From: Martin Mares <mj@ucw.cz>
Date: Mon, 2 Apr 2007 11:51:10 +0000 (+0200)
Subject: Pribyla kapitola o Strassenove algoritmu a hledani medianu.
X-Git-Url: http://mj.ucw.cz/gitweb/?a=commitdiff_plain;h=dd1eb7f63e685c7b61df29c4c52e6a698c430188;p=ads1.git

Pribyla kapitola o Strassenove algoritmu a hledani medianu.
---

diff --git a/3-strassen/3-strassen.tex b/3-strassen/3-strassen.tex
new file mode 100644
index 0000000..a500aa4
--- /dev/null
+++ b/3-strassen/3-strassen.tex
@@ -0,0 +1,179 @@
+\input ../lecnotes.tex
+\input epsf.tex
+
+\prednaska{3}{Rozdìl a panuj}{(zapsali Luká¹ Hermann, Vincent Krí¾, Oto Petøík)}
+
+\h{Násobení matic n$\times$n}
+
+Nejdøíve si pøipomeneme definici násobení dvou ètvercových matic typu $n \times n$. Platí, ¾e prvek v $i$-tém øádku a $j$-tém sloupci výsledné matice $Z$ se rovná standardnímu skalárnímu souèinu $i$-tého øádku první matice $X$ a $j$-tého sloupce druhé matice $Y$. Formálnì zapsáno:
+
+$$ Z_{ij} = \sum_{k=1}^n X_{ik} \cdot Y_{kj} $$
+
+\figure{nasobeni-matic.eps}{Násobení matic}{\epsfxsize}
+
+Algoritmus, který by násobil matice podle této definice, by mìl èasovou slo¾itost $ \Theta(n^3) $, proto¾e poèet prvkù ve výsledné matici je $n^2$ a jeden skalární souèin vektorù dimenze $n$ vy¾aduje lineární poèet operací.
+
+My se s touto èasovou slo¾itostí ov¹em nespokojíme a budeme postupovat podobnì jako pøi vylep¹ování algoritmu na násobení velkých èísel. Bez újmy na obecnosti pøedpokládejme, ¾e budeme násobit dvì matice typu $n \times n$, kde $n=2^k, k \in N$. Obì tyto matice rozdìlíme na ètvrtiny a tyto èásti postupnì oznaèíme u matice X - A, B, C a D, u matice Y - P, Q, R a S. Z definice násobení matic zjistíme, ¾e ètvrtiny výsledné matice Z mù¾eme zapsat pomocí èástí násobených matic. Levá horní ètvrtina bude odpovídat výsledku operací AP+BR, pravá horní ètvrtina bude AQ+BS, levá dolní CP+DR a zbylá CQ+DS (viz obrázek).
+
+\figure{nasobeni-matic-2.eps}{Násobení rozètvrcených matic}{\epsfxsize}
+
+\break 
+
+Pøevedli jsme tedy problém násobení ètvercových matic øádu $n$ na násobení ètvercových matic øádu ${n \over 2}$. Tímto rozdìlovánín bychom mohli pokraèovat, dokud bychom se nedostali na matice øádu 1, jejich¾ vynásobení je triviální. Dostali jsme tedy klasický algoritmus typu {\it rozdìl a panuj}. Pomohli jsme si ale nìjak ? V ka¾dém kroku provádíme 8 násobení matic polovièního øádu a navíc konstantní poèet operací na $n^2$ prvcích. Dostáváme tedy rekurentní zápis èasové slo¾itosti:
+
+$$ T(n) = 8T\left({n \over 2}\right) + \O(n^2) $$
+
+Po aplikaci Master Theoremu pak lehce dojdeme k závìru, ¾e slo¾itost je stále $O(n^3)$, tedy stejná jako pøi násobení matic z definice. Zdánlivì jsme si tedy nepomohli, ale stejnì jako tomu bylo u násobení velkých èísel, i teï mù¾eme zredukovat poèet násobení matic polovièního øádu, které nejvíce ovlivòuje èasovou slo¾itost algoritmu. Není to bohu¾el nic triviálního, a proto si radìji rovnou øekneme správné øe¹ení. Jedná se o Strassenùv algoritmus, který redukuje potøebný poèet násobení na 7, a je¹tì pøed tím, ne¾ si uká¾eme, jak funguje, doká¾eme si, jak nám to s èasovou slo¾itostí vlastnì pomù¾e:
+
+$$ T(n) = 7T\left({n \over 2}\right) + \O(n^2) \Longrightarrow \O(n^{log_2 7}) \doteq \O(n^{2.808}) $$
+
+Výsledná slo¾itost Strassenova algoritmu je tedy pøibli¾nì $\O(n^{2.808})$, co¾ je sice malé, ale pro velké matice znatelné zlep¹ení oproti algoritmu vycházejícího pøímo z definice. A nyní u¾ obrázkový dùkaz tohoto algoritmu:
+
+% zacatek slidu z prednasky
+\s{Strassenùv algoritmus: vzorce}
+
+\def\\{\noalign{\vskip 7pt}}
+
+$$
+\pmatrix{A & B \cr\\ C & D \cr}
+\cdot
+\pmatrix{P & Q \cr\\ R & S \cr}
+=
+\pmatrix{
+T_1 + T_4 - T_5 + T_7 &
+T_3 + T_5 \cr\\
+T_2 + T_4 &
+T_1 - T_2 + T_3 + T_6 \cr
+},$$
+
+kde:
+
+$$\vbox{\halign{$#$\hfil\qquad &$#$\hfil\qquad \cr
+T_1 = (A+D)\cdot(P+S)		& T_5 = (A+B)\cdot S \cr\\
+T_2 = (C+D)\cdot P		& T_6 = (C-A)\cdot (P+Q) \cr\\
+T_3 = A\cdot(Q-S)		& T_7 = (B-D)\cdot (R+S) \cr\\
+T_4 = D\cdot(R-P)					 \cr
+}}$$
+
+\medskip
+
+7 násobení místo 8 $\Rightarrow$ èasová slo¾itost $\O(n^{\log_2 7}) = \O(n^{2.808})$.
+
+\medskip
+
+[Zatím nejlep¹í výsledek: $\O(n^{2.376})$ \uv{s~opravdu velkým~$\O$.}]
+
+\break
+
+\s{Strassenùv algoritmus: dùkaz}
+
+Do ètvercù $4 \times 4$ si napí¹eme znaky $+$ nebo $-$ podle toho, jestli se pøi výpoètu dané matice pøièítá nebo odeèítá pøíslu¹ný souèin dvou matic. Øádky znamenají matice A, B, C a D a sloupce znaèí matice P, Q, R a S. Pokud se tedy v prvním øádku a prvním sloupci vyskytuje znak +, znamená to ¾e pøièteme souèin matic A a P. Nejdøíve si spoèítáme pomocné matice $T_1$ a¾ $T_7$ a z nich pak dopoèítáme, co bude na pøíslu¹ných místech ve výsledné matici.
+
+\def\bbb#1{\vbox to 10pt{\vss\hbox to 10pt{\hss\tenrm #1\hss}\vss}}
+\def\bb#1{\ifx#1.\bbb{$\cdot$}\else\bbb#1\fi}
+\def\zz#1#2#3#4{\bb#1\bb#2\bb#3\bb#4}
+\def\qq#1#2#3#4{{\offinterlineskip\vcenter{\halign{\vrule ##\vrule \cr\noalign{\hrule}\zz#1\cr\zz#2\cr\zz#3\cr\zz#4\cr\noalign{\hrule}}}}}
+
+$$
+T_1 = \qq{+..+}{....}{....}{+..+} \qquad
+T_2 = \qq{....}{....}{+...}{+...} \qquad
+T_3 = \qq{.+.-}{....}{....}{....} \qquad
+T_4 = \qq{....}{....}{....}{-.+.}
+$$
+$$
+T_5 = \qq{...+}{...+}{....}{....} \qquad
+T_6 = \qq{--..}{....}{++..}{....} \qquad
+T_7 = \qq{....}{..++}{....}{..--}
+$$
+
+\medskip
+
+\def\\{\noalign{\vskip 7pt}}
+$$
+\eqalign{
+T_1 + T_4 - T_5 + T_7 &= \qq{+...}{..+.}{....}{....} = AP + BR \cr\\
+T_3 + T_5 &= \qq{.+..}{...+}{....}{....} = AQ + BS \cr\\
+T_2 + T_4 &= \qq{....}{....}{+...}{..+.} = CP + DR \cr\\
+T_1 - T_2 + T_3 + T_6 &= \qq{....}{....}{.+..}{...+} = CQ + DS \cr
+}
+$$
+
+% konec slidu z prednasky
+
+Jak je vidìt, výsledná matice je tvoøena stejnými èástmi jako pøi obyèejném násobení. Tím je dùkaz dokonèen.
+
+\h{Hledání $k$-tého nejmen¹ího prvku (mediánu)}
+
+V tomto oddílu se budeme zabývat tím, jak co nejrychleji najít v jakékoli posloupnosti $n$ èísel $k$-tý nejmen¹í prvek popøípadì medián. Pro ty, co medián neznají, tu máme definici:
+
+\s{Definice:} Medián posloupnosti $a_1, a_2,\ldots , a_n$ je takové $m=a_i$, kde $\vert{j:a_j < m}\vert < {n \over 2}$ a $\vert{j:a_j > m}\vert < {n \over 2}$. kde $i,j \in {1, 2,\ldots , n} $
+
+Nejjednodu¹¹ím øe¹ením by urèitì bylo celou posloupnost nejdøíve setøídit a pak u¾ jednodu¹e vyhmátnout po¾adovaný prvek. To bychom dokázali v celkem slu¹ném èase $\O(n\ log(n))$, ale u¾ teï mù¾eme prozradit, ¾e to jde v èase $\O(n)$. Jak?
+
+Pou¾ívat budeme metodu {\it rozdìl a panuj}. Nìjakým zpùsobem si zvolíme jeden prvek posloupnosti, který nazveme {\it pivot}. Poté rozdìlíme zadanou posloupnost na tøi disjunktní mno¾iny. Do první dáme v¹echny prvky men¹í ne¾ pivot, do druhé stejné jako pivot a do tøetí vìt¹í ne¾ pivot. Tímto máme zaji¹tìno, ¾e prvky z první mno¾iny jsou urèitì men¹í ne¾ prvky z druhé a ty ne¾ prvky z tøetí. 
+
+O tom, jak jsou prvky uspoøádány uvnitø tìchto mno¾in ale nic nevíme. V posledním kroku na¹eho algoritmu se pak rozhodneme, na kterou mno¾inu ná¹ algoritmus rekurzivnì zavoláme. Pokud je $k$ men¹í ne¾ velikost první mno¾iny, pokraèujeme v první mno¾inì, pokud je $k$ men¹í ne¾ souèet velikostí první a druhé mno¾iny, pak hledaným prvkem je právì vybraný pivot a algoritmus skonèí, a nakonec pokud ani jedna podmínka splnìna nebyla, pustíme se do hledání v tøetí mno¾inì, ov¹em u¾ nehledáme $k$-tý nejmen¹í prvek, ale $l$-tý, kde $l$ se rovná $k$ minus velikost prvních dvou mno¾in. Pro vìt¹í názornost zapí¹eme tento algoritmus schematicky:
+
+
+{\bo Select($k,X$)}
+\algo
+\:if $\vert M\vert \leq 1 \Rightarrow $ triviální o¹etøení
+\:zvolíme pivota $p \in X$
+\:$M = \{x \in X; x < p\}, P = \{x \in X; x = p\}, V = \{x \in X; x > p\}$
+\:if $k \leq \vert M \vert \Rightarrow $ return {\bo Select($k,M$)}
+\:else if $k \leq\vert M\vert +\vert P\vert \Rightarrow $ return $p$
+\:else return \bo{Select($k-\vert M \vert -\vert P\vert , V$)}
+\endalgo
+
+Na první pohled je vidìt, ¾e se algoritmus zastaví (vstup se v¾dy zmen¹í alespoò o 1) a ¾e vydá v¾dy správný výsledek. Jak je to ov¹em s èasovou slo¾itostí? Rozdìlení do mno¾in a podmínky v druhém a tøetím kroku mají lineární slo¾itost, èemu¾ se nevyhneme. Pøi ne¹»astné volbì pivota se nám mù¾e stát, ¾e poèet rekurencí mù¾e být a¾ $n$, tedy celková slo¾itost v nejhor¹ím pøípadì je $\O(n^2)$, èím¾ jsme si oproti prostému setøídìní je¹tì pohor¹ili. Co s tím? Jak je vidìt, velmi dùle¾itá je volba pivota. Tu mù¾eme provést nìkolika zpùsoby:
+
+a) Pivot by se v setøídìné posloupnosti vyskytoval uprostøed, vstup by se tedy stále pùlil. Èasovou slo¾itost vypoèteme z rekurentního zápisu:
+
+$$ T(n) = T\left({n \over 2}\right) + \O(n) \Longrightarrow \O(n) $$
+
+To by bylo sice skvìlé, ale nalezení takového pivota je vlastnì vyøe¹ení úlohy hledání mediánu, o co¾ se sna¾íme. Tedy jsme si vùbec nepomohli.
+
+\break
+
+b) Pivot by se v setøídìné posloupnosti nacházel v prostøedních dvou ètvrtinách. Tím bychom v ka¾dém kroku urèitì odstranili mno¾inu velikosti ètvrtiny vstupu. Èasová slo¾itost tohoto øe¹ení by byla:
+
+$$ T(n) = T\left({3 \over 4}n\right) + \O(n) \Longrightarrow \O(n) $$
+
+Tímto bychom tedy také dosáhli lineární èasové slo¾itosti. Ale jak vybrat pivota tak, aby se nacházel v prostøedních dvou ètvrtinách, aby nám nám to nepokazilo lineární slo¾itost?
+
+c) Pivot bude v¾dy prostøední prvek zadané posloupnosti. V tomto pøípadì je zøejmé, ¾e existují urèité vstupy, pro které bude èasová slo¾itost $\O(n^2)$, ale v prùmìrném pøípadì, který je¹tì neumíme analyzovat, bude èasová slo¾itost $\O(n)$.
+
+d) Pivot bude náhodnì zvolený prvek zadané posloupnosti. Tímto dosáhneme nìèeho podobného jako v pøedchozím pøípadì, ale u¾ nebude existovat vstup, pro který by èasová slo¾itost byla $\O(n^2)$, jen nìkteré prùbìhy tohoto algoritmu na rùzných vstupech budou trvat takto dlouho. Prùmìrná èasová slo¾itost je tedy zase $O(n)$. 
+
+My se ale nespokojíme pouze s lineární prùmìrnou èasovou slo¾itostí. Chceme dosáhnout lineární slo¾itosti i v nejhor¹ím pøípadì. Ne¾ si ale prozradíme takové øe¹ení, rozmyslíme si, ¾e u¾ teï známe 3 èasové slo¾itosti, i kdy¾ ne v¹echny je¹tì umíme analyzovat. Je to èasová slo¾itost v nejhor¹ím pøídadì, v prùmìrném pøípadì pøes v¹echny vstupy a v prùmìrném pøípadì pøes v¹echny bìhy programu. V¹imnìte si hlavnì rozdílu posledních dvou.
+
+A nyní u¾ slibované øe¹ení s lineární nejhor¹í èasovou slo¾itostí. Vyjdeme z mo¾nosti b), tedy pokusíme se najít takového pivota, který by na pøí¹tí krok zaruèenì omezil velikost analyzované mno¾iny. Dosáhneme toho tímto algoritmem:
+
+{\bo Volba pivota}
+\algo
+\:rozdìlíme vstup na pìtice - $\O(n)$
+\:spoèteme medián ka¾dé pìtice - $\O(n)$
+\:spoèteme medián mediánù pìtic = Select(${n \over 10}$, {mediány pìtic}) a to je pivot
+\endalgo
+
+Abychom dokázali, ¾e tento algoritmus bude mít opravdu lineární èasovou slo¾itost, musíme si nejdøíve dokázat následující lemma:
+
+\s{Lemma:} V ka¾dém kroku vypadne alespoò ${3 \over 10}n$ prvkù.
+
+\proof Ten provedeme obrázkem. Pøedstavme si vybrané pìtice seøazené podle velikosti od nejvìt¹ího prvku a zakresleme je do sloupcù. Jejich mediány tedy vyplòují prostøední øadu. Tyto pìtice pak seøaïme podle velikosti jejich mediánù (nejmen¹í vlevo). Hledaný pivot se tedy nachází (pokud pøedpokládáme pro jednoduchost lichý poèet pìtic) pøesnì uprostøed. O prvcích nad pivotem a napravo od nìj mù¾eme urèitì øíct, ¾e jsou vìt¹í nebo rovny pivotu, prvky pod ním a nalevo od nìj jsou zase urèitì men¹í nebo rovny pivotu.
+
+Podle konstrukce algoritmu tedy zaruèenì vypadne jedna nebo druhá skupina prvkù. Obì tyto skupiny pøitom obsahují, jak je vidìt z obrázku, alespoò ${3 \over 10}n$ prvkù.
+
+\figure{petice.eps}{Pìtice}{125mm}
+
+Nyní u¾ se tedy mù¾eme pustit do výpoètu èasové slo¾itosti. V ka¾dém kroku funkce zavolá sama sebe nejdøíve na vstup velikosti ${n \over 5}$ a poté na vstup velikosti nejvý¹e ${7 \over 10}n$. Ostatní operace zùstávají lineární. Nejhor¹í èasovou slo¾itost tedy mù¾eme zapsat rekuretním vzorcem: 
+
+$$ T(n) = \O(n) + T\left({n \over 5}\right) + T\left({7 \over 10}n\right) $$
+
+Tento rekurentní vzorec ale zatím neumíme obecnì øe¹it. Mohli bychom ho postupnì rozepsat, ale trvalo by dlouho, ne¾ bychom z toho nìco vykoukali. Lep¹í bude rovnou dokázat, ¾e tento rekurentní vzorec implikuje lineární slo¾itost. Zkusíme tedy dosadit $T(n) = c.n$: 
+
+$$ cn = n + {cn \over 5} + {7 \over 10}cn \Leftrightarrow c = 10 $$
+
+Tímto jsme tedy dokázali, ¾e èasovná slo¾itost tohoto algoritmu v nejhor¹ím pøípadì je $\O(n)$.
+
+\bye
diff --git a/3-strassen/Makefile b/3-strassen/Makefile
new file mode 100644
index 0000000..bc44cc4
--- /dev/null
+++ b/3-strassen/Makefile
@@ -0,0 +1,3 @@
+P=3-strassen
+
+include ../Makerules
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diff --git a/3-strassen/petice.eps b/3-strassen/petice.eps
new file mode 100644
index 0000000..8faaf16
--- /dev/null
+++ b/3-strassen/petice.eps
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+@gr add add add 0 ne 0 0 0 1 @gs setcmykcolor currentcmykcolor @gr add add add
+0 ne and and and}ifelse}ifelse}bd/@PL{/LV where{pop LV 2 ge L2? not and{@np
+/Courier findfont 12 scalefont setfont 72 144 m
+(The PostScript level set in the Corel application is higher than)show 72 132 m
+(the PostScript level of this device. Change the PS Level in the Corel)show 72
+120 m(application to Level 1 by selecting the PostScript tab in the print)show
+72 108 m(dialog, and selecting Level 1 from the Compatibility drop down list.)
+show flush spg quit}if}if}bd/@BeginSysCorelDict{systemdict/Corel30Dict known
+{systemdict/Corel30Dict get exec}if systemdict/CorelLexDict known{1 systemdict
+/CorelLexDict get exec}if}bd/@EndSysCorelDict{systemdict/Corel30Dict known
+{end}if/EndCorelLexDict where{pop EndCorelLexDict}if}bd AutoFlatness{/@ifl{dup
+currentflat exch sub 10 gt{
+([Error: PathTooComplex; OffendingCommand: AnyPaintingOperator]\n)print flush
+@np exit}{currentflat 2 add setflat}ifelse}bd/@fill/fill ld/fill{currentflat{
+{@fill}stopped{@ifl}{exit}ifelse}bind loop setflat}bd/@eofill/eofill ld/eofill
+{currentflat{{@eofill}stopped{@ifl}{exit}ifelse}bind loop setflat}bd/@clip
+/clip ld/clip{currentflat{{@clip}stopped{@ifl}{exit}ifelse}bind loop setflat}
+bd/@eoclip/eoclip ld/eoclip{currentflat{{@eoclip}stopped{@ifl}{exit}ifelse}
+bind loop setflat}bd/@stroke/stroke ld/stroke{currentflat{{@stroke}stopped
+{@ifl}{exit}ifelse}bind loop setflat}bd}if L2?{/@ssa{true setstrokeadjust}bd}{
+/@ssa{}bd}ifelse/d/setdash ld/j/setlinejoin ld/J/setlinecap ld/M/setmiterlimit
+ld/w/setlinewidth ld/O{/$o xd}bd/R{/$O xd}bd/W/eoclip ld/c/curveto ld/C/c ld/l
+/lineto ld/L/l ld/rl/rlineto ld/m/moveto ld/n/newpath ld/N/newpath ld/P{11 rp}
+bd/u{}bd/U{}bd/A{pop}bd/q/@gs ld/Q/@gr ld/&{}bd/@j{@sv @np}bd/@J{@rs}bd/g{1
+exch sub/$k xd/$c 0 def/$m 0 def/$y 0 def/$t 1 def/$n _ def/$fil 0 def}bd/G{1
+sub neg/$K xd _ 1 0 0 0/$C xd/$M xd/$Y xd/$T xd/$N xd}bd/k{1 index type
+/stringtype eq{/$t xd/$n xd}{/$t 0 def/$n _ def}ifelse/$k xd/$y xd/$m xd/$c xd
+/$fil 0 def}bd/K{1 index type/stringtype eq{/$T xd/$N xd}{/$T 0 def/$N _ def}
+ifelse/$K xd/$Y xd/$M xd/$C xd}bd/x/k ld/X/K ld/sf{1 index type/stringtype eq{
+/s1t xd/s1n xd}{/s1t 0 def/s1n _ def}ifelse/s1k xd/s1y xd/s1m xd/s1c xd}bd/i{
+dup 0 ne{setflat}{pop}ifelse}bd/v{4 -2 roll 2 copy 6 -2 roll c}bd/V/v ld/y{2
+copy c}bd/Y/y ld/@w{matrix rotate/$ptm xd matrix scale $ptm dup concatmatrix
+/$ptm xd 1 eq{$ptm exch dup concatmatrix/$ptm xd}if 1 w}bd/@g{1 eq dup/$sdf xd
+{/$scp xd/$sca xd/$scf xd}if}bd/@G{1 eq dup/$SDF xd{/$SCP xd/$SCA xd/$SCF xd}
+if}bd/@D{2 index 0 eq{$dsf 3 1 roll 4 -1 roll pop}if 3 copy exch $Scra add exch
+load SetScr/$dsp xd/$dsa xd/$dsf xd}bd/$ngx{$SDF{$SCF SepMode_5 0 eq{$SCA}
+{$dsa}ifelse $SCP @ss}if}bd/@MN{2 copy le{pop}{exch pop}ifelse}bd/@MX{2 copy ge
+{pop}{exch pop}ifelse}bd/InRange{3 -1 roll @MN @MX}bd/@sqr{dup 0 rl dup 0 exch
+rl neg 0 rl @cp}bd/currentscale{1 0 dtransform matrix defaultmatrix idtransform
+dup mul exch dup mul add sqrt 0 1 dtransform matrix defaultmatrix idtransform
+dup mul exch dup mul add sqrt}bd/@unscale{}bd/wDstChck{2 1 roll dup 3 -1 roll
+eq{1 add}if}bd/@dot{dup mul exch dup mul add 1 exch sub}bd/@lin{exch pop abs 1
+exch sub}bd/cmyk2rgb{3{dup 5 -1 roll add 1 exch sub dup 0 lt{pop 0}if exch}
+repeat pop}bd/rgb2cmyk{3{1 exch sub 3 1 roll}repeat 3 copy @MN @MN 3{dup 5 -1
+roll sub neg exch}repeat}bd/rgb2g{2 index .299 mul 2 index .587 mul add 1 index
+.114 mul add 4 1 roll pop pop pop}bd/WaldoColor_5 where{pop}{/SetRgb
+/setrgbcolor ld/GetRgb/currentrgbcolor ld/SetGry/setgray ld/GetGry/currentgray
+ld/SetRgb2 systemdict/setrgbcolor get def/GetRgb2 systemdict/currentrgbcolor
+get def/SetHsb systemdict/sethsbcolor get def/GetHsb systemdict
+/currenthsbcolor get def/rgb2hsb{SetRgb2 GetHsb}bd/hsb2rgb{3 -1 roll dup floor
+sub 3 1 roll SetHsb GetRgb2}bd/setcmykcolor where{pop/LumSepsDict where{pop
+/SetCmyk_5{LumSepsDict/setcmykcolor get exec}def}{/AldusSepsDict where{pop
+/SetCmyk_5{AldusSepsDict/setcmykcolor get exec}def}{/SetCmyk_5/setcmykcolor ld
+}ifelse}ifelse}{/SetCmyk_5{cmyk2rgb SetRgb}bd}ifelse/currentcmykcolor where{
+pop/GetCmyk/currentcmykcolor ld}{/GetCmyk{GetRgb rgb2cmyk}bd}ifelse
+/setoverprint where{pop}{/setoverprint{/$op xd}bd}ifelse/currentoverprint where
+{pop}{/currentoverprint{$op}bd}ifelse/@tc_5{5 -1 roll dup 1 ge{pop}{4{dup 6 -1
+roll mul exch}repeat pop}ifelse}bd/@trp{exch pop 5 1 roll @tc_5}bd
+/setprocesscolor_5{SepMode_5 0 eq{SetCmyk_5}{0 4 $ink_5 sub index exch pop 5 1
+roll pop pop pop pop SepsColor true eq{$ink_5 3 gt{1 sub neg SetGry}{0 0 0 4
+$ink_5 roll SetCmyk_5}ifelse}{1 sub neg SetGry}ifelse}ifelse}bd
+/findcmykcustomcolor where{pop}{/findcmykcustomcolor{5 array astore}bd}ifelse
+/Corelsetcustomcolor_exists false def/setcustomcolor where{pop
+/Corelsetcustomcolor_exists true def}if CorelIsSeps true eq CorelIsInRIPSeps
+false eq and{/Corelsetcustomcolor_exists false def}if
+Corelsetcustomcolor_exists false eq{/setcustomcolor{exch aload pop SepMode_5 0
+eq{pop @tc_5 setprocesscolor_5}{CurrentInkName_5 eq{4 index}{0}ifelse 6 1 roll
+5 rp 1 sub neg SetGry}ifelse}bd}if/@scc_5{dup type/booleantype eq{dup
+currentoverprint ne{setoverprint}{pop}ifelse}{1 eq setoverprint}ifelse dup _ eq
+{pop setprocesscolor_5 pop}{findcmykcustomcolor exch setcustomcolor}ifelse
+SepMode_5 0 eq{true}{GetGry 1 eq currentoverprint and not}ifelse}bd/colorimage
+where{pop/ColorImage{colorimage}def}{/ColorImage{/ncolors xd/$multi xd $multi
+true eq{ncolors 3 eq{/daqB xd/daqG xd/daqR xd pop pop exch pop abs{daqR pop
+daqG pop daqB pop}repeat}{/daqK xd/daqY xd/daqM xd/daqC xd pop pop exch pop abs
+{daqC pop daqM pop daqY pop daqK pop}repeat}ifelse}{/dataaq xd{dataaq ncolors
+dup 3 eq{/$dat xd 0 1 $dat length 3 div 1 sub{dup 3 mul $dat 1 index get 255
+div $dat 2 index 1 add get 255 div $dat 3 index 2 add get 255 div rgb2g 255 mul
+cvi exch pop $dat 3 1 roll put}for $dat 0 $dat length 3 idiv getinterval pop}{
+4 eq{/$dat xd 0 1 $dat length 4 div 1 sub{dup 4 mul $dat 1 index get 255 div
+$dat 2 index 1 add get 255 div $dat 3 index 2 add get 255 div $dat 4 index 3
+add get 255 div cmyk2rgb rgb2g 255 mul cvi exch pop $dat 3 1 roll put}for $dat
+0 $dat length ncolors idiv getinterval}if}ifelse}image}ifelse}bd}ifelse
+/setcmykcolor{1 5 1 roll _ currentoverprint @scc_5/$ffpnt xd}bd
+/currentcmykcolor{GetCmyk}bd/setrgbcolor{rgb2cmyk setcmykcolor}bd
+/currentrgbcolor{currentcmykcolor cmyk2rgb}bd/sethsbcolor{hsb2rgb setrgbcolor}
+bd/currenthsbcolor{currentrgbcolor rgb2hsb}bd/setgray{dup dup setrgbcolor}bd
+/currentgray{currentrgbcolor rgb2g}bd/InsideDCS false def/IMAGE/image ld/image
+{InsideDCS{IMAGE}{/EPSDict where{pop SepMode_5 0 eq{IMAGE}{dup type/dicttype eq
+{dup/ImageType get 1 ne{IMAGE}{dup dup/BitsPerComponent get 8 eq exch
+/BitsPerComponent get 1 eq or currentcolorspace 0 get/DeviceGray eq and{
+CurrentInkName_5(Black)eq{IMAGE}{dup/DataSource get/TCC xd/Height get abs{TCC
+pop}repeat}ifelse}{IMAGE}ifelse}ifelse}{2 index 1 ne{CurrentInkName_5(Black)eq
+{IMAGE}{/TCC xd pop pop exch pop abs{TCC pop}repeat}ifelse}{IMAGE}ifelse}
+ifelse}ifelse}{IMAGE}ifelse}ifelse}bd}ifelse/WaldoColor_5 true def/$fm 0 def
+/wfill{1 $fm eq{fill}{eofill}ifelse}bd/@Pf{@sv SepMode_5 0 eq $Psc 0 ne or
+$ink_5 3 eq or{0 J 0 j[]0 d $t $c $m $y $k $n $o @scc_5 pop $ctm setmatrix 72
+1000 div dup matrix scale dup concat dup Bburx exch Bbury exch itransform
+ceiling cvi/Bbury xd ceiling cvi/Bburx xd Bbllx exch Bblly exch itransform
+floor cvi/Bblly xd floor cvi/Bbllx xd $Prm aload pop $Psn load exec}{1 SetGry
+wfill}ifelse @rs @np}bd/F{matrix currentmatrix $sdf{$scf $sca $scp @ss}if $fil
+1 eq{CorelPtrnDoFill}{$fil 2 eq{@ff}{$fil 3 eq{@Pf}{$fil 4 eq
+{CorelShfillDoFill}{$t $c $m $y $k $n $o @scc_5{wfill}{@np}ifelse}ifelse}
+ifelse}ifelse}ifelse $sdf{$dsf $dsa $dsp @ss}if setmatrix}bd/f{@cp F}bd/S{
+matrix currentmatrix $ctm setmatrix $SDF{$SCF $SCA $SCP @ss}if $T $C $M $Y $K
+$N $O @scc_5{matrix currentmatrix $ptm concat stroke setmatrix}{@np}ifelse $SDF
+{$dsf $dsa $dsp @ss}if setmatrix}bd/s{@cp S}bd/B{@gs F @gr S}bd/b{@cp B}bd/_E{
+5 array astore exch cvlit xd}bd/@cc{currentfile $dat readhexstring pop}bd/@sm{
+/$ctm $ctm currentmatrix def}bd/@E{/Bbury xd/Bburx xd/Bblly xd/Bbllx xd}bd/@c{
+@cp}bd/@P{/$fil 3 def/$Psn xd/$Psc xd array astore/$Prm xd}bd/tcc{@cc}def/@B{
+@gs S @gr F}bd/@b{@cp @B}bd/@sep{CurrentInkName_5(Composite)eq{/$ink_5 -1 def}
+{CurrentInkName_5(Cyan)eq{/$ink_5 0 def}{CurrentInkName_5(Magenta)eq{/$ink_5 1
+def}{CurrentInkName_5(Yellow)eq{/$ink_5 2 def}{CurrentInkName_5(Black)eq
+{/$ink_5 3 def}{/$ink_5 4 def}ifelse}ifelse}ifelse}ifelse}ifelse}bd/@whi{@gs
+-72000 dup m -72000 72000 l 72000 dup l 72000 -72000 l @cp 1 SetGry fill @gr}
+bd/@neg{[{1 exch sub}/exec cvx currenttransfer/exec cvx]cvx settransfer @whi}
+bd/deflevel 0 def/@sax{/deflevel deflevel 1 add def}bd/@eax{/deflevel deflevel
+dup 0 gt{1 sub}if def deflevel 0 gt{/eax load}{eax}ifelse}bd/eax{{exec}forall}
+bd/@rax{deflevel 0 eq{@rs @sv}if}bd systemdict/pdfmark known not{/pdfmark
+/cleartomark ld}if/wclip{1 $fm eq{clip}{eoclip}ifelse}bd
+% Copyright (c)1992-2003 Corel Corporation
+% All rights reserved.     v12 r0.0
+/z{exch findfont exch scalefont setfont}bd/ZB{9 dict dup begin 4 1 roll
+/FontType 3 def/FontMatrix xd/FontBBox xd/Encoding 256 array def 0 1 255{
+Encoding exch/.notdef put}for/CharStrings 256 dict def CharStrings/.notdef{}
+put/Metrics 256 dict def Metrics/.notdef 3 -1 roll put/BuildChar{exch dup
+/$char exch/Encoding get 3 index get def dup/Metrics get $char get aload pop
+setcachedevice begin Encoding exch get CharStrings exch get end exec}def end
+definefont pop}bd/ZBAddChar{findfont begin dup 4 1 roll dup 6 1 roll Encoding 3
+1 roll put CharStrings 3 1 roll put Metrics 3 1 roll put end}bd/Z{findfont dup
+maxlength 2 add dict exch dup{1 index/FID ne{3 index 3 1 roll put}{pop pop}
+ifelse}forall pop dup dup/Encoding get 256 array copy dup/$fe xd/Encoding exch
+put dup/Fontname 3 index put 3 -1 roll dup length 0 ne{0 exch{dup type 0 type
+eq{exch pop}{$fe exch 2 index exch put 1 add}ifelse}forall pop}if dup 256 dict
+dup/$met xd/Metrics exch put dup/FontMatrix get 0 get 1000 mul 1 exch div 3
+index length 256 eq{0 1 255{dup $fe exch get dup/.notdef eq{pop pop}{5 index 3
+-1 roll get 2 index mul $met 3 1 roll put}ifelse}for}if pop definefont pop pop
+}bd/CorelIsValidCharpath{pathbbox 3 -1 roll sub abs 0.5 ge 3 1 roll sub abs 0.5
+ge and}bd/@ftx{{currentpoint 3 -1 roll(0)dup 3 -1 roll 0 exch put dup @gs true
+charpath $ctm setmatrix CorelIsValidCharpath{@@txt}if @gr @np stringwidth pop 3
+-1 roll add exch m}forall}bd/@ft{matrix currentmatrix exch $sdf{$scf $sca $scp
+@ss}if $fil 1 eq{/@@txt/@pf ld @ftx}{$fil 2 eq{/@@txt/@ff ld @ftx}{$fil 3 eq
+{/@@txt/@Pf ld @ftx}{$fil 4 eq{/@@txt/CorelShfillDoFill ld @ftx}{$t $c $m $y $k
+$n $o @scc_5{show}{pop}ifelse}ifelse}ifelse}ifelse}ifelse $sdf{$dsf $dsa $dsp
+@ss}if setmatrix}bd/@st{matrix currentmatrix exch $SDF{$SCF $SCA $SCP @ss}if $T
+$C $M $Y $K $N $O @scc_5{{currentpoint 3 -1 roll(0)dup 3 -1 roll 0 exch put dup
+@gs true charpath $ctm setmatrix $ptm concat stroke @gr @np stringwidth pop 3
+-1 roll add exch m}forall}{pop}ifelse $SDF{$dsf $dsa $dsp @ss}if setmatrix}bd
+/@te{@ft}bd/@tr{@st}bd/@ta{dup @gs @ft @gr @st}bd/@t@a{dup @gs @st @gr @ft}bd
+/@tm{@sm concat}bd/e{/t{@te}def}bd/r{/t{@tr}def}bd/o{/t{pop}def}bd/a{/t{@ta}
+def}bd/@a{/t{@t@a}def}bd/t{@te}def/T{@np $ctm setmatrix/$ttm matrix def}bd/ddt
+{t}def/@t{/$stm $stm currentmatrix def 3 1 roll m $ttm concat ddt $stm
+setmatrix}bd/@n{/$ttm exch matrix rotate def}bd/@s{}bd/@l{}bd/_lineorientation
+0 def/_bitfont null def/_bitlobyte 0 def/_bitkey null def/_bithibyte 0 def
+% Copyright (c)1992-2003 Corel Corporation
+% All rights reserved.     v12 r0.0
+/@ii{concat 3 index 3 index m 3 index 1 index l 2 copy l 1 index 3 index l 3
+index 3 index l clip pop pop pop pop}bd/@i{@sm @gs @ii 6 index 1 ne{/$frg true
+def pop pop}{1 eq{s1t s1c s1m s1y s1k s1n $O @scc_5/$frg xd}{/$frg false def}
+ifelse 1 eq{@gs $ctm setmatrix F @gr}if}ifelse @np/$ury xd/$urx xd/$lly xd
+/$llx xd/$bts xd/$hei xd/$wid xd/$dat $wid $bts mul 8 div ceiling cvi string
+def $bkg $frg or{$SDF{$SCF $SCA $SCP @ss}if $llx $lly Tl $urx $llx sub $ury
+$lly sub scale $bkg{$t $c $m $y $k $n $o @scc_5 pop}if $wid $hei abs $bts 1 eq
+{$bkg}{$bts}ifelse[$wid 0 0 $hei neg 0 $hei 0 gt{$hei}{0}ifelse]/tcc load $bts
+1 eq{imagemask}{image}ifelse $SDF{$dsf $dsa $dsp @ss}if}{$hei abs{tcc pop}
+repeat}ifelse @gr $ctm setmatrix}bd/@I{@sm @gs @ii @np/$ury xd/$urx xd/$lly xd
+/$llx xd/$ncl xd/$bts xd/$hei xd/$wid xd $ngx $llx $lly Tl $urx $llx sub $ury
+$lly sub scale $wid $hei abs $bts[$wid 0 0 $hei neg 0 $hei 0 gt{$hei}{0}ifelse
+]$msimage false eq $ncl 1 eq or{/$dat $wid $bts mul $ncl mul 8 div ceiling cvi
+string def/@cc load false $ncl ColorImage}{$wid $bts mul 8 div ceiling cvi $ncl
+3 eq{dup dup/$dat1 exch string def/$dat2 exch string def/$dat3 exch string def
+/@cc1 load/@cc2 load/@cc3 load}{dup dup dup/$dat1 exch string def/$dat2 exch
+string def/$dat3 exch string def/$dat4 exch string def/@cc1 load/@cc2 load
+/@cc3 load/@cc4 load}ifelse true $ncl ColorImage}ifelse $SDF{$dsf $dsa $dsp
+@ss}if @gr $ctm setmatrix}bd/@cc1{currentfile $dat1 readhexstring pop}bd/@cc2{
+currentfile $dat2 readhexstring pop}bd/@cc3{currentfile $dat3 readhexstring pop
+}bd/@cc4{currentfile $dat4 readhexstring pop}bd/$msimage false def/COMP 0 def
+/MaskedImage false def L2?{/@I_2{@sm @gs @ii @np/$ury xd/$urx xd/$lly xd/$llx
+xd/$ncl xd/$bts xd/$hei xd/$wid xd/$dat $wid $bts mul $ncl mul 8 div ceiling
+cvi string def $ngx $ncl 1 eq{/DeviceGray}{$ncl 3 eq{/DeviceRGB}{/DeviceCMYK}
+ifelse}ifelse setcolorspace $llx $lly Tl $urx $llx sub $ury $lly sub scale 8
+dict begin/ImageType 1 def/Width $wid def/Height $hei abs def/BitsPerComponent
+$bts def/Decode $ncl 1 eq{[0 1]}{$ncl 3 eq{[0 1 0 1 0 1]}{[0 1 0 1 0 1 0 1]}
+ifelse}ifelse def/ImageMatrix[$wid 0 0 $hei neg 0 $hei 0 gt{$hei}{0}ifelse]def
+/DataSource currentfile/ASCII85Decode filter COMP 1 eq{/DCTDecode filter}{COMP
+2 eq{/RunLengthDecode filter}if}ifelse def currentdict end image $SDF{$dsf $dsa
+$dsp @ss}if @gr $ctm setmatrix}bd}{/@I_2{}bd}ifelse/@I_3{@sm @gs @ii @np/$ury
+xd/$urx xd/$lly xd/$llx xd/$ncl xd/$bts xd/$hei xd/$wid xd/$dat $wid $bts mul
+$ncl mul 8 div ceiling cvi string def $ngx $ncl 1 eq{/DeviceGray}{$ncl 3 eq
+{/DeviceRGB}{/DeviceCMYK}ifelse}ifelse setcolorspace $llx $lly Tl $urx $llx sub
+$ury $lly sub scale/ImageDataDict 8 dict def ImageDataDict begin/ImageType 1
+def/Width $wid def/Height $hei abs def/BitsPerComponent $bts def/Decode $ncl 1
+eq{[0 1]}{$ncl 3 eq{[0 1 0 1 0 1]}{[0 1 0 1 0 1 0 1]}ifelse}ifelse def
+/ImageMatrix[$wid 0 0 $hei neg 0 $hei 0 gt{$hei}{0}ifelse]def/DataSource
+currentfile/ASCII85Decode filter COMP 1 eq{/DCTDecode filter}{COMP 2 eq{
+/RunLengthDecode filter}if}ifelse def end/MaskedImageDict 7 dict def
+MaskedImageDict begin/ImageType 3 def/InterleaveType 3 def/MaskDict
+ImageMaskDict def/DataDict ImageDataDict def end MaskedImageDict image $SDF
+{$dsf $dsa $dsp @ss}if @gr $ctm setmatrix}bd/@SetMask{/$mbts xd/$mhei xd/$mwid
+xd/ImageMaskDict 8 dict def ImageMaskDict begin/ImageType 1 def/Width $mwid def
+/Height $mhei abs def/BitsPerComponent $mbts def/DataSource maskstream def
+/ImageMatrix[$mwid 0 0 $mhei neg 0 $mhei 0 gt{$mhei}{0}ifelse]def/Decode[1 0]
+def end}bd/@daq{dup type/arraytype eq{{}forall}if}bd/@BMP{/@cc xd UseLevel 3 eq
+MaskedImage true eq and{7 -2 roll pop pop @I_3}{12 index 1 gt UseLevel 2 eq
+UseLevel 3 eq or and{7 -2 roll pop pop @I_2}{11 index 1 eq{12 -1 roll pop @i}{
+7 -2 roll pop pop @I}ifelse}ifelse}ifelse}bd
+end
+%%EndResource
+%%EndProlog
+%%BeginSetup
+wCorel12Dict begin
+@BeginSysCorelDict
+2.6131 setmiterlimit
+1.00 setflat
+/$fst 128 def
+%%BeginResource: font TimesNewRomanPSMT-NormalItalic
+%!FontType1-1.0: TimesNewRomanPSMT-NormalItalic 001.003
%%Creator: Corel PostScript Engine
10 dict begin
/FontName /TimesNewRomanPSMT-NormalItalic def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array 0 1 255 {1 index exch /.notdef put} for
dup 77 /M put
dup 101 /e put
dup 100 /d put
dup 105 /i put
dup 225 /aacute put
dup 110 /n put
dup 121 /y put
dup 112 /p put
dup 116 /t put
dup 99 /c put
dup 72 /H put
dup 108 /l put
dup 97 /a put
dup 253 /yacute put
dup 118 /v put
dup 111 /o put
readonly def
/FontBBox {0 0 0 0} readonly def
currentdict end
currentfile eexec

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