∗
! " #$%
& %' () *++,
- ./0123 2/1456726 8/9:3;. /05< 2/=3 /0 543 93053; <= 543 ;<<>? @AA B;/.<03;.
.4CD3 543/; 3E3. =;<> 543 12C;3F 5430 =<97. <0 543 B;/.<0 GC;D30 .5C0D/01 6E 543
2/145.G/594F G/54 C .5C0DC;D 3H/2IB7JJ23;K. 12/05 /0 4/. 3E3? L3 631/0. 5< .B3C:M
NO0 <03 4<7;F E<7 G/22 C22 63 5C:30 5< E<7; 9322. 5< 63 :3B5 /0 .<2/5C;E
9<0P03>305F G/54 0< B<../6/2/5E <= 9<>>70/9C5/01 G/54 C0E <= E<7;
=322<G /0>C53.?
Q322F C2><.5 0< B<../6/2/5E??? 3H3;E 0/145 =;<> 0<G <0F O G/22 94<<.3
<03 <= E<7 C5 ;C0D<>F ;35;/3H3 E<7 =;<> E<7; 9322F C0D 5C:3 E<7 5<
54/. ;<<>F G43;3 E<7 >CE .33 /= 543 2/1456726 /. RS <; RT F C0D E<7
>CE 57;0 /5 RS <; RT C. E<7 G/.4?N
- >7;>7; ;/BB23. C;<70D 543 ;<<> C. 543 B;/.<03;. 9<0./D3; 543 B;<.B395 <=
4CH/01 .794 C0 3U395 <0 543/; 4/543;5< />B<5305 C0D 3V53;0C22E 9<05;<223D 3V/.I
53093.?
NO= C5 .<>3 B</05F O 5C:3 E<7 5< 54/. ;<<> C0D E<7 632/3H3 54C5 C22
@AA B;/.<03;. 4CH3 6330 94<.30 C0D 5C:30 43;3 C5 .<>3 5/>3F 5430
E<7 >CE 5322 >3 54/.?
O= E<7 C;3 9<;;395F O G/22 =;33 E<7 C22? O= <= 9<7;.3 E<7 C;3 /09<;;395???
G322 235K. .CE 0<03 <= E<7 G/22 2/H3 5< 8/B C0E ><;3 2/1456726 .G/5943.
/0 54/. G<;2D?N
L3 3V/5. G/54 C 8<7;/.4 <= 4/. 92<C:F 54<7145=722E 23CH/01 543 2/1456726 <0?
W43 B;/.<03;. C;3 /0 543 DC;: C. 5< 4<G 5< 135 =;33F 675 543E C;3 B3;=3952E
923C; C6<75 GC05/01 5< 63 C623 5< C5 23C.5 8/B 2/145 .G/5943. /05< <2D C13 XC0D
/5 2<<:. 2/:3 543E >/145 033D 5<YZ? [< 543E >7.5 9<>3 7B G/54 C .5;C531E 54C5
G/22 C00<7093 54C5 C22 @AA B;/.<03;. 4CH3 6330 94<.30 <02E /= 543E C957C22E 4CH3F
G/54 @AA\ 93;5C/05EF 4<B3=722E 63=<;3 543E C22 D/3 <= <2D C13?
-5 P;.5 /5 .33>. />B<../623 54C5 C0E <03 B;/.<03; 9<72D :0<G C6<75 G4C5
543 <543; ]] 4CH3 6330 7B 5<? ^<>/01 /05< 543 ;<<> C0D .33/01 543 2/1456726 /.
RS D<3.0K5 .33> 5< 1/H3 E<7 >794 /0=<;>C5/<0? _<7 D<0K5 :0<G G4< .35 /5F C0D
/= E<7 8/B 543 .G/594 E<7 4CH3 0< /D3C G4< G/22 .33 54C5 E<7 8/BB3D /5? W43;3
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@
.33>. 5< 63 0< GCE 5< .30D C >3..C13 5< C0E<03 /0 BC;5/972C;? O5 .33>. 4<B323..
54C5 543E G/22 135 <75 C5 C22? 75 /0 =C95M
S L3;3 /. 4<G? X_<7 >CE G/.4 5< B<0D3; <0 E<7; <G0 63=<;3 ;3CD/01 <0?Z
! " # $ ! %%&&" '''
()*+ ^75 543 .3 , 73093 <= DCE. /05< 62<9:. <= 230154 @AA? W43 P;.5
B;/.<03; 5< 3053; 543 ;<<> /0 C 1/H30 62<9: 57;0. 543 2/1456726 RT ? O= C B;/.<03;
3053;. 543 ;<<> C .39<0D 5/>3 /0 543 .C>3 @AA DCE 62<9:F 5430 43 57;0. 543
2/1456726 RS? O= C B;/.<03; 3053;. 543 ;<<> <0 543 2C.5 DCE <= C @AA DCE 62<9:F
C0D /5 /. 4/. P;.5 5/>3F C0D 543 2/1456726 /. .5/22 RT F 5430 54C5 B;/.<03; :0<G.
54C5 3H3;E B;/.<03; 4C. 6330 94<.30 3VC952E <093 /0 54/. @AA DCE 62<9:? L3 5430
9<;;3952E D392C;3. 54C5 C22 B;/.<03;. 4CH3 6330 /0 543 9305;C2 ;<<> C5 23C.5 <093?
O= 543 2/1456726 /. RS <0 543 2C.5 DCE /0 C 62<9: 5430 G3 4CH3 =C/23D 54/. 5/>3
.< G3 5;E C1C/0 /0 543 03V5 62<9: <= @AA DCE.F C0D :33B 5;E/01 705/2 .<>3<03
C00<7093.?
- ./ **0 )*1231 R ) 1)*+ W43 B;<6C6/2/5E <= .79933D/01 /0 C0E 1/H30
62<9: /. 543 07>63; <= <;D3;/01. <= 543 @AA B;/.<03;. D/H/D3D 6E 543 07>63;
<= B<../623 GCE. 543 B;/.<03;. 9<72D 4CH3 3053;3D 543 ;<<>? Q/54 n B;/.<03;.F
54C5 /. (n!/n ) ?
W43 3VB3953D 07>63; <= 62<9:. G4/94 >7.5 63 7.3D 63=<;3 .79933D/01 /.
3 , 7C2 5< 1/p G43;3 p /. 543 B;<6C6/2/5E <= .79933D/01 G/54 <03 62<9:? W< .33
54/.F .7BB<.3 p /. 543 94C093 <= .79933D/01 /0 C0E 1/H30 62<9:? W430 543 3VB3953D
07>63; <= 62<9:. 705/2 G3 .79933D /. 3 , 7C2 5< P kpq G43;3 q = 1 − p ? W4/.
/. 3 , 7C2 5<
D
p D (1/(1 − q)) = p/p = 1/p
q
<= 62<9:. /. ? 4 C94 62<9: 4C. 230154
W47. 543 3VB3953D 07>63;
.< 543 3VB3953D 07>63; <= DCE. 705/2 =;33D<>n /./n!n /n! F G4/94 /. O(n e n)
X7./01 [5/;2/01K. =<;>72CZ? 5<; 100 B;/.<03;.F 543 3VB3953D HC273 /. 100 /100!
<; CBB;<V/>C532E 10 ?
W43E 9C0 135 <75F C254<714 54/. /. C D/.CBB</05/012E 2C;13 07>63; =<; 543
B;/.<03;.M C6<75 10 E3C;.? [CD2E 543 70/H3;.3 >CE 4CH3 30D3D 2<01 63=<;3
543E C;3 =;33 678 ?
S 9 :; < =><? @?
n
k
k−1
2
n
n+1
1/2 n
101
44
41
A ! BC% BB&%D# E
()*+ 9 F 03 B;/.<03;F G4< G/22 63 :0<G0 C. G?? F G/22 63 ;3.B<0I
./623 =<; C00<709/01 54C5 3H3;E <543; B;/.<03; 4C. 3053;3D 543 ;<<> C5 .<>3
5/>3? W43 <543; (n − 1) B;/.<03;. G/22 63 <@=< ?
H
4 C94 <;D/0C;E B;/.<03; .5C;5. G/54 C ./0123 5<:30F 9C223D C ? F G4/94 543E
G/22 5;E 5< 23CH3 /0 543 9305;C2 ;<<>? W43 ^<7053.. G/22 9<22395 .<72. =;<> 543
9305;C2 ;<<> 705/2 .43 4C. C22 <= 543>? [43 >CE 5430 D392C;3 .7993..?
Q3 >CE C..7>3 54C5 543 2/1456726 .5C;5. /0 543 RT B<./5/<0 XC. 543 B;/.<03;
G4< 3053;. <0 543 P;.5 DCE >CE 57;0 /5 RT 63=<;3 D</01 C0E54/01 32.3Z?
Q430 C0 <;D/0C;E B3;.<0 3053;. 543 ;<<> C0D P0D. 543 2/1456726 RT F
543E >CE D;<B 543/; .<72 /0 543 ;<<>F /= 543E 4CH3 0<5 C2;3CDE D<03 .<F 6E
57;0/01 543 2/1456726 RS? O= 543 2/1456726 /. C2;3CDE RS 5430 543E 23CH3 /5
C2<03?
Q430 543 ^<7053.. 3053;. 543 ;<<>F /= .43 P0D. 543 2/1456726 RS 5430 .43
57;0. /5 RT C0D CDD. <03 5< 43; .<72 9<705? O= 43; 9<705 /. 0<G n − 1 5430
.43 :0<G. 54C5 3H3;E<03 >7.5 4CH3 3053;3D 543 ;<<> .< .43 9C0 D392C;3?
O= 543 2/1456726 /. RT G430 .43 3053;. .43 23CH3. /5 RT ?
- ./ **0 )*1231 R ) 1)*+ 9 5<; 543 .5;C531E 5< 9<>B2353 G3 033D 5<
4CH3 C .3 , 73093 <= 3H305. 4CBB30? Q3 P;.5 033D C .<72 D;<BB3D /0 543 ;<<>F
5430 =<; 543 ^<7053.. 5< B/9: /5 7BF 5430 C0<543; .<72 D;<BB3DF 359? -. 543
07>63; <= 709<7053D .<72. 1<3. D<G0F 543 B;<6C6/2/5E <= C 03G <03 57;0/01 7B
<0 543 03V5 DCE 1<3. D<G0 =;<> (n − 1)/n =<; 543 P;.5 .<72 5< 1/n =<; 543 2C.5
.<72? 3C0G4/23F 543 B;<6C6/2/5E =<; 543 ^<7053.. 5< .4<G 7B <0 543 03V5 DCE
/. 9<0.5C052E 1/n ?
[/093 543 3VB3953D 5/>3 033D3D =<; C0 3H305 <997;/01 G/54 B;<6C6/2/5E p <0
543 03V5 DCE /. 1/p F G3 />>3D/C532E 135 54C5 543 5/>3 /.
n
n−1
X
1
k
!
+ (n − 1)(n),
G4/94 /. 635G330 C0D ? W43;3=<;3 543 3VB3953D 07>63; <= DCE. 705/2 543
B;/.<03;. 3.9CB3 /.n O(n )2n?
W4/. /. >794 63553; 54C0 <7; B;3H/<7. 3VB<0305/C2 .<275/<0? W43 @AA B;/.<03;.
.4<72D 135 <75 /0 C;<70D @AF7 AA DCE.F <; C6<75 H ] E3C;.? W43E G/22 63 BC.5 543/;
63.5F 675 543E G/22 2/H3 5< .33 543 <75./D3 G<;2D? L<G3H3;F 543E 9C0 D< >794
63553; 54C0 54C5M
S :; < < = = =: ; < = @?
k=1
2
2
2
@ D" A% &
&
W4/. /. C1C/0 C >354<D =<; 9<22395/01 .<72.? W4/. 5/>3 543;3 /. 0< ./0123 9<7053;?
C543; 3H3;E<03 /. /0H<2H3D /0 C B;<93.. <= 9<22395/01 .<72. 5<13543;? W43 2/145I
6726 G/22 63 G<;54 D/U3;305 07>63;. <= .<72. <0 D/U3;305 0/145.?
()*+ - .3 , 73093 /. 1/H30 G4/94 D3.9;/63. 4<G >C0E .<72. 543 2/1456726
/. G<;54 <0 3C94 0/145F G4/94 /. C2GCE. C B<G3; <= 5G<? 35 V (n) D30<53 543
07>63; <= .<72. 54C5 543 2/1456726 /. G<;54 /= /5 /. 23=5 /0 543 RS B<./5/<0 <0
0/145 n F <; D/.9<H3;3D RS <0 0/145 n + 1 ?
-..7>3 54C5 543 07>63; <= B;/.<03;. /. C B<G3; <= 5G<? W4/. G/22 57;0 <75
0<5 5< >C553; /0 543 30D?
- B;/.<03; 3053;. 543 ;<<> <0 C 0/145 C0D 9<22395. 4<G3H3; >C0E .<72. 4CH3
6330 23=5 543;3 =;<> 2C.5 0/145 X.< /= /5 /. 0/145 n C0D 543 2/1456726 /. RS 43 B/9:.
7B V (n − 1) .<72.Z C0D 57;0. 543 2/1456726 RT ? L3 0<G 2<<:. C5 543 07>63; <=
.<72. 54C5 43 4C. 9<223953DF 675 ;3B;3.3053D /0 6C.3 ? O= 543 9<>/01 0/145 /.
G<;54MV (n) = 2 .<72. 5430 43 2<<:. C5 543 6/0C;E 6/5 2<= M G<;54 2 .<72.? O=
54/. 6/5 /. 1 5430 43 D;<B. 2 .<72. 6E 57;0/01 543 2/1456726 RS C0D .765;C95/01
=;<> M F 4/. <G0 5<5C2 <= .<72. 9<223953D? O= 543 2 I6/5 /. J3;< 5430 43 23CH3.
2543 2/1456726
RT ?
<5/93 54C5 54/. 4C. 543 3U395 54C5 .<72. C;3 1273D 5<13543;K /05< 27>B. <=
./J3 2 G4/94 9C0 63 5;C0.=3;;3D <0 0/145. G4/94 C;3 G<;54 2 ? Q4303H3; C
B;/.<03;
4C. 5G< 27>B. <= ./J3 2 43 1273. 543> 5<13543; /05< C 27>B <= ./J3
?
>CE <997; /= 43 4C. 7.5 B/9:3D 7B C 2 27>B <; 4C. 7.5 1273D 5G<
W4/.
2.>C223;
27>B. 5<13543;? Q430 C22 543 .<72. 4CH3 6330 1273D 5<13543; /05<
<03 27>B2 <= ./J3 n = 2 5430 543 B;/.<03; G4< 4<2D. 54/. 27>B D392C;3.
.7993..?
Q3 4CH3 E35 5< .CE G4C5 G<72D 63 C0 3 9/305 94</93 <= HC273. =<; 543 ?
[5C;5/01 G/54 C 62<9: <= 0/145. G<;54 @ /. C 1<<D /D3CF 5< 4<B3=722E 1273V (n)C22
543 ./0123 .<72. /05< BC/;.? W430 =<22<G G/54 C 62<9: <= 0/145. G<;54 H 5< 1273
/05< 62<9:. <= 7 .<72.F C0D .< <0? Q3 GC05 543 230154. <= 543 62<9:. 5< 63 2<01
30<714 5< 1/H3 C 1<<D 94C093 <= 127/01 C22 543 27>B. <= ./J3 2 /05< 27>B. <=
./J3 F 675 0<5 5<< 2<01 C. G3 D<0K5 GC05 5< GC.53 5/>3 <093 543E 4CH3 C22 6330
1273D?2
- ./ **0 )*1231 R ) 1)*+ O0 <;D3; 5< C94/3H3 C 1<<D C.E>B5<5/9.F G3
.5C;5 G/54 C 62<9: <= n + n log log n) 1I0/145.F 5430 (n log n + n log log n) 2I
0/145.F 5430 (n log n+n(n log
I0/145.F C22 543 GCE 7B 5<
log log
log n)
Xlog nZI0/145.? O= G3 4CH3
=C/23Dn) 4C=53; 54/. 07>63; <= DCE. (n5430log G3n+n9C0log./>B2E
54;<G 7B <7; 4C0D. C0D .5C;5 <H3; C1C/0? O0 <543; G<;D.F 543 .3 , 73093 V (n)
;3B3C5.?
W43 B;<6C6/2/5E <= 127/01 C22 27>B. <= 2 /05< 27>B. <= ./J3 2 G/54/0
0/145. XG43;3 /. .<>3 9<0.5C05Z /. 6<70D3D 6E e C.E>B5<5I
n
log(n)W4/.
+ cn/. :0<G0 C. 543c < ? <: 6 ? Q/54
/9C22E?
.<>3 9C;3=72
H8
3.5/>C5/<0 54/. ;3.725 9C0 63 3V530D3D 6E 94C01/01 c 5< C =7095/<0 c(n) ≤ log(n) ?
W4/. 1/H3. 7. C B;<6C6/2/5E <= .7993..=722E 9<>B235/01 3C94 .5C13 <= C5 23C.5
F G43;3 (n) /. C0 3;;<; =C95<; .794 54C5 (n) 530D. 5< @?
e
(n)
.5C13. /. C5 23C.5
?
W43 94C093 <= .7993..=722E 9<>B235/01 C22
1< 54;<714 543 G4<23 9E923 /.
W47. 543 3VB3953D 07>63; <= 5/>3. G3 033Dlog5< (n)
23.. 54C0 e
? W4/. 1/H3. 54C5 543 3VB3953D 07>63; <= DCE. /. <=
<;D3; e [n log(n)(log(n) + log(log(n)))] F G4/94 /. O(n log(n) ) ?
O5 9C0 63 B;<H3D 54C5 0< 94C013. 5< 543 230154. <= 62<9:. <= 0/145. V (n) 9C0
k
k
k
k
k
k
k
k
k+1
k
k−1
log2 n
k−1
k
2
k−1
k
−e−c
1
− log(n)
log2 (n)
elog2 (e)
(n)log2 (n)
2
log2 (e)
1
(n)log2 (n)
log2 (e)
2
7
/>B;<H3 543 C.E>B5<5/9.F 675 G4C5 /= G3 GC05 543 63.5 .3 , 73093 =<; B;39/.32E
@AA B;/.<03;.
F 7; C..7>B5/<0 54C5 543 07>63; <= B;/.<03;. /. C B<G3; <= 5G< 9C0 63
;32CV3D? W< CBB2E <7; .5;C531EF G3 7.5 033D 54C5 3H3;E<03 .5C;5. G/54 C5 23C.5
<03 .<72? [<F G/54 @AA B;/.<03;. =<; 3VC>B23F <03 B;/.<03; 9<72D 63 1/H30 H ]
.<72. C0D 543 <543; ]] B;/.<03;. 1/H30 @ .<72 5< .5C;5 G/54? W43 B;/.<03; G4<
P;.5 9<22395. @H .<72. D392C;3.?
W< 5;E 5< 135 C 1<<D 7BB3; 6<70D <0 543 3VB3953D 07>63; <= DCE. 5< =;33D<>
G3 7.3D C 9<>B753; ./>72C5/<0 5< .3C;94 54;<714 543 94</93. =<; V (n) ? F 7; 63.5
1/H3 C;<70D 77 AA DCE.F <; @H E3C;.? F 03 .3 , 73093 <= 62<9: 230154. G4/94 4C.
C6<75 54/. CH3;C13 /. 6 AF AF @AF AF H AF 7 AF AF H AF 7 ]AF AF AF
AF ]AF ]A8 F 54C5 /. 5< .CEM A @IDCE.F 5430 A HIDCE.F??? 5430 A 7 I
DCE.F 5430 H A @IDCE.F??? 5430 ]A 7 IDCE.F 5430 X6C9: 5< 543 .5C;5Z A @IDCE.F
??? W43 <B5/>/.C5/<0 C21<;/54> G<;:. 6E 5;E/01 5< <B5/>/.3 543 =<; 5G<
BC..3. 54;<714 543 5EB3. <= DCE. X@IDCE. 5430 HIDCE.F 5430 7 IDCE.V (n)359?Z 5430
7.5 ;3B3C5. 54C5 .3 , 73093 =<; 543 702/:32E 9C.3. /0 G4/94 543 B;/.<03;. 4CH3 .5/22
0<5 P0/.43D C=53; H BC..3.? O5 /. 0<5 305/;32E 923C; G4E 543 C6<H3 .3 , 73093 /.
1<<DF 54<714 /5 >C:3. .30.3 54C5 543 P;.5 ./V 53;>. C;3 D39;3C./01 ./093 23..
B3<B23 4CH3 5< >335K /0 2C53; .5C13.? O5 C2.< >C:3. .30.3 54C5 543 .3H3054 53;>
/. 2C;13;F ./093 <03 G<72D GC05 5< 1/H3 C 2<5 <= 5/>3 =<; 543 2C.5 5G< 62<9:. <=
7 .<72. 5< >335? /H/01 7B C5 54/. .5C13 >3C0. 4CH/01 5< 9<05/073 54;<714 543
03V5 .3H30 .5C13. 5< P0/.4 7B?
S *+ 0 R **) 1 R * R *) 1)*+ 5<; @AA B;/.<03;. G3
.7.B395 54C5 .<>3 .<;5 <= 4E6;/D C21<;/54> /. B;<6C62E 543 63.5F 5< 7.3 1<<D
B</05. <= ><;3 54C0 <03 .5;C531E? ^<22395/01 5<13543; .<72. C. /0 543 BE;C>/D
.943>3 /. 93;5C/02E C 1<<D /D3C 5< .5C;5 G/54F 675 .<>354/01 32.3 >CE 63 63553; /0
543 30D1C>3? - 4E6;/D 1/H30 6E ? 5321304C73; 6@8 7.3. 543 BE;C>/D .943>3 5<
.5C;5 G/54F 675 4C. C ^<7053.. .5C;5 9<22395/01 >/DGCE 54;<714? L/. .3 , 73093 <=
62<9: 230154. X94<.30 6E 4C0DZ 4C. 3VB3953D DCE. <= C;<70D ] 7 ]F C0D ;700/01
<7; <B5/>/.C5/<0 B;<1;C> <0 <0 543 HC;/C623. =<; 4/. .5;C531E 1/H3. C;<70D ]A?
-.E>B5<5/9C22EF 543E 9C00<5 D< 63553; 54C0 O(n log n) 3VB3953D 07>63; <=
DCE. 5< =;33D<>F 639C7.3 54C5 /. 543 3VB3953D 07>63;
<= DCE. =<; 3H3;E<03 5<
4CH3 H/./53D 543 ;<<>F /10<;/01 54C5 C22 B;/.<03;. C957C22E 4CH3 5< 9<>>70/9C53Y
L3;3 C;3 .<>3 HC;/C5/<0. E<7 >/145 2/:3 5< 54/0: C6<75? 4 C94 HC;/C5/<0 C..7>3.
C22 543 9<0D/5/<0. /0 543 <;/1/0C2 B;<623>F 675 G/54 .<>3 C.B395. C253;3D? O0
3C94 9C.3F E<7 >/145 2/:3 5< C.: E<7;.32= G43543; 543 B;/.<03;. 9C0 3.9CB3F C0D
/= .< G4C5 /. C0 3 9/305 GCE 5< D< 54/.? Q3 C..7>3 54C5 543 2/1456726 C2GCE.
.5C;5. RT ?
@? 23/ 3* 23 1 W43 9305;C2 ;<<> 9<05C/0. 5G< X<; ><;3Z 2/1456726.
X543 9<>>70/9C5/<0 94C0032 /. G/D3;Z?
23/ 3* ) RR 1
W43;3 C;3 5G< X<; ><;3Z /D305/9C2 ;<<>.? W43
B;/.<03;. C;3 5C:30 5< <03 C5 ;C0D<> 675 D<0K5 :0<G G4/94 543E C;3
/0?
? (*/)* ) S 1 *) )** *) W43 GC;D30 57;0. 543 2/1456726 RT
C5 @H - F 94<<.3. <03 B;/.<03; 5< H/./5 C5 @- F C0D 94<<.3. C1C/0 =<;
.<>3<03 5< H/./5 C5 H - ? W43 H/./5<;. <02E 5;C0.>/5 <; ;393/H3F 0<5 6<54?
W43 GC;D30 /. >C2/9/<7. C0D :0<G. 543 .5;C531E
7 ? 3 R 21 )0* S
54C5 543 B;/.<03;. G/22 7.3 X43 2/.530. 5< 543> C1;33/01 <0 G4C5 5< D<Z?
4 C94 DCE 43 G/22 94<<.3 G4/94 B;/.<03; 5< C22<G /05< 543 ;<<>? L/. 9<0I
.9/3093 D3>C0D. 54C5 43 C22<G. 3H3;E B;/.<03; 5< H/./5 543 ;<<> /0P0/532E
<=530?
? 33 /)1 RS *)1 * R SSR 2 S * W43 9<0D/5/<0 =<; 3H3;E<03 5< 63
=;33D /. 54C5 3H3;E B;/.<03; >7.5 9<;;3952E C00<7093 XC5 .<>3 5/>3Z? O0
<543; G<;D.M 3H3;E B;/.<03; >7.5 63 .7;3 54C5 C22 B;/.<03;. 4CH3 6330 5<
543 ;<<>?
? ( 23 S * R 21 SSR 2 S S -0E<03 9C0 C00<7093 <0 C0E DCEF 0<5
7.5 543 B;/.<03; G4< GC. .323953D 54C5 ><;0/01? W43 9<0D/5/<0 =<; 3H3;EI
<03 5< 63 =;33D /. 54C5 3H3;E<03 >7.5 9<;;3952E C00<7093 <0 543 ? DCE?
O= 543E C;3 /09<;;395 <; /= .<>3 C00<7093 G4/23 .<>3 D< 0<5 5430 543E C22
D/3?
? )1 RS *)1 )* )**0 * S *+ SSR 2 S * 4 H3;E<03 >7.5 9<;I
;3952E C00<7093 C5 .<>3 5/>3? Q430 .<>3<03 C00<7093.F 543E C;3 =;33D
X03H3; H/./5 543 ;<<> C1C/0ZF 675 543 <543;. .5CE 705/2 543E 5<< C00<7093?
/./5<;. C;3 94<.30 70/=<;>2E C><01 543 ;3>C/0/01 B;/.<03;.? <53 54C5
543 B;/.<03;. C;3 .5/22 ><.5 /053;3.53D /0 3H3;E<03 3.9CB/01F ;C543; 54C0
>/0/>/J/01 543/; <G0 5/>3 5< 3.9CB3?
? *0 32* *331 RS * SSR 2 S *) W43 B;/.<03;. C;3 C22<9C53D ;3D <;
6273 B;/.<0 9322.? W43 C00<709/01 B;/.<03; >7.5 9<;;3952E .CE 4<G >C0E
;3D 9322. 543;3 C;3 /0 <;D3; =<; 543> C22 5< 63 ;323C.3D?
]? 2 *)1 S *331 RS * SSR 2 S *) 4 C94 9322 4C. C 0C57;C2 07>63;
G;/5530 <0 543 GC22? W43 C00<7093; >7.5 1/H3 C22 543 07>63;.?
@A? 33 /)1 RS *)1 1* S 0 *11*1 R 33 /)1 RS *)1 X54/. /. C 9<>6/0C5/<0
<= C0D ]Z?
@@? S 0 R 1S *1 W43 B;/.<0 /. .7653;;C03C0F G/54 0< 92<9:.F
9C230DC;.F <; C0E <543; /0=<;>C5/<0 C. 5< G4C5 543 5/>3 /.? W43 B;/.<03;.
2<.3 C22 5;C9: <= 5/>3F C0D 543 GC;D30 94<<.3. B;/.<03;. C5 ;C0D<> 5/>3.?
O0 <543; G<;D.F 543 B;/.<03;. 4CH3 0< /D3C 4<G >C0E B3<B23 4CH3 H/./53D
543 ;<<> ./093 543E G3;3 2C.5 543;3F C0D 9C00<5 7.3 .5;C531/3. G4/94 9<705
DCE.? W43E <02E :0<G 543 <;D3; /0 G4/94 3H305. <997;?
H
?
@H ? S 0 R *1 33 /)1 RS *)1 21 SSR 2 S * ^<>6/03 HC;/C5/<0.
C0D @@? 4 H3;E B;/.<03; >7.5 C00<7093 54C5 3H3;E<03 4C. H/./53D C5 .<>3
5/>3F C0D 543E 9C00<5 7.3 DCE 9<705/01 .5;C531/3.?
@ ? S 0 R *1 *11* ) R RS * R RS * W43;3 C;3 <02E 5G<
B;/.<03;. C0D 543 5;C0.>/553; 4C. 5< .30D C >3..C13 XC 0C57;C2 07>63;Z
5< 543 ;393/H3;F 675 C. /0 @@ 543E 9C00<5 9<705 DCE.?
@7 ? S 0 R *1 *11*1 ) R S+ R RS * ^<>6/03 HC;/C5/<0.
] C0D @@? F 03 C00<7093; >7.5 1/H3 C22 07>63;. G;/5530 <0 543 GC22.F C0D
543 B;/.<03;. 2<.3 5;C9: <= 5/>3?
@ ? S 0 R *1 *11*1 ) R S+ R S+ 9 3 23 1
@ ? S 0 R *1 *11*1 ) R S+ R S+ 3 23
Q3 0<G 1/H3 C .B</23; =<; ><.5 <= 543.3 B;<623>.? O5 57;0. <75 54C5 543
.5;C531/3. 2/.53D C6<H3 X<; .2/145 ><D/P9C5/<0. <= 543>Z C;3 .7/5C623 =<; ><.5 <=
543.3 HC;/C5/<0.?
@? 23/ 3* 23 1 ^<705/01 .<72. X.5;C531E H Z G/22 .5/22 G<;:F C0D 9C0
63 >CD3 3H30 =C.53; C. 2 .<72. 9C0 63 23=5 /0 C ;<<> G4/94 4C. k D/.5/095
2/1456726.? log n 2/1456726. C22<G =<; 543 63.5 B<../623 5/>3 5< 3.9CB3
C. .<<0 C. 3H3;E<03 4C. C957C22E 6330 /0 543 ;<<> 5430 543 B;/.<03; /0 543
;<<> 9C0 D392C;3? [5;C531E 9C0 C2.< 63 />B;<H3D 6E C22<G/01 127/01. <=
.<72. /05< 2C;13; 27>B.F .794 C. 27>B. <= ./J3 (2 ) /= 543;3 C;3 k D/.5/095
2/1456726.?
^<705/01 .<72. X.5;C531E H Z G/22 .5/22 G<;:? O5 G/22
H ? 23/ 3* ) RR 1
63 .2<G3;F C254<714 543 3VB3953D 5/>3 705/2 3.9CB3 X=<; 07>63; <= ;<<>.
/0D3B30D305 <= nZ /. .5/22 O(n ) DCE.?
? (*/ *)* ) S 1 *) )** *) - .5;C531E ./>/2C; 5< .<72I9<22395/01
X.5;C531E H Z G<;:.? W43 ^<7053.. C2GCE. B/9:. 7B C0D 03H3; D;<B. .<72.?
4 H3;E<03 32.3 D;<B. .<72. C5 3H3;E <BB<;570/5E X54<714 543E C;3 =<;93D 5<
B/9: 543> 7B /= 543E P0D 543>Z? W4/. .5;C531E 4C. 3VB3953D 5/>3 635G330
C0D n ? XO= 543;3 C;3 k .<72. <75.5C0D/01 5430 543 94C093 <=
n
n
543log9<7053..
B/9:/01 <03 7B 543 9<>/01 0/145 /. 635G330 × C0D F
D3B30D/01 <0 4<G 54<.3 k .<72. C;3 D/.5;/6753D? W4/. 1/H3. C 5<5C2
3VB3953D
5/>3 <= 635G330 n log n C0D n ?Z
[5;C531E H G/22 G<;:F C254<714 543;3 /. 923C;2E 0<
7 ? 3 R 21 )0* S
6<70D <0 543 5/>3 705/2 3.9CB3 /5 D3B30D. <0 4<G >3C0 543 QC;D30
GC05. 5< 63?
? 33 /)1 RS *)1 * R SSR 2 S * W;EI5;EIC1C/0K X.5;C531E @Z G<;:.?
O053;23CH/01 9E923. <= .5;C531E G/22 C2.< G<;:M 4 C94 B;/.<03; 4C. <03 5EB3
<= .<72 =<; 3C94 B;/.<03; G4< G/22 4CH3 5< C00<7093? F 03 9E923 /. 1/H30 5<
3C94 B;/.<03;K. C553>B5. 5< 9<22395 543 .<72. D3.5/03D =<; 43;F 5430 C=53; n
k
2
k l
2
2
2
3
k
n
2
3
2
1
n
1
n2
<= 543.3 C .39<0D 9E923 /. D3H<53D 5< 3C94 B;/.<03; C0D .< <0? W4/. 1/H3.
C0 3VB3953D 5/>3 <= n log (n) ?
? ( 23 S * R 21 SSR 2 S S W43 B;/.<03;. 9C00<5 63 .7;3 <= 3.9CB/01?
[7BB<.3 543E G/22 C00<7093 <0 DCE ? W43;3 /. C P;.5 DCEF F <0 G4/94
543E C22 :0<G 54/.? W43 B;/.<03; G4< A3053;. <0 DCE D :0<G. 54C5D 543E 4CH3
3053;3D C0D 543 .5C53 <= 543 2/1456726? 4 H3;E <543; B;/.<03; <02E :0<G.
54C5 543E D/D 0<5 3053; 543 ;<<> <0 54C5 DCE? O= C D/U3;305 B;/.<03; 3053;3D
<0 DCE D 5430 C22 <= 543 <543; B;/.<03;. G4< D/D 0<5 3053; G<72D 4CH3 543
.C>3 /0=<;>C5/<0F C0D .< G<72D 4CH3 5< 9<>3 5< 543 .C>3 9<0927./<0M 54C5
543E .4<72D C00<7093 <0 DCE A XB;<H/D3D 543;3 C;3 C5 23C.5 B;/.<03;.Z?
W43;3=<;3 /5 9C00<5 >C553; G4< 3053;. <0 DCE F .< 543E >7.5 C22 :0<G
<0 DCE D − 1 ? Q4/94 9<05;CD/95. 543 C..7>B5/<0D54C5 D GC. 543 P;.5 DCE
543E C22 :03G 543E G<72D C00<7093 <0 DCE A ?
? )1 RS *)1 )* )**0 * S *+ SSR 2 S * [2<.4EK .<72I9<22395/01
XC. /0 543 C0.G3; 5< HC;/C5/<0 @H 632<GZ G/22 G<;:? Q430 C B;/.<03; 4C.
9<223953D @AA .<72. C0D 5430 1/H30 543> C22 CGCE C1C/0 543E >CE D392C;3
C0D 63 .35 =;33?
F ]F @A? *0 32* *331 R ) 2 *)1 S *331 RS * R ) 33 SSR 2 S *) 1 W43 B;/.<03;. 9C0 3.9CB3? [33 543 63;I543<;3> X632<GZ =<; C .5;C531E
C0D B;<<=?
@@? S 0 R 1S *1 ^<705/01 .<72. X.5;C531E H Z G/22 .5/22 G<;:?
@H ? S 0 R *1 33 /)1 RS *)1 21 SSR 2 S * [2<.4EK .<72 9<705I
/01 G/22 G<;:? W43 2/1456726 /. C2GCE. G<;54 <03 .<72? -0E B;/.<03; G4<
4C. 0<5 C00<7093D D<3. 543 =<22<G/01M O= 543 2/1456726 /. RS G430 543E
3053; 5430 543E 9<22395 543 .<72 C0D 57;0 543 2/1456726 RT ? O= 543 2/1456726
/. RT G430 543E 3053; C0D 543E 4CH3 <03 <; ><;3 .<72. 5430 543E D;<B <03
C0D 57;0 543 2/1456726 RS? -0E B;/.<03; G4< 4C. C2;3CDE C00<7093D C2I
GCE. D;<B. C0E .<72. G4/94 543E 4CH3F C0D 23CH3. C0E 54C5 C;3 /0 543 ;<<>?
W4/. .5;C531E 4C. 3VB3953D 5/>3 <;D3; 23.. 54C0 <; 3 , 7C2 5< ? W4/. 9C0
63 .4<G0 6E 9<0.5;795/01 C0 CBB;<B;/C53 C;:<H 94C/0 C0D e1/H/01 2<G3;
6<70D. =<; 543 94C093 54C5 C 1/H30 B;/.<03; G/22 C00<7093 /0 543 03V5 H AA
DCE.? <5/93 54C5 G430 543;3 /. <02E <03 B;/.<03; 23=5 5< C00<7093F 54/.
.5;C531E ;3D793. 5< .5;C531E H F .<72I9<22395/01 G/54 <03 ^<7053..?
<53 54C5 54/. .5;C531E G<72D C2.< G<;: X23.. 3 9/3052EZ /= B;/.<03;. G4<
4CH3 C2;3CDE C00<7093D 7.5 9<05/073 5< .2<.4 .<72. C;<70D X1/H3 C0D 5C:3
.<72. ;C543; 54C0 7.5 1/H3Z? W4/. /. 639C7.3 C ;C0D<> GC2: /0 C P0/53
.BC93 G/22 3H3057C22E 135 3H3;EB2C93? Q3 G/22 7.3 54/. =C95 3V530./H32E
2C53; <0?
-0<543; .5;C531E /. 54C5 3C94 B;/.<03; G4< /. 0<5 C .<72I9<22395<; 4C. C
XH3;E .>C22Z 94C093 3C94 DCE 543E 3053; <= 639<>/01 <03? -=53; C 07>63;
<= H/./5. 5< 543 ;<<> C. C .<72I9<22395<; 543E 1/H3 7B C0D 1< 6C9: 5< 63/01
2
2
n
C0 <;D/0C;E .<72I1/H3;? -0E B;/.<03; G4< 4C. C2;3CDE C00<7093D C2GCE.
1/H3. .<72. C0D 03H3; 9<22395.?
^C0 E<7 54/0: <= C HC;/C5/<0 G43;3 543 63.5 .5;C531E /. G<;.3 54C0 3VB<I
0305/C2 /0 543 07>63; <= B;/.<03;.
@ ? S 0 R *1 *11* ) R RS * R RS * Q3 4CH3 5G< B;/.<03;.F
<03 <= G4<> /. 5;E/01 5< .30D C >3..C13 5< 543 <543;? W43 5;C0.>/55/01
B;/.<03; 309<D3. 543 >3..C13 C. C 0C57;C2 07>63;F ? L3 5;/3. 5< 1/H3
543 ;393/H/01 B;/.<03; M .<72.? W43 B;<623> 0<G M/. 4<G 543 ;393/H3;
:0<G. G430 543 >3..C13 4C. 6330 .305 4<G D<3. .43 :0<G G430 .43
4C. ;393/H3D C22 <= 543 .<72. W< D< 54/. .43 <99C./<0C22E B75. C .<72 6C9:
/05< 543 ;<<> G430 .43 P0D. /5 3>B5E? L<B3=722E 543 5;C0.>/553;F 4CH/01
D;<BB3D C22 <= 4/. .<72.F G/22 5C:3 543 2C.5 .<72 6C9: 547. /0D/9C5/01
54C5 43 /. P0/.43D? W43 ;393/H3; G/22 5430 .33 54C5 543 .<72 4C. 6330 5C:30
C0D :0<G 54C5 C22 <= 543 .<72. 4CH3 6330 .305F 639C7.3 543 5;C0.>/553; G/22
<02E B/9: 7B C .<72 G430 43 /. D<03?
W< D< 54/. ;32/C62E 543 5G< B;/.<03;. 634CH3 C. =<22<G.M
W43 5;C0.>/553; D;<B. C22 <= 4/. .<72. 705/2 43 4C. 0<03 23=5? Q430 43 4C.
0< .<72. 23=5 43 G/22 5C:3 <03 .<72 =;<> 543 ;<<> /= 43 9C0? Q430 43 4C.
<03 .<72 43 G/22 D;<B /5 /0 543 ;<<> /= 43 9C0?
W43 ;393/H3; 5C:3. 3H3;E .<72 54C5 .43 9C0F C254<714 .43 <99C./<0C22E D;<B.
<03 6C9: /0 543 ;<<> XB/01.KZ? O= G430 .43 03V5 3053;. 543 ;<<> .43 P0D.
54C5 543 .<72 .43 4C. D;<BB3D 4C. 6330 5C:30F 5430 .43 :0<G. 54C5 543
5;C0.>/553; /. P0/.43D C0D .< :0<G. 543 5<5C2 07>63; <= .<72. .305?
@7 ? S 0 R *1 *11*1 ) R S+ R RS * 5<; n B;/.<03;.
5;C0.>/55/01 C0D <03 ;393/H/01F 543 5;C0.>/553;. C22 634CH3 C. /0 HC;/C5/<0
@ ? 5 /;.5 .7BB<.3 54C5 543 ;393/H3; GC05. 5< :0<G 543 .7> <= 543 07>63;.
<= 543 5;C0.>/553;.F M ?
W4/. 5/>3 543 ;393/H3; <99C./<0C22E 5;/3. 5< D;<B .<72. 6C9: C5 543 .C>3
5/>3? W43 <02E GCE 54C5 C22 n B/01. G/22 63 5C:30 n/. /= C22 n <= 543 5;C0.>/5I
5/01 B;/.<03;. C;3 P0/.43D? Q430 .43 .79933D. 5430 43; >CV/>7> HC273
GC. 543 .7> <= 543 07>63;. <= 543 5;C0.>/553;.F M ?
Q4C5 4CBB30. /. 54C5 543 ;393/H3;K. 5<5C2 9<223953D .<72. 7.7C22E /09;3C.3.F
675 03H3; =C22. 6C9: C. >794 C. n =;<> 543 97;;305 C22I5/>3 >CV/>7> 70I
23.. 54C5 >CV/>7> /. 543 5<5C2 07>63; <= .<72. 63/01 5;C0.>/553D? Q430
C 5;C0.>/553; P0/.43.F 543 ;393/H3;K. 5<5C2 /. C22<G3D 5< .2<.4 6C9: 6E <03
><;3 54C0 63=<;3? Q430 C22 5;C0.>/553;. C;3 P0/.43D 5430 543 ;393/H3;K.
5<5C2 G/22 .2<.4 635G330 M C0D M − n F C0D G430 .43 .33. 6<54 3V5;3>3.
/0 54C5 <;D3; 5430 .43 :0<G. /5 /. D<03?
0<G/01 543 5<5C2F F /. 30<714 5< C22<G C22 543 n 5;C0.>/553;. 5< .30D
C;6/5;C;E >3..C13.? M^4<<./01 6C.3 H F 1/H3 543 iI54 5;C0.>/553;
D/1/5. i, i +
/0
G4/94
5<
309<D3
4/.
>3..C13?
n, i + 2n, . . .
]
@ ? S 0 R *1 *11*1 ) R S+ R S+ 9 3 23 1 Q3
9C0 7.3 543 .<275/<0 5< HC;/C5/<0 @ 5<13543; G/54 C GCE 5< BC.. C;<70D
G4< /. 5;C0.>/553; C0D G4< /. ;393/H3;? W< 63 B;39/.3F 543E 7.3 2/1456726
<03 7.5 C. /0 @ ? [<>3 B;/.<03; /. 94<.30 5< 63 543 P;.5 5;C0.>/553;? Q3
C..7>3 2/1456726 5G< /. RS 5< .5C;5 G/54? Q4<3H3; 57;0. /5 RT XB/9:. 7B
543 2/.530/01 .5/9:KZ /. 543 P;.5 ;393/H3;? W43 5;C0.>/553; .33. 54C5 543
2/.530/01 .5/9: 4C. 6330 B/9:3D 7BF C0D .5C;5. 5;C0.>/55/01 <0 2/1456726
<03? Q430 543 ;393/H3; :0<G. 543 >3..C13 /. D<03F 43 B75. D<G0 543
2/.530/01 .5/9: C0D 639<>3. 543 03G 5;C0.>/553;? W43 03G ;393/H3; /.
G4<3H3; 03V5 B/9:. 7B 543 .5/9:? W43 B;/.<03;. :33B .30D/01 >3..C13.
C;<70D XG/54<75 :0<G/01 G4< 543E C;3 5;C0.>/55/01 5<Z C0D 3H3057C22E
3C94 B;/.<03; 9<22395. C22 543 >3..C13.?
@ ? S 0 R *1 *11*1 ) R S+ R S+ 3 23 [33
543 63;I 63;I543<;3> 632<G?
Q3 G/22 0<G 1/H3 <7; >354<D =<; HC;/C5/<0. F ] C0D @A?
O5 57;0. <75 54C5 3C94 B;/.<03; 9C0 5;C0.>/5 C0 C;6/5;C;E >3..C13 5< C22 <=
543 <543; B;/.<03;.F 7./01 <02E 543 <03 2/145?
Q3 G/22 .5C;5 G/54 <03 B;/.<03; 5;C0.>/55/01 <03 6/5 5< 3H3;E <543; B;/.<03;?
O= 543 5;C0.>/553; GC05. 5< .30D C AI6/5 5430 <0 C0E 3H30I07>63;3D DCE 43
23CH3. 543 2/1456726 RS C0D <0 C0E <DDI07>63;3D DCE 23CH3. 543 2/1456726 RT ?
O= 43 GC05. 5< .30D C @I6/5 5430 <0 C0E 3H30 DCE 43 23CH3. 543 2/1456726 RT
C0D <0 C0E <DD DCE 43 23CH3. 543 2/1456726 RS? 4 H3;E <543; B;/.<03; 23CH3. 543
2/1456726 RT ? <G C0E B;/.<03; G4< P0D. 543 2/1456726 RS G430 543E 3053;
543 ;<<> G/22 :0<G =<; .7;3 54C5 543 5;C0.>/553D 6/5 /. C A <; C @F D3B30D/01 <0
G43543; 543 B;3H/<7. DCE GC. 3H30 <; <DD? 4 H3;E B;/.<03; G/22 P0D 543 2/1456726
RS C5 .<>3 5/>3 XG/54 B;<6C6/2/5E @ZF C0D .< G/22 ;393/H3 543 >3..C13? F = 9<7;.3F
543;3 /. 0<54/01 .B39/C2 C6<75 3H30 C0D <DD DCE.? -0E 6/ 395/<0 635G330 N C0D
G<72D D< 7.5 C. G322? 5<; 3VC>B23F j ↔ (0, k) G<72D >3C0 54C5 DCE
{0,/.1}
× N AI6/5 DCE?
543
W4<.3 DCE. G4<.3 07>63; 9<;;3.B<0D 5< (0, n) C;3 3H30
jDCE.K C0D
k 54<.3 G4/94 9<;;3.B<0D 5<
C;3 <DD DCE.K?
W< .30D 5G< 6/5.F D/H/D3 543 DCE. (1,/05<m)=<7; .35.? O0 <543; G<;D.F B;<H/D3 C
6/ 395/<0 635G330 N C0D {0, 1} × {0, 1} × N ? W43 P;.5 6/5 /. ;3B;3.3053D 6E 543
P;.5 5G< 5EB3. <= DCEF
A C0D @ ><D 7 .CEF C0D 543 .39<0D 6/5 6E 543 <543; 5G<
5EB3. <= DCE.F H C0D ><D 7 .CE? -0E B;/.<03; G4< P0D. 543 2/1456726 RS G/22
:0<G =<; .7;3 <03 <= 543 6/5. 63/01 5;C0.>/553D?
W< 5;C0.>/5 C >3..C13 <= C;6/5;C;E 230154F B;<H/D3 C 6/ 395/<0 635G330 N
C0D {0, 1} × N × N ?
W< C22<G 3H3;E B;/.<03; 5< 5;C0.>/5 C >3..C13 5< 3H3;E <543; B;/.<03;F P;.5
D/H/D3 543 DCE. C><01 543 B;/.<03;. X.< 54C5 3C94 /. C22<9C53D C0 /0P0/53 07>63;Z
C0D 5430 ;70 543 C6<H3 C21<;/54> G/54 B;/.<03; k 5;C0.>/55/01 <0 DCE. G4/94
C;3 C22<9C53D 5< 43;? 5<; M B;/.<03;.F 54/. 9C0 63 54<7145 <= C. 1/H30 6E C
@A
6/ 395/<0 635G330 N C0D {1, · · · , M } × {0, 1} × N × N ?
W< .B33D 7B 5;C0.>/../<0F /= C0<543; B;/.<03; :0<G. C 1/H30 6/5 /0 <03 <= 543
>3..C13. 63/01 5;C0.>/553D 5430 543E 9C0 ;35;C0.>/5 54/. 6/5 6E C95/01 C. 543
5;C0.>/553; G<72D 394</01K 543 >3..C13?
&C &C
& &
Q3 G/22 0<G D/.97.. C >354<D 54C5 C22<G. 3C94 <= 543 B;/.<03;. 5< .30D C .35 <=
C;6/5;C;/2E 2<01 >3..C13.F <03 5< 3C94 <543; B;/.<03;? Q3 C..7>3 =7;543; 54C5
G3 C;3 /0 543 .355/01 <= HC;/C5/<0 @@ X C0D<> H/./5/01 5/>3.KZF C0D 43093 54C5
543 B;/.<03;. 4CH3 0< 5/>3 ;3=3;3093 <543; 54C0 543 <;D3; /0 G4/94 3H305. <997;?
02/:3 C22 543 HC;/C5/<0. G3 D/.97..3D 705/2 0<GF 54/. <03 9<72D 0<5 63 .<2H3D
7./01 D/;395 ><D/P9C5/<0. <= .5;C531/3. @ <; H ? F 03 <= 543 C754<;. X ?5 ?Z 9C>3
7B G/54 G4C5 G3 54/0: /. C0 <;/1/0C2 .5;C531E?
* 0* R * * R 0
W43 B;/.<03;. G/22 4CH3 C1;33D 7B<0 C0 <;D3;/01 C><01 543> C43CD <=
• 5/>3?n
;/.<03; @ G/22 63 543 :?<>< F 2<<:/01 C5 543 ?? =<;>3D 6E C22 543
• <543; B;/.<03;. XC0D 543 2/1456726Z? W4<.3 0<0I<6.3;H3;. G/22 63 9C223D
<:? 639C7.3 543E G/22 =<22<G C ./>B23 ;723?
3=<;3 .5C;5/01 4/. ;723F 543 P;.5 5;C0.>/553;F .CE B;/.<03; n F /05;<D793. A
• <; @ .<72. /05< 543 .E.53>?
W43 <6.3;H3; G/22 5;E 5< D3D793 4<G >C0E .<72. G3;3 <;/1/0C22E /05;<I
• D793D =;<> 543 634CH/<; <= 543 ;<6<5.? 5<; 54/.F B;/.<03; @ 4C. D/U3;305
B;<93D7;3. C5 4/. D/.B<.C2M
WG< ?= B;<93D7;3. , P 54C5 C22<G B;/.<03; @ 5< 9<0D795 3VB3I
;/>305.? L3 /. 5;E/01 5< PC0.G3;
B<./5/H32E 5< <03 <= 543 5G< , 73.5/<0.
M
=@ <=?< =<@ ? ? <
= ?? Q , Q? L<G3H3;F 6<54 n C0D
i <02E B;<D793
i = 0 B<./5/H3
1
9C0
;3.725.F
P B;/.<03;
P
<; 63 /09<0927./H3? L3093
@ G/22 <02E C0.G3; 031C5/H32E 5<
G430 P /. 9<0927./H3?
Q
- <?= B;<93D7;3 54C5 C22<G. B;/.<03; @ 5< .35 543 .E.53> 6C9:
5< /5. <;/1/0C2 B<./5/<0 X543 07>63; <= .<72. /0 543 .E.53> /. C. 543
5;C0.>/553; 23=5 /5Z? W4/. C22<G. 4/> 5< B;<933D G/54 CDD/5/<0C2 3VB3I
;/>305.?
W43 5G< 53.5/01 B;<93D7;3. G/22 3H3057C22E 1/H3 C0 C0.G3; 5< 543 <6.3;H3;?
<G B;/.<03; @ 5;/113;. B;/.<03; H /05< C0 <6.3;H/01 B4C.3? W4C5 /.F 543E
• X><;3 <; 23..Z 3V94C013 543/; ;<23.F C0D B;/.<03; 639<>3. C0 <6.3;H3;F
H
G4/23 B;/.<03; @ .5C;5. =<22<G/01 C ./>B23 ;723 C0D .< 639<>3. C ;<6<5?
4 H3057C22EF =;<> 543 3VB3;/>305. 43 G/22 9<0D795 <0 543 .E.53> =<;>3D
@@
0
0
1
1
1
0
0
1
6E 543 <543; B;/.<03;.F B;/.<03; H G/22 P0D <75 G4/94 6/5 B;/.<03; @ 23=5 /0
543 .E.53> C0D 5430 639<>3. C ;<6<5?
W4/. 9<05/073.F 9E92/01 54;<714 C22 543 B;/.<03;.? Q3 4CH3 3C94 B;/.<03;
• .30D/01 C P;.5 6/5 5< B;/.<03;
F 5430 C22 <= 543> .30D/01 C
i.39<0D 6/5F 359???
i + 1 mod n
./01 /053;>3D/C53.F C0E B;/.<03; 9C0 .30D C >3..C13 5< C0E <543;F C0D
• 0<5 <02E 5< 4/. =<22<G3; /0 543 <;D3;/01?
* 1 / 3*) 1* n = 3
Q3 0<G D3.9;/63 3C94 .53B /0 =722 =<; 543 9C.3 n = 3 ?
( / 3* )23*1 W43 634CH/<; <= 543 B;/.<03;. G4< C;3 0<5 97;;3052E <6.3;H/01
G/22 63 1/H30 6E 543 D/;3953D 1;CB4. ϕ F G/54 k C B<./5/H3 /05313; X.33 D/C1;C>
@Z? W43.3 1;CB4. D3.9;/63 543 07>63; <= .<72. 3C94 B;/.<03; /. 3C13; 5< 4CH3
C5 C0E 5/>3F C0D 43093 D353;>/03 G43543; 43 GC05. 5< D;<B <; 1;C6 C .<72
3C94 5/>3 43 3053;. /0 543 ;<<>? W43 1;CB4. C;3 5< 63 ;3CD 23=5 5< ;/145F C0D
9<0./D3;3D 5< ;3B3C5 X543 DC.43D 2/03Z? -5 C0E 5/>3 G43;3 ><;3 54C0 <03 <B5/<0
/. <U3;3DF 543 B;/.<03; 94<<.3. G4/94 <B5/<0 5< 5;E G/54 3 , 7C2 B;<6C6/2/5E?
k
Trigger
k
k-1
k-2
10
slosh
9
8
7
6
5
4
3
2
1
0
W< .5C;5 G/54F <03 <= 543 ;<6<5. G/22 =<22<G C0D <03 G/22 =<22<G
-..7>3 54C5 k /. 6/1F C0D k /. 6/113;? W4/. G/22ϕ 63 >CD3 B;39/.3 2C53;? ϕQ3
B2CE 543 ;<23 <= B;/.<03; @F C0D X=<; 0<GZ <02E <6.3;H3 B;/.<03; H XC0D Z ;700/01
543/; /0.5;795/<0. ϕ X;3.B? ϕ Z? <;3 B;39/.32EF G430 G3 135 C 94C093 5< 1<
/0 543 ;<<>F G3 0<53 G43543; 543 .5C53 <= 543 2/1456726 4C. 94C013D =;<> 543
2C.5 5/>3 G3 G3;3 543;3 XG4C5 G3 9C22 C = <= Z?
O= 543 5<5C2 07>63; <= .<72. /0 543 .E.53> /. 0 X;3>3>63; G3 /0927D3 543
2/1456726 /0 543 .E.53>YZF 0<54/01 9C0 4CBB30 639C7.3
6<54 B;/.<03;. C;3 6<54
3C13; 5< 135 ><;3 .<72.F 675 0<03 C;3 CHC/2C623? O= 543 5<5C2 /. @F 543 2/1456726
>/145 63 .G/5943D RS C0D RT .<>3 .>C22 07>63; <= 5/>3. X/= 543 B;/.<03; G4<
.5C;5. G/54 543 .<72 /. /0/5/C22E 3C13; 5< 135 ;/D <= /5ZF 675 3H3057C22E <03 <= 543
@H
k0
0
1
k0
k1
k1
H B;/.<03;. G/22 4CH3 @ .<72 C0D 63 3C13; 5< 4CH3 H F C0D 543 <543; G/22 4CH3 A
C0D 63 3C13; 5< 4CH3 @? [< 543 ./57C5/<0 G/22 .5C22 543;3 C=53; C P0/53 07>63; <=
8/9:3;/01.? [/>/2C; .5<B. G/22 <997; /= 543;3 C;3 7 <; .<72. /0 543 .E.53>?
F 0 543 <543; 4C0DF /= H .<72. C;3 CHC/2C623 /0 543 .E.53>F 543 .E.53> >/145
.5<B /0 C ./57C5/<0 G43;3 3C94 4CH3 @ .<72 C0D C;3 3C13; 5< 4CH3 H F 675 ><;3 />I
B<;5C052EF 543 2/1456726 = 63 57;03D RS C0D RT C0 C;6/5;C;/2E 2C;13 C><705
<= 5/>3.F /= 543E 6<54 :33B 1</01 54;<714 C .3 , 73093 2, 1, 0, 1, 2, 1, 0, ... XG/54 C
D32CE /0 543/; B4C.3.Z? W43 2/1456726 /. 5430 .C/D 5< = < =@ = ? W43
.C>3 54/01 9<72D 4CBB30 /= 543;3 C;3 F F??? .<72. /0 543 .E.53>? 5 /0C22EF /0 543
9C.3 <= .<72.F 543 .E.53> >/145 B;<D793 /0D3P0/53 8/9:3;/01 /0 543 2C>B /0 C
><;3 9<>B2/9C53D =C.4/<0?
W4/. 634CH/<; /. .7>>C;/J3D /0 543 =<22<G/01 94C;5?
7>63; <= .<72. O0D3P0/53
/0 543 .E.53> 8/9:3;/01
B<../623
6, 7, ..., k + k
/>B<../623
4, 5
B<../623
2, 3
/>B<../623
0, 1
O5 /. C2.< G<;54 0<5/01 54C5 543;3 3V/.5. C0 /05313; .794 54C5 /= 543;3 C;3
AF @F 7 <; .<72. /0 543 .E.53>F 543 .E.53> G/22 .5C22 /0M23.. 54C0 M 8/9:3;/01.?
L3093F <6.3;H/01 M + 1 8/9:3;/01. G/22 17C;C0533 54C5 G3 C;3 0<5 /0 C0E <= 543
9C.3. AF @F 7 <; F G4C5 G3 9C22 C ?==> ;3.725?
- ./ *) * S RS -..7>3 543 .E.53> 9<05C/0. 3/543; A <; @ .<72F C0D 9<0D795
<03 <= 543 =<22<G/01 B;<93D7;3.M
-DD @ .<72 5< 543 .E.53>? QC/5 =<; C B<./5/H3 ;3.725 =<; .<>3 5/>3? O= 54/.
P B<./5/H3 ;3.725 C;;/H3.F ;357;0 *1 F <543;G/.3 ;357;0 2 S SR S?
-DD .<72. 5< 543 .E.53>? QC/5 =<; C B<./5/H3 ;3.725? O= 54/. B<./5/H3 ;3.725
P C;;/H3.F ;357;0 *1 F <543;G/.3 ;357;0 2 S SR S?
W43 GC/5/01 5/>3. .4<72D 63 5C:30 .< 54C5 G3 9C0 B<5305/C22E <6.3;H3 C5 23C.5
8/9:3;/01. C0D 4<B3=722E 135 C B<./5/H3 ;3.725?
M+
Q31 4CH3 543 =<22<G/01 94C;5 <= <759<>3.M
0
1
1
0
P0
P0
P1
P1
!"
#
!"
L3093F C B<./5/H3 ;3.725 5< P 17C;C0533. C B<./5/H3 C0.G3; 5< Q ?
-..7>/01 G3 D/D 0<5 135 C 9<0927./H3 ;3.725 G3 G<72D 93;5C/02E 2/:3 5< ;70
=7;543; 3VB3;/>305.F 675 543 .E.53> 4C. B;<6C62E .5C223D? Q4C5 .4<72D G3 D<
0<G
i
i
@
S O= G3 9<72D ;357;0 543 .E.53> 5< /5. <;/1/0C2 .5C53 G/54 A <; @ .<72
XC. .35 7B 6E B;/.<03; ZF G3 9<72D 3VB3;/>305 =7;543;? W< D< 54/.F G3 G<72D
2/:3 5< 5C:3 .<72. <75 <= 543 .E.53>? O5 .33>. 4<B323..F /= =<; /0.5C093 <03 ;<6<5
4C. 0< .<72.F 543 <543; 4C. C0D 543E C;3 6<54 3C13; 5< 4CH3 ><;3? 75 /=
G3 C;3 ;3CDE 5< 1/H3 543> .<>3F 543E G/22 3H3057C22E 4CH3 C0D 63 G/22/01 5<
D;<B 543 .<72. C1C/0? Q3 9C0 5430 1;C6 54<.3 23=5<H3;.F 705/2 G3 C;3 6C9: 5< 543
/0/5/C2 07>63; XA <; @Z? W4/. C22<G. 7. 5< 9<0D795 <543; 3VB3;/>305.F C0D 43093
5< D353;>/03 3H3057C22E G43543; B;/.<03; 23=5 634/0D C .<72 <; 0<5? <53 54C5
G3 03H3; 4CH3 5< ;C/.3 543 07>63; <= .<72. CDD3D 5< 543 .E.53> 5< ><;3 54C0
@H 5< 135 /5 ><H/01 C1C/0F ./093 G/54 @H F C5 23C.5 <03 ;<6<5 B;/.<03; /. C5 543
.5C;5 <; /05< 4/. .2<.4K ;31/<0F C0D /. G/22/01 5< 3/543; 5C:3 <; 1/H3 .<72.?
)*)S <G 54C5 G3 :0<G G4C5 543 6/5 .305 6E B;/.<03; GC.F G3 B;3BC;3
<7; >3..C13 =<; B;/.<03; H 6E .355/01 543 5<5C2 07>63; <= .<72. 5< A <; @? -=53;
54C5F G3 G<72D 2/:3 5< ./10C2 B;/.<03; H 5< .5C;5 4/. ;<23 <= <6.3;H3;? W4/. /.
G43;3 543 07>63;. k 9<>3 /05< B2CE?
H 4C. C1;33D 63=<;34C0D 54C5 43 G/22 63 5;/113;3DK G430 43 4C.
;/.<03;
3VC952E
@ .<72. Xk = 18Z? <53 P;.5 54C5 G3 03H3; 033D3D 5< 1< 54C5 4/14
D7;/01 <7; 3VB3;/>305C5/<0 B4C.3 XG3 033D3D 5< 1< C5 ><.5 7B 5< @H Z? [< G3
9C0 63 .7;3 54C5 G3 4CH3 0<5 5;/113;3D B;/.<03; H 63=<;3 0<G? Q3 D;<B 54<.3
@ .<72. /0 543 .E.53>F C0D 5430 .5C;5 5< CBB2E 543 ;723 ϕ =<; .<>3 k C1;33D
<0 C43CD <= 5/>3F 6/113; 54C0 k ?
Q3 0<G 4CH3 @ <; @] .<72. /0 543 .E.53>F C0D 3C94 B;/.<03; /. ;700/01 C
;723 ϕ ? Q3 <02E 4CH3 ;<6<5. ;700/01 543 B2C93Y [< 543 G4<23 .E.53> 3H<2H3.
C99<;D/01 5< C ;C0D<> GC2:? [/093 543;3 C;3 <02E @ <; @] .<72. 543;3 C;3 P0/532E
>C0E B<../623 .5C53.? <;3<H3;F G3 :0<G 54C5 <03 <= 543 B;/.<03;. 4C. C5 23C.5
.<72.F C0D 43093 543 <B5/<0 <= /09;3C./01 <; D39;3C./01 4/. C><705 <= .<72.?
W4/. 17C;C0533. 54C5 <7; ;C0D<> GC2: 03H3; .5<B.F C0D 543;3 /. C 0<0IJ3;<
B;<6C6/2/5E <= 1355/01 =;<> C0E .5C53 5< C0E <543; .5C53? L3093 B;/.<03; H G/22
3H3057C22E 30D 7B G/54 @ .<72.?
<G 54C5 43 4C. 4/. @ 5;/113;/01 .<72.F B;/.<03; H 7.5 033D. 5< 3;C.3 543>
/0 4/. >305C2 9<705 <= .<72.? L3 /. 6C9: 5< A .<72.F C0D 543;3 >/145 63 @ .<72
23=5 .<>3G43;3 32.3 /0 543 .E.53>? L3 639<>3. C0 <6.3;H3; C0D 4/. ./57C5/<0 /.
./>/2C; 5< 543 <03 30 <E3D 6E B;/.<03; @ C5 543 .5C;5?
O0 543 9C.3 <= B;/.<03;.F G3 9C0 C957C22E 5C:3 k = 18, k = 20, k =
F C0D /0 1303;C2 543 K. G/22 /09;3>305 6E H 3C94 5/>3? W43 <02E
22,
k = 24, · · C;3
· 54C5 543E C;3 6/1k 30<714 54C5 G/54 54C5 >C0E .<72. /0 543
;3 , 7/;3>305.
.E.53> X<; <03 ><;3 /= 543 >3..C13 /. C Z 543 .E.53> 03H3; 135. .579: XG430 C22
B;/.<03;. C;3 ;<6<5.ZF C0D 54C5 B;/.<03;.1 C;3 0<5 5;/113;3D 5<< 3C;2E G430 <03
/. 5;E/01 5< 5;/113; .<>3<03 32.3? O09;3>305/01 6E H 1/H3. 7.5 30<714 233GCE .<
54C5 543 @ 6/5 >3..C13 D<3.0K5 .35 .<>3<03 32.3 <U 5<< 3C;2E?
+3 S <G 543 B;/.<03;. 7.5 4CH3 5< 9E923 54;<714 54C5 C21<;/54>F C0D 1/H3
=7;543; 6/5. 5< 543 B;/.<03; =<22<G/01 543> /0 543 <;D3;/01? W4/. G/22 3H3057C22E
C22<G 543> 5< 3V94C013 C;6/5;C;/2E 2<01 >3..C13. G/54 543 <543; B;/.<03;. 5<<?
*1*
i
0
k2
2
1
?
0
3
@7
1
2
* 1* R R )* /)1 RS *)1
Q3 G<72D 2/:3 P;.5 5< /D305/=E 543 />B<;5C05 B;<B3;5/3. 54C5 543 ;723. ϕ 4CH3
54C5 C22<G 543 C21<;/54> 5< G<;:? 3C22E C22 G3 9C;3 C6<75 /. 543 634CH/<7;
<= 543 .E.53> C. C G4<23? [B39/P9C22E G3 GC05 /5 5< 634CH3 /0 D/U3;305 GCE.
D3B30D/01 <0 543 07>63; <= .<72. /0 543 .E.53>F C. .4<G0 /0 D/C1;C> H ? O0 543
k
number of souls
trigger
will run indefinitely
upper bound
for resetting
procedure
can run indefinitely
will stop after at
most M flickers
test for P0
can run indefinitely
test for P1
will stop after at
most M flickers
0
9C.3 <= B;/.<03;.F 543 53.5 =<; P /. D<03 C5 543 6<70DC;E 635G330 @ C0D H
.<72. C0D 543 53.5 =<; C5 543 6<70DC;E 635G330 C0D 7 .<72.? W43 >CV/>7>
07>63; <= .<72. 54C5 P543 <6.3;H3; 033D. 5< CDD 5< 543 .E.53> 5< ;3.35 /5 /. @H ?
-2.<F /0 .30D/01 543 P;.5 6/5F 543 5;/113; HC273 k /. @ ?
<53 54C5 /0 543 9C.3 <= n = 3 F 543 =C95 54C5 C22 543 ;723. 7.3D C;3 <= 543
.C>3 5EB3 /. 0<5 ;3C22E />B<;5C05?
O0 543 1303;C2 9C.3F G3 G/22 4CH3 1 5EB3.
<= ;723F C22 G/54 D/U3;305 5;/113; HC273. k F C0D G3 ;3 , 7/;3 543 5;/113;/01n −B;/.<03;
5< CD<B5 543 .C>3 5EB3 <= ;723 C. 543 <03 543 5;/113;3D B;/.<03; /. ;700/01? L3
:0<G. G4/94 ;723 54/. /. 7.5 6E 9<705/01 543 07>63; <= 9E923. C22 543 B;/.<03;.
4CH3 1<03 54;<714?
* )23*1 - .35 <= ;723. 54C5 G/22 G<;: =<; 1303;C2 n C;3 .4<G0 /0 D/C1;C> ?
Q3 .35 h 5< H F C0D 543 <543; h C;3 D3P03D ;397;./H32EF .< 54C5 h ≥
P
? W43 HC273 T ;3=3;. 5< C 07>63; <= .<72 3V94C013. ;3 , 7/;3D 5< 9<H3;
254C5 .395/<0
h
<= 543 1;CB4F ;C543; 54C0 543 07>63; <= B3C:.? WC:3 543 07>63;
<= B3C:. s 5< 63 .794 54C5 543 5<5C2 230154K 2s h ≥ T ? W43 HC273 <= T G/22 63
.B39/P3D 2C53;?
-22 3VB3;/>305C5/<0 4CBB30. =<; HC273. <= .<72. 23.. 54C0 H F .< <093 C
B;/.<03; .5C;5. 7B <0 543 2<01 <7;03E 5<GC;D. H F 543E G/22 03H3;
63 C623 5<
9<>3 6C9: D<G0 705/2 543 <6.3;H3; GC05. 5< ;3.35 543 .E.53>? H 4C. 5< 63 .35
2C;13; 54C0 h + P h X543 .7> <= 543 4/143.5 B3C: /0 3C94 ϕ Z .< 54C5
1
0
i−1
j=1
1
i
i
j
i
i i
n−2
n−2
i=1
?,i
i
@
Trigger
k
Trigger
k
slosh
H
hn-2
hn-3
hn-4
h2
h1
0
k,0
T
T
H
H
H
hn-2
hn-2
hn-2
hn-3
hn-3
hn-3
hn-4
hn-4
hn-4
h2
h2
h2
h1
h1
h1
0
0
0
k,n-2
k,2
k,?
n−2
i=1
i
?,i
j
i
j
n−2
n−2
i=1
i
n−2
slosh
slosh
543 0<;>C2 ;700/01 <= 543 .E.53>F G/54 <543; B;/.<03;. <0 543/; J/1IJC1. D<G0
632<G G/22 0<5 6;/01 C0 3.9CB33 5< H C0D C22<G 4/> 5< 1< 6C9: D<G0? -1C/0 G3
D3P03 543 3VC95 HC273 <= H C 2/5523 2C53;F 675 C..7>3 =<; 0<G 54C5 /5 /. 6/1?
W43 5;/113; HC273. C;3 D/U3;305 =<; 3C94 B;/.<03;? W43 C21<;/54> G/22 G<;:
G/54 k ≥ (n − 1)H .<k 54C5 G/54 54C5 >C0E .<72. /0 543 .E.53>F C5 23C.5 <03
B;/.<03; /. /05< 4/. .2<.4 ;C013 C0D 543;3=<;3 543 .E.53> 9C00<5 135 .579: G430
54C5 >C0E C;3 CDD3D? W43E 033D <02E /09;3>305 6E H 3C94 5/>3F C. /0 543 9C.3
=<; B;/.<03;.? W< 5;/113; 543 B;/.<03; ;700/01 ;723 ϕ ./>B2E CDD k .<72.F
C0D E<7; >3..C13 XA <; @ZF C0D 639<>3 C ;<6<5? -. /0 543 9C.3 <= B;/.<03;.F
543 ;C0D<> GC2: G/22 3H3057C22E 30D 7B G/54 543 B;/.<03; 63/01 5;/113;3D <0 k
.<72.F C0D C22 <543; B;/.<03;. 4CH3 5;/113;. <= C5 23C.5 H ><;3 54C0 k .< G/22 0<5
63 5;/113;3D B;3>C57;32E?
^23C;2E 54/. .E.53> G/22 0<5 ;70 /0D3P0/532E G/54 A .<72.? O5 /. C2.< 923C;
54C5 /5 = ;70 /0D3P0/532E G/54 P h .<72.? L3;3 /. <03 .3 , 73093 54C5F /=
=<22<G3DF G/22 ;70 =<;3H3;M ^C22 543 B;/.<03; CBB2E/01 543 ;723 ϕ ;<6<5 i ? W<
.5C;5F .35 ;<6<5 A 5< 63 C5 543 6<55<> <= C0E HC223E <0 4/. 9E923F 7.5 63=<;3 C
B3C: <= C25/57D3 h F C0D 1/H3 5< 3C94 ;<6<5 i 3VC952E h .<72. X0393..C;/2E .5C;5/01
C5 543 B3C: <= 4/. 9E923Z? O= ;<6<5 j 1/H3. 4/. h .<72. 5< ;<6<5 A C0D 5430 5C:3.
543> 6C9:F G3 C;3 /0 C ./>/2C; B<./5/<0
5< 543 <03 G3 .5C;53D G/54? Q3 9C0
9<05/073 D</01 54/. /0D3P0/532E?
Q3 033D 5< .4<G 54C5 543 .E.53> 135. .579: =<; .<>3 4/143; HC273 <= .<72.?
W4/. G/22 ;3 , 7/;3 7. 5< B;<H3 54C5 0< B;<B3; .76.35 <= 543 ;<6<5. 9C0 ;70 /0D3=I
/0/532EF /= 543;3 C;3 23.. 54C0 H .<72. /0 543 .E.53>F G4/94 /. B;<H3D 2C53; <0?
/H30 54/.F /5 <02E 5C:3. <03 ;<6<5 <0 4/. GCE 7B 5< 5< .5C22 543 .E.53>? Q3
9C0 30.7;3 54/. G/22 4CBB30 6E B755/01 /0 h + PH h + 1 .<72.? Q3 9C0
0<G 5C:3 H 5< 63 C0E 07>63; 2C;13; 54C0 54/. 07>63;F .CE h + P h + 2 ?
* / R *) R * R 33* * Q3 0<G .4<G 54C5 G/54 23.. 54C0 H .<72.F
543 .E.53> 9C00<5 ;70 /0D3P0/532E /= 0<5 C22 ;<6<5. C;3 /0H<2H3D? -..7>3
543
.E.53> /. ;700/01 /0D3P0/532E G/54 C >/0/>C2 07>63; <= .<72. 94C01/01 4C0D.?
@
Trigger
k
slosh
k,1
T
Trigger
k
n−2
i=1
i
093 C ;<6<5 .5C;5. 7B 5<GC;D. F 43 9C0 <02E 5C:3 .<72. C0D 03H3; ;357;0 543>?
E >/0/>C2/5EF 43 03H3; 5C:3. HC0E 03G .<72. C0D >/145 C. G322 0<5 63 543;3?
[< G3 9C0 C..7>3 54C5 <7; .76.35 <= ;<6<5. >7.5 63 C623 5< ;70 /0D3P0/532E
G/54<75 C0E<03 23CH/01 5<GC;D. H <; 1/H/01 .<72. 5< C0E ;<6<5 1</01 5<GC;D.
?
H 5 /;.5F G3 G/22 C..7>3 54C5
/. >/../01? -..7>3 C .76.35 0<5 /0927D/01
ϕ .76.35 0<5 /0927D/01
;70.F
5430
543;3
/.
C
>/0/>C2
G4/94 ;70.?
ϕ
<GF 235 m 63 543 2C;13.5 07>63; .794 54C5 ϕ /. ϕ/0927D3D /0 54/. .76.35?
-. 543 .76.35 /. >/0/>C2 ϕ >7.5 9<>B2353 C =722 9E923F C. /= /5 D/D 0<5 5430
G3 9<72D ./>B2E 23CH3 ;<6<5
<75? W47.F C5 .<>3 5/>3 ;<6<5 m >7.5 4CH3
m
P
.<72.?
L<G3H3;F
6E
543
94</93
<= 543 ./J3. <= 543 B3C:.
F /5
h
h
/. 923C; 54C5 43 9C0 03H3; 135 ;/D <= 543> C22 G/54<75 B7.4/01h <03> <= 543 <543;
;<6<5. <05< /5. BC54 5<GC;D. H ?
-. 543;3 C;3 C P0/53 07>63; <= /0/5/C2 .5C53. X2<<:/01 <02E C5 543 ;<6<5.
632<G 543/; B3C:. h ZF 543;3 /. C 12<6C2 6<70D <0 543 07>63; <= 3V94C013.F L F
G4/94 9C0 <997; 63=<;3 543 .E.53> 4C25.? T /. 94<.30 5< 63 2C;13; 54C0 L ?
[< G3 C;3 23=5 G/54 543 9C.3 G430 ϕ /. /0927D3D /0 543 .76.35F 675 .<>3
/. >/../01? -. ϕ /. /0927D3D /5 >7.5 9<>B2353 C =722 9E923 XC. <543;G/.3
ϕ
/5 9<72D 63 23=5 <75 C0D
G3 G<72D 4CH3 543 B;3H/<7. 9C.3Z? F 093 /5 4C. ;3C943D
543 631/00/01 <= 543 .3;/3. <= B3C:. <= 43/145 h /5 G/22F =<; C5 23C.5 543 03V5
.<72 3V94C013.F 634CH3 3VC952E C. D/D /0 543 B;3H/<7. 9C.3? 75 54/.
T
.76.E.53> /. 17C;C0533D 5< .5<B 63=<;3ϕ ϕ P0/.43. /5. 43/145 h B3C:.F C. =<;
C5 23C.5 543 03V5 T 5;C0./5/<0. 54/. .76.E.53> 634CH3. 3VC952E C. /0 543 B;3H/<7.
9C.3? W47. ϕ G/22
03H3; 9<>B2353 /5. h B3C:.F C0D .< 03H3; 9<>B2353 C =722
9E923?
- /3 R 2* Q3 4CH3 0<G B;<H3D 3V/.53093 <= C .5;C531E? W< C957C22E CBB2E 54/.
.5;C531EF G3 G<72D 033D 5< 9C2972C53 B;39/.32E 543 HC273 <= .3H3;C2 9<0.5C05. 7.3D
/0 <7; C21<;/54>? 5<; /0.5C093F 543 9<0.5C05. T C0D M C0D 543 HC273. C5 G4/94
G3 53.5 =<; P F P C;3 4C;D 5< P0DF BC;5/972C;2E G/54/0 543 4<7; 54C5 543 GC;D30
4C. 1/H30 7.?
W43 <;/1/0. <= 54/. B;<623> C;3 0<5 923C;? -99<;D/01 5< 23130D 6 F 8 F ./>/2C;
B;<623>. 4CH3 6330 543 D32/145X6726Z <= L701C;/C0 >C543>C5/9/C0.? W43 P;.5
G;/5530 <997;3093 G3 9<72D P0D <= 543 B;<623> /0H<2H/01 @AA B;/.<03;. C0D C
70/, 73 2/1456726 /. <0 C0 <02/03 =<;7> 4<.53D C5 3;:323E 6 8 ? -0<543; HC;/C05
/0H<2H3. H <; H7 B;/.<03;.F 5G< 2/1456726. C0D 543 <62/1C5/<0 =<; 543 B;/.<03;. 5<
C2GCE. 94C013 3VC952E <03 <= 543 2/1456726.? W4/. <03 CBB3C;3D <0 543 @<
3.3C;94 6 8 ? W4<.3 5G< <02/03 <997;3093.F <0 543 3;:323E
=? G36./53 <= O
=<;7> /0 536;7C;E H AA H C0D 543 O G36./53 /0 72E H AA H F G3;3 =<22<G3D 6E
.3H3;C2 <543;.F /0 3/543; <= 543 5G< H3;./<0.? - 54;3CD GC. />>3D/C532E .5C;53D <0
6 8 F /5 GC. B762/.43D /0 543 5C22 /..73 <= 543 C543>C5/9C2 [9/3093.
3.3C;94
O0.5/5753 03G.23553; =??< 6 8 F C0D 543 B;<623> GC. P0C22E C.:3D
F
?,0
?,0
?,0
?,m
?,m
m
m−1
i=0
m
i
?,0
?,m
?,0
m
?,m
?,0
?,0
0
m
m
1
@
i
<0 543 B<B72C; ;CD/< .4<G G< 6 8 Y
Q3 G<72D 2/:3 5< 54C0: -0D;3G 300355. =<; /05;<D79/01 7. 5< 543 <;/1/0C2
B;<623>?
6 @8 3;5;C> 5321304C73;? @AA B;/.<03;. C0D C 2/1456726? 3G.1;<7B F CHC/2C623 54;<714 F 72E H H AA H ?
6H8 Q/22/C> 53223;? =<@= <::= = < @ =? ==?
F BC13. 7 F ]? [39<0D 3D/5/<0? <40 Q/23E [<0. O09?F @] ?
6 8 C543>C5/9C2 [9/3093. 3.3C;94 O0.5/5753? 4 >/..C;E 03G.23553;F <H3>63;
H AA H ? -2.< CHC/2C623 C5 ?
678 30C5C C22<.4 C0D -0D;3/ /0D3? C;: 303;1E C0D 543 =C53 <= 543 70/H3;.3?
H AA ? ?
6 8 NF 231N? @AA B;/.<03;. C0D C 2/1456726? 3G.1;<7B F CHC/2C623
54;<714 F 72E H7 H AA H ?
6 8 C5/<0C2 762/9
CD/<? ^C;5C2: CD/< [4<G? W;C0.9;/B5/<0
CHC/2C623 C5 ! ?
6 8 O
3.3C;94? <0D3; W4/. ^4C223013? " # $ % F 72E H AA H ?
6 8 Q/22/C> Q7? LC;D ;/DD23.? $ & ' () F 536;7C;E H AA H ?
@