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Mirkowska, ASD 04 SortowanieO = ALGORYTMY I STRUKTURY DANYCH WYKAAD 04 Problem Sortowania Gra|yna Mirkowska PJWSTK, 2003/2004 B  Plan wykBadu SformuBowanie problemu Sortowanie przez porwnywanie elementw Sortowanie przez wstawianie Sortowanie przez selekcj Operacja scalania cigw uporzdkowanych Sortowanie przez scalanie Szybkie sortowanie&?? & ,SformuBowanie problemu ' 2Sortowanie przez selekcj ( "Diagram przepBywu )Koszt algorytmu *Sortowanie przez wstawianie ,Schemat algorytmu +Algorytm Insertion_sort& -LPoprawno[ sortowania przez wstawianie '.!Koszt sortowania przez wstawianie "0Sortowanie przez scalanie 5PrzykBad 4Operacja scalania 6Algorytm scalania 8#Specyfikacja procedury scal(k,x,l) $3Sortowanie przez scalanie 9Koszt algorytmu Merge_Sort& < Szybkie sortowanie =!$PrzykBad wykonania >"Sortowanie szybkie - algorytm ?#Najgorszy przypadek @$Koszt [redni /A@  ` @ ff3Ιd332z` @ ff3Ιd332z` 999MMM` fffPP3f>?" dd@ ?4Zd@ d " @ ` n?" dd@   @@``@n?" dd@  @@``PV    @ ` ` p>> K0 f^ (  F   `  0PPf  c 6AminispirlB  <g   H@ 1Ȝ? p`  TKliknij, aby edytowa styl tytuBu z Wzorca++  c $8C P  Kliknij, aby edytowa style tekstu z Wzorca Drugi poziom Trzeci poziom Czwarty poziom Pity poziom,  c  c $I p    X* 2    c $N     Z*(2    c $R    Z*(2 Z  BsZ޽h))?? @ ff3Ιd332z Notatnik&  K0 0 <(   FF       XA StationeryPP`   S 0AminispirH  <x ?^~  ALGORYTMY I STRUKTURY DANYCHKliknij, aby edytowa styl tytuBu z WzorcaGG  c $T 3    ZKliknij, aby edytowa styl podtytuBu z Wzorca..  c $ ^#  X* 2f   c $첎 ^~#   Z*(2f   c $ ~#  Z*(2f Z  BsZ޽h))?? @ ff3Ιd332z| 0 P( F0E,   0l P    T*   0X     V* d  c $ ?    0   @  Kliknij, aby edytowa style wzorca tekstu Drugi poziom Trzeci poziom Czwarty poziom Pity poziom*  a  6 `P   T*   6Ŏ `   V* H  0޽h ? ̙3380___PPT10.iqh @(    0Ѓ P    R*    0     T*    6Č `P   R*    6 `   T*  H  0޽h ? ̙3380___PPT10.`jq K0  (  l  C Ž ^~  l  C Î 3    H  0޽h ? @ ff3Ιd332zl  K0 `@(  @l @ C T p`    @ S P`v   " PpXB @ 0DpH @ 0޽h ? @ ff3Ιd332z0  K0 NFp ( (7 l  C LK p`     0(M,$  0 zDany jest cig e elementw e1,e2,... , en nale|cych do pewnej liniowo uporzdkowanej przestrzeni <E, >.\k 2=&(A<  0U,$  0 Znalez tak permutacj i1, i2,... ,in liczb 1,..., n aby ei1ei2 ... ein P 2@#&N  0O` L ,$  0 Znalez tak funkcj r|nowarto[ciow i  na f : {1,2...,n} {e1,e2,... , en }, |e dla ka|dego i<n, f(i) f(i+1).x} 2E &T'  VZAPapeteria  ,$D  0 W5 3 1 6 7 2 8 2   VL_APapeteria ,$D  0 W1 2 3 5 6 7 8 2   3 W3yd?lubImpact` ( ,$D  0l    ,$D  0   6Gx: ,$D  0`B   0D @` @  0dd @  S 3 6 2 1 4 5 7 2   0h T S 1 2 3 4 5 6 7 2 XB  0DpH  0޽h ? @ ff3Ιd332z!!___PPT10!+D ' = @B D< ' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*k%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*kD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*kD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*PD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*PDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*}%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*}D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*}D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* ++0+ ++0+ ++0+ ++0+ ++0+  +D4  K0 YQ(  l  C h} p`     3 ZwG UNd)?Selection_sortArial Black$ k !20h:  04pz,$  0 Metoda Sortowanie odbywa si w n-1 przebiegach. W i-tym przebiegu szukamy i-tego najmniejszego elementu.j 2j j  00* ,$  0 Algorytm { for i := 1 to n-1 do min := i; j := i+1; while j < n+1 do if e[j] < e[min] then min := j fi od; swap(e[i],e[min]); od } 2I  Rl      ,$D  0`B  0D  `B  0D  `B   0D P `B   0D P fr   6P P p `B   0D `P `B B 0D `P   0 ` ,$D  0  69YHP 0,$D  0l  R   R ,$D  0fB B 6D    3 W3yd?minImpact R   0| 0 ,$  0 p*Odcinek uporzdkowany 2 l @@   @ @ ,$D  0fB  6D0 0   3 W3yd?e[i]Impact@@  XB  0DpH  0޽h ? @ ff3Ιd332z{$s$___PPT10S$+mӤD#' = @B DJ#' = @BA?%,( < +O%,( < +D{' =%(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<*%(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' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %()/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(./%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*++0+ ++0+ ++0+ +(  K0 UM( x l  C ( p`     s *̙`0  Ji := 1 2 2  0X̙@0  Ii < n ^B  6D @^B  6D   s *\̙Pp  `Znajdz x takie, |e x = minimum( e[i],..., e[n])1 21 1^B  6D   s *೒̙   x2ZamieD miejscami x i e[i] 2   s *̙`   Li := i+1 2     B(CDEFL@0((@  `    BChDE4F @hhP @        0 0   Qtak 2 ^B  6D0   0   Inie 2 V  0Œ0 0*,$  0 He[1] ... e[i-1] {e[i],...,e[n]}J% 2 %  0˒` @ 0Z ,$  0 &x {e[i],...,e[n]}, 2 ~  04Ғ `z,$  0 Ze[1] ... e[i-1] e[i] {e[i+1],...,e[n]}`. 2 .N  0lےpj,$  0 He[1] ... e[i-1] {e[i],...,e[n]}B% 2 %  ~ W3fԔ?NiezmienikTimes New RomanP,$D  0XB  0DpH  0޽h ? @ ff3Ιd332z___PPT10+,^D' = @B D' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*%D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*%D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*.%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*.D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*.D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*%D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*%D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*++0+ ++0+ ++0+ ++0+ + #  K0     (  l  C  p`   &  s *̙p,$D  0 Twierdzenie Algorytm Selection_sort jest poprawnym rozwizaniem problemu sortowania. W dowolnej strukturze danych, ktrej elementy s liniowo uporzdkowane przez relacj , algorytm zatrzymuje si dla dowolnych danych i daje w wyniku cig uporzdkowany niemalejco.8  2 a&"  3 Z"wG UNd)?Koszt algorytmuArial Black$ k !2L  0D  ,$  0 rA. Je[li operacj dominujc jest porwnywanie elementw:: 2: :`  0 @ ,$  0 dT(n) = n-1 + n-2 + ... +2 + 1 = n(n-1)/2 = Q(n2)83 2- 3@  0  ` ,$  0 fB. Je[li operacj dominujc jest zamiana elementw4 24 4"  0L ,$  0 8T(n) = 1*(n-1) = n-1 = Q(n) & 2 XB  0DpH  0޽h ? @ ff3Ιd332z___PPT10{+5 DG' = @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@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$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*3%(D' =-o6Bbox(out)*<3<*3D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*4D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*4D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*++0+ ++0+ ++0+ ++0+ ++0+ +u  K0 jb#(  C  l  C T p`     0T 0 *,$  0 2Sortowanie odbywa si w n -1 przebiegach. W i-tym przebiegu elementy na pozycjach 1...(i-1) s ju| uporzdkowane, a wstawiamy i-ty element przepychajc go do przodu na wBa[ciwe miejsce, tak by stworzyB wraz z innymi cig uporzdkowany dBugo[ci i. 2   3 ZwG UNd)?MetodaArial Black$ k !2`8 @ 0  ,$D  0 @ 0  ,$D  0 @ 0  ,$D  0l p  p  ,$D  0`B  0DP P fr  6K    0(  p*Odcinek uporzdkowany 2 `B  0D %P P `B   0Df  fB  B 6D   0-` p W  R i-ty element 2    0l2 0  EX 2   3 W3yd?PrzykBadImpact@ ,$D  0  P|6APapeteria  ,$D  0 c4 8 5 3 9 6  2   V:APapeteria0 * ,$D   0 a4 5 8 3 9 6  2   s * ,̙0  ,$D   0 S5" 2 7  V<A %PapeteriaP J ,$D   0 }4 5 3 8 9 6 4 2 % 7  VGAPapeteriap j ,$D  0 }4 3 5 8 9 6 4 2 % +  V+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<*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' =%(DH' =A@BBB B0B%(E' =1B B`BPB1:Bhidden*3>+B#style.visibility= `B<*!E' =4 B`BPB`B?<* %(//%,( < +D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =-o6Bbox(out)*<3<*!D' =%(D' =%(DH' =A@BBB B0B%(E' =1B B`BPB1:Bhidden*3>+B#style.visibility= `B<*"E' =4 B`BPB`B?<* %(4/%,( < +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<* ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+  ++0+! ++0+" +&  K0 @( K))3 l  C ` p`     s *  Pi := 2 2 2  6L  P p Oi < n+1   6l   :Umie[ e[i] w[rd elementw e[1],e[2],...e[i-1], przesuwajc elementy wiksze o jedno miejsce w prawo, tak by cig i-pierwszych elementw byB uporzdkowany 2  6P0 `  Mstart   0`p 0  STak  2   0P 0 GNie 2 ^B  6D @ @ ^B  6D@ @ ^B  6Dp@ @ ^B  6D @ @ P   6 P 0 p Ri := i+1   ^B  6DP 2  6,P@ Lstop ^B  6D 0 ^B  6D ^B  6D |  V ˜APapeteriaP ,$D  0 :e[1]... e[i-1] , i < n+2X 2 d  V@ʘAPapeteriap `W ,$D 0 8e[ 1]... e[i-1] , i<n +1 B 2   VTјAPapeteria0 p  ,$D 0 Fe[ 1]... e[i-1] e[i] , i<n +1 X$ 2 $  VژAPapeteria 0 T,$D 0 Fe[ 1]... e[i-2] e[i-1], i<n +2 X$ 2 $  3 W3yd?NiezmiennikImpact`@,$D   0XB  0DpH  0޽h ? @ ff3Ιd332zvn___PPT10N+mDR' = @B D ' = @BA?%,( < +O%,( < +D' =%(D' =%(DR' =A@BBB%B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B%barn(outVertical)*<3<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*DR' =%(D' =%(D' =4@BBBB%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*++0+ ++0+ ++0+ ++0+ +3  K0  p( x l  C  p`   4  3 Z4wG UNd)?Analiza ptli wewntrznejArial Black$ k !2  0 f{for i := 2 to n do j := i; pom := e[i]; while ( j>1 andif e[j-1]> pom ) do e[j] := e[j-1]; j := j-1 od; e[j] := pom od} 2  IR  VAPapeteria ,$D   0 &e[ 1]... e[i-1] B 2    VTAPapeteriap W,$D  0 De[ 1]... e[j-1] pom=e[i], j=iX# 2 &  V APapeteria  ,$D  0 Ppom< e[ j+1]... e[i], pom <e[j-1] B) 2 4 v  VAPapeteria  ,$D  0 <Pom < e[ j]e[j+1]... e[i] B 2 l  V APapeteria  ,$D  0 2pom< e[ j+1]... e[i] B 2   Vp"APapeteria@ @@' ,$D  0 *\e[ 1] ... e[j-1] pom< e[ j+1]... e[i]/ 2 &  VT.APapeteria  ,$D  0 `e[ 1] ... e[j-1] e[j] < e[ j+1]... e[i]1 2 1XB  0DpH  0޽h ? @ ff3Ιd332zK#C#___PPT10##+NEpD!' = @B D:!' = @BA?%,( < +O%,( < +D_' =%(D' =%(D' =A@BBBB0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* ++0+ ++0+ ++0+  ++0+  ++0+  ++0+  ++0+  +B  K0 j(  l  C O p`     BPA) DAlgorytm sortowania przez wstawianie poprawnie rozwizuje problem sortowania w ka|dej liniowo uporzdkowanej strukturze danych. 2   3 ZwG UNd)?TwierdzenieArial Black$ k !2h/   BV3fA)@  Algorytm sortowania przez wstawianie jest, w ka|dej liniowo uporzdkowanej strukturze danych, caBkowicie poprawny ze wzgldu na warunek pocztkowy n>0 i warunek koDcowy ("1<i n) e[i-1] e[i] . 8 2   XB  0DpH  0޽h ? @ ff3Ιd332zx   K0     (  l  C U p`     0`0@P FW(n) = Si=2...n (koszt maksymalny ptli wewntrznej) = Si=2...n (i-1) = n(n-1)/2 = O(n2)fY 2 (  Y(  0i `m  dA(n) = Si=2...n (koszt [redni ptli wewntrznej)= 43 2 $ 3  nDoGHdA Papeteria`P `0  Element pom z prawdopodobieDstwem 1/i mo|e zaj dowoln z pozycji od 1 do i.QQ&F(  VhuAdPapeteria`p ZOperacja dominujca - porwnywanie elementw.. 2.f3 .  0yP P0  = Si=2...n (Sj=1...i j*(1/i)) = Si=2...n (1/i)(i (i+1))/2 = (n+1)(n+2)/4 - 1.5 = (1/4)n2 +O(n)a 2    / a  3 8W3yd?Algorytm sortuje w miejscu.Impact p6XB  0DpH  0޽h ? @ ff3Ιd332z  K0 ,(  l  C  p`   8  0@  (1) Dzielimy zadanie posortowania caBego cigu na dwa podzadania: posortowania jego lewej i prawej poBowy. (2) Gdy obie cz[ci tworz ju| cigi uporzdkowane, wtedy scalamy je otrzymujc rozwizanie. 2   3 8W3yd?Idea : dziel i zwyci|aj !ImpactpXXB  0DpH  0޽h ? @ ff3Ιd332zr  K0 # *( <<3 l  C , p`     VAPapeteria`@ < 16 5 12 4 10 6 1 13 15 7 1 14 9 3 8 11= 2= =  VAPapeteria ,$D  0 c 16 5 12 4 10 6 1 13 2    V@APapeteria 0,$D  0 ` 15 7 1 14 9 3 8 11 2    VAPapeteria0PP,$D  0 R 16 5 12 4 2    VAPapeteria0 P,$D  0 Q 10 6 1 13 2   VtAPapeteria,$D 0 J 16 5 2   VAPapeteria@,$D 0 J 12 4 2   f䱛A$Niebieska lignina  ,$D 0 J 5 16 2   f8A$Niebieska lignina @ ,$D 0 J 4 12 2   fܹA$Niebieska lignina  ,$D   0 R 4 5 12 16 2 XB  0Dp  VԽAPapeteria ,$D   0 I10 6 2   fA$Niebieska lignina  ,$D   0 I6 10 2 !  lTśA$Niebieska lignina  ,$D  0 Q 1 6 10 13 2    \LɛAPapeteria 0 ,$D   0 I1 13 2   l̛A$Niebieska lignina 0 ,$D  0 I1 13 2 -  fЛA$Niebieska lignina  ,$D  0 c 1 4 5 6 10 12 13 16 2     VԛAPapeteria0 P,$D  0 P 15 7 1 14 2     V؛APapeteria0@P,$D  0 P 9 3 8 11 2   ! 3 W3yd?i.t.d.Impact h ,$D  0' " fܛA$Niebieska lignina P  ,$D  0 ]1 3 7 8 9 11 14 15 2 3 # VAPapeteria  ,$D  0 y5 1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 6 26 6H  0޽h ? @ ff3Ιd332zuYmY___PPT10MY+3D U' = @B DT' = @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<*%(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@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<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<* 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<*#+(+0+ ++0+ ++0+ ++0+ ++0+ ++0+  ++0+  ++0+  ++0+  ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+  ++0+" ++0+# +  K0 ** :E-*(  l  C  p`     0D`0,$  0 @Dane s dwa cigi X i Y, uporzdkowane niemalejco, x1,...xn i y1,...ym. Utworzy cig e=e1,...e n+m zBo|ony z elementw obu cigw uporzdkowany niemalejco. 26      :   0*`,$  0 tPWp = {n>0 m>0, x1... xn i y1... ym }) 2    N   0<0,$  0 JvWk = { e1... en+m , ("in+m)($ j)( ei= xj lub ei = yj)}< 2           VPAPapeteria p,$D  0 E1 2 l       ,$D  0  V  ,$D  0B  <D>0 0 ,$D  0  3 ZwG UNd)?PrzykBadArial Black$ k !2pB @ BDp P ,$D  0hl     0 ,$D  0rB  BD>  rB  BD>  B  BD@  ,$D   0  VaAPapeteria p,$D   0 E2 2 jz       ` ,$D  0lB ! <D>  lB " <D>   # VfAPapeteria  ,$D  0 E3 2 B $@ 6Dp P ,$D   0 % VkAPapeteria ,$D  0 E4 2 jz    &   ,$D 0lB ' <D>  lB ( <D>  B ) 6D  ,$D  0jz    *  `  ,$D 0lB + <D>  lB , <D>  B - 6D  ,$D  0 . VqAPapeteria 0 ,$D  0 E5 2 jz    /   ,$D 0lB 0 <D>  lB 1 <D>  B 2@ 6Dp  ,$D  0 3 VPvAPapeteria 0,$D  0 E6 2 jz    4  p ,$D 0lB 5 <D>  lB 6 <D>  B 7@ 6D@ @P ,$D  0 8 V{APapeteria ,$D  0 E7 2 hl @0 ; p0 ,$D 0rB 9 BDԔ@0rB : BD@0B ? 6D @pP ,$D  0 @ VAPapeteria @,$D  0 E8 2 vz @0 A   ,$D 0rB B BDԔ@0rB C BD@0B D 6D 0 P ,$D  0 E VąAPapeteria ,$D  0 E9 2 H  0޽h ? @ ff3Ιd332zۀӀ___PPT10+ aD}' = @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@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<*D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(Di' =4@BBB%B%(E' =1B B`BPB1:Bhidden*3>+B#style.visibility= `B<*D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B%barn(outVertical)*<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' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(Di' =4@BBB%B%(E' =1B B`BPB1:Bhidden*3>+B#style.visibility= `B<*D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B%barn(outVertical)*<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' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D' =%(D' =%(Di' =4@BBB%B%(E' =1B B`BPB1:Bhidden*3>+B#style.visibility= `B<*$D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-6B%barn(outVertical)*<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' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*&%(D' =-o6Bdissolve*<3<*&D' =%(D' =%(Di' =4@BBB%B%(E' =1B B`BPB1:Bhidden*3>+B#style.visibility= `B<*)D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =-6B%barn(outVertical)*<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' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<**%(D' =-o6Bdissolve*<3<**D' =%(D' =%(Di' =4@BBB%B%(E' =1B B`BPB1:Bhidden*3>+B#style.visibility= `B<*-D' =1:Bvisible*o3>+B#style.visibility<*-%(D' =-6B%barn(outVertical)*<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' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*/%(D' =-o6Bdissolve*<3<*/D' =%(D' =%(Di' =4@BBB%B%(E' =1B B`BPB1:Bhidden*3>+B#style.visibility= `B<*2D' =1:Bvisible*o3>+B#style.visibility<*2%(D' =-6B%barn(outVertical)*<3<*2D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*3%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*3D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*3D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-o6Bdissolve*<3<*4D' =%(D' =%(Di' =4@BBB%B%(E' =1B B`BPB1:Bhidden*3>+B#style.visibility= `B<*7D' =1:Bvisible*o3>+B#style.visibility<*7%(D' =-6B%barn(outVertical)*<3<*7D{' =%(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<*8D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*;%(D' =-o6Bdissolve*<3<*;D' =%(D' =%(Di' =4@BBB%B%(E' =1B B`BPB1:Bhidden*3>+B#style.visibility= `B<*?D' =1:Bvisible*o3>+B#style.visibility<*?%(D' =-6B%barn(outVertical)*<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' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*A%(D' =-o6Bdissolve*<3<*AD' =%(D' =%(Di' =4@BBB%B%(E' =1B B`BPB1:Bhidden*3>+B#style.visibility= `B<*DD' =1:Bvisible*o3>+B#style.visibility<*D%(D' =-6B%barn(outVertical)*<3<*DD' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*E%(D' =-o6Bdissolve*<3<*E++0+ ++0+ ++0+ ++0+  ++0+ ++0+# ++0+% ++0+. ++0+3 ++0+8 ++0+@ ++0+E +$  K0   0 v (  l  C  p`   XB  0Dpv  0` z,$D 0 {i:=1; j := 1; k :=1, while (i n and j m) do if x[i]< y[j] then e[k] := x[i]; i := i +1 else e[k] := y[j]; j := j +1 fi; k := k+1; od; B 2# A=  0hΝ ,$  0 "if ( j > m) then for i := i to n do e[k] := x[i]; k := k+1 od Else for j := j to m do e[k] := y[j]; k := k+1 od} 2N CDT  VtԝAPapeteria `,$D  0 ({k= i+j-1, e[1]... e[k-1] i wszystkie elementy x[1],...,x[i-1] oraz y[1],...,y[j-1] zostaBy ju| umieszczone na pozycjach od 1 do k-1 w cigu e .}B 2   3  W3yd?NiezmiennikGaramond` ,$D 0  3 W3yd?Koszt :Impact @(,$D  0  0ݝ ` ,$D 0 JO(n+m) 2 H  0޽h ? @ ff3Ιd332z[S___PPT103+`cD7' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(DR' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B%slide(fromBottom)*<3<*D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %( /%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D' =%(D' =%(D/' =4@BBB B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B+checkerboard(across)*<3<* D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bbox(out)*<3<* D ' =%(D' =%(DZ' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B-randombar(horizontal)*<3<* ++0+ ++0+ ++0+ ++0+  +   K0 @2( HdH l  C h p`     bܰA @Papeteria0   Wersja procedury scal (lewy,x,prawy) u|yta w algorytmie Sortowania przez scalanie ma nastpujc specyfikacj Wp = {lewy x prawy e[lewy] e[lewy+1] & e[x] e[x+1] e[x+2] & e[prawy]} Wk = {e[lewy] e[lewy+1] & e[x] e[x+1] & e[prawy] }  2j    @q]7  0 07 *Twierdzenie (*) Procedura scal(k, x,l) zastosowana do dowolnego cigu e[1],...,e[n] jest caBkowicie poprawna ze wzgldu na podan wy|ej specyfikacj.> 2  , H  0޽h ? @ ff3Ιd332z5  K0 P y(  l  C P p`   Y  bAA)Papeteria0J  procedure MS(lewy, prawy : integer); begin if prawy>lewy then x := (lewy+ prawy) div 2; MS(lewy,x); MS(x+1, prawy); scal (lewy, x, prawy) fi end MS; 2 {XB  0Dp  3 ZwG UNd)?Merge-sortArial Black$ k !2h   f()A$Niebieska lignina0 ,$D  0 Je[li lewy = prawy, to jest tylko jeden element w naszym cigu.B 2B B  fd.A$Niebieska ligninapP `*,$D  0 n W tym wywoBaniu rozwa|amy lew  poBow danego cigu *8 25 8  f03A$Niebieska lignina@ : ,$D  0 ^ Z zaBo|enia indukcyjnego : e[lewy] ... e[x]L0 2" 0  f:A$Niebieska lignina `,$D  0 n W tym wywoBaniu rozwa|amy praw  poBow danego cigu  8 27 8B  @ s *D`0 ,$D  0B  s *D  ,$D  0  f?A$Niebieska lignina  ,$D  0 b Z zaBo|enia indukcyjnego :e[x+1] ... e[prawy]L2 2   2  fhHA$Niebieska ligninap j ,$D  0 R Na mocy Tw (*) : e[lewy] ... e[prawy]L* 2  & H  0޽h ? @ ff3Ιd332zJ%B%___PPT10"%+44D#' = @B Dq#' = @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' =%(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@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+ ++0+ ++0+  ++0+  ++0+ ++0+ +I5  K0   `t ( ?$I l  C d p`   XB  0Dp  0Lf ,$  0 ~ZaB|my, |e n = 2 p. wtedy T(n) = T(n/2) + T(n/2) + cn T(1) = 0,@ 2 -&4 (  0Lm 0@,$  0 $T(n) = Q(n lg (n))& 2 & T  0r  ,$  0 8Po podstawieniu mamy T(2 0) = 0 T(2 p) = 2 T(2 p-1) +c nX9 2      9  3 W3yd?Ostatecznie : ImpactP  ,$D  0J  0|  ,$  0 T(2 p ) = 2 T(2 p-1) + c2 p = 2(2T(2 p-2) +c 2 p-1) + c2 p = 2 2 T(2p-2 ) + c2 p + c2 p = ...= 2 p T(2 0 ) + cp 2 p = Q( n lg n)P 2                  @n H  0޽h ? @ ff3Ιd332z**___PPT10*+,0D)' = @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@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*7%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*7D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*7D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*7@%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*7@D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*7@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<* 9%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* 9D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* 9D_' =%(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<*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+ ++0+ ++0+ ++0+ +  K0 L(  x  c $ p`     0l00 ,$  0 TKrok 1. Rozdzieli elementy danego cigu e1,e2,... ,en na dwie cz[ci wzgldem pewnego ustalonego elementu, tzw. mediany, tak by a lewo od niego znajdowaBy si elementy mniejsze, a na prawo elementy wiksze. f 2';Z&5`  0  ,$  0 ~Krok 3. Posortowa elementy znajdujce si na prawo od mediany.@ 2< @L  0츤P pp ,$  0 `Krok 2. Posortowa elementy na lewo od mediany.(1 2, 1  0䫤  LMetoda : 2    3 ,W3yd?Dziel i zwyci|ajImpactPptXB  0DpH  0޽h ? @ ff3Ιd332z{s___PPT10S+<D' = @B DJ' = @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<*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<*@++0+ ++0+ ++0+ +x  K0 `(  x  c $0ͤ p`     0xΤ 0@ `10 5 7 8 14 12 3 4 1 2 2  6Ԕ @p,$D  0.  0DҤppz,$  0 TRozdzielanie ze wzgldu na wybran median+ 2+ +  V֤APapeteria ,$D  0 V5 7 8 1 4 3  2   VۤAPapeteria,$D  0 J12 14 2   0ߤ ,$ 0 F10 2 B  0 : ,$  0 hStosujemy rekurencyjnie t sam zasad do obu cz[ci5 25 52  6Ԕ@ ,$D  0  V@APapeteria ,$D  0 K3 4 1 2   V APapeteria ,$D   0 I 8 7 2   0 ,$   0 E5 2 2  6Ԕ ,$D   0  V APapeteriap  ,$D   0 F 1 2   VAPapeteriap  ,$D   0 F 4 2   0hp  ,$  0 E3 2 2  6Ԕp ,$D  0  VAPapeteriap ,$D  0 E7 2   0p  ,$  0 E8 2 2  6Ԕ ,$D  0  VAPapeteriaP ,$D  0 F14 2   0  ,$  0 F12 2   3 W3yd?Ostatecznie :Impact  ,$D  0%  V APapeteria ,$D  0 k' 1 3 4 5 7 8 10 12 14( 2( (  <Gp`@ U 0U 0 [splitXB  0DpH  0޽h ? @ ff3Ιd332zT`L`___PPT10,`+ "D\' = @B DK\' = @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#' =%(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%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* 5%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* 5D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* 5D' =%(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$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(8/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bbox(out)*<3<* Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(O/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(T/%,( < +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?<* %(e/%,( < +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$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(p/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(u/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(z/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*++0+ ++0+ ++0+ ++0+ ++0+  ++0+  ++0+  ++0+  ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ +m  K0 h `   (  x  c $K p`     3 ZwG UNd)?Quick_sortArial Black$ k !2h   00OPJ BDane: n>0, cig e[1],..., e[n]." 2" "  0Qp@   J procedure QS(lewy, prawy) {if (prawy > lewy) then Split (lewy, prawy,j); QS(lewy,j-1); QS(j+1,prawy); fi } 2 B  VXAPapeteria0 * ,$D 0 d{e[lewy],..., e[j-1]}< e[j] {e[j+1],...,e[prawy]}B3 2 3  V`APapeteria  ,$D 0 fe[lewy] ... e[j-1] e[j] {e[j+1],...,e[prawy]}n4 2 4  VdjAPapeteria ` ,$D 0 2he[lewy] ... e[j-1] e[j] e[j+1] ... e[prawy]5 2  5&  VwAPapeteriap pj,$D  0 llewy prawy" 2 XB  0Dp  0Mg@ U 0U 0H  0޽h ? @ ff3Ιd332z___PPT10m+Eg-Dq' = @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@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*++0+ ++0+ ++0+ ++0+  +D.  K0 E =   (  x  c $ p`   z  0$  ,$  0 Koszt Operacji rozdzielania SPLIT dla n elementowego cigu wynosi n-1 porwnaD.Q 2Q Q  0T  ,$  0 BKoszt pesymistyczny algorytmu Quicksort mierzony liczb porwnaD wynosi : W(n) = Q(n 2) B\ 2T & 5  3 ZwG UNd)?TwierdzenieArial Black$ k !2@@   3 DW3yd?Jak wyglda najgorszy przypadek ?Impact@h,$D  0  0p,$  0 a1 2 3 4 5 6 7 8 9  2   0 ,$  0 Je[li Split jako median wybiera zawsze pierwszy element, to w wyniku rozdzielenia, jedna cz[  mBodsza bdzie pusta , a druga  starsza bdzie zawieraBa o jeden element mniej ni| w poprzednim kroku. 2&  0Ȩ `,$  0 \W(n) = (n-1) +W(n-1)= S i=2...n (i-1) = Q(n2)j/ 2    /  3 W3yd?Ostatecznie :Impact ,$D  0XB  0DpH  0޽h ? @ ff3Ιd332z  ___PPT10g +) D3' = @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' =%(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@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' =%(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+ ++0+ ++0+  +2  K0 (  x  c $æ p`     3 ZwG UNd)?TwierdzenieArial Black$ k !2`8  0(Ʀ `,$  0 HKoszt [redni algorytmu QuickSort, mierzony liczb porwnaD, wynosi A(n) = Q(n lg n) 4Y 2N @ 2R  0Φ 7 ,$  0 BA(n) = (n-1) + S j=1...n (1/n (A(j-1) + A(n-j))) A(0) = 0LC 2 * Cz P   P ,$D  0`B  0D  0֦P E1 2    0ڦ En 2    0ݦ `  Ej 2 `2   0 p fr   6 fr   6   0,P0  Gj-1 2   04   Gn-j 2 2  n$GHUA Papeteria0P,$D  0 `ZakBadamy, |e wszystkie ustawienia elementw w cigu i ka|dy podziaB w wyniku Split s jednakowo prawdopodobne. .t(2t&Q5  0  0` ,$  0 -A(0) = 0 A(n) = (n-1) + S j=1...n-1 A(j) 2/n D. 2   .  3 W3yd?Ostatecznie :Impact p ,$D  0  0 ,$  0 z A(n)=cn lg n 2 4Xr  0 @XB  0DpH  0޽h ? @ ff3Ιd332z  ___PPT10g +{m`D3' = @B D' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Y%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*YD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*YDn' =%(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$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*C%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*CD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*CD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*.%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*.D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*.D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %("/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* ++0+ ++0+ ++0+ ++0+ ++0+ +T$ 0  (  ^  S      c $x @   Przyjmijmy, |e A(i) <= cilg i = c lg e i ln i. 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