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Mirkowska, ASD_14 Progamowanie dynamiczneO = ALGORYTMY I STRUKTURY DANYCH WYKAAD 14 Techniki konstruowania algorytmw  programowanie dynamiczne Gra|yna Mirkowska PJWSTK, 2004&c  Plan wykBadu Paradygmat programowania dynamicznego PrzykBady Mno|enie macierzy NajdBu|szy wsplny podcig Problem plecakowy F0ZBZ0? r;!$Mno|enie macierzy <"Prosty Algorytm? >#%Paradygmat programowania dynamicznego &G%Liczymy minimalne koszty I& Algorytm MCM K(PrzykBad (n=6) L)6W jakiej kolejno[ci mno|y? )4NajdBu|szy wsplny podcig ( Analiza zadania 7 Algorytm2 3<Jak wyliczy dBugo[ nwp(X,Y)?&4Algorytm 5 Wypisanie nwp& O*Problem plecakowy Q+Algorytm   S,PrzykBad /PRTUV  ` @ ff3Ιd332z` @ ff3Ιd332z` 999MMM` fffPP3f>?" dd@ ?4Zd@ d " @ ` n?" dd@   @@``@n?" dd@  @@``PV    @ ` ` p>>  K0  D(  F   `  0PPf  c 6AminispirlB  <g   H| 1Ȝ? p`  TKliknij, aby edytowa styl tytuBu z Wzorca++  c $$ p  Kliknij, aby edytowa style tekstu z Wzorca Drugi poziom Trzeci poziom Czwarty poziom Pity poziom,  c  c $ p    X* 2    c $     Z*(2    c $    Z*(2 XB   0Dd$Z  BsZ޽h))?? @ ff3Ιd332z Notatnik&  K0 @ <(   FF       XA StationeryPP`   S 0AminispirH  <pm ?^~  ALGORYTMY I STRUKTURY DANYCHKliknij, aby edytowa styl tytuBu z WzorcaGG  c $Lp 3    ZKliknij, aby edytowa styl podtytuBu z Wzorca..  c $u ^#  X* 2f   c $x ^~#   Z*(2f   c $} ~#  Z*(2f Z  BsZ޽h))?? @ ff3Ιd332z| 0 P( F0E,   0 P    T*   0p     V* d  c $ ?    0  @  Kliknij, aby edytowa style wzorca tekstu Drugi poziom Trzeci poziom Czwarty poziom Pity poziom*  a  6 `P   T*   6H `   V* H  0޽h ? ̙3380___PPT10.BGh `(    0< P    R*    0@     T*    6F `P   R*    6+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@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bbox(out)*<3<* D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bbox(out)*<3<* D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* I%(D' =-o6Bbox(out)*<3<* ID' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(#/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bbox(out)*<3<* D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %((/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-o6Bbox(out)*<3<*L+P+0+ ++0+ ++0+ ++0+ ++0+ ++0+  +w1  K0     (  l  C l; p`     0< : ,$D 0 &Rozwa|y wszystkie mo|liwe ustawienia nawiasw w cigu, a potem wykona mno|enia zgodnie z ustawieniem nawiasw. 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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' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-o6Bbox(out)*<3<*(D' =%(D' =%(DP' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B#blinds(vertical)*<3<*D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %($/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bbox(out)*<3<* DY' =%(D' =%(D' =A@BBBB0B%(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?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<* +P+0+ ++0+ ++0+ ++0+ ++0+  ++0+  +(!  K0     (  l  C  p`   v  V́APapeteria0,$D  0 n1. Scharakteryzowa struktur rozwizania optymalnego.8 26 8  0 ,$  0 Rozwizanie optymalne zawiera w sobie optymalne rozwizania dla podproblemw.N 2N&@   VAPapeteria / ,$D  0 D2. Zdefiniowa rekurencyjnie warto[ rozwizania optymalnego, jako funkcj rozwizaD optymalnych dla podproblemw.s 2s&e *  BGHY:  3 ,$D  0 W naszym przykBadzie: dla pewnego k, policzymy najpierw optymalne ustawienie nawiasw dla iloczynu (A1,..., Ak) potem optymalne ustawienie nawiasw dla iloczynu (Ak+1..,An), a potem dodamy do tego koszt mno|enia 2 macierzy.v 2h  5  4&or  VDAPapeteriaf d`,$D  0 3. Skonstruowa rozwizanie problemu na bazie wyliczonych wielko[ci.F 2F FH  0޽h ?  @ ff3Ιd332z___PPT10+X'DR' = @B D ' = @BA?%,( < +O%,( < +D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %( /%,( < +D' =1:Bvisible*o3>+B#style.visibility<*N%(D' =-o6Bbox(out)*<3<*ND' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bbox(out)*<3<* D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bbox(out)*<3<* ++0+ ++0+ ++0+ ++0+  ++0+  +(  K0     (  r  S T p`     c $A ??0j8 $D  0  0H0z,$  0 LNiech m[i,j] bdzie minimaln liczb mno|eD skalarnych potrzebnych do policzenia iloczynu ( Ai..,Aj), oraz macierz Ai ma wymiar (p i-1 pi). 2]     tQE  0(  J ,$  0 7Niech s[i,j] = k, gdzie k realizuje minimum dla m[i,j].8 28@%B  s *Dp  ,$D  0  0 pp,$  0 \Rekurencyjny algorytm obliczania m[i,j] nie jest mo|liwy do zastosowania, bo jego koszt jest wykBadniczy, ale....s 2'%f'&!M  <@  ,$D  0  Przecie| mo|na zapamitywa policzone wcze[niej warto[ci m[i,j]! ,H(2H&@H  0޽h ? @ ff3Ιd332zd\___PPT10<+PD@' = @B D' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %( /%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =-o6Bbox(out)*<3<*8D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*s%(D' =-o6Bbox(out)*<3<*sDY' =%(D' =%(D' =A@BBBB0B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* D' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#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-$)CB?B*Y3>B ppt_y<* ++0+ ++0+ ++0+ ++0+  +!  K0    (  r  S , p`     00P ,$  0 for i :=1 to n do m[i,i] :=0; od; for x := 2 to n do for i := 1 to n-x+1 do j := i+x-1 m[i,j] := +; for k := i to j-1do q := m[i,k]+m[k+1,j] + pi-1*pk *pj; if q < m[i,j] then m[i,j] := q; s[i,j] :=k fi od od od return m,s;nI 2P   j^a_   05 @,$  0 y Koszt O(n3)> 2 fff T  BHFGiH' `,$D  0 hUstawienie pocztkowej warto[ci elementw tablicy m.55 5  hKGHA Papeteria@ p,$D  0 Dwie ptle odpowiedzialne za wypeBnienie wszystkich elementw m[i,j].GG&?  BOGw H  00 ,$D  0 ZWyszukiwanie minimum H  0޽h ??0 @ ff3Ιd332z___PPT10+x]DR' = @B D ' = @BA?%,( < +O%,( < +D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*I%(D' =-o6Bbox(out)*<3<*ID' =%(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' =-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@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bbox(out)*<3<* ++0+ ++0+ ++0+ ++0+ ++0+ +`[  K0 77FW{7(  r  S  p`   7l `  0mL ,$D 0B  <p` K15125  B  <0  K10500  B  <p`  J5375  B  <0   J3500 B  <L p `  J5000 B  <D 0  G0 B   <\0  K11875  B   <`p J7125  B   <(  0  J2500  B   <p `  J1000 B   <0   G0 B  <p` J9375  B  <0  J4375  B  <p`  I750 B  <8!0  G0 B  <$ 0 J7875  B  <(p`  J2625 B  <-0  G0 B  <+p`  K15750  B  <<40  G0 B   </0  G0  ! 0H6g G1 2  " 0P=@` ' G2 2  # 0@`   G3 2  $ 0D   G4 2  % 0B  g G5 2  & 0J @` '  G6 2  ' 0"@'  G1 2  ( 0PQg G2 2  ) 0T@ G3 2  * 0X@ G4 2  + 0<\@' G5 2  , 0_g G6 2  - 0c  0 ,$ 0 ; A1 A2 A3 A4 A5 A6< 2       < . 0Ln P ,$  0 M Tablica m 2   / 0Dr  ,$  0 M Tablica s 2  Zl ` @  S 0 p,$D 0B 2 B@v  G3  B 3 BzP`   G3  B 4 Bx   G3  B 5 B| P`  G5 B 6 Bd   G5 B 8 B` P  G3  B 9 B   G3  B : BP `  G3  B ; BH   G4 B = B   G3  B > B4 P`  G3  B ? B   G3 B A BP `  G1  B B B4   G2 B D B   G1   G 6@@@'  G1 2  H 6ܮ@   G2 2  I 6   G3 2  J 68 g  G4 2  K 6ܸ@ @'  G5 2  L 6@   G6 2  N 6$` @ '  G2 2  O 6 g  G3 2  P 6l   G4 2  Q 6 `  G5 2  R 6`@ '  G6 2  T 0@  ,$  0 em[2,5]= min { m[2,2] + m[3,5] + p1*p2*p5, m[2,3] + m[4,5] + p1*p3*p5, m[2,4] + m[5,5] + p1*p4*p5 } f 2"           fB U 0Df>O+ ,$D  0B V 0Df> ,$D  0h W 0,$  0 A1(3035) A2(35 15) A3(15 5) A4(5 10) A5(10 20) A6(20 25)D 2       DH  0޽h ? @ ff3Ιd332z""___PPT10"+krD!' = @B DX!' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*W%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*WD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*WD' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %( /%,( < +D' =1:Bvisible*o3>+B#style.visibility<*. %(D' =-o6Bbox(out)*<3<*. D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-o6Bdissolve*<3<*0D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*-%(D' =-6B'blinds(horizontal)*<3<*-D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*/ %(D' =-o6Bbox(out)*<3<*/ D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*S%(D' =-o6Bdissolve*<3<*SD' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %($/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*Te%(D' =-o6Bbox(out)*<3<*TeD' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %()/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*U%(D' =-o6Bbox(out)*<3<*UD' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(./%,( < +D' =1:Bvisible*o3>+B#style.visibility<*V%(D' =-o6Bbox(out)*<3<*V++0+- ++0+. ++0+/ ++0+T ++0+W +=#  K0    P (   r   S T p`      V APapeteria 0P,$D  0 *Zasada 3 paradygmatu programowania dynamicznego: Skonstruowa rozwizanie problemu na bazie wyliczonych wielko[ci.s 2s s8   04P,$  0 Tmn|(A,s,i,j) { If j>i then X := mn|(A,s,i,s[i,j]); Y:= mn|(A,s,s[i,j]+1, j); return wynik mno|enia macierzy (XY) else return Ai }B 2   .#F   3 ZwG UNd)?AlgorytmArial Black$ k !2pP ,$D  0$   0D  3 ,$  0 W naszym przykBadzie : Mn| (A,s,1,6) daje : (A1(A2 A3))((A4A5) A6)E 2-f ff ff ff ff ff ff EH   0޽h ? @ ff3Ιd332z ___PPT10+=6D!' = @B D' = @BA?%,( < +O%,( < +D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bbox(out)*<3<* D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %( /%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bbox(out)*<3<* D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*  %(D' =-o6Bbox(out)*<3<*  D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*  %(D' =-o6Bbox(out)*<3<*  D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* -%(D' =-o6Bbox(out)*<3<* -D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* -E%(D' =-o6Bbox(out)*<3<* -E++0+  ++0+  ++0+  +7  K0   0 I ( q@G@ l  C 7 p`     B9d @P,$  0 LNDane s dwa cigi znakw X i Y X = (x1,& xm) Y = (y1,& ,yn) Znalez taki cig Z=(z1,& zk), |e Z jest najdBu|szym podcigiem zarwno cigu X jak i Y, ozn. Z = nwp(X,Y) 2'       O@_  s *0Hfp  J,$  0 =X = aawbbsccpddlnvby Y = xxwsyyypzzlinsy nwp(X,Y) = wsplny> 2>Z D  3 ZwG UNd)?ProblemArial Black$ k !2P,$D  0  08N  7 ,$  0 \PrzykBad 2 f L  hlRAA)Papeteria ,$D  0 . Wygenerowa wszystkie podcigi cigu X. Dla ka|dego z tych cigw sprawdzi czy wystpuje w Y, zapamitujc jednocze[nie najdBu|szy z takich cigw. " 2 T  3 Z wG UNd)?Algorytm naiwnyArial Black$ k !2w,$D  0Hr  6X P,$D  0 BAle to zbyt du|o kosztuje O(2m)<"  "H  0޽h ? @ ff3Ιd332z**___PPT10*+{IDy)' = @B D4)' = @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|' =%(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' =-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@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' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(+/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bbox(out)*<3<* D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(0/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bbox(out)*<3<* D' =%(D|' =%(D$' =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+ +G  K0 80@(  l  C 5 p`     S X<ܰXAd @A) Papeteria*I' ,$D  0 Niech X= (x1,& xm) Y= (y1,& ,yn) oraz nwp(X,Y) = Z , gdzie Z= (z1,& zk). Je|eli xm= yn, wtedy zk=xm=yn oraz Z k-1= nwp(Xm-1, Yn-1). Je|eli xm yn, wtedy Z = nwp(Xm-1,Y) i zk xm lub Z = nwp(X,Yn-1) i zk yn.          "                                          %#'$n  0pZ<P \,$  0 8Oznaczenie: Xi = (x1,& xi) r 2    L  3 ZwG UNd)?TwierdzenieArial Black$ k !2z) ,$D  0  0d0 p<,$D  0 eAbrakadabrabarakuda 2F  3 ZwG UNd)?PrzykBadArial Black$ k !2` @p,$D  0  0 j0  <,$D  0 VAbrakadabr barakud 2   0dn0 `<,$D  0 aAbrakad barakud 2   VsAPapeteria@@ ` `,$D  0 Ea 2   V wAPapeteria@ 2 `,$D   0 Ed 2   00{0 `<,$D   0 ^ Abrakabaraku 2    VpAPapeteria  ,$D   0 Uitd 2  V8APapeteria@  `,$D   0 Ek 2 H  0޽h ? @ ff3Ιd332z33___PPT102+b=D0' = @B D^0' = @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$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D' =%(D' =%(D<' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B+checkerboard(across)*<3<*D' =%(D' =%(D<' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B+checkerboard(across)*<3<*D' =%(D' =%(D<' =A@BBB B0B%(E' =4 B`BPB`B?<* %($/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B+checkerboard(across)*<3<*D' =%(D' =%(D<' =A@BBB B0B%(E' =4 B`BPB`B?<* %()/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B+checkerboard(across)*<3<*D' =%(D' =%(D<' =A@BBB B0B%(E' =4 B`BPB`B?<* %(./%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B+checkerboard(across)*<3<*D' =%(D' =%(D<' =A@BBB B0B%(E' =4 B`BPB`B?<* %(3/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B+checkerboard(across)*<3<*D' =%(D' =%(D<' =A@BBB B0B%(E' =4 B`BPB`B?<* %(8/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B+checkerboard(across)*<3<*D' =%(D' =%(D<' =A@BBB B0B%(E' =4 B`BPB`B?<* %(=/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B+checkerboard(across)*<3<*+0+0+ ++0+  ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ +   K0   p(  r  S 8 p`     b`ߝAd @PapeteriaMB,$D  0 V string nwp(X,Y :string) : string; begin string Z; if x(m) = y(n) then Z := nwp(X m-1, Y n-1) x(m) else Z1 := nwp(X m-1, Y); Z2 := nwp(X, Y n-1); Z := dBu|szy z cigw Z1 i Z2; fi ; return Z; end; 2^     A  .B @ s *D# ,$D  0  0]W,$  0 R konkatenacja 2  R  0  ,$  0 vKoszt tego algorytmu bdzie w najgorszym razie wykBadniczy.< 2< <H  0޽h ? @ ff3Ιd332z___PPT10_+6D' = @B DV' = @BA?%,( < +O%,( < +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' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %( /%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bbox(out)*<3<* D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*<D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*<++0+ ++0+ ++0+ +9  K0 b(   l  C @ p`   H  6 ` ,$  0 hdBugo[ najdBu|szego wsplnego podcigu cigw X,Y.55 5:  bۯAd @Papeteriam,$D  0 t d(X,Y) = d( X m-1 ,Y n-1 ) +1, gdy xm=yn d(X,Y) = max (d(X m-1 ,Y),d(X ,Y n-1 ), gdy xmyn m 2          &   BA) ` ,$D   0 Koszt obliczenia d(X,Y) wynosi card(X) * card(Y) porwnaD.C 2CZ (z  @`  @ ,$D  0   BA) @` |(PrzykBad 1 2 3 4 5 6 B D C A B A 1 A 0 0 0 1 1 2 2 B 1 1 1 1 2 2 3 C 1 1 2 2 2 2 $ 2 f `B   0D @`B   0D @`B   0Dp p@`B   0D` `@`B  0DP P@`B  0D @`B  0DP p `B  0DP p `B  0DP p B @ s *D ,$D  0  0 G ,$  0 QKoszt 2f 2  6 f@ 0 ,$D  0 G0 2  6f0 @ ,$D  0 G1 2  6@i^0 ` ,$D  0 G1 H  0޽h ? @ ff3Ιd332zi'a'___PPT10A'+UBD%' = @B DX%' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %( /%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*5%(D' =-o6Bbox(out)*<3<*5Dn' =%(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' =-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@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    N (  l  C  p`     b#AA)Papeteria0p  Begin for i :=1 to m do d(i,0):=0 od; for j :=0 to n do d(0,j) := 0 od; for i :=1 to m do for j :=1 to n do if x(i)=y(j) then d(i,j) := d(i-1,j-1) +1; b(i,j) :=  \ else if d(i-1,j) d(i,j-1) then d(i,j) := d(i-1,j); b(i,j) :=   else d(i,j) := d(i,j-1); b(i,j) :=   fi fi od od end"p 2m  n  s *)̙P ,$D  0 Po przektnej zadanie, ktre trzeba rozwiza, |eby otrzyma optymalny wynik.N 2N Nb  s *4̙PP  ,$D  0  w gr zadanie, ktre trzeba rozwiza, |eby otrzyma optymalny wynikH 2H Hb  s *h ̙ `,$D  0  w lewo zadanie, ktre trzeba rozwiza, |eby otrzyma optymalny wynikH 2H H"  6jT + F,$D  0"  6xc ,$D  0"  6$v   ,$D  0H  0޽h ? @ ff3Ιd332z___PPT10+(aD' = @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#' =%(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<* 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+ +nI  K0 ;( B@L@ l  C xK p`      b+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+ +(;  K0 ##37 "(  r  S  p`     6("`,$D 0 ZDane jest n obiektw oraz liczba caBkowita M (pojemno[ plecaka). Niech s(i) oznacza rozmiar obiektu itego, a w(i) warto[ tego obiektu. Zarwno M jak rozmiary i warto[ci s liczbami naturalnymi. Znalez takie liczby naturalne x1, x2,...xn, by Sxi s(i) M, a suma Sxi w(i) miaBa najwiksz warto[.. 2       J L  C ZwG UNd)?PrzykBadArial Black$ k !2% ,$D  0  0W 2 ,$ 0 Powiedzmy, |e w banku s obiekty A,B,C,D,E, ktrych rozmiary i warto[ci s jak na rysunku.[ 2[ [Vl W  1 W,$D 0@ W J W Jf2  6  g9 f2  69 : f2   6:9  f2   6  f2   6  g f2   6g    0W J KA,3,4 2 N W J  J =`2  0  g9 `2  09 : `2  0:9  `2  0  `2  0  g `2  0g    0W J KB,4,5 2 @ j   j f2  6 z f2  6 M g f2  6M g f2  6f f2  6 f z f2  6z f   6j  LC,7,10 2 N j  !  3 `2 " 0 z `2 # 0 M g `2 $ 0M g `2 % 0f `2 & 0 f z `2 ' 0z f  ( 0Lj  LD,8,11 2 N j  )  `2 * 0 z `2 + 0 M g `2 , 0M g `2 - 0f `2 . 0 f z `2 / 0z f  0 0j  LE,9,13 2 4l   7 ,$D 0 3 BC DEFAss`ehpw 3FXsB  2@ FaWhXx0ay (IzJg`Hn4}- u|hx":3X(;$# &<HhEHK&@=/ ( !( 7APB@H}Uexpswhy`xOv@m0hac`T\hs~x[=  G`jbxOP@                                      S" H  4 BcCDEF\AA==H@7-P(0--+@(K&]*` cQH9 hE((#Z0)KS?03.8;DHLMP{[_c@x;700Jc|pph^XRPP+,@                     3`  5 0$  L17 2  6 04`  ,$ 0 ZBodziej mo|e zabra 5 obiektw A o warto[ci 20 lub 4 obiekty B o warto[ci 16 lub 2C i 1A o warto[ci 24.i 2i iH  0޽h ? @ ff3Ιd332z___PPT10z.S`#+YLbD' = @B Da' = @BA?%,( < +O%,( < +D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*Dn' =%(D' =%(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?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*1%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*1D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*1D' =-g6B fade*<3<*1D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*7%(D' =-o6Bdissolve*<3<*7D' =%(D' =%(DP' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6%(D' =-6B#blinds(vertical)*<3<*6++0+ ++0+ ++0+6 +#  K0 $    (   r   S  p`      0H! ,$ 0 &for j := 1 to n do for i := 1 to M do if i  s[j] 0 then if koszt[i]< (koszt[i-s[j]] + w[j])then koszt [i] := koszt[i-s[j]] + w[j]; b[i] :=j fi fi od od, 2<D0         0 @ H,$ 0 koszt[i] = najwiksza warto[ jaka mo|e by osignita je[li pojemno[ plecaka wynosi iX 2XQ   0,s  ,$ 0 b[i] ostatni element, ktry zostaB dodany do plecaka by osign warto[ maksymaln.U 2UR   Vx APapeteria" /p),$D 0 "Rozwizanie polega na wyliczeniu najlepszych kombinacji elementw dla wszystkich pojemno[ci plecaka od 1 do M.o 2o o   08S s ,$D  0 ^Koszt: n* M operacji 2    BG1Hz ZM F~,$D 0 `Niezmiennik: dla x <i, koszt[x] = maksymalna warto[ plecaka o pojemno[ci x, je[li mamy do dyspozycji tylko obiekty o numerach <j&dH   0޽h ?  @ ff3Ιd332z ___PPT10.p_+͇NsD' = @B D[' = @BA?%,( < +O%,( < +D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* D' =%(D' =%(DP' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B#blinds(vertical)*<3<* D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* ++0+  ++0+  ++0+  ++0+  ++0+  +  K0 (5(  (r ( S ( p`    8 d]w (/dw ( 6)" `t] G2  ( 6T+" `] G3  ( 6-" `] G4  ( 60" `] G5  ( 6 4" `] G6   ( 67" `]  G7   ( 6 ;" ` ]  G8   ( 6B" ` ]  G9   ( 6A" `d]t G1   ( 6E" ` ]  H10  ( 6H" ` ] H11  ( 6dL" `]& H12  ( 6O" `&]6 H13  ( 6TS" `6]F H14  ( 6V" `F]V H15  ( 6TZ" `V]f H16  ( 6]" `g]w H17 )!l dw} M(dw},$D 0`N d]w ( dw ( \TaAPapeteria" t] G0  ( \\jAPapeteria" ] G4  ( \hAPapeteria" ] G4  ( \lAPapeteria" ] G4  ( \@pAPapeteria" ] G8  ( \$xAPapeteria" ]  G8  ( \vAPapeteria"  ]  G8  ( \zAPapeteria"  ]  H12   ( \~APapeteria" d]t G0  !( \APapeteria"  ]  H12  "( \APapeteria"  ] H12  #( \APapeteria" ]& H16  $( \APapeteria" &]6 H16  %( \APapeteria" 6]F H16  &( \APapeteria" F]V H20  '( \APapeteria" V]f H20  (( \APapeteria" g]w H20 EN d]w ;( dw} <( \APapeteria" t] >  =( \ APapeteria" ] GA  >( \8APapeteria" ] GA  ?( \ܨAPapeteria" ] GA  @( \lAPapeteria" ] GA  A( \ȬAPapeteria" ]  GA  B( \4APapeteria"  ]  GA  C( \xAPapeteria"  ]  GA  D( \APapeteria" d]t >  E( \APapeteria"  ]  GA  F( \APapeteria"  ] GA  G( \APapeteria" ]& GA  H( \APapeteria" &]6 GA  I( \tAPapeteria" 6]F GA  J( \,APapeteria" F]V GA  K( \APapeteria" V]f GA  L( \APapeteria" g]w GA 8!z dw} N( dw,$D 0aN d]w O( dw P( \APapeteria" t] G0  Q( \APapeteria" ] G4  R( \APapeteria" ] G5  S( \APapeteria" ] G5  T( \APapeteria" ] G8  U( \dAPapeteria" ]  G9  V( \`APapeteria"  ]  H10  W( \APapeteria"  ]  H12  X( \DAPapeteria" d]t G0  Y( \APapeteria"  ]  H13  Z( \8APapeteria"  ] H14  [( \APapeteria" ]& H16  \( \( APapeteria" &]6 H17  ]( \APapeteria" 6]F H18  ^( \XAPapeteria" F]V H20  _( \APapeteria" V]f H21  `( \xAPapeteria" g]w H22 EN d]w a( dw} b( \APapeteria" t] >  c( \ APapeteria" ] GA  d( \ %APapeteria" ] GB  e( \#APapeteria" ] GB  f( \@,APapeteria" ] GA  g( \'APapeteria" ]  GB  h( \3APapeteria"  ]  GB  i( \L.APapeteria"  ]  GA  j( \9APapeteria" d]t >  k( \<APapeteria"  ]  GB  l( \@APapeteria"  ] GB  m( \DAPapeteria" ]& GA  n( \XBAPapeteria" &]6 GB  o( \HGAPapeteria" 6]F GB  p( \OAPapeteria" F]V GA  q( \MAPapeteria" V]f GB  r( \QAPapeteria" g]w GB 9!z dw} s( Cdw ,$D 0bN d]w t( dw u( \YAPapeteria" t] G0  v( \TAPapeteria" ] G4  w( \l\APapeteria" ] G5  x( \_APapeteria" ] G5  y( \gAPapeteria" ] G8  z( \8fAPapeteria" ]  H10  {( \4jAPapeteria"  ]  H10  |( \mAPapeteria"  ]  H12  }( \qAPapeteria" d]t G0  ~( \tAPapeteria"  ]  H14  ( \xAPapeteria"  ] H15  ( \{APapeteria" ]& H16  ( \APapeteria" &]6 H18  ( \APapeteria" 6]F H20  ( \APapeteria" F]V H20  ( \APapeteria" V]f H22  ( \APapeteria" g]w H24 EN d]w ( dw} ( \APapeteria" t] >  ( \ APapeteria" ] GA  ( \8APapeteria" ] GB  ( \ܛAPapeteria" ] GB  ( \lAPapeteria" ] GA  ( \ȟAPapeteria" ]  GC  ( \4APapeteria"  ]  GB  ( \xAPapeteria"  ]  GA  ( \APapeteria" d]t >  ( \APapeteria"  ]  GC  ( \APapeteria"  ] GC  ( \APapeteria" ]& GA  ( \APapeteria" &]6 GC  ( \tAPapeteria" 6]F GC  ( \,APapeteria" F]V GA  ( \APapeteria" V]f GC  ( \APapeteria" g]w GC 9!z dw} (  dwF ,$D  0bN d]w ( dw ( \APapeteria" t] G0  ( \APapeteria" ] G4  ( \APapeteria" ] G5  ( \APapeteria" ] G5  ( \APapeteria" ] G8  ( \dAPapeteria" ]  H10  ( \`APapeteria"  ]  H11  ( \APapeteria"  ]  H12  ( \DAPapeteria" d]t G0  ( \APapeteria"  ]  H14  ( \4APapeteria"  ] H15  ( \APapeteria" ]& H16  ( \4APapeteria" &]6 H18  ( \APapeteria" 6]F H20  ( \@APapeteria" F]V H21  ( \APapeteria" V]f H22  ( \LAPapeteria" g]w H24 EN d]w ( dw} ( \APapeteria" t] >  ( \<APapeteria" ] GA  ( \PAPapeteria" ] GB  ( \,APapeteria" ] GB  ( \APapeteria" ] GA  ( \APapeteria" ]  GC  ( \t%APapeteria"  ]  GD  ( \$APapeteria"  ]  GA  ( \(APapeteria" d]t >  ( \/APapeteria"  ]  GC  ( \3APapeteria"  ] GC  ( \L2APapeteria" ]& GA  ( \(6APapeteria" &]6 GC  ( \=APapeteria" 6]F GC  ( \t<APapeteria" F]V GD  ( \EAPapeteria" V]f GC  ( \ GAPapeteria" g]w GC 9!z dw} ( s dw ,$D  0bN d]w ( dw ( \LAPapeteria" t] G0  ( \PAPapeteria" ] G4  ( \|OAPapeteria" ] G5  ( \XSAPapeteria" ] G5  ( \[APapeteria" ] G8  ( \YAPapeteria" ]  H10  ( \]APapeteria"  ]  H11  ( \eAPapeteria"  ]  H13  ( \`APapeteria" d]t G0  ( \gAPapeteria"  ]  H14  ( \oAPapeteria"  ] H15  ( \4nAPapeteria" ]& H17  ( \0rAPapeteria" &]6 H18  ( \TzAPapeteria" 6]F H20  ( \}APapeteria" F]V H21  ( \ APapeteria" V]f H23  ( \APapeteria" g]w H24 EN d]w ( dw} ( \APapeteria" t] >  ( \APapeteria" ] GA  ( \ APapeteria" ] GB  ( \ĎAPapeteria" ] GB  ( \TAPapeteria" ] GA  ( \APapeteria" ]  GC  ( \APapeteria"  ]  GD  ( \`APapeteria"  ]  GE  ( \APapeteria" d]t >  ( \APapeteria"  ]  GC  ( \APapeteria"  ] GC  ( \̯APapeteria" ]& GE  ( \lAPapeteria" &]6 GC  ( \\APapeteria" 6]F GC  ( \APapeteria" F]V GD  ( \APapeteria" V]f GE  ( \APapeteria" g]w GC  ( 0f d M,$  0 t.Najwiksza warto[ = 24 2 @ ( 09 0 ,$  0 fZawarto[ plecaka: b[17]= C, b[17-7]= C, b[10-7]=A4 24 4B ( s *D<dm,$D  0 ( 0,$  0 Utablica kosztw 2   ( 0<,$  0 ytablica b-ostatni element 2&   ( <HPCI)$zw},$D  0H ( 0޽h ? @ ff3Ιd332z((___PPT10n(.@0F+6D'' = @B DU'' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*M(%(D' =-o6Bdissolve*<3<*M(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*N(%(D' =-o6Bdissolve*<3<*N(D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*(D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*(Dn' =%(D' =%(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?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*(Dn' =%(D' =%(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?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*(Dn' =%(D' =%(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?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*s(%(D' =-o6Bdissolve*<3<*s(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-o6Bdissolve*<3<*(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-o6Bdissolve*<3<*(D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-6B'blinds(horizontal)*<3<*(D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-6B'blinds(horizontal)*<3<*(++0+( ++0+( ++0+( +* 0 |t (  X  C    t  S   @    lub B. Jest wiele komercyjnych zastosowaD w ktrych wystpuje problem plecakowy: pakowanie kontenerw, samochodw. tH  0޽h ? ̙3380___PPT10.0R+ 0 $b(  $X $ C     $ S  @   d Koszt tego algorytmu n*M. Dla maBych M jest on akceptowalny ale dla dy|ych nie. Zauwa|my, |e ten algorytm nie dziaBa poprawnie je[li M lub rozmiary lub warto[ci nie jest liczbami naturalnymi. Co wicej nie jest znany |aden dobry algorytm dla tego problemu.@,H $ 0޽h ? ̙3380___PPT10.o, 0 .&,(  ,X , C    & , S  @   b[M] jest w plecaku i mog to odczyta. pozostaBa zawarto[ jest taka sama jak dla optymalnego rozwizania dla plecaka o rozmiarze M-s[b[M]]. Zatem nastpny obiekt to b[M-s[b[M]]]. Oznaczmy M-s[b[M]]= k wtedy nastpny element bior jako b[k]. Teraz musz sprawdzi optymalne rozwizanie dla plecaka o rozmiarze k- s[b[k]] (odejmuj rozmiar ostatnio wlozonego obiektu) itd.  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