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Mirkowska, ASD_03 Wyszukiwanie c.d.O =ALGORYTMY I STRUKTURY DANYCH jWYKAAD 03 cd. Wyszukiwanie Gra|yna Mirkowska PJWSTK& )  Plan wykBadu L Problem min-max algorytm naiwny algorytm rekurencyjny optymalno[ Drugi co do wielko[ci element Algorytm turniej K-ty co do wielko[ci element cigu Algorytm Hoare Z2ZZZ#ZZ2#  &9Minimum i maksimum :Modyfikacja algorytmu naiwnego ;0Analiza kosztu [redniego <"Koszt [redni c.d. = Koszt [redni c.d& >DMin-max rekurencyjny -- przykBad #?NMin-max rekurencyjny -- przykBad c.d. (@Algorytm min-max rekurencyjny A&Poprawno[ min-max  BHCzy algorytm min-max zatrzymuje si? %C.c.d.  stop dla min-max D:Poprawno[ algorytmu min-max  E!Koszt rekurencyjnego alg. min_max&F0Czy mo|na zrobi lepiej? Y0Z1\3]4^5_6LAlgorytm rozdzielania a8K Cwiczenia  Na wiczeniach Inna wersja algorytmu min-max oraz Analiza algorytmu naiwnego wyszukiwania elementu k-tego co do wielko[ci. Przedyskutowa znajdowanie mediany (tzn elementu, ktry powinien znalez sie na [rodku cigu gdy zostanie on uporzdkowany np. rosnco.) Mo|na zaBo|yc, |e elementy nie powtarzaj si , i |e jest ich 2k+1. Mo|na te| rozwa|y algorytm Partition (odmiana Split).Fq4$qe6"CS  [2T+U,V-W.X/GAlgorytm min-max 3 H Analiza kosztu I&Analiza poprawno[ci J0Analiza poprawno[ci c.d. `7/8  ` @ ff3Ιd332z` @ ff3Ιd332z` 999MMM` fffPP3f>?" dd@ ?4Zd@ d " @ ` n?" dd@   @@``@n?" dd@  @@``PV    @ ` ` p>> K0  F(  F   `  0PPf  c 6AminispirlB  <g   HG 1Ȝ? p`  TKliknij, aby edytowa styl tytuBu z Wzorca++  c $J ``  Kliknij, aby edytowa style tekstu z Wzorca Drugi poziom Trzeci poziom Czwarty poziom Pity poziom,  c  c $`Q p    X* 2    c $LV     Z*(2    c $@[    Z*(2 ZB   c $D"p0Z  BsZ޽h))?? @ ff3Ιd332z Notatnik&  K0 @ <(   FF       XA StationeryPP`   S 0AminispirH  <d  ?^~  ALGORYTMY I STRUKTURY DANYCHKliknij, aby edytowa styl tytuBu z WzorcaGG  c $@  3    ZKliknij, aby edytowa styl podtytuBu z Wzorca..  c $ ^#  X* 2f   c $ ^~#   Z*(2f   c $ ~#  Z*(2f Z  BsZ޽h))?? @ ff3Ιd332z| 0 P( F0E,   0Xj P   j T*   0j    j V* d  c $ ?  j  0\  @ j Kliknij, aby edytowa style wzorca tekstu Drugi poziom Trzeci poziom Czwarty poziom Pity poziom*  a  6Hj `P  j T*   6Lj `  j V* H  0޽h ? ̙3380___PPT10.0Kbp @,(  , , 0j P   j T*   , 0j    j V*   , 6\ `P  j T*   , 6 `   V*  H , 0޽h ? ̙3380___PPT10.Pby K0 0(  l  C % ^~  l  C & 3    H  0޽h ? @ ff3Ιd332zy___PPT10Y+D=' = @B +  K0  @4(  @l @ C 0G p`    @ S 8xI   " PpH @ 0޽h ? @ ff3Ιd332zy___PPT10Y+D=' = @B +.  K0 | t P  (  x  c $w p`      W3fԔ?ProblemTimes New Romanr,$D  0  <70Pj,$ 0 LDany jest cig elementw e[i] i = 1,...,n nale|cych do pewnego zbioru E liniowo uporzdkowanego przez relacj . Znalez najwikszy i najmniejszy element tego cigu.d 2o"   . W3fԔ?Specyfikacja algorytmuTimes New Roman ,$D  06  <P;` ,$ 0 Wp = { e[i] e[j] dla i j} Wk = { e[min] e[i] oraz e[max] e[i] dla wszystkich i =1,2...n}.xd 2  !4C  W3fԔ?Algorytm naiwnyTimes New Roman 6 ,$D 0  N A) ` ,$D 0 Znajdz minimum metod sekwencyjnego przegldania cigu. Znajdz maksimum, t sam metod.Y 2Y Y   <hP @p,$   0 R T(A,n)= 2n-2 2  H  0޽h ? @ ff3Ιd332z&""___PPT10!+:\D!' = @B D ' = @BA?%,( < +O%,( < +Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =%(DX' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B+checkerboard(across)*<3<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*d%(D' =-o6Bbox(out)*<3<*dD' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<*%( /%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D ' =%(D' =%(DZ' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B-randombar(horizontal)*<3<*D_' =%(D' =%(D' =A@BBBB0B%(E' =4 B`BPB`B?<*%(*/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*  %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*  D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*  ++0+ ++0+ ++0+ ++0+  +1  K0 `(  x  c $( p`   /  b)APapeteria``,$D  0 i#Algorytm naiwny nie jest optymalny!$ 2$ $  b.APapeteria J,$D  0 Je[li e[i] jest wiksze od e[max], to nie ma potrzeby porwnywa e[i] z e[min].Q 2Q QB @ BDP ,$D  0lz @ :   @ : ,$D  0   W3fԔ?NiezmiennikTimes New RomanP   <؛Q@ @:  N(" 1j<i)(e[max] e[j] e[min]e[j]),( 2& (z 0    00 ,$D  0/   <\Q0  { min :=1; max :=1; for i := 2 to n do if e[i] >e[max] then max := i else if e[i]< e[min] then min := i fi fi od; } 21 $    Zw?Algorytm 2Arial Black08b   TGOHI0 P@ R   TZGHeI: `h ,$D 09  <QP 0 ,$  0 Koszt W(n) = 2n-2 A(n) = O(n)  2 @B  HQ =@_U 0U 0 :Czy ten wynik mo|na poprawi? H  0޽h ?/@   @ ff3Ιd332z""___PPT10"+MBD!' = @B D!' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =%(D3' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bbox(out)*<3<* Du' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<*D' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<*++0+Q ++0+Q ++0+Q +3  K0   p  (  x  c $ G  <PQ `J,$  0 *Uwaga 2 Drugie porwnanie jest wykonywane tylko wtedy, gdy odpowiedz w pierwszym porwnaniu jest negatywna."m 2 d m<  <QP0 ,$ 0 (Dn - zbir wszystkich danych rozmiaru n.6) 2$'  <Qp@ ,$  0 2A(n) = S d Dn p(d)* t(d)t 2   &    NGDHP 0@ ,$D 0  <0GQ  ,$  0 aZbir wszystkich permutacji 2 `   <KQ `: ,$  0 zZaBo|ymy, |e wszystkie permutacje s jednakowo prawdopodobne.> 2> >   <DPQ @,$  0 Q p(d) = 1/n! 2     HG ;H*0 p,$D  0H  0޽h ? @ ff3Ιd332z ''___PPT10&+)7zD?%' = @B D$' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*G%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*GD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*GD' =%(D?' =%(D' =A@BBBB0B@B%(D' =1:Bvisible*o3>+B#style.visibility<*m%(D' ,=+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*mD' ,=+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*mD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* >%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* >D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* >D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<*%(-/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bbox(out)*<3<* D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* ++0+Q ++0+Q ++0+Q ++0+Q ++0+Q ++0+ Q ++0+ Q +U$  K0    (  x  c $kQ p`  Q   <lQMi,$  0 ZNiech d = a1...an bdzie dan permutacj liczb 1...n i b = b1,...bn bdzie cigiem takim, |e n_ 2   +   _&  <wQ]9,$  0 bi = liczba elementw wikszych ni| ai na lewo od ai , tzn. o mniejszych indeksach w cigu d.X_ 2 #   )N# *H  <|Q,$  0 bPrzykBad Niech d = 6752314, wtedy b = 0023353.2 22 2  <QM Y ,$  0 PObserwacja (a) Cig b jest jednoznacznie wyznaczony przez d. (b) 0 bi < iN] 2 H &UT  <̍Q Y,$  0 (Drugiego porwnania w algorytmie nie wykonujemy tylko wtedy, gdy na lewo od i-tego elementu nie byBo liczby wikszej, czyli tylko wtedy, gdy bi =0.B 2!_ &H  0޽h ? @ ff3Ιd332z___PPT10+ 2TD' = @B DP' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*_%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*_D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*_D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*_%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*_D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*_D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*2%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*2D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*2D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*]%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*]D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*]D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*++0+Q ++0+Q ++0+Q ++0+Q ++0+Q +'  K0     (  x  c $|6Q p`  Q   <7Q`` hPrawdopodobieDstwo tego, |e bi =0 wynosi 1/i, poniewa| bi przyjmuje tylko warto[ci od 0 do i-1.Bb 2  '@(  <,Qp0`P,$  0 Std prawdopodobieDstwo wykonania drugiego porwnania wynosi 1-1/i. I 2C I   W3fԔ?OstatecznieTimes New Roman` 0^ ,$D 0  <Q0` ,$  0 Zredni koszt dla danych d: t(d) = (n-1) + S i =2...n 1* p(drugie porwnanie) =BQ 2  QX  <xQ` 0 Z ,$  0 , (n-1) + S i=2...n (1-1/i) = 2n-2 - lg n +c B- 2  &$I  <LQ ` @ ,$  0 'A(n) = S d 1/n! t(d) = 2n - lg n + c 8( 2 & 6   NQA) ` ,$D 0 "1/x dx= O(lg|x|) 2& H  0޽h ? @ ff3Ιd332z___PPT10+6D' = @B Dx' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*I%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*ID' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*ID{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Q%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*QD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*QD' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bbox(out)*<3<* D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*-%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*-D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*-D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(++0+Q ++0+Q ++0+Q ++0+Q ++0+ Q +`  K0 %%(  x  c $HQ p`  Q 8  64Q"`.n,$D  0 (ZaBo|enie: n = 2 k8 2 '  H QBH,$  0 {7 4, 5, 1, 8, 33, 11, 44, 10, 7, 2, 13, 22, 3, 55, 9, 6 88 8  <Q̙ |,$  0 s+ 4, 5, 1, 8, 33, 11, 44, 10 min= max=, 2, ,  6TuL̙ ,$  0 z( 7, 2, 13, 22, 3, 55, 9, 6 min= max=") 2% )  <lL,$  0 ZDany cig: 2    6hL̙ _l ,$  0 ^4, 5, 1, 8 min= max=  2    6L̙  l ,$   0 k33, 11, 44, 10 min= max=  2    6L̙  < ,$D  0 ^7, 2, 13, 22 min= max= 2    6L̙ _O< ,$D  0 \3, 55, 9, 6 min= max= 2 2   Hz5He ,$D  02   H HB  ,$D  02  HgHLq ,$D   02  BEM ,$D  0  6L̙B ?b,$   0 I4,5 2   6L̙r ,$  0 I1,8 2   6pL̙B O b,$  0 K33,11 2   6L̙B  b,$  0 K44,10 2   6ȼL̙B  /b,$D  0 I7,2 2   6lL̙B b,$D   0 K13,22 2   6L̙r /o,$D"  0 J3,55 2   6L̙r /,$D$  0 I9,6 2 2  B* O ,$D   02  B*b o Z ,$D  02  B*  ,$D  02  Bcg / ,$D   02  Bcg  ,$D  02  Bcg O ,$D  02  Bcg j ,$D#  02  B*  ,$D!  0   <Lr l ,$  0 S1" 2  ! <|Lr l ,$  0 S8" 2  " < Lr _ l ,$  0 T10" 2  # <dLr  l ,$  0 T44" 2  $ <LO|,$  0 S1" 2  % <LL |,$  0 T44" 2 H  0޽h ? @ ff3Ιd332z___PPT10׾+CLD' = @B Dn' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* D' =%(D+' =%(D' =A@BBBB0B@B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*7%(D' K=+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*7D' K=+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*7D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*,D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*,D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*)D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*)D' =%(DH' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D#' =+4 8?nCB!#ppt_y-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* D' =+4 8?\CB#ppt_wBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(DH' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D#' =+4 8?nCB!#ppt_y-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =+4 8?\CB#ppt_wBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* D' =%(DH' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D#' =+4 8?nCB!#ppt_y-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =+4 8?\CB#ppt_wBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(DH' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D#' =+4 8?nCB!#ppt_y-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =+4 8?\CB#ppt_wBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*!D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*!D' =%(DH' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D#' =+4 8?nCB!#ppt_y-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =+4 8?\CB#ppt_wBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(DH' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D#' =+4 8?nCB!#ppt_y-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =+4 8?\CB#ppt_wBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*"D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*"D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*#D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*#D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*$D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*%D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*%D' =%(DH' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D#' =+4 8?nCB!#ppt_y-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* D' =+4 8?\CB#ppt_wBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* D' =%(DH' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D#' =+4 8?nCB!#ppt_y-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =+4 8?\CB#ppt_wBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* D' =%(DH' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D#' =+4 8?nCB!#ppt_y-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =+4 8?\CB#ppt_wBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(DH' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D#' =+4 8?nCB!#ppt_y-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =+4 8?\CB#ppt_wBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(DH' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D#' =+4 8?nCB!#ppt_y-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =+4 8?\CB#ppt_wBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(DH' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D#' =+4 8?nCB!#ppt_y-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =+4 8?\CB#ppt_wBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*++0+L ++0+L ++0+L ++0+L ++0+L ++0+L ++0+ L ++0+ L ++0+ L ++0+L ++0+L ++0+L ++0+L ++0+L ++0+L ++0+L ++0+L ++0+ L ++0+!L ++0+"L ++0+#L ++0+$L ++0+%L +  K0 %%^(  x  c $gL p`  L   HdL@$ {7 4, 5, 1, 8, 33, 11, 44, 10, 7, 2, 13, 22, 3, 55, 9, 6 88 8  6L̙0  s+ 4, 5, 1, 8, 33, 11, 44, 10 min= max=, 2, ,  6tL̙0  }+ 7, 2, 13, 22, 3, 55, 9, 6 min=2 max=55", 2( ,  <L`  ZDany cig: 2    6L̙   ^4, 5, 1, 8 min= max=  2   60L̙ P0  l33, 11, 44, 10 min= max=  2    6(kL̙   `7, 2, 13, 22 min=2 max=22 2    6hL̙ p  _3, 55, 9, 6 min=3 max=55 2 p2   Hz5H? p2   H H@ p2   HgHB j2  B   6L̙  I4,5 2   6@L̙  I1,8 2   6L̙   K33,11 2   60L̙ `  K44,10 2   6lL̙ p I7,2 2   6L̙  K13,22 2   6yL̙ p J3,55 2   6}L̙ p I9,6 2 j2  B*  j2  B* ( j2  B* X j2  Bcg0 p(j2  Bcg [ j2  Bcg  j2  Bcg0 @(j2  B*0 `(  <؂L   S1" 2    <L 0  S8" 2  ! <HL   T10" 2  " <L  `  T44" 2  # <L S1" 2  $ <aL`  T44" 2  % <Fkp ,$ 0 Zmin = 1 max = 55 2 H  0޽h ? @ ff3Ιd332znf___PPT10F.D9]+0D' = @B D' = @BA?%,( < +O%,( < +D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =-o6Bbox(out)*<3<*%+8+0+%k +,*  K0 ^V(  x  c $Tk p`  k t  <k@ obiekt function min_max (int i, j); //deklaracja i tworzenie obietktw result, lewy, prawy; { if ( i+1=j ) then if e[j] > e[i] then result.max := j; result.min := i else result.min := j; result.max :=i fi else x:= (i+j-1) div 2; lewy := min_max( i, x); prawy := min_max( x+1, j); if e[prawy.min]+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(D,' =%(D' =4@BBBB%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D#' =+4 8?nCB!#ppt_y+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =+4 8?\CB#ppt_wBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =%(D,' =%(D' =4@BBBB%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D#' =+4 8?nCB!#ppt_y+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =+4 8?\CB#ppt_wBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*+#   K0 " (   x   c $@S p`   f   <#k @ `wp = {( $m) n = 2 m, e[i] <E, > dla i=1,...n}\1 2 /   <Zk  &hwk = {("i n) e[min] e[i] , ("i n) e[i] e[max]}5 2   3   NLA J NJe|eli speBniony jest warunek pocztkowy wp , to po wykonaniu nastpujcych instrukcji {wynik := min_max(1,n); min := wynik.min; max := wynik.max;} speBniony bdzie warunek koDcowy wk. 2@)    ZwGS)?8|f 'P<]TwierdzenieArial Blackdh@H   0޽h ? @ ff3Ιd332z___PPT10i.`,%+D=' = @B +r8  K0  S(  x  c $ p`   T  bAPapeteria ,$D  0 W ka|dym wywoBaniu funkcji min_max(i,j) parametry aktualne i, j speBniaj warunek j-i+1 = 2 k dla pewnego k, 1 k mXw 2] & Q`  <8ɕp ,$  0 zDla pierwszego wywoBania wBasno[ jest oczywi[cie speBniona.> 2> >  N$G1Hm 0 P,$D 0 U j = n, i =1   <p0  ,$  0 Je|eli w cigu s tylko 2 elementy, tzn. k=1. Wtedy nie wywoBujemy powtrnie funkcji min-max.^ 2^ ^  N(G8H.  ,$D 0 Pbo i+1=j     <+ ` ,$  0 <Je|eli k>1 oraz j-i+1 = 2 k , to funkcja zostanie wywoBana dla nowych argumentw: i, x oraz x+1, j, ale wtedy,q 2 U qL   <- p ,$   0 Zx-i+1 = (i+j-1)/2  i +1 = (j-i+1)/2 = 2 k-1 ". 2)  .f   <2 z,$   0 jj-(x+1)+1 = j  ((i+j-1)/2 +1) +1 =(j-i+1)/2 = 2 k-1 ,6 21  6L   S Z wG UNd)?LematArial Black$ k !2`8,$D 0"   T6G@ 0@RU 0U 0 PMin- max   H  0޽h ?/  @ ff3Ιd332z((___PPT10'.+ D&' = @B D%' = @BA?%,( < +O%,( < +D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bbox(out)*<3<* D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*>%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*>D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*>D' =%(D' =%(DV' =A@BBB*B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B)barn(outHorizontal)*<3<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*^%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*^D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*^D' =%(D' =%(DV' =A@BBB*B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B)barn(outHorizontal)*<3<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*q%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*qD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*qD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* .%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* .D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* .D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* 6%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* 6D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* 6++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+  ++0+  +%  K0 2*(  x  c $ p`     <2b` L,$  0 DBugo[ rozwa|anego przez funkcj min-max cigu maleje w kolejnych wywoBaniach.P 2P P  N$bGHi p,$D 0 \jest 2 razy mniejsza F  <PaP`V ,$  0 bW konsekwencji proces musi si zakoDczy dla k=1.2 22 2  <d P ,$  0 @Je[li speBniony jest warunek pocztkowy wp, to algorytm min_max zatrzymuje si dla dowolnych danych.e 2e@(&  S ZwG UNd)?WniosekArial Black$ k !2`8T H  0޽h ? @ ff3Ιd332z{___PPT10[.+E4DO' = @B D ' = @BA?%,( < +O%,( < +D_' =%(D' =%(D' =A@BBBB0B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*PD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*PD' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<*%( /%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*2%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*2D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*2D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*e%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*eD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*e++0+ ++0+ ++0+ ++0+b +";  K0  W(  x  c $ m p`     <q`,$  0 Dowd przez indukcj ze wzgldu na liczb wywoBaD rekurencyjnych funkcji min_max.RR&Ix  <xZZ,$  0 Dla jednego tylko wywoBania , tzn. gdy m=1 wynik algorytmu jest poprawny.J 2J Jz  <`: ,$  0 ZaB|my, |e wynik algorytmu jest poprawny dla wewntrznych wywoBaD, czyliJ 2J J  <d` ` ,$  0 ~e[k] e[lewy.max] dla i k x oraz e[k] e[prawy.max] dla x+1 k jN 2 N  <  ,$  0 bPo wykonaniu instrukcji : 2   < ,$   0 .^Mamy : e[k] e[result.max] dla i k j 0 2&      tkAA)Papeteria p0:,$D  0 Hrif e[lewy.max]+B#style.visibility<*R%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*RD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*RD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*J%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*JD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*JD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*J%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*JD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*JD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*N%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*ND' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*ND{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D@' =%(D' =%(D' =4@BBB B%(E' =4 B`BPB`B?<*%(#/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<* D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*0D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*0D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<*%(5/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bbox(out)*<3<* ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+  ++0+  +-  K0     (  x  c $ ϑ p`   B  <Б`00 ,$  0 ZOperacja dominujca - porwnywanie elementw. 2. .s  <pՑ0 @,$ 0 9T(min_max, 2) = 1 T(min_max, 2 k)= 2* T(min_max,2 k-1) +2B: 24   t  NhݑGHO @,$D 0 zJe[li w cigu s tylko 2 elementy, to wykonujemy 1 porwnanie>> >"  TGN  ` @SU 0U 0 PMin- max   R  NGbaHi   ,$D 0 XDla k>1 wykonujemy 2 wywoBania rekurencyjnie-- -5  N$G%H  ` ,$D 0 ;Dwa dodatkowe porwnania dla ustalenia ostatecznego wyniku<< <    < 0 @,$   0 mT(min_max, n) = 3/2 n - 2 2     W3fԔ?Ostatecznie:Times New Roman 0:,$D 00   <d 0 ,$ 0 T(2k) = 3*2 k-1 - 2L 2    H  0޽h ??0 @ ff3Ιd332z  ___PPT10 .0K<_+ D' = @B D' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*.%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*.D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*.D' =%(D' =%(D>' =A@BBB B0B%(E' =4 B`BPB`B?<*%( /%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B-randombar(horizontal)*<3<*D' =%(D' =%(DV' =A@BBB*B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B)barn(outHorizontal)*<3<*D' =%(D' =%(DV' =A@BBB*B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B)barn(outHorizontal)*<3<*D' =%(D' =%(DV' =A@BBB*B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B)barn(outHorizontal)*<3<*D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bbox(out)*<3<* D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<*%($/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bbox(out)*<3<* D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* ++0+ ++0+ ++0+ ++0+ ++0+ ++0+  ++0+  +   K0 *"  (   x   c $O p`       W3fԔ?TwierdzenieTimes New Roman  Z,$D  0   <QP ,$  0 Ka|dy algorytm wyznaczajcy minimum i maksimum dowolnego cigu e[1],...e[n] elementw dowolnej liniowo uporzdkowanej przestrzeni musi wykona w przypadku pesymistycznym co najmniej [3/2 n]  2 porwnania. 2 r   NYG5Hx `  ,$D 0 jmin_max jest optymalnym rozwizaniem problemu min-max66/H   0޽h ?  @ ff3Ιd332zsk___PPT10K.s{+/ZD' = @B Dj' = @BA?%,( < +O%,( < +DR' =%(D' =%(D' =4@BBBB%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* Dk' =%(D' =%(D' =A@BBB B0B%(E' =4 B`BPB`B?<*%( /%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D%' =+4 8?pCBB+#ppt_w*sin(2.5*pi*$)CB?B*Y3>B ppt_w<* D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<* D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bbox(out)*<3<* +p+0+  ++0+  + K0 F>D(  D D <d.K@|,$  0 bProblem: Dany jest cig n-elementw pewnej przestrzeni liniowo uporzdkowanej <E, >. Znalez k-ty najwikszy element tego cigu., 2S/  D S W3yd?PrzykBadImpactP0,$D  0  D <$5K ,$  0 j$2, 4, 6, 12, 78, 45, 3, 33, 17, 22% 2% % D <9KP `",$  0 Element najwikszy = 78 element drugi co do wielko[ci = 45 3-ci najwikszy = 33 4-ty najwikszy = 22e 2e e D S >K p`  K d.k-ty najwikszy elementH D 0޽h ? @ ff3Ιd332z ___PPT10+uD!' = @B D' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*DD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*DDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*DD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*DD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*D%%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D%D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D%D' =%(D?' =%(D' =A@BBBB0B@B%(D' =1:Bvisible*o3>+B#style.visibility<*De%(D' ,=+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*DeD' ,=+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*De++0+DK ++0+DK ++0+DK + K0 QIH(  H H b$HKAPapeteria`I ,$  0  Krok1. Wyszukaj element e[max] najwikszy w[rd elementw e[i],...,e[n]; Krok 2. ZamieD elementy na pozycjach i-tej oraz max Krok 3. Powtrz postpowanie dla nastpnego i. ( 2:  H S W3yd?Koszt :Impact  ,$D 0 H <OK p  ,$  0 {5T(n) = (n-1) + (n-2) +... +(n-k) = k*n - k*(k+1)/26 26 6 H S TK p`  K P(Pierwsze rozwizanie H H 0޽h ? @ ff3Ιd332z___PPT10+&1ND5' = @B D' = @BA?%,( < +O%,( < +D' =%(D?' =%(D' =A@BBBB0B@B%(D' =1:Bvisible*o3>+B#style.visibility<*HZ%(D' ,=+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*HZD' ,=+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*HZD' =%(D?' =%(D' =A@BBBB0B@B%(D' =1:Bvisible*o3>+B#style.visibility<*HZ%(D' ,=+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*HZD' ,=+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*HZD' =%(D?' =%(D' =A@BBBB0B@B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' ,=+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*HD' ,=+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*HD' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-o6Bbox(out)*<3<*HD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*H6%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*H6D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*H6+p+0+HK ++0+HK +\" K0    Py (  P P <aK)a,$  0 &Rozdziel wszystkie elementy na wiksze od pewnego elementu M(cz[ starsza) i na mniejsze od M (cz[ mBodsza).q 2q q P TgKG(HU td,$D 0 [ M = mediana   \ P <8lKPlJ,$  0 vUmie[ median tak by oddzielaBa cze[ mBodsz od starszej.< 2< < P <oK   > 2 v P <sKyu ,$  0 Wynikiem jest mediana, je[li w cz[ci starszej jest tylko k-1 elementw.I 2I I P < xK  ,$ 0 XW przeciwnym przypadku: je[li elementw starszych jest >k-1, to szukaj k-tego elementu w cz[ci starszej. Je[li elementw starszych jest mniej ni| k-1, to szukaj elementu k-(liczba elementw starszych+1) w[rd elementw mBodszych.  2   P S zK p`  K ^6Czy mo|na to zrobi taniej? H P 0޽h ?P @ ff3Ιd332z___PPT10+ MD' = @B DJ' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Pq%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*PqD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*PqD' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<*%( /%,( < +D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-o6Bbox(out)*<3<*PD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P<%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*P<D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*P<D' =%(D?' =%(D' =A@BBBB0B@B%(D' =1:Bvisible*o3>+B#style.visibility<*PI%(D' ,=+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*PID' ,=+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*PID' =%(D' =%(DP' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-6B#blinds(vertical)*<3<*P++0+PK ++0+PK ++0+PK ++0+PK ++0+PK +e K0 T(  T T <KP s+10 , 5, 7, 9, 11, 4 , 3, 2, 12, 6, 1, 2, ,T T <lK00*,$  0 nW podanym cigu szukamy 7tego co do wielko[ci elementu8 28 8 T <TK@,$  0 Mmediana 2 B T 6Dp,$D 0nz 0 z T 0 z,$D  0 T <hK0 z `Cz[ mBodsza 2   T rhKA$Niebieska lignina0  ^5 7 9 4 3 2 6 1 2 `z @z  T @z,$D  0  T <̥K@z bCz[ starsza 2   T r$KA$Niebieska lignina N11 12 2 B  T 6D ,$D 0 T <K @,$  0 J10 2 4 T rKA$Niebieska lignina  ,$   0 ^5 7 9 4 3 2 6 1 2 B T 6D P ,$D  0 T <\K z ,$   0 Mmediana 2 bz  p  T  p ,$D   0 T <+B#style.visibility<*T8%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*T8D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*T8D' =%(D' =%(DI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =-6B)barn(outHorizontal)*<3<*TD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*TDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*TDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* T%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* TD' =%(D' =%(DI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<* T%(D' =-6B)barn(outHorizontal)*<3<* TD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*TDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*TD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*TD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*TD' =%(D' =%(DI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =-6B)barn(outHorizontal)*<3<*TD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*TDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*TDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*TD' =%(D' =%(DI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =-6B)barn(outHorizontal)*<3<*TD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*TDY' =%(D' =%(D' =A@BBBB0B%(E' =4 B`BPB`B?<*%(a/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*TD' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*TD' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<*TD' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<*T++0+TK ++0+TK ++0+TK ++0+TK ++0+TK ++0+TK ++0+TK ++0+TK +n+ K0    X (  X; X <\Op ,$ 0 ! function Hoare(l, p, k) { j := SPLIT(l, p);//lub Partition(l,p) if ( p-j = k-1) then wynik := e[j] else if p-j>k-1 then Hoare(j+1, p, k) else Hoare(l,j-1, k-(p-j+1)) fi fi } ," 2*%  29 4 X bOAPapeteria@ ,$D  0 n({e[1]...,e[j-1]}< e[j]<{e[j+1],...,e[n]}) 2) ) X b 2> > X b OAPapeteria : ,$D  0 zK-ty najwikszy znajduje si w[rd elementw e[l],... e[j-1]> 2> >B X@ 6D ,$D  0B X 6DF  ,$D  0  X <<O0p`,$ 0 6ZakBadam, |e elementy w cigu nie powtarzaj si i, |e algorytm zwraca jako wynik warto[ k-tego najwikszego elementu.x 2x x  X S O p`  O PAlgorytm Hoare& H X 0޽h ? @ ff3Ιd332z___PPT10+qD' = @B DR' = @BA?%,( < +O%,( < +D' =%(D' =%(DX' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* X%(D' =-6B+checkerboard(across)*<3<* XD' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-6B'blinds(horizontal)*<3<*XD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*XD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*XDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*XD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*XD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*XD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*XDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*XD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*XD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*XD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*X++0+XK ++0+XO ++0+XO ++0+XO ++0+ XO +P K0 _ W \ (  \ \ <h,O0  |>int function SPLIT(lewy,prawy){ mediana := e[lewy]; i := lewy+1; j := prawy; bool := true; while (bool) do while (j>lewy andif e[j] mediana ) do j := j-1 od; while (i<j andif e[i] < mediana) do i := i+1 od; if (i<j) then swap(e[i], e[j]); i := i+1; j := j-1; else bool := false fi od; swap(e[lewy],e[j]); return j; }, 2O ,-(    \ b 3OAPapeteriap 0* ,$D 0 @ ("k, lewy< k <j) e[k] < e[j] ("k, j < k prawy) e[j] e[k]A 2  A \ S GO p`  O =Algorytm rozdzielania " \ HIOz@U 0U 0 Npowrt H \ 0޽h ? @ ff3Ιd332zy___PPT10Y+^0D' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(DG' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =+v4 ,0B*O3> B<*\+8+0+\O +x  K0 <`(  <r < S 0j p`  j  < <4jM* Bint Partition (l, p: int){ x := e[p]; i := l - 1; j := l; while (j < p) do if e[j] d" x then swap(e[i+1], e[j]); i := i+1; fi; j := j+1 od ; swap(e[i+1],e[p]); result := i+1; return result }"V 2sR   '/;= H < 0޽h ? @ ff3Ιd332z80___PPT10.(`etA K0    dZ (  dh d <pO pz,$  0 vKa|dy element jest co najwy|ej raz porwnywany z median. "< 2; < d <O ,$  0 [Koszt algorytmu SPLIT 2 F d <O0pP,$ 0 >Czyli T(SPLIT, n ) = n-1 = Q(n)8 2333  d <O0 ,$  0 ^W( n,k) = n-1 +W( n-1,k) 2  d <DO ,$  0 ~8Czyli W(n,k)= k*n  k(k+1)/2 2  d <+B#style.visibility<*d%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*dD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*dD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*d<%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*d<D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*d<D' =%(D' =%(D@' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*d %(D' =-o6Bbox(out)*<3<*d D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*dD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*dD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*dD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*dD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*dR%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*dRD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*dRD' =%(D' =%(DI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<* d%(D' =-6B)barn(outHorizontal)*<3<* dD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* d%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* dD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* dD' =%(D' =%(DI' =4@BBB*B%(D' =1:Bvisible*o3>+B#style.visibility<* d%(D' =-6B)barn(outHorizontal)*<3<* dD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* d%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* dD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* dDa' =%(D ' =%(D' =A@BBBB0B%(E' =4 B`BPB`B?<*%(>/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* d%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* dD' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* dD' =+4 8?RCB?BCB#ppt_xB*Y3>B ppt_x<* dD' =+4 8?RCB?BCB#ppt_yB*Y3>B ppt_y<* d++0+df ++0+df ++0+df ++0+df ++0+df ++0+df ++0+ df ++0+ df ++0+ df +  K0 8:(  8r 8 S  f p`  f  8 S f`` f "PpH 8 0޽h ? @ ff3Ιd332z___PPT10i.0b+D=' = @B +# K0    Lp (  L L <${j000* hZakBadam, |e elementy w cigu e nie powtarzaj si.5 25 5b L <j  <,$  0 { x := 1; while x k do max := x; for i := x+1 to n do if e[i] > e[max] then max := i fi od; swap(e[x], e[max]); x := x+1; od; wynik := e[k] }, 2Q D- L bTjAPapeteria@ ,$D  0 g!e[1]>...>e[x-1] >{e[x],...,e[n] }" 2" "Z L bjAPapeteria P  ,$D  0 .e[max] {e[x],...,e[n]}6 2 - L bjAPapeteria @  ,$D  0 g!e[1]>...>e[x-1] >{e[x],...,e[n] }" 2" " L  W3fԔ?NiezmiennikTimes New Roman,$D  0  L S xj p`  j HAlgorytm naiwny- kty co do wielko[ci&H L 0޽h ? @ ff3Ιd332z___PPT10+ D' = @B DC' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*LD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*LD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*LD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*LD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*LD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*LD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*LD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*LDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*LD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*L++0+Lj ++0+Lj ++0+Lj ++0+Lj + K0 1 ) P0(  0& 0 <f  <Problem Dany jest cig r|nych elementw e[1],...,e[n] pewnej przestrzeni liniowo uporzdkowanej < E, >. Znalez drugi co do wielko[ci element tego cigu. N 2G6 B 0 <o,$  0 WP = { e[i] e[j] dla i j , n>0}, WK = {1 wynik n, e[j] e[wynik] < e[max] dla j=1,2,...,n } s 2  ! s9 0 T|GHe  p ,$D 0 ;e[wynik] = maximum({e[1]...,e[n]} - maximum{e[1],...,e[n]})<< < 0 < 0J,$  0 FAlgorytm naiwny : 1. Znalez element maksymalny. 2. Usun go z rozwa|aD . 3. Znalez element maksymalny w pozostaBym zbiorze. 2  0 S ,7'   d<Drugi najwikszy element cigu H 0 0޽h ?0 @ ff3Ιd332zWO___PPT10/+G Dk' = @B D&' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0'%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*0'D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*0'D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0's%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*0'sD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*0'sD' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-o6Bbox(out)*<3<*0D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*0D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*0++0+0 ++0+0 ++0+0 +4 K0 ` 4%(  4$ 4 <̜ B{ i := 2; max :=1; while i n do if e[i] > e[max] then max := i fi; i := i+1; od; pom := e[1]; e[1] := e[max]; e[max] := pom; i:=3; wynik := 2; while i n do if e[i] > e[wynik] then wynik := i fi; i := i+1; od; } B" 2%X #$ " 4 bpAPapeteria0,$D  0 YSwap(e[1], e[max]); 2 B 4@ BD ,$D  0 4 S W3yd?KosztImpact p ,$D 0 4 <h0 `P ,$  0 R T(n) = 2n -3 2  | 4 B ]},$D  0 BMax: = 1; for ( i =2; i<= n; i++){ if (e[i]>e[max]) {max:=i;} }CCZ& W  4 Bx W ,$D  0 Gwynik := 2; for ( i =3; i<= n; i++){ if (e[i]>e[wynik]){wynik:=i;} } HG&(6  4 bĶAPapeteria/R,$D  0 pe[max] := maksimum(1,n);8  4 bLAPapeteria  ,$D  0 re[wynik] := maksimum(2,n);  4 S  p`   7Algorytm naiwny H 4 0޽h ? @ ff3Ιd332z%%___PPT10%+ QD]$' = @B D$' = @BA?%,( < +O%,( < +Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*4D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*4D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*4D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*4D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*4D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*4D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* 4%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* 4D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* 4D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* 4%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* 4D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* 4D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* 4%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* 4D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* 4D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<*%()/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-o6Bbox(out)*<3<*4D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4 %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*4 D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*4 +P+0+4 ++0+4 ++0+4 ++0+ 4 ++0+ 4 ++0+ 4 +c K0 p+,8A(  8. 8 S Z"wG UNd)?Metoda - TurniejArial Black$ k !2P0  8 <׌`p > 2  8 <ٌz_f 8Metoda polega na porwnywaniu ssiednich elementw cigu. Elementy wiksze (wygrywajce) przechodz do nastpnej  rundy .z 2z zz  @ 8  @,$D  02 8 BDߌ  G2 2 8 B   G3 2  8 B4  G4 2  8 B @ G6 2  8 B 0 G1 2  8 B@ P G8 2  8 BP  G7 2 8 B$p   G5 vz `  8  ,$D  0rB 8 BD`  rB 8B BDP 2 8 B   ,$D  0 G4 2 8 Bk ,$D  0 G5 2 8 Bk @ ,$D  0 G8 2 8 Bk   ,$D   0 G6 vz `  8 P ,$D  0rB 8 BD`  rB 8B BDP vz `  8 ` ,$D  0rB 8 BD`  rB 8B BDP vz `  8 P ,$D  0rB 8 BD`  rB 8B BDP vz P  8 P  ,$D   0rB  8 BDP 0 rB !8B BD0 P vz  `0  "8  `0 ,$D   0rB #8 BD 0 rB $8B BD `0 2 %8 B|k p ,$D   0 G5 2 &8 Bk0 p ,$D   0 G8 2 '8 Bk 0 ,$D 0 G8 vz   (8  ,$D  0rB )8 BD 0 rB *8B BDP  +8 BG` P@@U 0U 0 ,8 S k,  k ^6Czy mo|na to zrobi lepiej? H 8 0޽h ? @ ff3Ιd332zpEhE___PPT10HE+1 DC' = @B D_C' = @BA?%,( < +O%,( < +Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*8D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*8Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*8D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*8D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*8D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*8Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*8D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*8D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*8D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*8Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*8D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*8D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*8D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*8Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*8D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*8D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*8D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*8Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*8D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*8D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%8%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*%8D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*%8Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*"8%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*"8D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*"8D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*&8%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*&8D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*&8Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(8%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(8D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(8D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<*%(Y/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*'8%(D' =-o6Bbox(out)*<3<*'8++0+8k ++0+8k ++0+8k ++0+8k ++0+%8k ++0+&8k ++0+'8k + K0 (  <(  <h < <`,$  0 Ka|dy element, z wyjtkiem maksymalnego przegraB co najwy|ej raz.B 2B Bt < <0@6,$  0 Element drugi co do wielko[ci przegraB jedynie z elementem maksymalnym.H 2H HD < < 04 ,$  0 XW[rd elementw, ktre graBy z najwikszym!- 2-߳ - < F W3fԔ?Gdzie szuka elementu "drugiego" ?Times New Roman0 `z ,$D 0" < BG p `pP @,U 0U 0 < <   `Por. przykBad 2   < S  p`   6Analiza metody H < 0޽h ? @ ff3Ιd332zJB___PPT10"+vQD^' = @B D' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*<B%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*<BD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*<BD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*<H%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*<HD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*<HD' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =-o6Bbox(out)*<3<*<D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*<-%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*<-D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*<-++0+< ++0+< ++0+< +; K0    @ (  @  @ <0,$  0 i#Krok1. Zbudowanie drzewa turnieju.$ 2$ $  @ <0 *,$  0 $ZaB|my, |e n= 2k., 2  F @ <0,$  0 `Krok 2. Wybranie elementu drugiego najwikszego.1 21 1C @  GʿHA R|owa lignina ,$D 0 [lg n -1@ @ <K  : ,$  0 RA ile elementw przegraBo z najwikszym?*) * @ <8K  ,$  0 r,Tyle, ile byBo  rund ! 2 B  @ BDP   ,$D  0@  @  z KG*Hh=A Papeteria0`,$D 0 bn -1 porwnaD   @ S W3yd?Razem :Impact` 0 ,$D   0  @ <K`  ,$   0 qT(n)= n + lg n -2 2&   @ S HK p`  K ?Koszt algorytmu Turniej H @ 0޽h ?/ @ @ @ ff3Ιd332z..___PPT10.+LD,' = @B Dw,' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*@D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*@D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*@$%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*@$D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*@$D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* @%(D' =-o6Bbox(out)*<3<* @D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*@1%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*@1D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*@1D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*@*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*@*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*@*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* @%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* @D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* @D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*@D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*@D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<*%(./%,( < +D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =-o6Bbox(out)*<3<*@Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* @%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* @D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* @D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* @%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* @D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* @++0+@ ++0+@K ++0+@K ++0+@K ++0+@K ++0+@K ++0+ @K ++0+ @K +  K0   0""$ (  $x $ c $)f p`  f  $ <d,f`P~ ,$  0 \Idea algorytmu: Porwnaj parami elementy danego cigu. Mniejszy z ka|dej pary wstaw do cigu M, a wikszy z ka|dej pary do cigu W.& 2߳t \ $ </f` ` ,$ 0 vDany cig: 4, 5, 1, 8, 34, 11, 44, 4, 7, 2, < 2< < $ 63fP ,$D  0 P Mniejsze:  2   $ 65fP ,$D  0 XWiksze:  2   $ <(f 0 I,$D  0 G  2  $ <?f ,$  0 G4 2   $ <LCfp ,$  0 G5 2   $ <Gf 0 @ I,$D   0 G  2   $ 6hKf 0 I,$D  0 G  2   $ <Hf ,$   0 G1 2   $ <Rfp ,$   0 G8 2  $ 6|Vf 0 @ I,$D   0 G  2  $ <4Zf @ y,$D   0 G  2  $ <XUfP ,$  0 H11 2  $ <afp ,$  0 H34 2  $ 6tef @ y,$D  0 G  2  $ <[f ,$D  0 G  2  $ <0lfp ,$  0 G4 2  $ <lpfp0 ,$  0 H44 2  $ 6tf I,$D  0 G  2  $ <mf ,$D  0 G  2  $ <D{f ,$  0 G2 2  $ <fp ,$  0 G7 2  $ 6Hf  ,$D  0 G  2  $ <Xf @ 0 ,$  0 VWybierz minimum  2  $ <f `0 ,$  0 WWybierz maksimum  2 r $ BHH  @ ,$D 0r $ BHH @@ ,$D  0 $ <hf @,$  0 s-4, 5, 1, 8, 34, 11, 44, 4, 7, 2,. 2. .  $ < `p@U 0U 0 !$ 6df ,$D 0 Uwaga. Nie musimy przenosi samych elementw do tablic M i W. Wystarczy zapamita pozycje tych elementw.k 2k k  "$ H$fG,@WU 0U 0 Kdrugi H $ 0޽h ? @ ff3Ιd332z___PPT10`+GpςD$' = @B Dߊ' = @BA?%,( < +O%,( < +D' =%(D?' =%(D' =A@BBBB0B@B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' ,=+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*$D' ,=+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*$D' =%(D?' =%(D' =A@BBBB0B@B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' ,=+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*$D' ,=+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*$D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*$<%(D' =-o6Bbox(out)*<3<*$<D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*$D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*$D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*$D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*$D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* $%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* $D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* $D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* $%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* $D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* $D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* $%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* $D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* $D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* $%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* $D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* $D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* $%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* $D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* $D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*$D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*$D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*$D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*$D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*$D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*$D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*$D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*$D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*$D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*$D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*$D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*$D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*$D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*$Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*$D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-o6Bbox(out)*<3<*$D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*$D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$.%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*$.D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*$.D' =%(D' =%(D<' =A@BBB B0B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*!$%(D' =-6B+checkerboard(across)*<3<*!$+ +0+$f ++0+$f ++0+$f ++0+$f ++0+$f ++0+$f ++0+ $f ++0+ $f ++0+ $f ++0+ $f ++0+ $f ++0+$f ++0+$f ++0+$f ++0+$f ++0+$f ++0+$f ++0+$f ++0+$f ++0+$f ++0+$f ++0+$f ++0+$f ++0+$f ++0+$f ++0+$f ++0+$f ++0+!$f +,  K0   P ,z (  ,x , c $f p`  f  , HfA)@ :,$D 0 { // min-max i:=1; k:=1; while i< n do if e[i] >e[i+1] then w[k] :=i; m[k] :=i+1 else w[k] :=i+1; m[k] :=i fi i := i+2; k := k+1; od; min := m[1]; max :=w[1]; for i := 2 to k-1 do if e[m[i]] < e[min] then min := m[i] fi; if e[w[i]]> e[max] then max := w[i] fi; od; } 2*#33f  , , <f`P 0f,$  0 FOperacja dominujca : porwnywanie.$ 2$ $ , <Dj `,$  0 z41szy krok = n/2 porwnaD  2 . , <j0 ,$  0 HZaB|my, |e n jest liczb parzyst. % 2% %. , <8j @ ,$   0 HWybranie minimum = n/2 - 1 porwnaD% 2% %< , <f p ,$   0 LWybranie maksimum = n/2 - 1 porwnaD. ' 2& '  , <D j ,$  0 x2Razem = 3n/2 -2 porwnaD 2 H , 0޽h ? @ ff3Ιd332zK C ___PPT10# +C/{nD' = @B D:' = @BA?%,( < +O%,( < +D' =%(D' =%(DV' =A@BBB*B0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =-6B)barn(outHorizontal)*<3<*,D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,$%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*,$D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*,$D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*,%D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*,%D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*,D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*,D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*,%D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*,%D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,'%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*,'D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*,'D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<*%((/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* ,%(D' =-o6Bbox(out)*<3<* ,++0+,j ++0+,j ++0+,j ++0+,j ++0+,j ++0+,j ++0+ ,j +6/  K0   ` 0P (  0x 0 c $%j p`  j < 0 HD'jA) ` ,$D 0 { //rozdzielanie i:=1; k:=1; while i< n do if e[i] >e[i+1] then w[k] :=i; m[k] :=i+1 else w[k] :=i+1; m[k] :=i fi i := i+2; k := k+1; od; } 2$''& 0 <.j ,$  0 Dla ka|dego j=1...k-1 istnieje takie u<k, |e e[m[j]] e[w[u]]2A 27 A 0 <5jp J ,$   0 Dla ka|dego j=1...k-1 istnieje takie u<k, |e e[w[j]] e[m[u]]2A 27 A 0 <:j p,$  0 V{aden indeks w tablicy w[i] nie mo|e wskazywa minimum cigu e. {aden indeks w tablicy m[i] nie mo|e wskazywa maksimum cigu e." 2  0  W3fԔ?NiezmiennikTimes New Roman` ,$D  0B 0@ BD @ 0,$D  0"  0 <@ P0 @TU 0U 0H 0 0޽h ? @ ff3Ιd332z$#___PPT10#+;|KD"' = @B D"' = @BA?%,( < +O%,( < +D' =%(D' =%(DP' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-6B#blinds(vertical)*<3<*0Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*0D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*0Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*0D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*0D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0/%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*0/D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*0/D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0/A%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*0/AD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*0/AD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0/%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*0/D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*0/D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0/A%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*0/AD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*0/AD' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<*%(./%,( < +D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-o6Bbox(out)*<3<*0++0+0j ++0+0j ++0+0j ++0+0j +Y  K0 p4"(  4x 4 c $PSj p`  j  4 6Tj ,$D 0  min := m[1]; max :=w[1]; for i := 2 to k-1 do if e[m[i]] e[max] then max := w[i] fi; od; 28   4 <]j``p,$  0 W[rd indeksw m[1], ...m[k] istnieje taki indeks min, |e e[min] e[i] dla wszystkich i =1,...n W[rd indeksw m[1], ...m[k] istnieje taki indeks max, |e e[i] e[max] dla wszystkich i =1,...nL 2Ac#  4 <fj p,$  0 @ e[min] e[i] dla wszystkich i =1,...n e[i] e[max] dla wszystkich i =1,...n`Y 2*" Y" 4 <  @UU 0U 0H 4 0޽h ? @ ff3Ιd332zWO___PPT10/+Dk' = @B D&' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4c%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*4cD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*4cD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4c%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*4cD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*4cD' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-o6Bbox(out)*<3<*4D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4Y%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*4YD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*4Y++0+4j ++0+4j ++0+4j +] K0 (  &'`(  ` ` 6[O 0 ^  ` 0^Of@ m)10, 5, 9, 8, 14, 7, 12, 3, 11** *^B ` 6D^B ` 6D ` 0TOp Ei 2  ` 0|fO  Ej 2   ` 0,$D  0  ` 0@ ,$D  0  ` 0`0 ,$D  0  ` < lkOp `,$D  0 @   ` <nOHE<Ifp,$D  0 Mmediana  ` 0rOf P @,$D 0 F14  ` 0vOf @,$D 0 E3 z   `  ,$D   0 ` 0{Ofp `  m)10, 5, 9, 8, 3, 7, 12, 14, 11** *fB ` 6D0  fB ` 6D 0  ` 0O@ `  Ei 2  ` 0}O  Ej 2  ` <OHE<If  ,$D  0 Mmediana z @  `  ,$D   0 ` 0@Of@   n*10, 5, 9, 8, 3, 7, 12, 14, 11++ +fB ` 6D fB ` 6D0 0  ` 0qO  Ei 2  ` 0̌O  Ej 2  ` 0Of@ P ,$D   0 O7  ` 0|Of@ P P ,$D  0 F10 z  P  ` 0 ,$D  0  ` BOA)P  N i < j 2  fB !` 6D 0B "`@ 6D  ,$D   0z P 0`  #` p ` ,$D   0 $` B(OA)p 0`  N i > j 2  fB %`B 6DP p R &` HOGw `0@U 0U 0 Ppowrt   '` S O p`  O 4Jak dziaBa algorytm SPLIT?@H ` 0޽h ? @ ff3Ιd332z$CC___PPT10B+"$DA' = @B DKA' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* `%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* `D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* `D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* `%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* `D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* `Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* `%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* `D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* `Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* `%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* `D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* `Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* `%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* `D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* `Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*`%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*`D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*`D_' =%(D' =%(D' =A@BBBB0B%(E' =4 B`BPB`B?<*%()/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*`%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*`D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*`D_' =%(D' =%(D' =A@BBBB0B%(E' =4 B`BPB`B?<*%(//%,( < +D' =1:Bvisible*o3>+B#style.visibility<*`%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*`D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*`Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*`%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*`D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*`Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*"`%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*"`D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*"`Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*`%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*`D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*`Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#`%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*#`D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*#`D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*`%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*`D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*`D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*`%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*`D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*`+P+0+ `O ++0+ `O ++0+`O ++0+`O ++0+`O ++0+`O +G 0 @(d(  (^ ( S    j ( c $j @  j ZWspomnij o r|nych wersjach tego algorytmu: Np.. bez pomocniczych tablic ale z zamianami , z innym ustawieniem par, np.:pierwszy-ostatni itd&H ( 0޽h ? 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