nazývají se {\I kombinaèní obvody.}
-
-Ka¾dá hradlová sí» má nìjaké vstupy, nìjaké výstupy a uvnitø jsou propojovaná
-hradla. Ka¾dý vstup hradla je pøipojen buïto na~nìkterý ze~vstupù sítì nebo
-na~výstup jiného hradla. Výstupy hradel mohou být propojeny na~vstupy dal¹ích
-hradel (mohou se vìtvit), nebo na výstupy sítì. Pøitom máme zakázáno vytváøet
-cykly. 
-
-Ne¾ si øekneme formální definici, podívejme se na obrázek.
-
-\figure{hradlova_sit.eps}{Hradlová sí» -- tøívstupová verze funkce {\I majorita}.}{3in}
-
-Obrazek znázoròuje hradlovou sí», která poèítá, zda je alespoò na~dvou ze~vstupù
-jednièka. Pojïme si ale {\I hradlovou sí»} definovat formálnì.
-
-\s{Definice:} {\I Hradlová sí»} je urèena:
-\itemize\ibull
-\:{\I abecedou} $\Sigma$ (to je nìjaká koneèná mno¾ina symbolù, obvykle $\Sigma=\{0,1\}$);
-\:po dvou disjunktními koneènými mno¾inami $I$~({\I vstupy}), $O$~({\I výstupy}) \hfil\break a~$H$~({\I hradla});
-\:acyklickým orientovaným multigrafem~$(V,E)$, kde~$V = I \cup O \cup H$;
-\:zobrazením~$F$, které ka¾dému hradlu $h\in H$ pøiøadí nìjakou funkci~$F(h):
-  \Sigma^{a(h)} \rightarrow \Sigma$, co¾ je funkce, kterou toto hradlo vykonává.
-  Èíslu $a(h)$ øíkáme {\I arita} hradla~$h$;
-\:zobrazením~$z: E \rightarrow {\bb N}$, které ka¾dé hranì vedoucí do~nìjakého
-  hradla pøiøazuje nìkterý ze vstupù tohoto hradla.
-\endlist
-
-\>Pøitom jsou splnìny následující podmínky:
-
-\itemize\ibull
-\:$\forall i \in I: \deg^{in}(i)=0$ (do~vstupù nic nevede);
-\:$\forall o \in O: \deg^{in}(o)=1~\land~\deg^{out}(o)=0$ (z~výstupù nic nevede a do~ka¾dého vede právì jedna hrana);
-\:$\forall h \in H: \deg^{in}(v)=a(v)$ (do~ka¾dého hradla vede tolik hran, kolik je jeho arita);
-\:$\forall h \in H~\forall j: 1\le j\le a(h)$ existuje právì jedena hrana~$e$ taková, ¾e~$e$ konèí v~$h$ a~$z(e)=j$, 
-  (v¹echny vstupy hradel jsou zapojeny).
-\endlist
-
-\s{Pozorování:} Kdybychom pøipustili hradla s~libovolnì vysokým poètem vstupù, mohli
-bychom libovolný problém se vstupem délky~$n$ vyøe¹it jedním hradlem o~$n$~vstupech,
-kterému bychom pøiøadili funkci, která by na¹i úlohu rovnou vyøe¹ila. Tento model
-v¹ak není ani realistický, ani pìkný. Proto pøijmìme omezení, ¾e~arity v¹ech hradel
-budou omezeny nìjakou pevnou konstantou $k$ (uká¾e se, ¾e nám bude staèit dvojka
-a~vystaèíme si tedy pouze s nulárními, unárními a binárními hradly). 
-Následující obrázky ukazují, jak hradla o~více vstupech nahradit dvouvstupovými:
-
-\twofigures{hradlo_ternor.eps}{Trojvstupové hradlo \sc or.}{0.5in}{hradlo_ternbior.eps}{Jeho nahrazení 2-vstupovými hradly.}{0.6in}
-
-
-Nyní bychom je¹tì mìli definovat, co taková hradlová sí» vlastnì poèítá a~jak
-její výpoèet probíhá.
-
-\s{Definice:} {\I Výpoèet sítì} probíhá po~{\I taktech.} V nultém taktu jsou definovány pouze 
-hodnoty na~vstupech sítì a na~výstupech hradel arity 0. Mù¾eme si to pøedstavit
-tak, ¾e na~zaèátku nemá ¾ádné hradlo definovánu výstupní hodnotu (a¾ na ji¾ 
-zmínìná hradla nulární). V~ka¾dém dal¹ím taktu pak vydají výstup hradla, která
-na~konci minulého taktu mìla definovány v¹echny hodnoty na vstupech. Jakmile
-budou po~nìjakém koneèném poètu taktù definované i hodnoty v¹ech výstupù, sí»
-se zastaví a~vydá výsledek.
-
-\s{Pozorování:} Proto¾e je sí» acyklická, je jasné, ¾e jakmile jednou nìjaké hradlo vydá
-výstup, tak se tento výstup bìhem dal¹ího výpoètu sítì ji¾ nezmìní.
-
-\figure{vypocet_site.eps}{Výpoèet hradlové sítì.}{6cm}
-
-\>Podle toho, jak sí» poèítá, si ji mù¾eme rozdìlit na~vrstvy:
-
-\s{Definice:} {\I $i$-tá vrstva $S_i$} obsahuje právì takové vrcholy~$v$, pro~které nejdel¹í cesta ze~vstupù
-sítì do $v$ má délku rovnou~$i$.
-
-\s{Pozorování:} V¹imnìme si, ¾e v $i$-tém taktu vydají hodnoty právì hradla z $S_i$.
-Dává tedy smysl prohlásit za~{\I èasovou slo¾itost} sítì poèet
-jejích vrstev. Podobnì {\I prostorovou slo¾itost} definujeme jako poèet hradel v~síti.
-
-\s{Pøíklad:} Sestrojme sí», která zjistí, zda se mezi jejími~$n$ vstupy
-vyskytuje alespoò jedna jednièka.
-
-\>{\I První øe¹ení:} zapojíme hradla za~sebe (sériovì). Èasová i prostorová
-slo¾itost odpovídají~$\Theta(n)$. Zde ov¹em vùbec nevyu¾íváme toho, ¾e by mohlo poèítat více
-hradel souèasnì.
-
-\figure{hloupy_or.eps}{Hradlová sí», která zjistí, zdali je na vstupu alespoò jedna jednièka.}{0.7in}
-
-\>{\I Druhé øe¹ení:} Hradla budeme spojovat do~dvojic, pak výsledky z~tìchto dvojic opìt
-do~dvojic a tak dále. Díky paralelnímu zapojení dosáhneme èasové slo¾itosti $\Theta(\log n)$,
-prostorová slo¾itost zùstane lineární.
-
-\figure{chytry_or.eps}{Chytøej¹í øe¹ení stejného problému pro vstup velikosti 16.}{3in}
-
-\bye
diff --git a/old/4-hradla/Makefile b/old/4-hradla/Makefile
deleted file mode 100644 (file)
index 65a25ae..0000000
+++ /dev/null
@@ -1,3 +0,0 @@
-P=4-hradla
-
-include ../Makerules
diff --git a/old/4-hradla/chytry_or.eps b/old/4-hradla/chytry_or.eps
deleted file mode 100644 (file)
index 3d88cff..0000000
+++ /dev/null
@@ -1,2324 +0,0 @@
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diff --git a/old/5-addsort/1_9_deleni_bloku.eps b/old/5-addsort/1_9_deleni_bloku.eps
deleted file mode 100644 (file)
index 2c96b1c..0000000
+++ /dev/null
@@ -1,3225 +0,0 @@
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-    x1 y1 moveto x1 y2 lineto x2 y2 lineto x2 y1 lineto\r
-  mat setmatrix\r
-} bind def\r
-/Rr {\r
-  mat currentmatrix pop\r
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-    x1 xrad add y2 moveto\r
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-    0 0 1 90 -180 arc setmatrix\r
-    matrix currentmatrix x1 xrad add y1 yrad add translate xrad yrad scale\r
-    0 0 1 180 270 arc setmatrix\r
-    matrix currentmatrix x2 xrad sub y1 yrad add translate xrad yrad scale\r
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-    matrix currentmatrix x2 xrad sub y2 yrad sub translate xrad yrad scale\r
-    0 0 1 0 90 arc setmatrix\r
-    closepath\r
-  mat setmatrix\r
-} bind def\r
-/Ov {\r
-  mat currentmatrix pop\r
-    concat translate scale 1 0 moveto 0 0 1 0 360 arc closepath\r
-  mat setmatrix\r
-} bind def\r
-end\r
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-/rightdiagonal true definepattern pop\r
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-{ %definepattern\r
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diff --git a/old/5-addsort/5-addsort.tex b/old/5-addsort/5-addsort.tex
deleted file mode 100644 (file)
index 36b35cc..0000000
+++ /dev/null
@@ -1,312 +0,0 @@
-\input ../lecnotes.tex
-
-\prednaska{5}{Paralelní sèítání, bitonické tøídìní}{(zapsal: Petr Jankovský)}
-
-Minule jsme si zavedli paralelní výpoèetní model, ve kterém si nyní nìco naprogramujeme \dots
-
-\h{Sèítání binárních èísel}
-
-Mìjme dvì èísla $x$ a $y$ zapsané ve~dvojkové soustavì. Jejich èíslice oznaème
-$x_{n-1}\ldots x_0$ a $y_{n-1}\ldots y_0$, kde $i$-tý øád má váhu $2^i$. Nyní budeme chtít tato èísla
-seèíst.
-
-K tomuto úèelu se~ihned nabízí pou¾ít starý dobrý \uv{¹kolní algoritmus}, který
-funguje ve~dvojkové soustavì stejnì dobøe jako v~desítkové. Zkrátka sèítáme èísla
-zprava doleva. V¾dy seèteme pøíslu¹né èíslice pod~sebou a~pøièteme pøenos z~ni¾¹ího
-øádu. Tím dostaneme jednu èíslici výsledku a~pøenos do~vy¹¹ího øádu. Formálnì
-bychom to mohli zapsat tøeba takto:
-$$
-z_i=x_i \oplus y_i \oplus c_i,
-$$
-kde $z_i$ je $i$-tá èíslice souètu, $\oplus$ znaèí operaci {\sc xor} (souèet modulo~2) a~$c_i$ je {\I pøenos}
-z~$(i-1)$-ního øádu do~$i$-tého. Pøenos pøitom nastane tehdy, pokud se~nám potkají
-dvì jednièky pod~sebou, nebo kdy¾ se~vyskytne alespoò jedna jednièka a~k~tomu
-pøenos z~ni¾¹ího øádu. Jinými slovy tehdy, kdy¾ ze~tøí xorovaných èíslic jsou alespoò
-dvì jednièky (pomocí obvodu pro majoritu z minulé pøedná¹ky lehce zkonstruujeme):
-$$
-\eqalign{
-c_{0} &= 0, \cr
-c_{i+1} &= (x_i~\&~y_i)\lor(x_i~\&~c_i)\lor(y_i~\&~c_i).\cr
-}
-$$
-
-Takovéto sèítání sice perfektnì funguje, nicménì je bohu¾el pomìrnì pomalé.
-Pokud bychom stavìli hradlovou sí» podle tohoto pøedpisu, byla by slo¾ená z~nìjakých
-podsítí (\uv{krabièek}), které budou mít na~vstupu $x_i$, $y_i$ a~$c_i$ a jejich výstupem
-bude $z_i$ a~$c_{i+1}$. 
-
-\figure{hloupe_scitani.eps}{Sèítání ¹kolním algoritmem.}{1.5in}
-
-V¹imìme si, ¾e~ka¾dá krabièka závisí na~výstupu té pøedcházející. Jednotlivé
-krabièky tedy musí urèitì le¾et na~rùzných hladinách. Celkovì bychom museli pou¾ít
-$\Theta{(n)}$ hladin a~jeliko¾ je ka¾dá krabièka konstantnì velká, také $\Theta{(n)}$ hradel. To dává
-lineární èasovou i~prostorovou slo¾itost, èili oproti sekvenènímu algoritmu jsme si nepomohli.
-
-Zamysleme se nad tím, jak by se proces sèítání mohl zrychlit.
-
-\h{Pøenosy v~blocích}
-Jediné, co nás pøi sèítání brzdí, jsou pøenosy mezi jednotlivými øády. Ka¾dý øád,
-aby~vydal souèet, musí poèkat na~to, a¾~dopoèítají v¹echny pøedcházející øády.
-Teprve pak se~toti¾ dozví pøenos. Kdybychom ov¹em pøenosy dokázali spoèítat
-nìjakým zpùsobem paralelnì, máme vyhráno. Jakmile známe v¹echny pøenosy, souèet
-u¾~zvládneme dopoèítat na~konstantnì mnoho hladin -- tedy v~konstantním èase.
-
-Podívejme se na~libovolný {\I blok} v~na¹em souètu. Tak budeme øíkat èíslùm
-$x_j\ldots x_i$ a $y_j\ldots y_i$ v~nìjakém intervalu indexù $\left<i,j\right>$. Pøenos $c_{j+1}$ vystupující z~tohoto bloku závisí mimo hodnot sèítancù u¾ pouze na~pøenosu $c_{i}$, který do bloku vstupuje.
-
-\figure{blok_scitani.eps}{Blok souètu.}{3in}
-
-Z tohoto pohledu se mù¾eme na blok také dívat jako na nìjakou funkci, která
-dostane jednobitový vstup a vydá jednobitový výstup. To je nám pøíjemné, nebo» 
-takových funkcí existují jenom ètyøi typy:
-\numlist\ndotted
-\:$f(x) = 0$,  ~~~~(0)
-\:$f(x) = 1$,  ~~~~(1)
-\:$f(x) = x$,  ~~~~($<$ -- kopírování)
-\:$f(x) = \neg{x}$.
-\endlist
-Jak se za~chvíli uká¾e, poslední pøípad, kdy by~nìjaký blok pøedával opaèný
-pøenos, ne¾ do~nìj vstupuje, navíc nikdy nemù¾e nastat. Pojïme si to rozmyslet.
-Jednobitové bloky se chovají velice jednodu¹e:
-
-\figure{bloky_1bit.eps}{Tabulka triviálních blokù.}{1.1in}
-
-Z prvního bloku evidentnì v¾dy vyleze 0, a»~do~nìj vstoupí jakýkoli pøenos.
-Poslední blok naopak sám o~sobì pøenos vytváøí, a»~ji¾ do~nìj vleze jakýkoliv.
-Bloky prostøední se chovají stejnì, a~to tak, ¾e~samy o~sobì ¾ádný pøenos nevytvoøí,
-ale~pokud do~nich nìjaký pøijde, tak~také odejde.
-
-Mìjme nyní nìjaký vìt¹í blok~$C$ slo¾ený ze~dvou men¹ích podblokù $A$ a~$B$, jejich¾
-chování u¾ známe. Z~toho mù¾eme odvodit, jak se chová celý blok:
-
-\figure{tabulka_skladani_bloku.eps}{Skládání chování blokù.}{1.3in}
-
-Pokud vy¹¹í blok pøenos pohlcuje, pak a»~se u¾~ni¾¹í blok chová jakkoli, slo¾ení
-tìchto blokù musí v¾dy pohlcovat. V~prvním øádku tabulky jsou tudí¾ nuly. Analogicky,
-pokud vy¹¹í blok generuje pøenos, tak~ten ni¾¹í na~tom nic nezmìní. V~druhém
-øádku tabulky jsou tedy samé jednièky. Zajímavìj¹í pøípad nastává, pokud vy¹¹í blok
-kopíruje -- tehdy zále¾í èistì na~chování ni¾¹ího bloku.
-
-V¹imnìme si, ¾e~skládání chování blokù je vlastnì úplnì obyèejné skládání
-funkcí. Nyní bychom mohli prohlásit, ¾e~budeme poèítat nad~tøíprvkovou abecedou,
-a~¾e~celou tabulku doká¾eme spoèítat jedním jediným hradlem. Pojïmì si v¹ak
-rozmyslet, jak~bychom takovouto vìc popsali èistì binárnì. Jak tedy tyto tøi stavy
-popisovat pouze nìkolika bity?
-
-Evidentnì nám k tomuto binárnímu zakódování tøí stavù budou staèit bity dva.
-Oznaème si je jako $p$ a $q$. Tato dvojice mù¾e nabývat hned ètyø mo¾ných hodnot,
-kterým pøiøadíme tøi mo¾ná chování bloku. Toto kódování mù¾eme zvolit zcela
-libovolnì, ale pokud si ho zvolíme ¹ikovnì, u¹etøíme si dále práci pøi kompozici.
-Zvolme si tedy kódování takto:
-
-\itemize\ibull
-\:$(1,*) = <$,
-\:$(0,0) = 0$,
-\:$(0,1) = 1$
-\endlist
-
-Tomu, ¾e blok kopíruje, odpovídá dvojice $p = 1$; $q =$ \<cokoliv>. V~ostatních
-pøípadech bude~$p$ nulové a~$q$ nám bude øíkat, co je na~výstupu pøíslu¹ného bloku.
-Jinými slovy $p = 0$ znamená, ¾e funkce je konstanta, pøièem¾ $q$ øíká jaká; naproti
-tomu $p = 1$~znamená, ¾e funkce je identita, a»~u¾~je $q$ cokoli.
-
-Pojïme si nyní ukázat, jak bude celé skládání blokù vypadat. Rozmysleme si,
-kdy je~$p$ celého dvojbloku jednièkové, tedy kdy celý dvojblok kopíruje. To nastane
-tehdy, pokud kopírují obì jeho èásti, a tedy  $p = p_a~\&~p_b$. Dále $q$ bude rovno jednièce,
-pokud $q = (\neg{p_a}~\&~q_a) \lor (p_a~\&~q_b)$.
-
-Skládání chování blokù lze tedy popsat buï ternárnì -- tabulkou, ale lze to
-i~binárnì vý¹e uvedeným pøedpisem.
-
-Nyní si tedy mù¾eme dopøedu vypoèítat chování bloku velikosti jedna, poté
- z~nich skládáním blokù velikosti dva, dál velikosti ètyøi, osm, atd \dots
-
-\h{Paralelní sèítání}
-
-Paralelní algoritmus na~sèítání u¾~zkonstruujeme pomìrnì snadno. Bez újmy
-na~obecnosti budeme pøedpokládat, ¾e~poèet bitù vstupních èísel je mocnina dvojky,
-jinak si vstup doplníme nulami, co¾ výsedný èas bìhu algoritmu zhor¹í maximálnì
-konstanta-krát.
-
-\algo
-\:Spoèteme chování blokù velikosti~1. ($\O(1)$ hladin)
-\:Postupnì poèítáme chování blokù\foot{myslíme \uv{pøirozenì zarovnané} bloky, tedy takové, jejich¾ poloha je násobkem velikosti} velikosti 2, 4, 8, ..., $2^k$. 
-  ($\O(\log n)$ hladin, na~nich¾ se skládají bloky)
-\:$c_0 \leftarrow 0$ (pøenos do nejni¾¹ího øádu je v¾dy 0)
-\:Urèíme $c_n$ podle $c_0$ a chování (jediného) bloku velikosti~$n$.
-\:Postupnì dopoèítáme pøenosy na~hranicích dìlitelných $2^k$ \uv{zahu¹»ováním}:
-  jakmile víme $c_{2^k}$, mù¾eme dopoèítat $c_{2^k+2^{k-1}}$ podle
-  chování bloku $\left< 2^k, 2^k+2^{k-1}\right>$. ($\O(\log n)$ hladin,
-  na~nich¾ se dosazuje)
-\:Seèteme: $\forall i: z_i = x_i \oplus y_i \oplus c_i$.
-\endalgo
-
-\figure{1_9_deleni_bloku.eps}{Výpoèet pøenosu.}{2.5in}
-
-Algoritmus pracuje v~èase $\O(\log n)$. Hradel je pou¾ito lineárnì: na~jednotlivých
-hladinách kroku~2 poèet hradel exponenciálnì klesá od~$n$ k~1, na~hladinách kroku~5
-exponenciálnì stoupá od~1 k~$n$, tak¾e dohromady se seète na~$\Theta(n)$.
-
-\h{Tøídìní}
-
-Nyní se podíváme na~druhý problém, a~to na~problém tøídìní. Ji¾ známe pomìrnì efektivní sekvenèní algoritmy, které dovedou tøídit v~èase $\O(n\log n)$. Byli bychom jistì rádi, kdybychom to zvládli je¹tì rychleji. Pojïme se podívat, zda by nám v tom nepomohlo problém paralelizovat.
-
-Budeme pøi tom pracovat ve~výpoèetním modelu, kterému se øíká komparátorová sí». Ta je postavená z~hradel, kterým se øíká komparátory.
-
-\s{Definice:} {\I Komparátorová sí»} je kombinaèní obvod, jeho¾ hradla jsou
-komparátory.
-
-\figure{sortnet.0}{Komparátor}{0.7in}
-
-Komparátor umí porovnat dvì hodnoty a~rozhodnout, která z~nich je vìt¹í
-a~která men¹í. Nevrací v¹ak booleovský výsledek jako bì¾né hradlo, ale má dva
-výstupy, pøièem¾ na~jednom vrátí men¹í ze~vstupních hodnot a~na~druhém vìt¹í.
-
-Výstupy komparátorù se nevìtví. Nemù¾eme tedy jeden výstup \uv{rozdvojit} a~pøipojit ho na~dva vstupy. (Vìtvení by dokonce ani nemìlo smysl, proto¾e zatímco rozdvojit bychom mohli, slouèit u¾~ne. Pokud tedy chceme, aby sí» mìla $n$~vstupù i~$n$~výstupù, rozdvojení stejnì nesmíme provést, i kdybychom jej mìli povolené.)
-
-\s{Pøíklad:} {\sl Bubble sort}
-
-Obrázek Bubble 1 ilustruje pou¾ití komparátorù pro tøídìní Bubble sortem.
-©ipky pøedstavují jednotlivé komparátory. Výpoèet v¹ak je¹tì mù¾eme vylep¹it.
-
-\twofigures{sortnet.1}{Bubble 1}{143pt}{sortnet.2}{Bubble 2}{143pt}
-
-Sna¾íme se výpoèet co nejvíce paralelizovat (viz obrázek Bubble 2).
-Jak je vidìt, komparátory na sebe nemusejí èekat. Tím mù¾eme výpoèet urychlit a místo èasu $\Theta{(n^2)}$ docílit èasové slo¾itosti $\Theta{(n)}$. V obou pøípadech je zachován kvadratický prostor.
-
-Nyní si uká¾eme je¹tì rychlej¹í tøídící algoritmus. Pùjdeme na nìj v¹ak trochu \uv{od lesa}. Nejdøíve vymyslíme sí», která bude umìt tøídit jenom nìco -- toti¾ bitonické posloupnosti. Bez újmy na obecnosti budeme pøedpokládat, ¾e ka¾dé dva prvky na vstupu jsou navzájem rùzné.
-
-\medskip
-\s{Definice:} Øekneme, ¾e posloupnost $x_0,\dots,x_{n-1} $ je {\I èistì bitonická} právì tehdy, kdy¾
-pro nìjaké $x_k, k\in\{0, \dots, n-1\} $ platí, ¾e~v¹echny prvky pøed ním (vèetnì jeho samotného)
-tvoøí rostoucí poslopnost, kde¾to prvky stojící za~ním tvoøí poslopnost klesající.
-Formálnì zapsáno musí platit, ¾e:
-$$x_0\leq x_1\leq \dots \leq x_k \geq x_{k+1}\geq\dots \geq x_{n-1}.$$
-
-\s{Definice:} Posloupnost $x_0 \dots x_{n-1}$ je {\I bitonická}, právì kdy¾ $\exists~j\in \{0,\dots ,n-1\}$, pro
-které je rotace pùvodní posloupnosti o $j$ prvkù, tedy posloupnost
-$$x_j,x_{(j+1) \bmod n},\dots, x_{(j+n-1) \bmod n},$$ èistì bitonická.
-
-\s{Definice:} {\I Separátor $S_n$} je sí», ve které jsou v¾dy~$i$-tý a~$(i+{n/2})$-tý prvek vstupu
-(pro $i=0,\dots, {n/2}-1$) propojeny komparátorem. Minimum se pak stane~$i$-tým,
-maximum  $(i+{n/2})$-tým prvkem výstupu.
-\figure{sortnet.3}{$(y_i, y_{i+{n/2}}) = \<CMP>(x_i, x_{i+{n/2}})$} {300pt}
-
-\s{Lemma:} Pokud separátor dostane na~vstupu bitonickou posloupnost, pak jeho výstup $y_0, \dots, y_{n-1}$
-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 $y_i < y_{j + {n/2}}$.
-
-Separátor nám tedy jednu bitonickou posloupnost na~vstupu rozdìlí na~dvì
-bitonické posloupnosti, pøièem¾ navíc ka¾dý prvek první posloupnosti ($y_0,\dots, y_{n/2 -1}$)
-je men¹í nebo roven prvkùm druhé posloupnosti ($y_{n/2}, \dots, y_{n-1}$).
-
-\>Ne¾ pøistoupíme k dùkazu lemmatu, uka¾me si, k èemu se nám bude hodit.
-
-
-\s{Definice:} {\I Bitonická tøídièka $B_n$} je obvod sestavený ze separátorù, který dostane-li na vstupu bitonickou posloupnost délky $n$ (BÚNO konstruujeme tøídièku pro $n=2^k$), vydá setøídìnou zadanou posloupnost délky $n$. 
-
-Tøídièka dostane na~vstupu bitonickou posloupnost. Separátor~$S_n$ ji pak dle lemmatu rozdìlí na~dvì bitonické posloupnosti, kdy je ka¾dý prvek z~první posloupnosti men¹í ne¾ libovolný prvek z~druhé. Tyto poloviny pak dal¹í separátory rozdìlí na~ètvrtiny, ..., a¾~na~konci zbudou pouze jednoduché posloupnosti délky jedna (zjevnì setøídìné), které mezi sebou mají po¾adovanou nerovnost -- tedy ka¾dá posloupnost (nebo spí¹e prvek) nalevo je $\leq$ ne¾ prvek napravo od~nìj.
-
-\centerline{\epsfbox{sortnet.5}}
-
-Jak je vidìt, bitonická tøídièka nám libovolnou bitonickou posloupnost délky~$n$
-setøídí na~$\Theta(\log n)$ hladin.
-
-Nyní se dá odvodit, ¾e pokud umíme tøídit bitonické posloupnosti, umíme setøídit v¹echno.
-Vzpomeòme si na~tøídìní sléváním -- Merge sort. To funguje tak, ¾e zaène s~jednoprvkovými posloupnostmi, které jsou evidentnì setøídìné, a~poté v¾dy dvì setøídìné posloupnosti slévá do~jedné. Kdybychom nyní umìli paralelnì slévat, mohli bychom vytvoøit i~paralelní Merge sort. Jinými slovy, potøebujeme dvì rostoucí posloupnosti nìjak efektivnì slít do~jedné setøídìné. Uvìdomme si, ¾e to zvládneme jednodu¹e -- staèí druhou z~posloupností obrátit a~\uv{pøilepit} za~první, èím¾ vznikne bitonická posloupnost, kterou poté mù¾eme setøídit na¹í bitonickou tøídièkou.
-
-\s{Pøíklad:} {\sl Merge sort}
-
-Bitonická tøídièka se tedy dá pou¾ít ke~slévání setøídìných posloupností. 
-Uka¾me si, jak s~její pomocí sestavíme souèástky {\I slévaèky} $M_n$:
-
-Setøídìné posloupnosti $x_0,\dots, x_{n-1}$ a~$y_0,\dots, y_{n-1}$ spojíme do jedné bitonické
-posloupnosti $x_0,\dots, x_{n-1},y_{n-1},\dots, y_0$. Z~této posloupnosti vytvoøíme pomocí bitonické tøídièky
-$B_{2n}$ setøídìnou posloupnost. Vytvoøíme tedy blok $M_{2n}$, který se ov¹em sestává de~facto pouze z~bloku $B_{2n}$, jeho¾ druhá polovina vstupù je zapojena v~obráceném poøadí.
-\figure{sortnet.6}{Paralelní MergeSort.}{3in}
-
-Nyní se pokusme odhadnout èasovou slo¾itost. Ná¹ MergeSort bude mít øádovì hloubku blokù $\log n$.
-V ka¾dém bloku $M_n$ je navíc ukryta bitonická tøídièka s takté¾ logaritmickou hloubkou.
-Celková hloubka tedy bude $\log 2 + \log4 + \dots + \log 2^k + \dots + \log n$.
-Po seètení nakonec dostáváme výslednou èasovou slo¾itost $\Theta (\log^2 n)$.
-
-Dodejme je¹tì, ¾e existuje i~tøídicí algoritmus, kterému staèí jen ${\O}(\log n)$ hladin.
-Jeho multiplikativní konstanta je v¹ak pøíli¹ veliká, tak¾e je v~praxi nepou¾itelný.
-
-Vra»me se nyní k dùkazu lemmatu, který jsme na pøedminulé stránce vynechali.
-
-\h{Dùkaz Lemmatu:}
-(i) Nejprve nahlédneme, ¾e lemma platí, je-li vstupem èistì bitonická
-posloupnost. Dále BÚNO pøedpokládejme, ¾e vrchol posloupnosti je v~první polovinì (kdyby
-byl vrchol za~polovinou, staèilo by zrcadlovì obrátit posloupnost i~komparátory a~øe¹ili
-bychom stejný problém). Nyní si definujme $k := \min j: x_j > x_{j+n/2}$. (Pokud
-by~takové~$k$ neexistovalo, znamenalo by~to, ¾e vstupní posloupnost je monotónní.
-Separátor by tedy nic nedìlal a~pouze zkopíroval vstup na~výstup, co¾ jistì lemma splòuje.) Nyní si v¹imìme,
-¾e jakmile jednou zaène platit, ¾e prvek na~levé stranì je men¹í ne¾ na~pravé, bude
-nám tato relace platit a¾~do~konce. Oznaème vrchol vstupní posloupnosti jako~$x_m$.
-Pak~$k$ bude jistì men¹í ne¾~$m$ a~$k+{n/2}$ bude vìt¹í ne¾~$m$. Mezi~$k$ a~$m$ je tedy
-vstupní posloupnost neklesající, mezi $k+{n/2}$ a~$n-1$ nerostoucí.
-
-Do pozice~$k$ tedy separátor bude pouze kopírovat vstup na~výstup, od~pozice~$k$
-dál u¾~jen prohazuje. 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. Podobnì úsek $k+{n/2}$ a¾~$n-1$ nahradíme èistì
-bitonickou posloupností. Obì poloviny tedy budou bitonické.
-
-Dostaneme-li na vstupu obecnou bitonickou posloupnost, pøedstavíme si,
-¾e je to èistì bitonická posloupnost zrotovaná o~$r$ prvkù (BÚNO
-doprava). 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 výstupní
-posloupnosti tedy budou zrotované o~$r$ prvkù, ale na jejich
-bitoniènosti se nic nezmìní.
-
-(ii) Z dùkazu (i) pro èistì bitonickou posloupnost víme, ¾e $y_0\dots y_{n/2-1}$ je èistì bitonická a bude rovna $x_0\dots x_{k-1},x_{k+n/2}\dots x_{n-1}$ pro vhodné $k$ a navíc bude mít maximum v $x_{k-1}$ nebo $x_k+{n/2}$. Mezi tìmito body ov¹em ve vstupní posloupnosti urèitì nele¾el ¾ádný $x_i$ men¹í ne¾ $x_k-1$ nebo $x_k+{n/2}$ (jak je vidìt z obrázku) a posloupnost  $x_k \dots x_{k-1+{n/2}}$ je rovna $y_{n/2}\dots y_{n-1}$. Pro obecné bitonické posloupnosti uká¾eme stejnì jako v (i).
-\qed
-
-\medskip
-
-\centerline{\epsfbox{sortnet.7}}
-
-
-\h{Paralelní násobení}
-
-Podobnì jako u~sèítání si vzpomeneme na~¹kolní algoritmus -- tentokráte v¹ak pro násobení.
-To fungovalo tak, ¾e~jsme si jedno ze~dvou binárních èísel na~vstupu (øíkejme mu tøeba $x$) $n$-krát posouvali. Tam kde pak byly v~èísle~$y$ jednièky, pøíslu¹né kopie $x$ jsme seèetli. Jinými slovy tedy násobení umíme pøevést na~nìjaké posuny (ty lze realizovat pouze \uv{pøedrátováním} -- nic nás nestojí), násobení jedním bitem (co¾ je 
-{\sc and} ) a~nakonec potøebujeme výsledných~$n$ èísel seèíst.
-\figure{skolni_scitani.eps}{©kolní sèítání.}{2in}
-
-Jak nyní seèíst $n$ $n$-bitových èísel..? Nabízí se vyu¾ít osvìdèený \uv{stromeèek} -- sèítat dvojice èísel, výsledky pak opìt po~dvojicích seèíst, a¾ na~konci vyjde jediný výsledek.
-\figure{stromecek.eps}{Stromeèek}{1.4in}
-
-Toto øe¹ení by v¹ak vedlo na~èasovou slo¾itost $\Theta (\log^2 n)$, nebo» sèítat umíme v~èase $\Theta (\log n)$. To je sice dle na¹ich mìøítek
-docela efektivní, ale pøekvapivì to jde je¹tì lépe -- toti¾ na~$\Theta (\log n)$ hladin. Této 
-slo¾itosti dosáhneme malým trikem.
-
-Vymyslíme obvod konstantní hloubky, který na~vstupu dostane tøi èísla. Odpoví pak dvìma èísly
-takovými, ¾e budou mít stejný souèet jako pùvodní tøi èísla. Jinými slovy pomocí
-tohoto obvodu budeme umìt seètení tøí èísel pøevést na seètení dvou èísel.
-
-\figure{obvod.eps}{$x+y+z = p+q$}{0.3in}
-
-V¹imnìme si, ¾e~kdy¾ sèítáme tøi bity, mù¾e být pøenos do~vy¹¹ího øádu nula èi~jednièka. Vezmeme si tedy bity $x_i$, $y_i$, $z_i$ a~ty seèteme. To nám dá dvoubitový výsledek, pøièem¾ ni¾¹í bit z~tohoto výsledku po¹leme do~èísla~$p$, vy¹¹í do~èísla~$q$.
-
-\figure{obvod_real.eps}{Redukování sèítání}{1.2in}
-
-Toto zredukování sèítání nám nyní umo¾ní opìt stavit strom, by» o malièko slo¾itìj¹í.
-
-\figure{sl_stromecek.eps}{Slo¾itìj¹í stromeèek}{0.9in}
-
-Pokud jsme mìli na~zaèátku $n$ èísel, po~první redukci nám jich zbývá jen $2/3 \cdot n$ a~obecnì v~$k$-té hladiné $(2/3)^k \cdot n$. Znamená to, ¾e èísel nám ubývá exponenciálnì, tak¾e poèet hladin bude logaritmický. Redukující obvod je pøi tom jen konstantnì hluboký, tak¾e celé redukování zvládneme v~èase $\Theta (\log n)$. Na~konci Slo¾itìj¹ího stromeèku pak máme umístìnou jednu obyèejnou sèítaèku, která zbývající dvì èísla seète v~logaritmickém èase.
-
-Seètení v¹ech $n$ èísel tedy zabere $\Theta (\log n)$ hladin.
-
-Kdy¾ se nyní vrátíme k~násobení, zbývá nám vyøe¹it posouvání a~{\sc and}ování. Uvìdomme si, ¾e to je plnì paralelení a~zvládneme ho za~konstantnì mnoho hladin. Celé násobení tedy zvládneme v~logaritmickém èase.
-
-\bye
diff --git a/old/5-addsort/Makefile b/old/5-addsort/Makefile
deleted file mode 100644 (file)
index 4f7be80..0000000
+++ /dev/null
@@ -1,3 +0,0 @@
-P=5-addsort
-
-include ../Makerules
diff --git a/old/5-addsort/blok_scitani.eps b/old/5-addsort/blok_scitani.eps
deleted file mode 100644 (file)
index 225a323..0000000
+++ /dev/null
@@ -1,1673 +0,0 @@
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-130.36052 39.68497 lineto stroke
- 0 0.3985 dtransform truncate idtransform setlinewidth pop
- [0 3.49998 ] 1.74998 setdash
-newpath 9.92125 9.92125 moveto
-29.76373 9.92125 lineto stroke
-newpath 52.48499 96.9054 moveto
-52.48499 97.30173 52.32755 97.68184 52.04729 97.9621 curveto
-51.76703 98.24236 51.38692 98.3998 50.99059 98.3998 curveto
-50.59425 98.3998 50.21414 98.24236 49.93388 97.9621 curveto
-49.65363 97.68184 49.49619 97.30173 49.49619 96.9054 curveto
-49.49619 96.50906 49.65363 96.12895 49.93388 95.8487 curveto
-50.21414 95.56844 50.59425 95.411 50.99059 95.411 curveto
-51.38692 95.411 51.76703 95.56844 52.04729 95.8487 curveto
-52.32755 96.12895 52.48499 96.50906 52.48499 96.9054 curveto closepath fill
-newpath 131.85492 96.9054 moveto
-131.85492 97.30173 131.69748 97.68184 131.41722 97.9621 curveto
-131.13696 98.24236 130.75685 98.3998 130.36052 98.3998 curveto
-129.96419 98.3998 129.58408 98.24236 129.30382 97.9621 curveto
-129.02356 97.68184 128.86612 97.30173 128.86612 96.9054 curveto
-128.86612 96.50906 129.02356 96.12895 129.30382 95.8487 curveto
-129.58408 95.56844 129.96419 95.411 130.36052 95.411 curveto
-130.75685 95.411 131.13696 95.56844 131.41722 95.8487 curveto
-131.69748 96.12895 131.85492 96.50906 131.85492 96.9054 curveto closepath fill
-17.35184 28.21677 moveto
-(0) cmr10 9.96265 fshow
-48.24048 29.76646 moveto
-(k) cmmi10 9.96265 fshow
-73.83423 33.03938 moveto
-(n) cmmi7 6.97385 fshow
- [] 0 setdash 0 setlinecap
-newpath 73.83423 31.60748 moveto
-78.75914 31.60748 lineto stroke
-74.31104 25.68127 moveto
-(2) cmr7 6.97385 fshow
-82.16853 29.11678 moveto
-(\000) cmsy10 9.96265 fshow
-92.13113 29.11678 moveto
-(1) cmr10 9.96265 fshow
-117.86421 28.21677 moveto
-(k) cmmi10 9.96265 fshow
-125.57831 28.21677 moveto
-(+) cmr10 9.96265 fshow
-136.7364 32.13937 moveto
-(n) cmmi7 6.97385 fshow
-newpath 136.7364 30.70747 moveto
-141.66132 30.70747 lineto stroke
-137.21321 24.78127 moveto
-(2) cmr7 6.97385 fshow
-167.0135 28.21677 moveto
-(n) cmmi10 9.96265 fshow
-175.2073 28.21677 moveto
-(\000) cmsy10 9.96265 fshow
-185.1699 28.21677 moveto
-(1) cmr10 9.96265 fshow
-32.76373 7.43059 moveto
-(p) cmr10 9.96265 fshow
-38.57533 7.43059 moveto
-(osloupnost) cmr10 9.96265 fshow
-87.94583 7.43059 moveto
-(prohozen\023) cmr10 9.96265 fshow
-127.27063 7.43059 moveto
-(a) cmr10 9.96265 fshow
-135.57283 7.43059 moveto
-(separ\023) cmr10 9.96265 fshow
-158.34863 7.43059 moveto
-(atorem) cmr10 9.96265 fshow
-showpage
-%%EOF
diff --git a/old/5-addsort/sortnet.mp b/old/5-addsort/sortnet.mp
deleted file mode 100644 (file)
index 081f9f4..0000000
+++ /dev/null
@@ -1,675 +0,0 @@
-%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);
-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,z96);
-label.top(btex $x_{n-1}$ etex,z106);
-
-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);
-
-% ve skutecnosti dle znaceni 
-% by melo byt z6* ale uz 
-% obsazeno
-z815=(1.5v,2v);
-z855=(5.5v,2v);
-z895=(9.5v,2v);
-z813=(13.5v,2v);
-
-z715=(1.5v,1v);
-z755=(5.5v,1v);
-z795=(9.5v,1v);
-z713=(13.5v,1v);
-
-z9=(4v,0v);
-
-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);
-drawarrow(z815--z715);
-drawarrow(z855--z755);
-drawarrow(z895--z795);
-drawarrow(z813--z713);
-
-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);
-label.rt(btex Bitonick\'a t\v r\'\i di\v cka $B_{n}$ etex,z9);
-
-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/old/5-addsort/sortnet.mpx b/old/5-addsort/sortnet.mpx
deleted file mode 100644 (file)
index 9814d6d..0000000
+++ /dev/null
@@ -1,548 +0,0 @@
-% Written by DVItoMP, Version 0.64/color (Web2C 7.5.4)
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n0="cmr10";
-_s("a",_n0,1.00000,0.0000,0.0000,);
-setbounds _p to (0,-3.4869)--(4.9813,-3.4869)--
- (4.9813,8.4682)--(0,8.4682)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n0="cmr10";
-_s("b",_n0,1.00000,0.0000,0.0000,);
-setbounds _p to (0,-3.4869)--(5.5348,-3.4869)--
- (5.5348,8.4682)--(0,8.4682)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n0="cmr10";
-_s("min",_n0,1.00000,0.0000,0.0000,);
-setbounds _p to (0,0.0000)--(16.6045,0.0000)--
- (16.6045,6.6536)--(0,6.6536)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n0="cmr10";
-_s("max",_n0,1.00000,0.0000,0.0000,);
-setbounds _p to (0,0.0000)--(18.5416,0.0000)--
- (18.5416,4.2895)--(0,4.2895)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n0="cmr10";
-_s("x1",_n0,1.00000,0.0000,0.0000,);
-setbounds _p to (0,0.0000)--(10.2394,0.0000)--
- (10.2394,6.4204)--(0,6.4204)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n0="cmr10";
-_s("x2",_n0,1.00000,0.0000,0.0000,);
-setbounds _p to (0,0.0000)--(10.2394,0.0000)--
- (10.2394,6.4204)--(0,6.4204)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n0="cmr10";
-_s("x3",_n0,1.00000,0.0000,0.0000,);
-setbounds _p to (0,0.0000)--(10.2394,0.0000)--
- (10.2394,6.4204)--(0,6.4204)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n0="cmr10";
-_s("x4",_n0,1.00000,0.0000,0.0000,);
-setbounds _p to (0,0.0000)--(10.2394,0.0000)--
- (10.2394,6.4204)--(0,6.4204)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n0="cmr10";
-_s("x5",_n0,1.00000,0.0000,0.0000,);
-setbounds _p to (0,0.0000)--(10.2394,0.0000)--
- (10.2394,6.4204)--(0,6.4204)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n0="cmr10";
-_s("x1",_n0,1.00000,0.0000,0.0000,);
-setbounds _p to (0,0.0000)--(10.2394,0.0000)--
- (10.2394,6.4204)--(0,6.4204)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n0="cmr10";
-_s("x2",_n0,1.00000,0.0000,0.0000,);
-setbounds _p to (0,0.0000)--(10.2394,0.0000)--
- (10.2394,6.4204)--(0,6.4204)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n0="cmr10";
-_s("x3",_n0,1.00000,0.0000,0.0000,);
-setbounds _p to (0,0.0000)--(10.2394,0.0000)--
- (10.2394,6.4204)--(0,6.4204)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n0="cmr10";
-_s("x4",_n0,1.00000,0.0000,0.0000,);
-setbounds _p to (0,0.0000)--(10.2394,0.0000)--
- (10.2394,6.4204)--(0,6.4204)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n0="cmr10";
-_s("x5",_n0,1.00000,0.0000,0.0000,);
-setbounds _p to (0,0.0000)--(10.2394,0.0000)--
- (10.2394,6.4204)--(0,6.4204)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n1="cmmi10";
-_s("x",_n1,1.00000,0.0000,0.0000,);
-_n2="cmr7";
-_s("0",_n2,1.00000,5.6939,-1.4944,);
-setbounds _p to (0,-1.4944)--(10.1633,-1.4944)--
- (10.1633,4.2895)--(0,4.2895)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n1="cmmi10";
-_s("x",_n1,1.00000,0.0000,0.0000,);
-_n2="cmr7";
-_s("1",_n2,1.00000,5.6939,-1.4944,);
-setbounds _p to (0,-1.4944)--(10.1633,-1.4944)--
- (10.1633,4.2895)--(0,4.2895)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n1="cmmi10";
-_s("x",_n1,1.00000,0.0000,0.0000,);
-_n2="cmr7";
-_s("2",_n2,1.00000,5.6939,-1.4944,);
-setbounds _p to (0,-1.4944)--(10.1633,-1.4944)--
- (10.1633,4.2895)--(0,4.2895)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n1="cmmi10";
-_s(":",_n1,1.00000,0.0000,0.0000,);
-_s(":",_n1,1.00000,4.4278,0.0000,);
-_s(":",_n1,1.00000,8.8556,0.0000,);
-setbounds _p to (0,0.0000)--(13.2834,0.0000)--
- (13.2834,1.0516)--(0,1.0516)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n1="cmmi10";
-_s("x",_n1,1.00000,0.0000,0.0000,);
-_n3="cmmi7";
-_s("n",_n3,1.00000,5.6939,-1.4944,);
-_n4="cmsy7";
-_s(char0,_n4,1.00000,10.6188,-1.4944,);
-_n2="cmr7";
-_s("2",_n2,1.00000,16.8454,-1.4944,);
-setbounds _p to (0,-2.3246)--(21.3148,-2.3246)--
- (21.3148,4.2895)--(0,4.2895)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n1="cmmi10";
-_s("x",_n1,1.00000,0.0000,0.0000,);
-_n3="cmmi7";
-_s("n",_n3,1.00000,5.6939,-1.4944,);
-_n4="cmsy7";
-_s(char0,_n4,1.00000,10.6188,-1.4944,);
-_n2="cmr7";
-_s("1",_n2,1.00000,16.8454,-1.4944,);
-setbounds _p to (0,-2.3246)--(21.3148,-2.3246)--
- (21.3148,4.2895)--(0,4.2895)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n1="cmmi10";
-_s("n",_n1,1.00000,0.0000,0.0000,);
-setbounds _p to (0,0.0000)--(5.9799,0.0000)--
- (5.9799,4.2895)--(0,4.2895)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n1="cmmi10";
-_s("S",_n1,1.00000,0.0000,0.0000,);
-_n3="cmmi7";
-_s("n",_n3,1.00000,6.1090,-1.4944,);
-setbounds _p to (0,-1.4944)--(11.5320,-1.4944)--
- (11.5320,6.8078)--(0,6.8078)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n1="cmmi10";
-_s("S",_n1,1.00000,0.0000,0.0000,);
-_n5="cmmi5";
-_s("n",_n5,1.00000,7.3045,1.1829,);
-interim linecap:=0;
-vardef _r(expr _a,_w)(text _t) =
-  addto _p doublepath _a withpen pencircle scaled _w _t enddef;
-_r((7.3045,0.2491)..(11.6921,0.2491), 0.3985,);
-_n6="cmr5";
-_s("2",_n6,1.00000,7.8033,-3.8949,);
-setbounds _p to (0,-3.8949)--(13.3857,-3.8949)--
- (13.3857,6.8078)--(0,6.8078)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n1="cmmi10";
-_s("S",_n1,1.00000,0.0000,0.0000,);
-_n5="cmmi5";
-_s("n",_n5,1.00000,7.3045,1.1829,);
-interim linecap:=0;
-vardef _r(expr _a,_w)(text _t) =
-  addto _p doublepath _a withpen pencircle scaled _w _t enddef;
-_r((7.3045,0.2491)..(11.6921,0.2491), 0.3985,);
-_n6="cmr5";
-_s("2",_n6,1.00000,7.8033,-3.8949,);
-setbounds _p to (0,-3.8949)--(13.3857,-3.8949)--
- (13.3857,6.8078)--(0,6.8078)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n1="cmmi10";
-_s("S",_n1,1.00000,0.0000,0.0000,);
-_n5="cmmi5";
-_s("n",_n5,1.00000,7.3045,1.1829,);
-interim linecap:=0;
-vardef _r(expr _a,_w)(text _t) =
-  addto _p doublepath _a withpen pencircle scaled _w _t enddef;
-_r((7.3045,0.2491)..(11.6921,0.2491), 0.3985,);
-_n6="cmr5";
-_s("4",_n6,1.00000,7.8033,-3.8949,);
-setbounds _p to (0,-3.8949)--(13.3857,-3.8949)--
- (13.3857,6.8078)--(0,6.8078)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n1="cmmi10";
-_s("S",_n1,1.00000,0.0000,0.0000,);
-_n5="cmmi5";
-_s("n",_n5,1.00000,7.3045,1.1829,);
-interim linecap:=0;
-vardef _r(expr _a,_w)(text _t) =
-  addto _p doublepath _a withpen pencircle scaled _w _t enddef;
-_r((7.3045,0.2491)..(11.6921,0.2491), 0.3985,);
-_n6="cmr5";
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