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B65 1C;0 4>2545=0 =0ABC?=0 B5>@5<0. 9;5@>289 F8:; 2 3@0DV VA=CT B>4V V BV;L:8 B>4V :>;8: 1) 3@0D - 72 O7=89; 2) 2AV 9>3> 25@EV2:8 <0NBL ?0@=V ABC?5=V. 1.2.2. 5@52 O=89 0;3>@8B<. "5?5@ <>6=0 >13>2>@8B8 0;3>@8B< @VH5==O  H;OE>< 1C4C20==O =09:>@>BH>3> :VABO:>2>3> 45@520. 04;O AB8A;>ABV 1C45<> =07820B8 F59 0;3>@8B< 45@52 O=8<. !?>G0B:C >13>2>@8<> 2;0AB82>ABL A?@O<;V==O. >74828<>AO O:89-=51C4L ;0=FN3, =0?@8:;04, =0 @8A. 5. /:I> A?@0254;820 =5@V2=VABL B@8:CB=8:0, B> d[1,3] d" d[1,2] + d[2,3] V d[3,5] ( d[3,4] + d[4,5]. !:;02H8 FV 42V =5@V2=>ABV, >45@68<> d[1,3] + d[3,5] ( d[1,2] + d[2,3] + d[3,4] + d[4,5]. 0 =5@V2=VABN B@8:CB=8:0 >45@68<>: d[1,5] ( d[1,3] + d[3,5]. AB0B>G=>: d[1,5] ( d[1,2] + d[2,3] + d[3,4] + d[4,5] B65, O:I> A?@0245;820 =5@V2=VABL B@8:CB=8:0, B> 4;O :>6=>3> ;0=FN30 2V@=>, I> 2V4AB0=L 2V4 ?>G0B:C 4> :V=FO ;0=FN30 <5=H5 01> 4>@V2=NT AC<0@=V9 4>268=V 2AVE @515@ ;0=FN30. >25@=5<>AO 4>  9 >?8H5<> @>72 O7CNG89 WW 45@52 O=89 0;3>@8B<. >1C4CT<> =0 2E>4=><C ;0=FN3>2V  =09:>@>BH5 :VABO:>25 45@52> V ?>42>T<> CAV 9>3> @51@0. 45@68<> 3@0D G  72 O7=89, 7 25@EV2:0<8, I> <0NBL BV;L:8 ?0@=V ABC?5=V. >1C4CT<> 59;5@>289 F8:; G, ?>G8=0NG8 7 25@EV2:8 1, F8:; 704020B8<5<> ?5@5;V:>< 25@EV2>:. @>3;O=5<> ?5@5;V: 25@EV2>:, ?>G8=0NG8 7 1 9 1C45<> 70:@5A;N20B8 :>6=C 25@EV2:C, O:0 ?>2B>@NT 265 7CAB@V=CBC 2 ?>A;V4>2=>ABV. 0;8H8BLAO BC@, I> V T @57C;LB0B>< 0;3>@8B<0. @8:;04 1. 0=0 ?>2=0 AVBL, I> ?>:070=0 =0 @8A. 5. =09B8 BC@ 604V1=8< V 45@52 O=8< 0;3>@8B<0<8. -1234561-64871426-711710347-431048114-51157735-76141010117-B01;. 1  VH5==O. 04V1=89 0;3>@8B< (948 4> =091;86G>3> <VAB0 7 <VAB0 1) 40T BC@ 1 (4) 3-(3) 5-(5) 4 (11) 6 (10) 2 (6) 1, 45 157 4C6>: 2:070=V =><5@8 25@EV2>:, 0 2 4C6:0E  4>268=8 @515@. >268=0 BC@0 4>@V2=NT 39, BC@ ?>:070=89 =0 @8A. 5. 2. 5@52 O=89 0;3>@8B< A?>G0B:C 1C4CT :VABO:>25 45@52>, I> ?>:070=5 =0 @8A. 6 HB@8E>2>N ;V=VTN, ?>BV< 59;5@>289 F8:; 1-2-1-3-4-3-5-6-5-3-1, ?>BV< BC@ 1-2-3-4-5-6-1 4>268=>N 43, I> ?>:070=89 ACFV;L=>N ;V=VTN =0 @8A. 5a. "5>@5<0. >E81:0 45@52 O=>3> 0;3>@8B<C 4>@V2=NT 1. >2545==O. V7L<5<> <V=V<0;L=89 BC@ 4>268=>N fB 9 CAC=5<> V7 =L>3> =094>26H5 @51@>. >268=0 >B@8<0=>3> 30<V;LB>=>2>3> ;0=FN30 LHC <5=H5 fB. ;5 F59 6 ;0=FN3 <>6=0 CO28B8 O: :VABO:>25 45@52>, B><C I> F59 ;0=FN3 4>AO30T CAVE 25@EV2>: 9 =5 <0T F8:;V2. >268=0 =09:>@>BH>3> :VABO:>2>3> 45@520 LMT <5=H5 01> 4>@V2=NT LHC. 0T<> ;0=FN6>: =5@V=>AB59: fB > LHC ( LMT(6);5 ?>42>T=5 45@52>  2>=> 6 59;5@>289 3@0D  <8 725;8 4> BC@C H;OE>< 28?@O<;5=V9, >B65, 4>268=0 >B@8<0=>3> 70 0;3>@8B<>< BC@C 704>2V;L=NT =5@V2=>ABV: 2LMT > fA(7)><=>6CNG8 (6) =0 420 V 7 T4=CNG8 A (7), >B@8<CT<> ;0=FN6>: =5@V2=>AB59: 2fB > 2LHC ( 2LMT ( fA,(8)01> 2fB > fA, 72V4:8 fA/fB > 1 + (; ( = 1. "5>@5<C 4>2545=>. "0:8< G8=><, <8 4>25;8, I> 45@52 O=89 0;3>@8B< ?><8;OTBLAO <5=H =V6 2 420 @078. "0:V 0;3>@8B<8 265 =07820NBL ?@81;87=8<8, 0 =5 ?@>AB> 52@VAB8G=8<8. V4><V I5 :V;L:0 ?@>AB8E 0;3>@8B<V2, I> 30@0=BCNBL C 3V@H><C 28?04:C ( = 1. ;O B>3>, I>1 7=09B8 A5@54 =8E 0;3>@8B< 1V;LH B>G=VH89, 70945<> 7 4@C3>3> :V=FO V A?>G0B:C >?8H5<> brute-force enumeration - ?5@51V@ A8;>N 72V@0, O: 9>3> 72CBL 2 0=3;><>2=V9 ;VB5@0BC@V. @>7C<V;>, I> ?>2=89 ?5@51V@ ?@0:B8G=> 70AB>A>2CTBLAO BV;L:8 2 7040G0E <0;>3> @>7<V@C. 03040T<>, I>  7 n <VAB0<8 ?>B@51CT 70 C<>2 ?>2=>3> ?5@51>@C @>73;O40==O (n - 1)!/2 BC@V2 2 A8<5B@8G=V9 7040GV V (n-1)! BC@V2 2 =5A8<5B@8G=V9, 0 D0:B>@V0;, O: ?>:070=> C =0ABC?=V9 B01;8FV, 7@>AB0T 4>2>;V H284:>: 5!10!15!20!25!30!35!40!45!50!~102~106~1012~1018~1025~1032~1040~1047~1056~1064 )>1 ?@>2>48B8 ?>2=89 ?5@51V@ 2 , ?>B@V1=> =02G8B8AO (157 ?>2B>@5=L) 35=5@C20B8 2AV ?5@5AB0=>2:8 =040=>3> G8A;0 m M;5<5=BV2. &5 <>6=0 7@>18B8 :V;L:><0 A?>A>10<8, 0;5 =09?>H8@5=VH89 (B>1B> ?@8:;040T<89 4;O ?5@51>@=8E 0;3>@8B<V2 @VH5==O V=H8E 7040G)  F5 ?5@51V@ 2 ;5:A8:>3@0DVG=><C ?>@O4:C. %09 <0T<> 45O:C 015B:C V =01>@8 A8<2>;V2 015B:8 (;VB5@), I> WE =07K20NBL A;>20<8. VB5@8 2 015BFV 2?>@O4:>20=V: =0?@8:;04, 2 @>AV9AL:V9 015BFV ?>@O4>: ;VB5@: 0 ( 1 ( & ( O (A8<2>; ( G8B05BLAO ?>?5@54=NT). /:I> 7040= ?>@O4>: ;VB5@, <>6=> 2?>@O4:C20B8 V A;>20. !:06V<>, 40=5 A;>2> u = (u1,u2,& ,um)  I> A:;040TBLAO 7 ;VB5@ u1,u2,& ,um,  V A;>2> v = (v1,v2,& ,vb). ">4V O:I> u1 ( v1, B> 9 u ( v, O:I> 6 u1= v1, B> ?>@V2=NNBL 4@C3V ;VB5@8 B>I>. &59 ?>@O4>: A;V2 V =07820TBLAO ;5:A8:>3@0DVG=8<. "><C 2 @>AV9AL:8E A;>2=8:0E A;>2> 0106C@ AB>WBL @0=VH5 A;>20 010:0. !;>2> 1C@ AB>WBL @0=VH5 A;>20 1C@0, B><C I> ?@>1V; 22060TBLAO B0:8<, I> ?>?5@54=NT 1C4L-O:C ;VB5@C 015B:8. >73;O=5<>, A:06V<>, ?5@5AB0=>28 7 ? OB8 5;5<5=BV2, I> ?>7=0G5=V F8D@0<8 1 & 5. 5:A8:>3@0DVG=> ?5@H>N ?5@5AB0=>2>N T 1-2-3-4-5, 4@C3>N  1-2-3-5-4, & , >AB0==L>N  5-4-3-2-1. "@510 CA2V4><8B8 7030;L=89 0;3>@8B< ?5@5B2>@5==O 1C4L-O:>W ?5@5AB0=>28 2 =0ABC?=C. @028;> B0:5: A:06V<>, =040=0 ?5@5AB0=>20 1-3-5-4-2. "@510 ?5@5<VIC20B8AO 274>26 ?5@5AB0=>28 A?@020 =0;V2>, ?>:8 2?5@H5 =5 ?>10G8<> G8A;>, I> <5=H5 =V6 ?>?5@54=T (2 ?@8:;04V F5 3 ?VA;O 5). &5 G8A;> (Pi-1) B@510 71V;LH8B8, ?>AB0282H8 70<VABL =L>3> O:5AL G8A;> 7 B8E, I> @>7B0H>20=V ?@02>@CG, >B Pi 4> Pn. '8A;> 1V;LH5 =V6 Pi-1, >1>2 O7:>2> 7=0945BLAO, B><C I> Pi-1 < Pi. /:I> T :V;L:0 25;8:8E G8A5;, B>, >G5284=>, B@510 AB028B8 =09<5=H5 7 =8E. 5E09 F5 1C45 Pj (j > i  1). >BV< G8A;> Pi-1 B0 2AV G8A;0 2V4 Pi 4> Pn, =5 2@0E>2CNG8 Pj, B@510 2?>@O4:C20B8 70 7@>AB0==O<.  @57C;LB0BV >B@8<0T<> =0ABC?=C ?5@5AB0=>2C, 2 ?@8:;04V  1-4-2-3-5. >BV< >45@68<> 1-4-2-5-3 (B>9 65 0;3>@8B<, 0;5 A?@>I5=89 28?04>:) B>I>. "@510 CA2V4><;N20B8, I> 2  7 n <VAB0<8 =5 ?>B@V1=V 2AV ?5@5AB0=>28 7 n 5;5<5=BV2, B><C I> ?5@5AB0=>28, A:06V<>, 1-3-5-4-2 V 3-5-4-2-1 (>AB0==V9 5;5<5=B 7 T4=0=89 7 ?5@H8<) 7040NBL >48= V B>9 A0<89 BC@. B65 B@510 70DV:AC20B8 ?>G0B:>25 <VAB> 1 B0 ?@8T4=C20B8 4> =L>3> 2AV ?5@5AB0=>28 7 V=H8E 5;5<5=BV2. &59 ?5@51V@ 40AB (n - 1)! @V7=8E BC@V2, B>1B> ?>2=89 ?5@51V@ 2 =5A8<5B@8G=V9  (<8, O: V @0=VH5, 1C45<> @>7@V7=OB8 BC@8 1-3-5-4-2 V 1-2-4-5-3). 0=89 0;3>@8B< >?8A0=89 =0 <>2V 0A:0;L (A<. >40B:8). @8:;04 2. >72 O65<> , I> ?>AB02;5=0 2 ?@8:;04V 1 ;5:A8:>3@0DVG=8< ?5@51>@><. 02545=0 28I5 ?@>3@0<0 =04@C:CT <VAB0, I> A:;040NBL :@0I89 BC@: 1-2-6-5-4-3 B0 9>3> 4>268=C 36. 060=> C4>A:>=0;8B8 ?5@51V@, 70AB>AC202H8 @>7C<.  =0ABC?=><C ?C=:BV 1C45 >?8A0=89 0;3>@8B<, I> @50;V7CT ?@>ABC, 0;5 H8@>:> 70AB>A>20=C V 4C65 :>@8A=C V45N. 1.2.3. 5B>4 3V;>: B0 <56 > V45W <5B>40 3V;>: B0 <56 ?@8E>48;> 1030B> 4>A;V4=8:V2, 0;5 VBB; 7 A?V202B>@0<8 =0 >A=>2V 2:070=>3> <5B>40 @>7@>18;8 240;89 0;3>@8B< @>72 O7:C  9 F8< A?@8O;8 ?>?C;O@870FVW ?>4E>40.  B>3> G0AC <5B>4 3V;>: B0 <56 1C2 7 CA?VE>< 70AB>A>20=89 4> 1030BL>E 7040G. ;O @VH5==O  1C;> ?@84C<0=> 45:V;L:0 V=I8E <>48DV:0FV9 <5B>4C, 0;5 2 1V;LH>ABV ?V4@CG=8:V2 2<:;04TBLAO @>1>B0 VBB;0. 030;L=0 V45O B@82V0;L=0: B@510 ?>4V;8B8 25;8G57=5 G8A;> 20@V0=BV2, H> ?5@518@0NBLAO, =0 :;0A8 9 >B@8<0B8 >FV=:8 (7=87C  2 7040GV <V=V<V70FVV, 725@EC  2 7040GV <0:A8<V70FVV) 4;O F8E :;0AV2, I>1 <0B8 <>6;82VABL 2V4:8=CB8 20@V0=B8 =5 ?> >4=><C, 0 FV;8<8 :;0A0<8. !:;04=VABL T 2 B><C, I>1 7=09B8 B0:5 ?>4V;5==O =0 :;0A8 (3V;:8) V B0:V >FV=:8 (<568), I>1 ?@>F54C@0 1C;0 5D5:B82=>N. -1234561-64871426-711710347-431048114-51157735-76141010117-B01;. 28:;045<> 0;3>@8B< VBB;0 =0 ?@8:;04V 1 ?>?5@54=L>3> @>74V;C. >2B>@=> 70?8H5< <0B@8FC: 0< 1C45 7@CG=VH5 B@0:BC20B8 !ij O: 20@BVABL ?@>W74C 7 <VAB0 i 2 <VAB> j. -1234561-204310420-15146314-10734470-17454402-43673340-7B01;. 3@8?CAB8<>, I> 4>1@89 <5@ <VAB0 j =04@C:C202 =0:07 28?;0GC20B8 :>6=><C :><V2>O65@>2V, I> 2 WE02 2 <VAB>, 5 4>;0@V2. &5 >7=0G0T, I> 2AV;O:89 BC@ ?>45H52H5T =0 5 4>;0@V2, >A:V;L:8 2> 2AV;O:><C BC@V B@510 2 WE0B8 2 <VAB> j. ;5 >A:V;L:8 2AV BC@8 @V2=><V@=> ?>45H52H0;8, B> :>;8H=V9 <V=V<0;L=89 BC@ 1C45 V B5?5@ :>HBC20B8 <5=H 70 2AV. 4>1@89 2G8=>@: <5@0 <>6=0CO28B8 O: 7<5=LH5==O 2AVE G8A5; j-3> AB>21FO <0B@8FV ! =0 5. /:I> 6 <5@ 7010602 18 A?@>2048B8 :><V2>O65@0 7 j-3> <VAB0 V 2AB0=>282 =03>@>4C 70 28W74 C @>7<V@V 10 4>;;0@V2, F5 <>6=0 1C;> 1 28@078B8 2V4=V<0==O< 10 7 CAVE 5;5<5=BV2 j-W AB@>:8. &5 7=>2 18 7<V=8;> 20@BVABL :>6=>3> BC@C, 0;5 <V=V<0;L=89 BC@ 7>AB02AO 1 <V=V<0;L=8<. 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B65 281@0=5 =C;L>25 @51@> 1,2). >7V1 T<> 2AV BC@8 =0 420 :;0A8: BV, I> 2<VICNBL @51@> (1,2), V BV, I> 9>3> =5 2<VICNBL. @> 4@C389 :;0A <>6=0 A:070B8, I> ?@8945BLAO ?@8?;0B8B8 I5 1, >B65 BC@8 FL>3> :;0AC :>HBCNBL 35 01> 1V;LH5. !B>A>2=> ?5@H>3> :;0AC, B> 2 =L><C ?>B@V1=> @>73;O=CB8 <0B@8FN (B01;. 6) 7 28:@5A;5=>W ?5@H>N AB@>:>N V 4@C38< AB>21F5<. 1234561-010336201-1410312-010344501-1354201-06713301-B01;. 513456201141031-01034401-135401-0673301-B01;. 6134562011410303-01034301-135301-0663301-B01;. 7345621410401-13501-063301-B01;. 8 >40B:>2> 2 7<5=H5=V9 <0B@8FV ?>AB02;5=0 701>@>=0 2 :;VBFV (2,1), B><C I> 281@0=5 @51@> (1,2) 9 70<8:0B8 ?5@54G0A=> BC@ @51@>< (2,1) =5<>6=0. <5=H5=C <0B@8FN <>6=0 ?@825AB8 =0 >48= ?> ?5@H><C AB>21FN, >B65 :>6=89 BC@, I> 2V4?>2V40T W9, :>HBCT =5 <5=H5 35. 57C;LB0B =0H8E 30;C65=L 9 >B@8<0==O >FV=>: ?>:070=89 =0 @8A. 6. >;0 CO2;ONBL :;0A8: 25@E=5 :>;>  :;0A 2AVE BC@V2; =V6=T :>;> ;V2>@CG  :;0A 2AVE BC@V2, I> 2<VICNBL @51@> (1,2); =86=T :>;> ?@02>@CG  :;0A 2AVE BC@V2, I> =5 2<VICNBL @51@> (1,2). '8A;0 =04 :>;0<8  >FV=:8 7=87C. @>4>268<> 30;C65==O 2 4>40B=89 1V:: ;V2>@CG-4>=87C. ;O FL>3> >FV=5<> =C;V 2 7<5=H5=V9 <0B@8FV C[1,2] 2 B01;. 7. 0:A8<0;L=0 >FV=:0 2 :;VBFV (3,1) 4>@V2=NT 3. "0:8< G8=><, >FV=:0 4;O 25@EV2:8, I> 7=0E>48BLAO 7=87C ?@02>@CG =0 @8A. 7 T 35 + 3 = 38. ;O >FV=:8 25@EV2:8, I> 7=0E>48BLAO 2=87C ;V2>@CG =0 @8A. 7 B@510 28:@5A;8B8 7 <0B@8FV C[1,2] I5 AB@>:C 3 B0 AB>215FL 1. B@8<0T<> <0B@@8FN C[(1,2),(3,1)] 2 B01;. 8.  FN <0B@8BFN B@510 ?>AB028B8 701>@>=C 2 :;VB:C (2,3), >A:V;L:8 265 ?>1C4>20=89 D@03<5=B BC@C 7 @515@ (1,2) V (3,1), B>1B> [3,1,2], V B@510 701>@>=8B8 ?5@54G0A=5 70<:=5==O (2,3). &O <0B@8FO ?@82>48BLAO ?> AB>21FN =0 1 (B01;. 9), B0:8< G8=>< :>6=89 BC@ 2V4?>2V4=>3> :;0AC (B>1B> BC@, I> <VAB8BL @51@0 (1,2) V (3,1)) :>HBCT 36 9 1V;LH5. 345621310401-135002-063203-B01;. 934621303403-350030B01;. 103440-500B01;. 11FV=NT<> B5?5@ =C;V 2 ?@82545=>9 <0B@8FV C[(1,2),(3,1)] =C;L 7 <0:A8<0;L=>N >FV=:>N 3 7=0E>48BLAO C :;VBFV (6,5). V4 T<=89 20@V0=B <0T >FV=:C 38 + 3 = 41. ;O >B@8<0==O >FV=:8 4>40B=L>3> 20@V-0=BC CAC20T<> AB@>:C 6 8 AB>215FL 5, AB028<> 701@>=C C :;VB:C (5,6), 482. B01;. 10. &N <0B@8FN =5<>6=0 ?@825AB8. B65, >FV=:0 4>40B=L>3> 20@V0=B0 =5 71V;LHCTBLAO (@8A. 8).  FV=NG8 =C;V C <0B@8FV 2 B01;. 10, >B@8<0T<> 30;C65==O 70 281>@>< @51@0 (2,6), 2V4 T<=89 20@V0=B >45@6CT >FV=:C 36 + 3 = 39, 0 4;O >B@8<0==O >FV=:8 4>40B=L>3> 20@V0=BC 28:@5A;NT<> 4@C3C AB@>:C B0 H>AB89 AB>215FL, >45@6CNG8 <0B@8FN 2 B01;. 11.  <0B@8FN B@510 4>1028B8 701>@>=C 2 :;VB:C (5,3), B><C I> 265 ?>1C4>20=89 D@03<5=B BC@C [3,1,2,6,5] V B@510 701>@>=8B8 ?5@54G0A=5 ?>25@=5==O (5,3). "5?5@, :>;8 70;8H8;0AO <0B@8FO 2E2 V7 701>@>=0<8 ?> 4V03>=0;OE, 4>1C4>2CT<> BC@ @51@0<8 (4,3) V (5,4). 8 =5 40@5<=> 30;C78-;8AO, ?> 4>40B=V< 20@V0=B0<. 0@07 >B@8<0=89 BC@: 1 ! 2 ! 6 ! 5 ! 4 ! 3 ! 1, O:89 :>HBCT 36. # @07V 4>AO3=5==O =87C ?> 45@52C ?5@51>@C :;0A BC@V2 72C782AO 4> >4=>3> BC@C, 0 >FV=:0 7=87C ?5@5B2>@8;0AO 2 B>G=C 20@BVABL. B65 2AV :;0A8, I> <0NBL >FV=:C 36 V 28I5 :@0I>3> BC@C =5 <VABOBL. "><C 2V4?>2V4=V 25@EV2:8 28:@5A;NNBLAO. 8:@5A;NNBLAO B0:>6 25@EV2:8, 2 O:8E 28:@5A;5=V >18420 =0I04:8. 8 :>;>A0;L=> 7<5=H8;8 ?>2=89 ?5@51V@. 0;8H8;>AO ?5@52V@8B8 G8 =5 2<VICT :@0I>3> BC@C :;0A, I> 2V4?>2V40T <0B@8FV ![Not(1,2)], B>1B> ?@82545=>W <0B@8FV ! 7 701>@>=>N 2 :;TBFV (1,2), O:0 ?@82545=0 =0 1 ?> AB>21FN (I> 40;> >FV=:C 34 + 1 = 35). FV=:0 =C;59 40T 3 4;O =C;O 2 :;VBFV (1,3). B65 >FV=:0 2V4 T<=>3> 20@V0=BC 35 + 3 ?5@525@HCT 20@BVABL 265 >B@8<0=>3> BC@C 36, 72V4:8 2V4 <=89 20@V0=B 2V4AV:0TBLAO. ;O >B@8<0==O >FV=:8 4>40B=L>3> 20@8V0=BC 28;CG0T<> 7 <0B@8FV ?5@HC AB@>:C V B@5BV9 AB>215FL, 2AB0=>2;NT<> 701>@>=C (3,1) 9 >B@8<CT<> <0B@8FN. &O <0B@8FO ?@82>48BLAO ?> G5B25@BV9 AB@>FV =0 1, >FV=:0 :;0AC 4>AO30T 36 9 :>;> 70:@5A;NTBLAO. A:V;L:8 C 25@EV2:8  2AV 218BV >18420 =0I04:8, 2>=0 21820TBLAO B56. 5@EV2>: =5 70;8H8;>AO. 5@51V@ 70:V=G5=>. 8 >B@8<0;8 B>9 65 <V=V<0;L=89 BC@, I> ?>:070=89 ?V4:@5A;N20==O< 2 B01;. 2. 04>2V;L=8E B5>@5B8G=8E >FV=>: H284:>4VW 0;3>@8B<C VBB;0 B0 A?>@V4=5=8E 0;3>@8B<V2 =5<0T, 0;5 ?@0:B8:0 ?>:07CT I> ACG0A=8E & 2>=8 G0AB> 4>72>;ONBL 28@VH8B8  7 n = 100. &5 25;8G57=89 ?@>3@5A C ?>@V2=O==V 7 ?>2=V< ?5@51>@><. @V< B>3>, 0;3>@8B<8 B8?C 3V;>: B0 <56 O2;ONBLAO, O:I> =5<0 <>6;82>ABV 4>25AB8 WE 4> :V=FO, 5D5:B82=8<8 52@VAB8G=8<8 ?@>F54C@0<8. 1.2.4. ;3>@8B< 59:AB@8 4=8< 7 20@V0=BV2 @VH5==O  T 20@V0=B 7=0E>465=O =09:>@>BH>3> ;0=FN30, I> 2<VICT 2AV <VAB0. B@8<0=89 ;0=FN3 ?>BV< 4>?>2=NTBLAO 28EV4=8< <VAB><  28=8:0T HC:0=89 BC@. >6=0 CO28B8 1030B> ?@>F54C@ @VH5==O FVTW 7040GV, =0?@8:;04 DV78G=5 <>45;N20==O. 0 ?;>A:V9 4>HFV <0;NTBLAO :0@B0 <VAF52>ABV, 2 <VAB0, I> ;560BL =0 @>728;FV H;OEV2, 701820NBLAO F2OE8, =0 :>65= F2OE >4O30NBLAO :V;LFO, H;OE8 C:;040NBLAO <>BC7:0<8, O:V ?@82 O7CNBL 4> 2V4?>2V4=8E :V;5FL. )>1 7=09B8 =09:>@>BHC 2V4AB0=L <V6 i B0 k, B@510 27OB8 V 2 >4=C @C:C B0 k 2 4@C3C 9 @>7BO3=CB8. "V <>BC7:8, I> =0BO3=CBLAO V =5 4>72>;OBL @>72>48B8 @C:8 H8@H5, 9 CB2>@OBL =09:>@>BH89 H;OE <V6 i B0 k. 4=0: <0B5<0B8G=0 ?@>F54C@C, O:0 ?@><>45;NT FN DV78G=C, 283;O40T 4>A8BL A:;04=>. V4><V ?@>ABVHV 0;3>@8B<8. 48= 7 =8E - 0;3>@8B< 59:AB@8, O:89 70?@>?>=>20=89 02B>@>< I5 2 1959 @. &59 0;3>@8B< 28@VHCT 7030;L=C 7040GC:  >@VT=B>20=>9, =5>@VT=B>20=>9 01> <VH0=>9 (B>1B> B0:>9, 45 G0AB8=0 H;OEV2 <0T @CE 2 >48= 1V:) <5@56V 7=09B8 =09:>@>BH89 H;OE <V6 42><0 =040=8<8 25@EV2:0<8. ;3>@8B< 28:>@8AB>2CT B@8 <0A828 7 n (n 4>@V2=NT :V;L:>ABV 25@EV2>: <5@56V) G8A5; :>6=89. 5@H89 <0A82 a 2<VICT ?>7=0G:8 7 42><0 7=0G5==O<8: 0 (25@EV2:0 I5 =5 @>73;O=5=0) V 1 (25@EV2:0 265 @>73;O=5=0); 4@C389 <0A82 b <VAB8BL 2V4AB0=V  ?>B>G=V =09:>@>BHV 2V4AB0=V 2V4 vi 4> 2V4?>2V4=>W 25@EV2:8; B@5BV9 <0A82 A <VAB8BL =><5@8 25@EV2>: (k-89 5;5<5=B c T =><5@ ?5@54>AB0==L>W 25@EV2:8 =0 ?>B>G=><C =09:>@>BH><C H;OEC 7 vi 2 vk. 0B@8FO 2V4AB0=59 Dik 7040T 4>268=8 4C3 dik; O:I> B0:>W 4C38 =5<0T, B> dik ?@8A2>NTBLAO 25;8:5 G8A;> , I> 4>@V2=NT  <0H8=>W =5A:V=G5==>ABV . 02312"""""23025"22""351225018""""""180"20"""22""02314""""2023024"""""1424016"35""""160B01;. 12 "5?5@ <>6=0 >?8A0B8 9 A0<89 0;3>@8B< 59:AB@8. ;3>@8B< 59:AB@8 1. =VFV0;V70FVO. # FV:;V 2V4 >4=>3> 4> n 70?>2=8B8 =C;O<8 <0A82 0; 70?>2=8B8 G8A;>< i <0A82 A: ?5@5=5AB8 i-BCN AB@>:C <0B@8FV D C <0A82 b; a[i]:=1; c[i]:=0; {i-=><5@ AB0@B>2>W 25@EV2:8} 2. 030;L=89 :@>:. =09B8 <V=V<C< A5@54 =52V4<VG5=8E (B>1B> B8E k, I> 4;O =8E a[k] = 0); =5E09 <V=V<C< 4>AO30TBLAO =0 V=45:AV j, B>1B> bj ( bk; a[j]:=1; O:I> bk > bj + djk B> (bk:=bj + djk; ck:=j) {#<>20 >7=0G0T, I> H;OE vi & vk 4>2H5 =V6 H;OE vi & vj, vk. /:I> CAV a[k] 2V4<VG5=V, B> ;>268=0 H;OEC vi & vk 4>@2V=NT b[k]. "5?5@ B@510 ?5@5;VG8B8 25@EV2:8, I> 7 =8E A:;040TBLAO =09:>@>BH89 H;OE} 3. 840G0 2V4?>2V4V. {(;OE vi & vk 2840TBLAO C 72>@>B=L><C ?>@O4:C =0ABC?=>N ?@>F54C@>N:} 3.1. z:=c[k]; 3.2. 840B8 z; 3.3. z:=c[z]; /:I> z = 0, B> :V=5FL, V=0:H5 9B8 4> 3.2. 12345678a00100000b1225018""""c33033333B01;. 13;O 28:>=0==O 0;3>@8B<C B@510 n @07V2 ?5@53;O=CB8 <0A82 b 7 n 5;5<5=BV2, B>1B> 0;3>@8B< 59:AB@8 <0T :204@0B8G=C 206:VABL. @>V;NAB@CT<> @>1>BC 0;3>@8B<C 59:AB@8 G8A5;L=8< ?@8:;04>< (4;O 1V;LH>W A:;04=>ABV 1C45<> 22060B8, I> 45O:V <VAB0 (25@EV2:8) i,j =5 7 T4=0=V <V6 A>1>N, B>1B> D[i,j] = "). 5E09, =0?@8:;04, I = 3. "@510 7=09B8 =09:>@>BHV H;OE8 7 25@EV2:8 3. !:;04>25 <0A82V2 a, b, c ?VA;O 28:>=0==O ?5@H>3> ?C=:BC ?>:070=5 2 B01;. 13. 12345678min bk=12a10100000b1225018""""c33033333min bk=18a10110000b1225018"38""c33033433min bk=25a11110000b12250184738"60c33032432min bk=38a11110100b122501847386260c33032462min bk=47a11111100b122501847386160c33032452min bk=60a11111101b122501847386160c33032452B01;. 14G5284=>, A:;04>25 B01;8FV 7<V=NTBLAO ?V4 G0A 28:>=0==O 7030;L=>3> :@>:C. &5 284=> 7 B01;8FV 14. "0:8< G8=><, 4;O @VH5==O  ?>B@V1=> n @07V2 70AB>AC20B8 0;3>@8B< 59:AB@8 B0:, O: =040=> =86G5. V7L<5<> 4>2V;L=C ?0@C 25@EV2>: j, k. #AC=5<> @51@> C[j,k]. 0 4>?><>3>N 0;3>@8B<0 59:AB@8 7=0945<> =09:>@>BHC 2V4AB0=L <V6 <VAB0<8 j & k. %09 FO 2V4AB0=L 2<VICT 45O:5 m. B>6, <0T<> G0AB8=C BC@C j, m, k. "5?5@ 4;O :>6=>W ?0@8 ACAV4=VE <VAB (C =040=><C ?@8:;04V - 4;O j, m B0 m ,k) 28;CG8<> 2V4?>2V4=5 @51@> 9 7=0945<> =09:>@>BHC 2V4AB0=L. @8 FL><C =09:>@>BH0 2V4AB0=L =5 ?>28==0 2<VIC20B8 265 28:>@8AB0=5 <VAB>. 0;V 70 0=0;>3VTN 7=0E>48<> =09:>@>BHC 2V4AB0=L <V6 ?0@0<8 25@EV2>: 0;3>@8B<>< 59:AB@8, 4>:8 2AV 25@EV2:8 =5 1C4CBL 28:>@8AB0=V.  T4=0T<> >AB0==N 25@EV2:C 7 ?5@H>N V >B@8<0T<> BC@. 09G0ABVH5 F5 >AB0==T @51@> 28O2;OTBLAO 4C65 25;8:8<, V BC@ E0@0:B5@87CTBLAO 2V4GCB=>N ?>E81:>N. 572060NG8 =0 F5, 0;3>@8B< 59:AB@8 <>6=0 2V4=5AB8 4> =01;865=8E 0;3>@8B<V2. 1.2.5. 2B>@AL:89 <5B>4 @VH5==O 7040GV :><V2>O65@0 / 70?@>?>=CN A2V9 <5B>4 @VH5==O .  >73;O=5<> @8A. 9 - 10 9 A?@>1CT<> 7=09B8 2 =8E =09:>@>BHV BC@8. 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