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=-o6Bbox(out)*<3<*?Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*$D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(n/%,( < +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<*: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$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' 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=%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*5%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*5D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*5Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*>%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*>D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*>Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*6%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*6D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*6D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*9%(D' =-o6Bbox(out)*<3<*9++0+M ++0+M ++0+M ++0+ M ++0+!M ++0+"M ++0+#M ++0+.M ++0+4M ++0+9M ++0+?M ++0+@M +  K0 ''99 1'(   x   c $M p`  M    0+B#style.visibility<* ;%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* ;D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* ;D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* ;X%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* ;XD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* ;XD{' =%(D#' =%(D' 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P jest pocztkowym podziaBem zbioru V, 3. T pustym stosem (bd tam krawdzie drzewa). 2& <  t ܌_ж_жAԔPapeteria:,$D  0 :{ while (not empty (pq) and card(P)>1) do kr := min(pq); pq := delmin(pq); A := Find(P,kr.pocztek); B := Find(P,kr.koniec); if A B then P := Union(P,B,A); T := push(T,kr) fi od }, 2^x     <  f_ж_жGrHJ[?  0O ,$D  0 lFind (P,x) = ten zbir podziaBu P, do ktrego nale|y x773 < # l0_ж_жGHb 1?  ,$D  0 $Union(P,B,A) = podziaB powstajcy z P przez usunicie zbiorw A, B i doBczenie zbioru A B.,^Y SH < 0޽h ?/ << @ ff3Ιd332z___PPT10+qoD' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D6' =A@BBB%B0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =-6B%barn(outVertical)*<3<*<D' =%(D' =%(D6' =A@BB BB0B%(E' =4 B`BPB`B?<* %( /%,( < +D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =-6B%slide(fromBottom)*<3<*<D' =%(D' =%(D8' =A@BBBB0B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =-6B'checkerboard(down)*<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+< +?I  K0 P@v(  @x @ c $( p`   n @ N_ж_ж 1Z,$ 0 vNiech U bdzie skoDczonym zbiorem, a P podziaBem zbioru U. < 2< < @ N_ж_ж 1WD,$ 0 Rozwa|amy struktur, w ktrej okre[lone s dwie operacje Find(P,x), Union(P,A,B):R 2R@9  @  t _ж_жA 1Papeteria" P* ,$D  0 lFind(P,x) zwraca ten zbir podziaBu P, do ktrego nale|y x, Union(P,A,B) zwraca nowy podziaB zbioru U, ktry powstaje z P przez zastpienie zbiorw A, B ich suma teoriomnogo[ciow . 245 op @  Z_ж_жwG UNd)?PrzykBadArial Black$ k !2',$D 0' @ N_ж_ж 1s cm ,$ 0 u/Niech U={1,2,3,4,5,6} i P ={{1,2},{3},{4,5,6}}.0 20 0 @  @_ж_ж0W>1  ?Jak zapamita podziaB?Arial Black5HY?fHc ` F ,$D  0  @ Nt_ж_ж 1V S,$ 0 Do zapamitania podziaBu wystarczy tablica np. R = [1,1,2,3,3,3].E 2E EB  @ Np_ж_ж 19  ,$   0 Wtedy Find(P,i)= R(i)  2@  @ ND_ж_ж 1 ,$   0 a Union(P,A,B)  2   @  f_ж_жGHq 1 Pc) ,$D  0 ZmieD nazw zbioru na np B, dla wszystkich elementw, ktre nale|aBy do zbioru o nazwie A . ``&G&  @ NP%_ж_ж 1 Y+,$   0 tKoszt Union -liniowy 2& T @  fL*_ж_жGHx 1  t ,$D  0 *Koszt Find jest staBy& H @ 0޽h ?/  @@ @ ff3Ιd332z44___PPT104.+oq+jD2' = @B D1' = @BA?%,( < +O%,( < +D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =-6B'blinds(horizontal)*<3<*@D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =-6B'blinds(horizontal)*<3<*@D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*@D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*@D' =%(D' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*@D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*@D' =-g6B fade*<3<*@Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*@D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*@D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =-6B'blinds(horizontal)*<3<*@D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* @%(D' =-6B'blinds(horizontal)*<3<* @D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* @%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* @D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* @D' =%(D' =%(DX' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =-6B+checkerboard(across)*<3<*@D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* @%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* @D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* @D' =%(D' =%(DX' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* @%(D' =-6B+checkerboard(across)*<3<* @D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* @%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* @D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* @D' =%(D' =%(DV' =A@BBB B0B%(D' =-6B+checkerboard(across)*<3<*@D' =1:Bhidden*o3>+B#style.visibility<*@%(+h+0+@ ++0+@ ++0+@ ++0+@ ++0+ @ ++0+ @ ++0+ @ ++0+ @ ++0+ @ ++0+@ ++0+@ +v  K0 x$p$`55D$(  Dx D c $ 5 p`  5 @ D 3 Z wG UNd)?KosztArial Black$ k !26 z ,$D  0 D N 5_ж_ж 1 ,$  0 Koszt Algorytmu Kruskala w tej implementacji: Koszt utworzenia kolejki priorytetowej = O(m lg m). Koszt 2m operacji Find = O(mlgn), ka|da operacja Find O(lg(n)) Koszt m operacji Union = O(m) , ka|da operacja Union O(1) 2C+ D H5_ж_жi^ 1 G ,$  0 Razem O(m lg n)$ 2f & L D N5_ж_ж 1,$  0 TDrzewa z balansowaniem i kompresj [cie|ek+ 2+ +1 D N(5_ж_ж 1< K,$  0 c.d. Struktura Find-Union 2f& az  0  D  0 ,$D   0~B  DB N_ж_жD1 ~B  D N_ж_жD1   D N$5_ж_ж10  EB 2 B  D N_ж_жD 10 0 ,$D  0yz `   D ` ,$D  0 D N|(5_ж_ж 1 EA 2 ~B DB N_ж_жD 1`p~B D N_ж_жD 1p~B DB N_ж_жD 1@p~B D N_ж_жD 1p~B DB N_ж_жD 1Pp@ ~B D N_ж_жD 1@p oz  @  D  @ ,$D  0tT ` D#  p p~B DB N_ж_жD 1` ~B D N_ж_жD 1P~B DB N_ж_жD 1@P~B D N_ж_жD 1 D Nd/5_ж_ж 1@ 0  EA 2 ~B DB N_ж_жD1 `P ~B D N_ж_жD1P `@~B D N_ж_жD 1 pP `~B DB N_ж_жD 1 pp ~B  D N_ж_жD 1p p0 2 !D T_ж_ж 1 ` ,$D 0B "D N_ж_жD21@p 0 ,$D   0B #D@ N_ж_жD21p p,$D   0B $D N_ж_жD21p ,$D   0B %D N_ж_жD 1 ,$D  02 &D N_ж_ж3 1@ p,$D   02 'D N_ж_ж2 1 0 ,$D   0z  (D ,$D  0~B )DB N_ж_жD1` ~B *D N_ж_жD21`PP~B +DB N_ж_жD21`~B ,D N_ж_жD1`~B -DB N_ж_жD1~B .D N_ж_жD21``2 /D T_ж_ж 10 0D N8:5_ж_ж 10P EA 2 ~B 1D N_ж_жD 1`~B 2D N_ж_жD1@~B 3DB N_ж_жD1~2 4D N_ж_ж3 1P~2 5D N_ж_ж2 1  H D 0޽h ? @ ff3Ιd332z7Q/Q___PPT10Q+CUDP' = @B DO' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*D+%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D+D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D+D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*DD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*DDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* D%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* DD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* DDR' =%(D' =%(D' =4@BBBB%(E' =4 B`BPB`B?<* %(/%,( < +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' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* D%(D' =-o6Bbox(out)*<3<* DDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*DD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*DD' =%(D' =%(D3' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*!D%(D' =-o6Bbox(out)*<3<*!DDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*"D%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*"DD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*"DDR' =%(D' =%(D' =4@BBBB%(E' =4 B`BPB`B?<* %(3/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*&D%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*&DD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*&DDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#D%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*#DD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*#DDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*'D%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*'DD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*'DDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$D%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*$DD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*$DDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%D%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*%DD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*%DDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(D%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(DD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(DD' =%(Do' =%(D' =4@BBB B%(E' =4 B`BPB`B?<* %(W/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =-o6Bbox(out)*<3<*DD' =%(D?' =%(D' =A@BBBB0B@B%(D' =1:Bvisible*o3>+B#style.visibility<*D.%(D' ,=+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D.D' ,=+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D.D' =%(D?' =%(D' =A@BBBB0B@B%(D' =1:Bvisible*o3>+B#style.visibility<*D.%(D' ,=+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D.D' ,=+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D.D' =%(D|' =%(D$' =A@BBB B0B%(E' =4 B`BPB`B?<* %(h/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =-o6Bbox(out)*<3<*D++0+D5 ++0+D5 ++0+D5 ++0+D5 +*/ 0 0z(  0^ 0 S     0 c $q @   pFDowd przez sprowadzenie do sprzeczno[ci: Przypu[my, |e istnieje takie drzewo do ktorego krawdz e nie nalezy i ktrego koszt jest minimalny. 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