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Zcie|ki Eulera&( Wie|e Hanoi 0Algorytm 2 Permutacje 3Problemy decyzyjne 5.Pierwsza  klasyfikacja  6Klasy problemw decyzyjnych 8(Problem komiwoja|era W3PCzy programowanie dynamiczne mo|e pomc?)) )=*Problem speBnialno[ci& >"Zcie|ki Hamiltona& C$Kolorowanie grafw D% Inne problemy :P = NP ? <Wszystko albo nic @!HRozstrzygalno[ i nierozstrzygalno[&B#FNierozstrzygalno[ problemu  Stopu E'Kolorowanie grafw planarnych J, Algorytm 1 K- Algorytm 2 /p479;  ` @ ff3Ιd332z` @ ff3Ιd332z` 999MMM` fffPP3f>?" dd@ ?4Zd@ d " @ ` n?" dd@   @@``@n?" dd@  @@``PV    @ ` ` p>>  K0  D(  F   `  0PPf  c 6AminispirlB  <g   HJ 1Ȝ? p` J TKliknij, aby edytowa styl tytuBu z Wzorca++  c $J p J Kliknij, aby edytowa style tekstu z Wzorca Drugi poziom Trzeci poziom Czwarty poziom Pity poziom,  c  c $J p   J X* 2    c $J    J Z*(2    c $J   J Z*(2 XB   0D`Z  BsZ޽h))?? @ ff3Ιd332z Notatnik&  K0 @ <(   FF       XA StationeryPP`   S 0AminispirH  <?e ?^~ e ALGORYTMY I STRUKTURY DANYCHKliknij, aby edytowa styl tytuBu z WzorcaGG  c $tBe 3   e ZKliknij, aby edytowa styl podtytuBu z Wzorca..  c $0Ge ^# e X* 2f   c $Ke ^~#  e Z*(2f   c $Oe ~# e Z*(2f Z  BsZ޽h))?? @ ff3Ιd332z| 0 ( F0E,   0h P    T*   0     V* d  c $ ?    0  @  Kliknij, aby edytowa style wzorca tekstu Drugi poziom Trzeci poziom Czwarty poziom Pity poziom*  a  68 `P   T*   6H `   V* H  0޽h ? ̙3380___PPT10.0Ly K0 0(  l  C Ze ^~ e l  C [e 3   e H  0޽h ? @ ff3Ιd332zy___PPT10Y+D=' = @B +  K0 P@4(  @l @ C e p`  e  @ S pe@  e " PpH @ 0޽h ? @ ff3Ιd332zy___PPT10Y+D=' = @B +$  K0 `z( q@G@ l  C e p`  e   0Xe Z,$  0 Powiemy, |e problem jest wielomianowy, tzn. jest rozwizywalny w czasie wielomianowym, tzn. istnieje algorytm rozwizujcy ten problem dla danych rozmiaru n w czasie O(n k), dla pewnego ustalonego k.4 2    0ew ,$  0 \Wyszukiwanie Sortowanie w tablicy Sortowanie z u|yciem struktur drzewiastych Kompresja danych NajdBu|szy wsplny podcig Najkrtsze [cie|ki 2 H  3 ZwG UNd)?PrzykBadyArial Black$ k !2h ,$D  0H  0޽h ? @ ff3Ιd332z/'___PPT10+:D{' = @B D6' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(D?' =%(D' =A@BBBB0B@B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' ,=+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' ,=+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* D' =%(D?' =%(D' =A@BBBB0B@B%(D' =1:Bvisible*o3>+B#style.visibility<* M%(D' ,=+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* MD' ,=+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* MD' =%(D?' =%(D' =A@BBBB0B@B%(D' =1:Bvisible*o3>+B#style.visibility<*My%(D' ,=+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*MyD' ,=+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*MyD' =%(D?' =%(D' =A@BBBB0B@B%(D' =1:Bvisible*o3>+B#style.visibility<*y%(D' ,=+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*yD' ,=+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*y+p+0+e ++0+e +8}  K0 89(  r  S ,  p`      0` ,$  0 ,Dla danego grafu niezorientowanego zbada, czy istnieje [cie|ka Eulera, tzn. Droga lub cykl w grafie przechodzcy przez ka|d krawdz i to tylko raz. 2&@Q,  0` ` ,$  0 8Nie istnieje [cie|ka Eulera. 2&l    $`0@P ,$D  0  0Pp < 2 TB B c $DPp TB  c $DpP TB  c $D  TB B c $D  TB   c $DpP `2   0@ `2   0`  `2   0p  `2   0``  TB  c $D  B @ s *DfԔP ,$D  0, # 0!  ,$  0 0Istnieje [cie|ka Eulera. 2&B % s *DfԔ` P ,$D  0B & s *DfԔ P @ ,$D   0B ' s *DfԔ` P ,$D   0B ( s *DfԔp 0 ,$D   0B ) s *DfԔPp,$D   0B * s *DfԔ ,$D   0B + s *DfԔ ,$D  0B , s *DfԔ ,$D  0l 0 @p  40 @ ,$D  0TB B c $D` TB  c $D` p@ TB  c $D` TB B c $Dp` @ TB  c $D  @TB  c $DP P TB B c $D@ P TB B c $Dp@ TB B c $DP  TB B c $Dp@ p `2  00 `2  0Pp`2  00 @ `2  0 P `2  0@ p `2   0 `2 ! 00  `2 " 0  TB -B c $DP  B . s *DfԔ P,$D  0B /@ s *DfԔP ` ,$D  0B 0 s *DfԔ ` ,$D  0B 1@ s *DfԔ`P  ,$D  0B 2@ s *DfԔ0pP ,$D  0B 3@ s *DfԔ  ,$D  0 5 010 ,$  0 B1. Zbada, czy graf jest spjny 2. Zbada, czy graf wszystkie, z wyjtkiem co najwy|ej dwch wierzchoBkw, maj rzd parzysty. 2 L 6 C ZwGH*UNd)?AlgorytmArial Black$ k !2P @@,$D  0l 7 V|7APapeteria` ,$D  0 \Koszt O(m), gdzie m jest liczb krawdzi grafu&/ 2 f# /" 8 6G@U 0U 0H  0޽h ? @ ff3Ιd332z\\___PPT10\+GsD|[' = @B D7[' = @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<*D' =%(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<*#%(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<*,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?<* %(1/%,( < +D' =1:Bvisible*o3>+B#style.visibility<**%(D' =-o6Bbox(out)*<3<**D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(6/%,( < +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' =%(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?<* %(E/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*&%(D' =-o6Bbox(out)*<3<*&D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(J/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D' =%(Do' =%(D' =4@BBB B%(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' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(Y/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-o6Bbox(out)*<3<*0D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(^/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*1%(D' =-o6Bbox(out)*<3<*1D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(c/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*2%(D' =-o6Bbox(out)*<3<*2D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(h/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*3%(D' =-o6Bbox(out)*<3<*3D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(m/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*6%(D' =-o6Bbox(out)*<3<*6D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(r/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*5%(D' =-o6Bbox(out)*<3<*5D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(w/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*7%(D' =-o6Bbox(out)*<3<*7++0+ ++0+ ++0+# ++0+5 ++0+7 +  K0  p0> (  l  C o p`     0$q0,$  0 f Danych jest n kr|kw, umieszczonych w porzdku rosncych [rednic, na dr|ku A. Zadanie polega na przeniesieniu wszystkich kr|kw na dr|ek B z wykorzystaniem pomocniczego dr|ka C (oba dr|ki B i C s pocztkowo puste), ale mniejszy kr|ek musi zawsze le|e na wikszym. 2 D  3 ZwG UNd)?ProblemArial Black$ k !2 p,$D  0l  p  p,$D  0f  600 @ `  0f 0 `   0kz  `   05    6y@P `p OA 2 `  0  p `  0   `  0 `  lB  <D>  lB  <D> lB  <D>0 0   0p P p OB 2   0P p OC 2   s *5` p ,$D  0 ! s *  ,$D  0 " s *   ,$D  0 # s *kz` ,$D  0 $ s *` p ,$D  0 % s *5 @ ` ,$D   0 & s * 0 ,$D   0 ' s *f`  ,$D   0 ( s * @ ` ,$D   0 ) s *5 0 ,$D   0 * s *` ,$D  0 + s *kz ` ,$D  0 , s * 0 ,$D  0 - s *5@ p ,$D  0 . s *0 0@ ,$D  0 / 00  ,$D  0 1 s *@ p ,$D  0 2 s *5 @ 0 ,$D  0 3 s *kz`   ,$D  0 4 s * ` ,$D  0 5 s * @ 0 ,$D  0 6 s *5 @` ,$D  0 7 s *`  ,$D  0 8 s *f 0 ,$D  0 9 s * @` ,$D  0 : s *5` @ ,$D  0 ; s *`   ,$D  0 < s *kz P ,$D  0 = s *` @ ,$D   0 > s *5 @  ,$D!  0H  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@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<*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<*$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<*)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<*.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<*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<*1Dn' =%(D' =%(D' =4@BBBB%(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<*2Dn' =%(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<*4Dn' =%(D' =%(D' =4@BBBB%(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<*3Dn' =%(D' =%(D' =4@BBBB%(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<*5Dn' =%(D' =%(D' =4@BBBB%(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<*6Dn' =%(D' =%(D' =4@BBBB%(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<*7Dn' =%(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<*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<*9Dn' =%(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<*>+8+0+ +-  K0   = (  l  C  p`   ^  0x@ ,$  0 Procedure przenies(n, A,B, C); {przenie[ n kr|kw z A na B wykorzystujc C} begin if (n<>0) then przenies(n-1, A,C,B); przeloz (A,B); {przeB| jeden kr|ek z A na B} przenie[(n-1, C, B, A) fi end 2  9 %I$  <  ,$  0 2 Koszt T(1) = 1 T(n) = T(n-1) +1 +T(n-1)"3f# 32  00  ],$  0 8Rozwizanie : T(n) = 2 n -16 2 f    VAPapeteriaP p,$D  0 [T(64) = 0.5 miliona lat 2 B @ BDop  P ,$D  0  0p0,$  0 n"Koszt wykBadniczy 2f H  0޽h ? @ ff3Ιd332z##___PPT10#+PzDu"' = @B D0"' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*f%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*fD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*fD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*f%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*fD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*fD{' =%(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$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*3%(D' =-o6Bbox(out)*<3<*3Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(D|' =%(D$' =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<*++0+ ++0+ ++0+ ++0+ ++0+ +6  K0  `( x l  C 0 p`   |  0 ,$  0 Dla danej liczby naturalnej n wygenerowa wszystkie permutacje liczb {1,2,...,n}.R 2R RD  3 ZwG UNd)?ProblemArial Black$ k !2 8p,$D  0  00 ,$  0 Procedure generuj(k : integer); var t : integer; begin now := now +1; tab[k] := now; if now =n then wypisz(tab);fi; for t:= 1 to n do if tab[t] = 0 then generuj(t); od; now := now-1; tab[k] := 0; end; 2  !   V APapeteria@J,$D  0 tWywoBanie:generuj(0) z now =-1 i tab[i]=0 dla i=1..n daje; 2;&!  VAPapeteria ,$D  0 c1234 1243 1324 1423 1342 1432 2   VAPapeteria0 0 ,$D  0 c2134 2143 3124 4123 3142 4132 2   VAPapeteria p,$D  0 c2314 2413 3214 4213 3412 4312 2 %  V APapeteria j,$D  0 k2341 2431 3241 4231 3421 4321 2   0@ ,$  0 rKoszt rzdu n!$ 2f H  0޽h ? @ ff3Ιd332z''___PPT10'+.D%' = @B De%' = @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<*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$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(D|' =%(D$' =A@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@BBB B0B%(E' =4 B`BPB`B?<* %(0/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bbox(out)*<3<* ++0+ ++0+ ++0+ ++0+  ++0+  ++0+  ++0+  ++0+  +AO  K0 ''O'(  l  C T< p`     00>`,$  0 PProblem, ktrego rozwizanie ma dawa odpowiedz binarn tak lub nie nazywa bdziemy problemem decyzyjnym. Jl 28' l  0tFP ,$  0 BDanych jest n kart, na ktrych wydrukowane s kolorowe obrazki. Czy mo|na z nich uBo|y kwadrat tak, by wszystkie obrazki pasowaBy do siebie ksztaBtem i kolorem? 2 d   W# Tp ,$D  0` X 0 ` Y 0Zckz0` Z 0WQ5`B [ 0Dp`B \ 0D`Z2 ] s *5`B ^B 0DP  `B _B 0DP`B ` 0DPP `B a 0DPP`B b 0D  p`B c 0D PZ2 d s *kz`@ `B eB 0D0``B f 0D`B g 0Dpp`B h 0DppF { 3 ZwG UNd)?PrzykBadArial Black$ k !2 8) ,$D  0& | 0 Qp ,$  0 LAlgorytm naiwny: przegldamy wszystkie mo|liwe uBo|enia. Odpowiadamy TAK, je[li jakie[ uBo|enie jest poprawne, odpowiadamy NIE, gdy |adne uBo|enie nie byBo poprawne. 2 d   }# T p! ,$D  0` ~ 0 `  0Zckz0`  0WQ5`B  0Dp`B  0D`Z2  s *5`B B 0DP  `B B 0DP`B  0DPP `B  0DPP`B  0D  p`B  0D PZ2  s *kz`@ `B B 0D0``B  0D`B  0Dpp`B  0Dppd   # T  ,$D  0`  0 `  0Zckz0`  0WQ5`B  0Dp`B  0D`Z2  s *5`B B 0DP  `B B 0DP`B  0DPP `B  0DPP`B  0D  p`B  0D PZ2  s *kz`@ `B B 0D0``B  0D`B  0Dpp`B  0Dppd   # T ! ,$D  0`  0 `  0Zckz0`  0WQ5`B  0Dp`B  0D`Z2  s *5`B B 0DP  `B B 0DP`B  0DPP `B  0DPP`B  0D  p`B  0D PZ2  s *kz`@ `B B 0D0``B  0D`B  0Dpp`B  0Dpp,  s *4fpp ,$D   0 ( Koszt Q(n!) F 2555 H  0޽h ? @ ff3Ιd332zA'9'___PPT10'+Ү/D&' = @B D%' = @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<*lDn' =%(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' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*W%(D' =-o6Bbox(out)*<3<*WD{' =%(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' =%(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<*D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(1/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*++0+ ++0+ ++0+| ++0+ +V  K0 H@)(   l  C  p`   2  dA(Jasny uko[ny w grP XB  0D    0, p ,$  0 j$Algorytmy rozsdne 2 2  0 :,$  0 XAlgorytmy wymagajce nierozsdnie du|o czasu- 2- -2  0 0 ,$D  0B @ 6D 0 ,$D  0  0d 0PJ,$  0 ZAlgorytmy sortowania 2 2  0 p ,$D  0B  6D ,$D   0  0P P 0,$   0 \Algorytmy wyszukiwania 2 2  0 0 ,$D  0B !@ 6D p` ,$D  0 " 0P  ,$  0 VKompresja danych 2 2 # s *f0 ` ,$D  0B $ 6D0 `,$D  0 % 0@,$  0 j$ MaBpia ukBadanka  2 R '  BCDEdF,ԔXd080@PP`PXh@        P ` ,$D  0 ( VAPapeteriaj ,$D  0 LAle ... Ktry z dwch algorytmw o koszcie Q(n 100) i Q(2 n) dla maBych n, jest lepszy?nX 2+   XH  0޽h ? @ ff3Ιd332zEE___PPT10E+LD0D' = @B DC' = @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<*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<* 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' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(4/%,( < +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' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(E/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =-o6Bbox(out)*<3<*#D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(J/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-o6Bbox(out)*<3<*$D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(O/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =-o6Bbox(out)*<3<*%D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(T/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-o6Bbox(out)*<3<*(D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(Y/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =-o6Bbox(out)*<3<*'++0+ ++0+ ++0+ ++0+ ++0+" ++0+% ++0+( +f   K0  m(  l  C H p`     0$ ',$  0 P - klasa problemw decyzyjnych rozwizywalnych w czasie wielomianowym6G 2fB G  0P0 ,$  0 tNP = klasa problemw decyzyjnych, dla ktrych dowd, |e podane rozwizanie (algorytm) jest poprawne mo|na zweryfikowa w czasie wielomianowym.& 2f l p@  p@,$D  02  B&<p@ L NP   2   6f   OP   0 pJ,$  0 Tzn. rozwizywalnych przez algorytm niedeterministyczny w czasie wielomianowym.P 2O Pb  0 p` ,$D  0H  0޽h ? @ ff3Ιd332z___PPT10+@D-' = @B D' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*G%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*GD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*GD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* P%(D' =-o6Bbox(out)*<3<* PD' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bbox(out)*<3<* ++0+ ++0+ ++0+  +$\  K0 .&,/( 6La l  C t p`     0j,$  0 bZadanie komiwoja|era polega na odwiedzeniu wszystkich miast z danego zbioru i powrt do punktu wyj[cia, tak by pokonana droga byBa najkrtsza. 2 &  0p 7,$  0 <W wersji decyzyjnej Czy dla danego k istnieje cykl przechodzcy przez wszystkie wierzchoBki danego grafu taki, |e suma kosztw jego krawdzi nie przekracza k.& 2f l  `   *`P p ,$D  0`2  0@ `2  0` `2  0 p`2   0 P `2   0   `2   0 `B   0D PfB   6DoPPfB  6DoPP fB  6DoPP fB  6Do P 0 `B B 0DPp`B  0DPP `B  0DP `B  0D`B B 0D   0| G6 2   0 0 PP 7 G4 2   0 ` 0 G  G7 2   0pP7 G8 2   0xg G9 2   0  G3 2   0 G4 2   0   G3 2   0$`` G  G5 2   0! w  G7 2  " 0+@ `  H10 2  # B CDELF$XXpPX(`8 pxx0h@@@        B $ 6DpPP,$D  0B % 6DpP ` ,$D  0B & 6Dp0 P ` ,$D  0B ' 6Dp0 @ ,$D  0B ( 6Dp @ ,$D   0B )@ 6DpP,$D   0 + 01P pJ ,$   0 NKoszt=28 2  p , V6APapeteriaP3 ,$D   0 pAlgorytm naiwny : wygenerowa wszystkie mo|liwe cykle. 9 29 9L - 3 ZwG UNd)?Problem NPArial Black$ k !2 ,$D  0, . Vt;APapeteria` Z ,$D  0 rKoszt Q(n!), 2 T / BAG,)H#S  ,$D 0 hA mo|e zastosowa metod programowania dynamicznego?55 5H  0޽h ?/ @ ff3Ιd332z<~<___PPT10^<++!lxD:' = @B D:' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %( /%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-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<**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<*$Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*%D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*%Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*&%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*&D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*&Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*'D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*'Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*)D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*)D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*+ %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*+ D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+ D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*,D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*,D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(E/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*.%(D' =-o6Bbox(out)*<3<*.D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(J/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*-%(D' =-o6Bbox(out)*<3<*-DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*/%(+P+0+ ++0+ ++0++ ++0+, ++0+. ++0+/ +E,  K0 X|(  Xr X S l] p`    X 0^d,$ 0 Niech C(i,j) bdzie kosztem drogi od i do j, gdzie i, j {1,2,...n}.,E 28 &:H X 0e7-,$ 0 Niech T(i;j1,...jk) bdzie kosztem optymalnej drogi z i do 1, ktra prowadzi przez wierzchoBki j1,...,jk dokBadnie raz w dowolnym porzdku.n 2   N  $ T X s *p"`W Przy tym oznaczeniu koszt optymalnej drogi komiwoja|era jest rwny T(1; 2,3,...n).S 2S S  X 0uS  ,$ 0 Mamy nastpujc zale|no[ rekurencyjn: T(i;j1,...,jk) = min 1 m k {C(i,jm) + T(jm;j1,...,jk/jm)} T(i;j) = C(i,j) + C(j;1) 2.          tH &l G 3  X G3 ,$D 02 X <G  Gi 2  X <:  bjm" 2  X <~ 3  G1 ZB  X s *D :IZB  X s *D Z ZB  X s *DZf j ZB X s *Djf ZB X s *D @ ZB X s *D@ ~ I X 0` n  przez j1,..., jk8 2  &. X <8c"`p ,z,$D 0 Ta rekursja wymaga jednak zapamitania rozwizaD dla dowolnego podzbioru j1,...,jk, tzn. wykBadniczo du|o miejsca i czasu.B{ 2J  )&pH X 0޽h ? @ ff3Ιd332z___PPT10.^+ywD' = @B Dp' = @BA?%,( < +O%,( < +D$' =%(D' =%(Dt' =A@BB7BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*XD' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*XD' =-g6B fade*<3<*XD$' =%(D' =%(Dt' =A@BB7BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*XD' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*XD' =-g6B fade*<3<*XD$' =%(D' =%(Dt' =A@BB7BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*XD' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*XD' =-g6B fade*<3<*XD' =%(D' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*XD' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*XD' =-g6B fade*<3<*XD' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-6B'blinds(horizontal)*<3<*X++0+X ++0+X ++0+X ++0+X +lI  K0  "(  l  C ( p`   F  0  ,$   0 BRozwizanie Metoda zero-jedynkowa&" 2 f&  0 G ,$   0 SProblem 2f j  0 @| ,$   0 Czy dla danej formuBy istnieje warto[ciowanie, ktre speBnia t formuB? J 2J J2  6 P,$D  0 N Jzyk 2  6,  ,$D  0 M Semantyka   2  6P,$D  0 jSpeBnialno[     6,$D  0 <   0pp`,$D  0d  hGOHzA Papeteria`,$D  0 ((p q) r) N 2 Tl  P0  P 0 ,$D  0  h\G)HA Papeteria P0  < N p       0x  b p q r 1 0 1 2 TB  c $D` @ @  0p {  Hv: 2 TB  c $DP P TB  c $D  f  hGv9HA Papeteria ` ,$D  0 (((p q) r) (v) = 1B  >  0 P J,$   0 NKoszt : 2 n dla formuBy o dBugo[ci n,( 2   (  3 W3yd?Ale ...Impact ,$D   0H  0޽h ??0 @ ff3Ιd332z&66___PPT105+`9D3' = @B Dm3' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %( /%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-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' =%(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$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(3/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*J%(D' =-o6Bbox(out)*<3<*JD' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(8/%,( < +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$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(B/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-o6Bbox(out)*<3<*(+0+0+ ++0+ ++0+ ++0+ ++0+  ++0+  ++0+  ++0+ ++0+ ++0+ +o  K0 &&0NO%(  r  S  p`     0@P,$  0 Czy w danym niezorientowanym grafie istnieje [cie|ka przechodzca przez ka|dy wierzchoBek dokBadnie raz?i 2i i$  0@ ` : ,$  0 0Nie ma [cie|ki Hamiltona 2& .  6(Ԕ@ p: ,$  0 4Istnieje [cie|ka Hamiltona 2&  l PP  "PP ,$D  0`2  00 `2  0 P `2  0`2   0@`2   0 0 `2   0p`2   0 0P `2   0  PP `2  0  `2  0P0 ` `2  0P@`2  0@ppTB  c $DTB  c $DPTB  c $DP PP TB  c $DP@P` TB  c $D@ TB B c $DPTB  c $DpTB  c $DP@` TB  c $DpTB  c $D0 TB B c $D P TB  c $D  TB B c $D@TB  c $DpTB   c $D@ TB !B c $Dp B @ s *DfԔP,$D  0B A s *DfԔp,$D  0B B s *DfԔp,$D   0B C@ s *DfԔP,$D  0 l pP  KP P ,$D  0f2 $ 6 PPf2 % 6@Ppf2 & 6f2 ' 6pf2 ( 6`  f2 ) 60 f2 * 6  P f2 + 6@ p f2 , 6  f2 - 6p  f2 . 6ppf2 / 6` ZB 0 s *D PZB 1 s *D PpPZB 2 s *D  p ZB 3 s *Dpp ZB 4 s *D` ZB 5B s *DPpZB 6 s *Dp ZB 7 s *Dp ` ZB 8 s *DpZB 9 s *D0  ZB :B s *D   ZB ; s *D @ ZB <B s *D` ZB = s *DZB > s *D`  ZB ?B s *D0 TB DB c $Dp`  B E@ s *DfԔ PP ,$D   0B F s *DfԔP PP ,$D   0B G@ s *DfԔ P ,$D   0B H s *DfԔ@ ,$D  0B I s *DfԔP @` ,$D  0B J s *DfԔ@P P ` ,$D  0B L s *DfԔ ,$D   0^ M VT@APapeteria @ ,$D  0 ^Algorytm naiwny : sprawdzi wszystkie [cie|ki.0 20 0 N 04E ,$  0 nKoszt Q(n!)( 2  O <IG   @U 0U 0 ]EulerH  0޽h ? @ ff3Ιd332zHwH___PPT10WH+VD#G' = @B DF' = @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<*iDn' =%(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$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*K%(D' =-o6Bbox(out)*<3<*KD' =%(D|' =%(D$' =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' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(%/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*C%(D' =-o6Bbox(out)*<3<*CD' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(*/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*A%(D' =-o6Bbox(out)*<3<*AD' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(//%,( < +D' =1:Bvisible*o3>+B#style.visibility<*B%(D' =-o6Bbox(out)*<3<*BD' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(4/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-o6Bbox(out)*<3<*LD' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(9/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*E%(D' =-o6Bbox(out)*<3<*ED' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(>/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*F%(D' =-o6Bbox(out)*<3<*FD' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(C/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*G%(D' =-o6Bbox(out)*<3<*GD' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(H/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-o6Bbox(out)*<3<*HD' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(M/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*I%(D' =-o6Bbox(out)*<3<*ID' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(R/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*J%(D' =-o6Bbox(out)*<3<*JD' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(W/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*M%(D' =-o6Bbox(out)*<3<*MD' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(\/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*N %(D' =-o6Bbox(out)*<3<*N ++0+ ++0+ ++0+ ++0+M ++0+N +  K0  @(  r  S o p`     VpAPapeteria ,$D  0 ,Zadanie Pokolorowa wierzchoBki niezorientowanego grafu G , tak by wierzchoBki ssiednie miaBy r|ne kolory. "n 2g n  0v  Z ,$  0 `Najmniejsz liczb kolorw jakich trzeba u|y do pokolorowania grafu G nazywamy liczb chromatyczn grafu , ozn. c(G),v 2q&l  V}APapeteria0 ,$D  0 Problem decyzyjny: Dany jest graf G. Ustali, czy k kolorw wystarczy do pokolorowania tego grafu."d 2S dv   W3fԔ?Nie jest znany |aden algorytm wielomianowy znajdowania liczby chromatycznej grafu.Times New Roman` ,$D  0  0c ) @ U 0U 0H  0޽h ? @ ff3Ιd332zg____PPT10?+q靑D{' = @B D6' = @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<*v%(D' =-o6Bbox(out)*<3<*vD' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D' =%(D' =%(D)' =4@BB BB%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B%slide(fromBottom)*<3<*++0+ ++0+ ++0+ +)  K0 =5   (  r  S  p`     0ԑ 0,$  0 UProblem plecakowy 2   0 ,$  0 VProblem pakowania  2   0`  ,$  0 \Problem planowania pracy 2 h  PABukiet< 6,$D   0 hDany jest cig obiektw s1,...sn (0) oraz pojemno[ plecaka C. Problem polega na znalezieniu podzbioru T {1,2,...,n} aby Ssi dla i T przyjmowaBa warto[ najwiksz oraz Ssi C. 2  G & @{.  VAPapeteriaZ ,$D  0 ZDana jest nieograniczona liczba kontenerw o pojemno[ci 1 oraz n obiektw rozmiaru s1,...sn, gdzie 0 si 1. Jaka jest najmniejsza liczba kontenerw, potrzebna do zapakowania wszystkich obiektw? 2T   Z&g^   PABukiet `,$D  0 XNiech J1,...,Jn bd zadaniami do wykonania, t1,...,tn - czasem koniecznym do wykonania zadania, a d1,...,dn terminami wykonania zadaD, p1,...,pn kar za przekroczenie terminu. Znalez tak kolejno[ wykonywania zadaD, by zminimalizowa kary.B 2      0S @ U 0U 0H  0޽h ? @ ff3Ιd332zME___PPT10%+KD' = @B Dt' = @BA?%,( < +O%,( < +D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D_' =%(D' =%(D' =A@BBBB0B%(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?dCB1+#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@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<*+P+0+ ++0+ ++0+ ++0+ ++0+ ++0+ +7  K0 WOP( k !k  l  C  p`   (  0z ,$  0 >Klasa NPC = problemy NP-zupeBne& 2 f   0,$  0 FProblem p jest NP-zupeBny, je[li 1. nale|y do klasy NP i 2. ka|dy inny problem z tej klasy jest wielomianowo redukowalny do p. 2   0hS OM  O Problem p 2    0S ~ M  ZProblem p  2  b @ ZZ\G7HxJI*  M redukcja   iz     ] W ,$D  0fB  6D` `   0  R wielomianowo 2    0   Gf 2   0 ,$  0 BOdpowiedzi dla danych x jest TAK" 2" "  0  ,$  0 Xwttw 2"  0 ,$  0 HOdpowiedzi dla danych f(x) jest TAK% 2% %*2  b APapeteria - ,$D  0 dDane do problemu p (2 @2  bTAPapeteria M ] ,$D  0 z*Dane do problemu p  (2   BA)CF ,$D  0 PrzykBad: Problem [cie|ek Hamiltona redukuje si do problemu komiwoja|era.L 2L& (H  0޽h ? @ ff3Ιd332z'&___PPT10&+_AjD%' = @B D$' = @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$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D' =%(D|' =%(D$' =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$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(*/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =-o6Bbox(out)*<3<*%D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(//%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ +  K0 pH(  l  C 2 p`   n  n 4GHA Papeteria" S ,$D  0 <Gdyby istniaBo wielomianowe rozwizanie dla jakiegokolwiek problemu z klasy NPC, to istniaBby wielomianowy algorytm dla wszystkich problemw klasy NP." \  n:GHn>A Papeteria"  , ,$D  0 |2Gdyby udowodniono wykBadnicze dolne ograniczenie dla jakiego[ problemu klasy NPC, to |adnego z problemw NPC nie mo|na by rozwiza wielomianowo.(2   C W3yd?klasa NPCImpactMH  0޽h ?/  @ ff3Ιd332z  ___PPT10 + iFD ' = @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<*+p+0+ ++0+ +%  K0    4 (  r  S HH p`     0I ,$  0 p*Powiemy, |e problem jest rozstrzygalny, je[li istnieje algorytm, ktry dla dowolnych danych x po skoDczonej liczbie krokw daje rozwizanie problemu. 2   <O@@  ,$D  0 |6W przeciwnym przypadku problem jest nierozstrzygalny77 7p  VTAPapeteria` ,$D  0 FTwierdzenie Problem stopu jest nierozstrzygalny (halting problem). &G 2 f;&4   0Y@ z ,$  0 8Dany jest dowolny algorytm i dane do tego algorytmu. Pytamy, czy ten algorytm koDczy obliczenia dla tych danych czy nie?z 2z zD  3 ZwG UNd)?ProblemArial Black$ k !2p ,$D  0  0^  ,$  0 &Czy istnieje algorytm Q, ktry dla dowolnego algorytmu A napisanego w pewnym ustalonym jzyku programowania i dla ustalonych danych x, po skoDczonej liczbie krokw odpowiada na pytanie, czy A zaptla si dla danych x, czy nie. 2&H  0޽h ? @ ff3Ιd332z}u___PPT10U+CD!' = @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<*%(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$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* z%(D' =-o6Bbox(out)*<3<* zD' =%(D|' =%(D$' =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+  +A  K0 "&%(  r  S v p`   H  BC6DEF:iwuh`QE5-&($$\z -8DQv"[JwsZ%'zCH'c  /&A2lOo + AcEc42""G=X`s+soO"=4$Mh| k!!+46FxFx>0>XzoJ)I1.6?()&  @LCPsJk#0a . 2>DPPH75#&'W,*6LY:\`@                                            `   <x`p  EW RB @ s *D    0 }$ h"Program wej[ciowy 2   6܁U  EW    65 @  EW ^   6~v /@ RB   s *D h   @BCDElFf[[{l{l+sg1:I[tQpyR8hPr)ATwwtR:  %/8@a._Xmmjlda$   d2'( l$$K= 4"" m ZJ8'??ov@D@                              `|    0 0  EQ 2 RB  s *D@RB @ s *DP pRB @ s *DP  p RB  s *D p   6p   Z TAK (stop)     6p  \NIE (ptla)   RB @ s *Dpp 0   0ܙ  NHipotetyczny program Q dla problemu stopu Odpowiada Tak, gdy program dany zatrzymuje si i Nie, je[li program ma nieskoDczon ptl 2 RB  s *D `    0H z Twyj[cie 2   VXAPapeteriap0 O Program S 2  F  BCnDETF|UU%nmPIFG@:-%|jqq=e'vR4 zh _MM==Z< ;50n)b'T(J CE A3%c\n+QQrwwnb\RJ~J~MuLjScsCp;<@                             x   s *P ,$D  0 GS 2   s * ,$D  0 ES 2   s * @ ,$D  0 ES 2    0  ,$  0 \Sprzeczno[ 2  B ! 0D5o @ ,$D  0B " 0D5o @`0 ,$D  0 # 0, 0 ,$  0 \Sprzeczno[ 2  B $ 0D5o @ p0 ,$D  0B %@ 0D5Ԕ @ ,$D   0 & 0 j  while true do od; 24H  0޽h ? @ ff3Ιd332z$$___PPT10$+jD#' = @B DI#' = @BA?%,( < +O%,( < +D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %( /%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*  %(D' =-o6Bbox(out)*<3<*  D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =-o6Bbox(out)*<3<*"D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =-o6Bbox(out)*<3<*!D' =%(D|' =%(D$' =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' =%(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+  ++0+# +<  K0 %%**q%(  ~  s * p`   z C   C,$D 0  3 r_ж_ж8c?"`S   IA   3 r_ж_жf38c? ,  EB   3 r_ж_жf38c? c  EC   3 r_ж_ж8c?c   ED   C x_ж_ж8c?   EE    3 r_ж_ж8c?   EF    3 rH_ж_ж8c?c   IG    # l_ж_ж8c?C.    f_ж_ж8c?d,$ 0 dTwierdzenie o 5 barwach Ka|d map mo|na pokolorowa co najwy|ej 5 kolorami w taki sposb, |e dwa ssiadujce obszary maj r|ne kolory." 2r f    fl_ж_ж8c?d] ,$ 0 V1922 < 26 paDstw 1971 < 96 paDstw, 2, ,   f_ж_ж8c? 7 ,$ 0 &1977 Appel, Haken, 2@   f_ж_ж8c? * ,$ 0 Twierdzenie Dowolny graf planarny mo|na pokolorowa co najwy|ej 4 kolorami.0L 2 @ L z  Zw   Zw,$D 02  C x`_ж_ж8c?66   GA 2  C xL_ж_ж8c?6   GE 2  C xD_ж_ж8c?6 w GG 2  C x_ж_ж8c?6 :  GB 2  C x_ж_ж8c?6   GC 2  C xL!_ж_ж8c?   GD 2  C xD&_ж_ж8c? :  GF 2  C x"_ж_ж8c? Z  GH B   `_ж_жD8c? 6 B   `_ж_жD8c?  B   `_ж_жD8c?p p B   `_ж_жD8c?  B   `_ж_жD8c?  B B  `_ж_жD8c? c B   `_ж_жD8c? ) B    `_ж_жD8c?) 6) B !  `_ж_жD8c?V V B "  `_ж_жD8c?9 F B #B  `_ж_жD8c?9 V B $  `_ж_жD8c?V 6 B %  `_ж_жD8c?  B &B  `_ж_жD8c? ^ ' s _ж_жBC{DEF8c?{l]{)S@  "` I^ ( s _ж_жBACDEF8c?k <rA@  "`b  n ) s 0_ж_жBC?DE(F8c? /\<N@yh? @   "` YF " * 0 # c @U 0U 0H  0޽h ? ff___33h`___PPT10@.7j+w |D4' = @B D' = @BA?%,( < +O%,( < +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' =%(DP' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B#blinds(vertical)*<3<*D' =%(D' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(DP' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B#blinds(vertical)*<3<*++0+  ++0+  ++0+ ++0+ +A  K0 vBnB@^^0B(  0~ 0 s *;w )    0  f`=_ж_ж8c?M ,$ 0 Sekwencyjne wierzchoBkowe kolorowanie grafu Wej[cie: Dany jest graf G = <V, E>, gdzie V ={v1,v2,...,vn} definiujca poprawne kolorowanie Wyj[cie: funkcja f : V{1,2,3...} for i :=1 to n do f(vi):= najmniejszy numer koloru, ktry jeszcze nie zostaB u|yty do pokolorowania ssiadw wierzchoBka vi o mniejszych indeksach; od;8I 2,s&z ` &  0  ,$D 02 0  f_ж_ж8c?&2 0  f_ж_ж8c?t2 0  f_ж_ж8c?`  P2 0  f_ж_ж8c?S"}2  0  f_ж_ж8c?FP2  0  f_ж_ж8c? m2  0  f_ж_ж8c?)22  0  f_ж_ж8c?pC2  0  f_ж_ж8c?c& 2 0  f_ж_ж8c?gpB 0B  `_ж_жD8c?B 0  `_ж_жD8c?FV2B 0  `_ж_жD8c?&B 0  `_ж_жD8c?F}B 0  `_ж_жD8c?"B 0B  `_ж_жD8c?CB 0  `_ж_жD8c? 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