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Mirkowska, ASD_01 WprowadzenieO =ALGORYTMY I STRUKTURY DANYCH `WYKAAD 01 Wprowadzenie Gra|yna Mirkowska PJWSTK 1  Plan wykBadu hOrganizacja wykBadu i wiczeD z ASD. Cel wykBadu. Zapoznanie studentw z podstawowym zestawem algorytmw realizujcych zadania typu wyszukiwanie, sortowanie, oraz z najcz[ciej wykorzystywanymi strukturami danych: stosami, kolejkami, sBownikami, kolejkami priorytetowymi i drzewami. Przedstawione zostan rwnie| zasadnicze problemy algorytmiki zwizane z analiz poprawno[ci i kosztu algorytmw. O czym bdzie mowa w tym wykBadzie? .5"j&O [FO czym bdzie mowa w tym wykBadzie? $ Jak formuBowa (specyfikowa) zadania i algorytmy? Jak porwnywa algorytmy? Co to jest struktura danych? Jak weryfikowa algorytm (program)? Czy zawsze mo|na znalez lepsze rozwizanie? Czy zawsze istnieje algorytm rozwizujcy dany problem?  >Od problemu do jego rozwizania  Co to jest algorytm? <Algorytm - metoda postpowania   0Jak zapisywa algorytmy? 2Jak porwnywa algorytmy?  Co to jest struktura danych?  PrzykBad1  (Poprawno[ algorytmu <CaBkowita poprawno[ algorytmu Koszt algorytmu  PrzykBady  6ZBo|ono[ czasowa algorytmu Notacja asymptotyczna 6Porwnywanie rzdw funkcji HPorwnanie szybko[ci wzrostu funkcji %4ZBo|ono[ a rozmiar i czas HCzy szybko[ mo|e pokona zBo|ono[?% 2% %PrzykBad 2   Niezmiennik "PrzykBad 3 #<Czy algorytm P zatrzymuje si? /T!$%:  ` @ ff3Ιd332z` @ ff3Ιd332z` 999MMM` fffPP3f>?" dd@ ?4Zd@ d " @ ` n?" dd@   @@``@n?" dd@  @@``PV    @ ` ` p>> K0 `X (  F   `  0PP`  S 0AminispirlB  <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 $lJ p   J X* 2    c $J    J Z*(2    c $J   J Z*(2 Z  BsZ޽h))?? @ ff3Ιd332z Notatnik&  K0 0 <(   FF       XA StationeryPP`   S 0AminispirH  <P8 ?^~ 8 ALGORYTMY I STRUKTURY DANYCHKliknij, aby edytowa styl tytuBu z WzorcaGG  c $,8 3   8 ZKliknij, aby edytowa styl podtytuBu z Wzorca..  c $@8 ^# 8 X* 2f   c $Ħ8 ^~#  8 Z*(2f   c $l8 ~# 8 Z*(2f Z  BsZ޽h))?? @ ff3Ιd332z| 0 @( 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*   6  `   V* H  0޽h ? ̙3380___PPT10.  ny K0  (  l  C 8 ^~ 8 l  C t8 3   8 H  0޽h ? @ ff3Ιd332zy___PPT10Y+D=' = @B +u  K0 |P@(  @l @ C P p`    @ C Pp  " Pp @ nG2DokBadny program wykBadu @ U 0U 0H @ 0޽h ? @ ff3Ιd332zy___PPT10Y+D=' = @B +  K0 `$( 8,/ r  S X p`   r  S ,Pp  H  0޽h ? @ ff3Ιd332zy___PPT10Y+D=' = @B +#  K0 xpp T( u@@U@ Tl T C t$ p`    T 0 'G ,$  0 r.SformuBowanie problemu. 2 r T <G9HG ,$D  0 T 0+  ,$  0 n*Rozwizanie problemu. 2   T xZw?ALGORYTMArial Black@ ,$D  0r  T V`!8APapeteriaG  ,$D  0 tPrzykBad: Dany jest cig liczb. Znalez najwiksz z nich.; 2; ;N  T V/APapeteria@ `z,$D 0 NNiech max ma warto[ rwn pierwszemu elementowi cigu. Porwnaj max z kolejnymi elementami cigu i je[li spotkasz warto[ wiksz, przyjmij j jako now warto[ max. 2 H T 0޽h ? @ ff3Ιd332z___PPT10+PD' = @B D' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*TDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*TD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*TDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* T%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* TD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* T%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* TDA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* T%(++0+T ++0+T ++0+ T ++0+ T +"  K0 ) ! D ( q Dl D C C p`   z D 0lE R,$  0 Algorytm to metoda postpowania, ktra prowadzi do rozwizania jakiego[ problemu.R 2R R D 0J p ,$  0 .Algorytm, to skoDczony cig etapw, ktre pozwalaj przeksztaBci dane informacje wej[ciowe w informacje wyj[ciowe. v 2v v  D 0H0` ,$ 0 s!Al-Khowrizm (Persja, 8-9w.n.e.)" 2"  D hSAAPapeteria ,$D  0 R WBcz gaz; Zagotuj wod; Wsyp do szklanki kaw rozpuszczaln; Zalej kaw wrzc wod; Dosyp cukru, je[li lubisz; Poczekaj kilka minut; 2  D h@YAAPapeteriaP  ,$D  0 r(Euklides) Dopki x r|ne od y wykonuj: Je|eli x>y, to odejmij y od x i wynik podstaw na x; W przeciwnym przypadku od y odejmij x i wynik podstaw na y; koniec dopki wynikiem jest y 2 H D 0޽h ? @ ff3Ιd332zJB___PPT10"+<JD' = @B D' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*DR%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*DRD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*DRD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*DD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*DD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*DD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*DD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Dv%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*DvD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*DvD' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* D"%(D' =-o6Bbox(out)*<3<* D"++0+D ++0+D ++0+ D ++0+D ++0+D +.  K0    p ( x pl p C hm p`    p bDoAA)Papeteria,$D 0 4Funkcje rekurencyjne, Algorytmy Markova, Maszyny Turinga, Automaty, Wyra|enia regularne_ 2_@# "z p @@ p G ,$D 0 p 6tp @ HDane  p 6x0 @ LAlgorytm     p 6`}`p @@ JWyniki    p <HA)B,$D  0 k Teza Churcha 2&  p 0s q,$ 0 (PrzykBad dla algorytmu Euklidesa Dane x =21, y =12. (x,y) (21,12) (9,12) (9,3) (6,3) (3,3) Wynik 3"l 2d l p xG(HbA$ Niebieska lignina@@,$D  0 Pstan pamici przed wykonaniem algorytmu *( 2( )~ p xG(HbA$ Niebieska lignina@@,$D  0 Hstan pamici po wykonaniu algorytmu *$ 2$ % p <Ԕ"` ,',$D 0 Dopki x r|ne od y, od wikszej z liczb x, y odejmuj mniejsz. Wynikiem jest y.R 2R RH p 0޽h ?/ pp @ ff3Ιd332z  ___PPT10 +EDX' = @B D' = @BA?%,( < +O%,( < +Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*pD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*pD_' =%(D' =%(D' =A@BBBB0B%(E' =4 B`BPB`B?<*%( /%,( < +D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*pD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*pD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*pD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*pD' =%(D' =%(DX' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =-6B+checkerboard(across)*<3<*pD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* p%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* pD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* pD' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<*%("/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =-o6Bbox(out)*<3<*pD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* p%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* pD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* p+P+0+p ++0+ p ++0+ p ++0+p ++0+p ++0+p +V  K0 jb<(    <l < C |8 p`  8 b < 0Ȱ/5,$ 0 Najcz[ciej formuBujemy zadania i problemy w jzyku naturalnym ale...F 2F F < 0l Y ,$ 0 8instrukcja warunkowa: if test then {Instrukcje} else{Instrukcje} fi instrukcja iteracji(ptla): while test do{Instrukcje}od instrukcja zBo|enia: {Instrukcja1; Instrukcja 2;...Instrukcja n-ta; } instrukcja przypisania: x := wyra|enie algebraiczne;d 202 U9$ < 00mY,$  0 Ljzyk naturalny nie jest jednoznaczny.' 2' ' <  W3fԔ?Konstrukcje:Times New RomanjO,$D 0H < 0޽h ? @ ff3Ιd332z|t___PPT10T+KD' = @B DK' = @BA?%,( < +O%,( < +D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =-6B'blinds(horizontal)*<3<*<D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*<D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*<D' =%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =-u6Bdiamond(in)*<3<*<D' =%(D' =%(DR' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =-6B%slide(fromBottom)*<3<*<++0+< ++0+< ++0+< +'  K0 P(  FHz Pl P C x8 p`   ~ P 0Ѕ` ,$  0  prostota czytelno[ dBugo[ kodu poprawno[ czas realizacji zajto[ pamiciS 2S S P hՅAA)Papeteria ` ,$D 0 Idealny algorytm to taki, ktry ma prosty kod, jest napisany w oglnie dostpnym jzyku programowania, Batwo go zrozumie, liczy szybko, nie wymaga du|o miejsca w pamici i zawsze daje poprawne wyniki. 2  P z W3fԔ?KryteriaTimes New Roman 0 jH P 0޽h ? @ ff3Ιd332zP!H!___PPT10(!+ٺD ' = @B DW ' = @BA?%,( < +O%,( < +D' =%(D?' =%(D' =A@BBBB0B@B%(D' =1:Bvisible*o3>+B#style.visibility<*P %(D' ,=+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*P D' ,=+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*P D' =%(D?' =%(D' =A@BBBB0B@B%(D' =1:Bvisible*o3>+B#style.visibility<*P %(D' ,=+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*P D' ,=+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*P D' =%(D?' =%(D' =A@BBBB0B@B%(D' =1:Bvisible*o3>+B#style.visibility<*P$%(D' ,=+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*P$D' ,=+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*P$D' =%(D?' =%(D' =A@BBBB0B@B%(D' =1:Bvisible*o3>+B#style.visibility<*P$0%(D' ,=+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*P$0D' ,=+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*P$0D' =%(D?' =%(D' =A@BBBB0B@B%(D' =1:Bvisible*o3>+B#style.visibility<*P0A%(D' ,=+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*P0AD' ,=+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*P0AD' =%(D?' =%(D' =A@BBBB0B@B%(D' =1:Bvisible*o3>+B#style.visibility<*PAS%(D' ,=+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*PASD' ,=+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*PASD' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<*%()/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-o6Bbox(out)*<3<*P+p+0+P ++0+P +#  K0    t8 (  tr t S < p`    t 0,$  0 SformuBowanie problemu algorytmicznego wymaga zwykle okre[lenia [rodowiska, ktrego problem dotyczy. f 2f fF t bAԔPapeteria" `gG ,$  0 r,Problem  Znalez najwikszy element w danym cigu wymaga okre[lenia czym s elementy (np.. Liczbami, zbiorami, dokumentami) i jak si je porwnuje. 2  t 0%  ,$  0 &Algorytm, ktry ma realizowa pewn metod rozwizania problemu musi zna to [rodowisko i mc si nim posBugiwa.r 2r rv t tdA APapier gazetowy"  ,$D  0 LStruktur danych bdziemy nazywali system relacyjny, ktrego uniwersum okre[la warto[ci zmiennych a operacje i relacje dostarczaj narzdzi do realizacji algorytmu. 2 N  t C ZwG UNd)?DefinicjaArial Black$ k !2% 7,$D  0H t 0޽h ? @ ff3Ιd332z___PPT10+D' = @B DC' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*tf%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*tfD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*tfD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*tD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*tD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*tr%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*trD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*trDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* t%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* tD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* tD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*tD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*t++0+t ++0+t ++0+t ++0+t +)  K0  z  L ( xBxB Ll L C   p`   z L V APapeteria @ ,$D  0 while (x y) { if x >y then x := x  y else y := y  x fi } return x&n 2 d L 0P0 0J,$  0 ~Je[li rozwa|ymy ten algorytm w strukturze liczb caBkowitych podajc jako pocztkowe warto[ci x=a i y=b, to algorytm zwrci jako wynik nwd(a,b)  2&L L 0  ,$  0 rJe[li rozwa|ymy ten algorytm w strukturze, ktrej uniwersum skBada si z odcinkw na prostej, relacja > pozwala porwna dBugo[ci odcinkw, natomiast operacja  daje w wyniku r|nic odcinkw, to algorytm zwraca jako wynik najdBu|szy odcinek, ktry mie[ci si caBkowit ilo[ razy w danych pocztkowo odcinkach.: 2: : L t W3fԔ?UwagaTimes New Roman  ,$D 05  L xlGHUA$ Niebieska lignina` @ ,$D  0 YCzasami daje wynik! 9  L xh&GmHgA$ Niebieska lignina0 ,$D  0 ]Zawsze daje daje wynik! H L 0޽h ?/  L L @ ff3Ιd332z___PPT10+UDd' = @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<*LD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*LD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*LD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L:%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*L:D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*L:D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-o6Bbox(out)*<3<*LD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* L%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* LD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* LD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* L%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* LD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* L++0+L ++0+L ++0+L ++0+ L ++0+ L +9  K0  ~  $ (  $l $ C P p`   B $ 0,$  0 hIntuicyjnie, poprawno[ = zgodno[ z zamierzeniami.5 25 5 $  W3fԔ?SpecyfikacjaTimes New Roman Z,$D  0^ $ 00 ,$  0 Specyfikacj algorytmu nazywa bdziemy par warunkw (wBasno[ci) C 2C C $ HXȞGwHf  ,$D  0 j$Warunek pocztkowy  $ HGHk 0 p ,$D  0 dWarunek koDcowy   $ 00 T,$ 0  < wp , wk > 2 $@b  $@ <n7G;`@,$D  0H  $ V APapeteria@ `z,$D  0 Algorytm Alg dziaBajcy w strukturze danych S jest cz[ciowo poprawny ze wzgldu na specyfikacj <wp, wk> wttw dla wszystkich danych speBniajcych warunek pocztkowy, je|eli algorytm zatrzyma si, to uzyskane wyniki speBniaj warunek koDcowy., 23Zbc  $ fA$Niebieska lignina p  ,$D   0 S |= {wp} Alg {wk}" 2@H $ 0޽h ?/ $$ @ ff3Ιd332z++___PPT10h++fD)' = @B D)' = @BA?%,( < +O%,( < +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<*$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<* $Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(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<*$C%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*$CD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*$CD' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* $ %(D' =-o6Bbox(out)*<3<* $ D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*$D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*$D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* $%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* $D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* $D_' =%(D' =%(D' =A@BBBB0B%(E' =4 B`BPB`B?<*%(4/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* $%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* $D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* $++0+$ ++0+$ ++0+$ ++0+$ ++0+ $ ++0+ $ ++0+ $ +  K0  X( 7J8 Xl X C . p`   X X 0 @ X 0/P@ `wp  2$ X 0<5P P `wk  2$ X 09 t SAlg  2$  X V>APapeteria ,$D  0  Powiemy, |e algorytm Alg dziaBajcy w strukturze danych S jest caBkowicie poprawny ze wzgldu na specyfikacj <wp,wk> wttw dla wszystkich danych w strukturze S speBniajcych warunek pocztkowy wp, algorytm zatrzymuje si i daje wyniki speBniajce warunek koDcowy wk. &  2@tpGD  X fHA$Niebieska ligninaP @,$D   0 (S |= {wp} Alg{wk}H 2 @H X 0޽h ? @ ff3Ιd332z  ___PPT10| + D ' = @B D ' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*XD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*XD_' =%(D' =%(D' =A@BBBB0B%(E' =4 B`BPB`B?<*%( /%,( < +D' =1:Bvisible*o3>+B#style.visibility<* X%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* XD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* X+p+0+X ++0+ X +=7  K0 nf@  (   l  C < p`     0,$  0 RMiary kosztu:  2   . W3fԔ?pami i czasTimes New RomanPD,$D  0r  0` ` ,$  0  Liczba instrukcji liczba operacji arytmetycznych liczba wywoBaD proceduryM 2M Mr  <HdII ,$D 0b  BG9HKI5 0 ,$D 0D  0   ,$  0 j Liczba zmiennych ilo[ miejsca potrzebna dla danych6 26 6^  0` p0f,$  0 Oglnie: wybr miary zale|y od typu problemu, rodzaju rozwizania. D 2D DH  0޽h ? @ ff3Ιd332z_.W.___PPT107.+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<* 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<* 3%(D' ,=+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* 3D' ,=+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* 3D' =%(D?' =%(D' =A@BBBB0B@B%(D' =1:Bvisible*o3>+B#style.visibility<* 3M%(D' ,=+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* 3MD' ,=+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* 3MD' =%(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@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<* 6%(D' ,=+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* 6D' ,=+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* 6D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* D%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* DD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* D++0+  ++0+  ++0+  ++0+  +FL  K0 !P4 (  4l 4 C - p`    4 0/ ,$  0 h$ Mno|enie macierzy 2  4 010@P,$  0 cWyszukiwanie elementu w tablicy 2   4 04P ,$  0 O Sortowanie  2   4 V3APapeteria` f ,$D   0 Majc dany algorytm, konkretne [rodowisko i konkretne dane mo|emy policzy liczb operacji dominujcych. j 2j j 4 ( W3fԔ?Operacja dominujcaTimes New Roman p,$D  0 4 0(? ,$  0 ROperacje + , * 2   4 0B 0@,$  0 P porwnywanie 2    4 6H00 0 ,$D  0  4 x W3fԔ?ProblemTimes New Roman`,$D  0  4 0G @ ,$   0 aKoszt algorytmu dla danych d: 2  4 ~ W3fԔ?t (Alg, d)Times New Roman J ,$D   0 4 H8Gk`HD  @ ,$D   0 Lalgorytm    4 H`PGH  ,$D   0 Hdane H 4 0޽h ?/ 44 @ ff3Ιd332z==___PPT10=+l>EDy;' = @B D4;' = @BA?%,( < +O%,( < +Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* 4%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* 4D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* 4Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*4D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*4D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*4D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*4D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*4D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*4D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4 %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*4 D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*4 D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4 %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*4 D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*4 Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* 4%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* 4D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* 4D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* 4 %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* 4 D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* 4 D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*4D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*4D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* 4%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* 4D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* 4Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*4D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*4D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*4D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*4D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*4D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*4++0+48 ++0+48 ++0+4 ++0+4 ++0+4 ++0+ 4 ++0+ 4 ++0+4 ++0+4 +*  K0 ` HT( > Hl H C (n p`   , H VpoAPapeteriaP LDefinicja ZBo|ono[ czasowa to liczba operacji dominujcych (podstawowych) wykonanych przez algorytm w czasie jego realizacji, wyra|ona jako funkcja rozmiaru danych. * 2   H z W3fԔ?T(Alg,n)Times New Roman@,$D 0 H 0hw@pj ,$  0 LNiech Dn bdzie zbiorem danych rozmiaru n dla pewnego problemu P oraz A algorytmem rozwizujcym problem P.,l 2 d&dl  p0   H p0 ,$D  0l H V rAPapeteria p0  BW(Alg,n) = sup {t(Alg,d) : d Dn}8" 2@  H 0 W3fԔ?zBo|ono[ pesymistycznaTimes New Roman0 l @` p  H` @p,$D  0p H V8APapeteria@P p NA(Alg,n) = S{ p(d) * t(Alg,d) : d Dn}J( 2 &$  H $ W3fԔ?zBo|ono[ [redniaTimes New Roman0`   H 0x : ,$ 0 Uwaga1.Faktyczny czas wykonania algorytmu jest proporcjonalny do zBo|ono[ci czasowej."V 2Q V  H 00 j,$ 0 Uwaga2 Czas wykonania algorytmu jest bardziej interesujcy dla du|ych n ni| dla maBych. *Y 2R YH H 0޽h ? @ ff3Ιd332z___PPT10+;DD' = @B D' = @BA?%,( < +O%,( < +D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-o6Bbox(out)*<3<*HD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Hl%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*HlD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*HlDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* H%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* HD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* HDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* H%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* HD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* HD' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* HV%(D' =-o6Bbox(out)*<3<* HVD' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<*%(!/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* HY%(D' =-o6Bbox(out)*<3<* HY++0+H ++0+ H ++0+ H +S'  K0 , $ p ( (  (l ( C  p`    ( 0t @@ ,Niech f , g : N R+. 0 2  ( VhAPapeteria,$D  0 FPowiemy, |e g jest co najwy|ej rzdu f wttw ($c>0)($noN)("n>no) g(n) c f(n). \Q 2- &'& ( VAPapeteriapv ,$D  0 XPowiemy, |e g jest co najmniej rzdu f wttw ($c>0)($noN)("n>no) c*f(n) g(n). nQ 2-&'& ( 00@p ,$ 0 g = W (f )> 2       ( 0@=,$ 0 x g = O (f )> 2      ( 0  p ,$ 0 g = Q (f )> 2      ( VAPapeteria 0 ,$D  0 Powiemy, |e rzdy funkcji f i g s takie same, wttw g= O(f) i f = O(g). J 2J&0H ( 0޽h ? @ ff3Ιd332z___PPT10+D#' = @B D' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(D' =%(D|' =%(D$' =A@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$' =A@BBB B0B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*( %(D' =-o6Bbox(out)*<3<*( D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* (%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* (D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* (D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<*%(!/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* ( %(D' =-o6Bbox(out)*<3<* ( +P+0+( ++0+( ++0+( ++0+ ( ++0+ ( ++0+ ( +-  K0   , (   ,l , C D p`    , 0 P@J,$  0 |PrzykBad 1 Niech f(n)=100n, g(n)= 2n+100, h(n) = 0.1 n2 +n.@? 2 /  ?F , 0@0,$  0 Mamy f = O(n) f =W (n) g= O(n2) g = Q(n) h = O(n2) = O(n3 ) h O(n) h = W (n) f 2       f , bAA)Papeteria,$D 0 Lemat (O porwnywaniu rzdw funkcji) Niech lim n f(n)/g(n) = c. Wtedy 1. Je|eli c 0 to f i g s tego samego rzdu. 2. Je|eli c= 0, to f = O(g) oraz f W (g). 3. Je|eli c=+ , to f ma rzd wikszy ni| g, g = O(f) i g W (f).  2/ "E*&0 , 0P,$  0 T&PrzykBad 2 f(n) = 0.3 n3 + 10n + 100 g(n)= n3 h(n) = log n lim n f(n)/g(n)= 0.3 Czyli f = Q(g) lim f(n)/h(n) = + Czyli h = O(f), h W(f). 2    @=$-H , 0޽h ? @ ff3Ιd332z!!___PPT10 +D' = @B D' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,?%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*,?D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*,?D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,f%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*,fD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*,fD' =%(D|' =%(D$' =A@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<*,=T%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*,=TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*,=TD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,Td%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*,TdD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*,TdD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,d%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*,dD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*,d++0+, ++0+, ++0+, ++0+, +a'  K0 vn\(  \l \ C X8 p`  8  \ h[8AA)PapeteriaP  ,$D  0 Powiemy, |e algorytm Alg ma zBo|ono[ czasow wielomianow wttw T(Alg,n)= Q(na) a N wykBadnicz wttw T(Alg,n) = Q(an) a R+ liniow wttw T(Alg,n)= Q(n) kwadratow wttw T(Alg,n)= Q(n2) logarytmiczn wttw T(Alg,n)= Q(lg n)  2K  !  ;   \ 0s8p  ,$  0 <lg n! = S i=1..n lg i n lg nB 2Nl    \ p` ,$D  0 \ 6H|8   < fB \ 6D @ fB \ 6D @@`  \ 0 @` \ 0P @` \ 0 @` \ 0 P@` \ 0 ` p@ \  BCDE@F  L`H8(p H p @     P @ \ 0,80 ph  Vlg n 2 \ c $A I?? 08 I$D  0 \ 4 W3fԔ?A co to za funkcja lg n!?Times New Roman `. ,$D 0 \ 6G 0@ U 0U 0H \ 0޽h ? @ ff3Ιd332z{s___PPT10S+9D' = @B D' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*\D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*\D' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<*%( /%,( < +D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-o6Bbox(out)*<3<*\D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*\D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*\Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*\D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*\Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*\D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*\+p+0+\8 ++0+\8 + K0 )!!`(  `dB ` <DԔ dB ` <DԔ 0P XB ` 0D00 XB ` 0DPP XB ` 0Dpp XB ` 0D XB  ` 0D XB  ` 0D XB  ` 0D @ XB  ` 0D` @` XB  ` 0D@@@XB ` 0D @ XB ` 0D@^B ` 6Dp  ` B CDE4Fp 8$p`Pppx  h0 @    @p  `  B`CDE(Fp `l( 8@ 0` @   p XB ` 0D XB ` 0D@@ XB ` 0D XB ` 0D@XB ` 0D@ ` s *?0   O f(n)= log n 2   ` s *C @ Jf(n)=n 2  ` s *8Bp  _ f(n) = 2n& 2  `  BC DELF$p `x ` 0  @p `P P@x@        ` s *L0  c f(n) =0.25 n2& 2    ` 0Q P@ HPorwnanie szybko[ci wzrostu funkcji% 2$ %" !` 6G/ P0@U 0U 0H ` 0޽h ? @ ff3Ιd332zy___PPT10Y+D=' = @B +H  K0 //6@d0 >/(  d d C 8X p`<$D 0    4d 0Y pJ ,$  0 $ Jaki jest maksymalny rozmiar problemu, ktry mo|na rozwiza w ustalonym czasie, znajc zBo|ono[ algorytmu?p 2p pt 5d 0_,$  0 Ile czasu potrzeba na rozwizanie zadania o ustalonym rozmiarze i zBo|ono[ci?N 2N Nl p    +d bĚAPapeteria@  b63*103,   ,d bAPapeteria @  _106,   -d bdAPapeteria   _103,   .d bhAPapeteria   H19  /d bAPapeteria @  b36*108,   0d bAPapeteria@  Y13* 107"   1d b\APapeteria  Y60* 103"   2d bAPapeteria  H31 TB 9d c $Dp `  :d 00p D  LT(A,n) 2  ;d 0L t  Jczas 2 ^l `  @d` ,$D 0 d ZA Pergamin`  d `A Pergamin `@ an=102,    d `A Pergamin@``  bn= 104,    d `A Pergamin   ^n3,    d `A Pergamin`  Xlg n  d `hA Pergamin@  Gn   d `A Pergamin@   fn lg n& d `A Pergamin    ^n2,   d `A Pergamin  ^2n,   d bAPapeteria  @ H1s  d bXAPapeteria @`  a11dni,  d b\APapeteria` @ a6.6ms,  d bAPapeteria`@`  }13.3 ms,& d bDAPapeteria@  @ K0.6ms  d b\APapeteria @ @ K0.1ms  d bAPapeteria  @ J10ms  d b( APapeteria @ b106lat,   d bAPapeteria@@ `  J10ms  d bpAPapeteria@ @ `  J0.1s  d bAPapeteria @ `  J100s  d b`APapeteria@`  c10100 l,  TB =d c $D`  >d 0 LT(A,n) 2  ?d 0"`@4 Lwymiar 2 H d 0޽h ? @ ff3Ιd332zsk___PPT10K.+fD{' = @B D6' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*dD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*dD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*5dN%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*5dND' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*5dNDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*@d%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*@dD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*@dD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4dp%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*4dpD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*4dpDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*B ppt_x<*B ppt_y<*APapeteria`M  Vlg n  l b@CAPapeteriaM :  En   l b|GAPapeteria: & Xn2&   l bhKAPapeteria X2n&   l bDPAPapeteria& Ps4  l bXTAPapeteria&0  R2*s4  l b$YAPapeteria`M  Ps1  l bT]APapeteria`M 0  ms1 108  l baAPapeteriaM :  Ps2  l beAPapeteria: & Ps3  l biAPapeteria Ps5  l bmAPapeteriaM : 0  S10*s2  l bhqAPapeteria: &0  d 10*s3(  l buAPapeteria0  E?  l S ~w0e0e  p`   0 l 0y@`,$  0 Mamy 5 algorytmw A1, A2, A3, A4, A5 rozwizujcych ten sam problem. Niech si oznacza maksymalny rozmiar problemu, ktry mo|na rozwiza na komputerze 1 przy pomocy algorytmu Ai w ustalonych t jednostkach czasu. Jaki jest maksymalny rozmiar problemu, ktry mo|na rozwiza w tym samym czasie na komputerze 10 razy szybszym?E 2     ( c @Lb  l 0؂ ,$  0 PrzykBad A5. Dla komputera 1: T(A5,s5) = 2 s5 = t . Dla komputera 2 : T(A5,s5) = 2 s5 = t /10. Szukamy takiego x, |e T(A5,x) = t. Mamy wic t = 10* 2 s5 = 2 x = 2 s5+lg10 . Czyli x = 3.2 + s5. 2    &   $ !     H l 0޽h ? @ ff3Ιd332z6.___PPT10+4D' = @B D=' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*lE%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*lED' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*lEDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*l%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*lD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*lD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* l%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* lD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* l+p+0+l ++0+ l +C*  K0 8X( 2z4) 8l 8 C W p`     8 VXAPapeteria@`  ,$D  0 f{ i:= 1; k := 1; x:=0; while (i n){ x := x + k; k := k + 2; i := i + 1; } return x }&o 2"L@E6 8 0_ 0@,$  0 ^Problem: Obliczy kwadrat liczby naturalnej n. 0 20 0 8 0`b n ,$  0 :Specyfikacja: warunek pocztkowy: nN, n>0 warunek koDcowy: x jest kwadratem liczby n.Ja 2" a 8  W3fԔ?TwierdzenieTimes New RomanP 0,$D 0 8 blkA Papier gazetowy ` 0,$D  0 Algorytm jest caBkowicie poprawny w strukturze liczb naturalnych.B 2B BH 8 0޽h ? @ ff3Ιd332z !!___PPT10 +9D' = @B D' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*80%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*80D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*80D{' =%(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#' =%(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#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*80%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*80D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*80D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*80a%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*80aD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*80aD' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<*%(#/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =-o6Bbox(out)*<3<*8D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*8D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*8++0+8 ++0+8 ++0+8 ++0+8 +4  K0    0T (  0l 0 C  p`   & 0 VAPapeteria  ,$D  0 l{i := 1; k := 1; x :=0; while (i n){ x := x + k; k := k + 2; i := i + 1; } return x }&r 2#N@G* 0 `AR|owa ligninam3 g,$D  0 fk =2i-1, x =(i-1)2" 2   0 0T p  < 2  0 ` AR|owa ligninaS Z* M ,$D  0 Nk = 2i+1 2   0 f<A$Niebieska lignina ,$  0 bi n+1, k = 2i-1, x = S j=1..i-1 (2j-1) = (i-1)2L2 2 2,  0 `pAR|owa lignina`@ Z,$D  0 h x = i2 , 2   p  0 `\AR|owa lignina  ,$D  0 :x = (i-1)2, k = 2i-1, i n+1B 2   J  0 B觥A S` ,$D 0 ^Niezmiennikiem ptli nazywa bdziemy wBasno[ (formuB), ktra je[li jest prawdziwa na pocztku wykonania tre[ci ptli, to jest rwnie| prawdziwa po wykonaniu tre[ci ptli. 2   0  W3fԔ?niezmiennikTimes New Roman  ,$D  0H 0 0޽h ? @ ff3Ιd332zs&k&___PPT10K&+;iD$' = @B Db$' = @BA?%,( < +O%,( < 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Od problemu do jego rozwizaniaCo to jest algorytm?Algorytm - metoda postpowaniaJak zapisywa algorytmy?Jak porwnywa algorytmy?Co to jest struktura danych? Przykad1Poprawno algorytmuCakowita poprawno algorytmuKoszt algorytmu PrzykadyZoono czasowa algorytmuNotacja asymptotycznaPorwnywanie rzdw funkcji%Porwnanie szybkoci wzrostu funkcji Slajd 19Zoono a rozmiar i czas%Czy szybko moe pokona zoono? Przykad 2 Niezmiennik Przykad 3Czy algorytm P zatrzymuje si? 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