Lower Bounds and Algorithms for Dominating Sets in Web … · Lower Bounds and Algorithms for...
Transcript of Lower Bounds and Algorithms for Dominating Sets in Web … · Lower Bounds and Algorithms for...
Lower Bounds and Algorithms for Dominating Sets in
Web Graphs
Colin Cooper, Ralf Klasing, Michele Zito
To cite this version:
Colin Cooper, Ralf Klasing, Michele Zito. Lower Bounds and Algorithms for Dominating Setsin Web Graphs. [Research Report] RR-5529, INRIA. 2006, pp.26. <inria-00070478>
HAL Id: inria-00070478
https://hal.inria.fr/inria-00070478
Submitted on 19 May 2006
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INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
Lower Bounds and Algorithms for Dominating Setsin Web Graphs
Colin Cooper — Ralf Klasing — Michele Zito
N° 5529
Mars 2005
Unité de recherche INRIA Sophia Antipolis2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex (France)
Téléphone : +33 4 92 38 77 77 — Télécopie : +33 4 92 38 77 65
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H"1 R )]2H<)MhK4LHK
εHMh 4K
|(Zn)j − (µn)j | ≤ Knj−ε ^
½¬½ Zn
u kw]-wzomou t\Dwx£_D¡9moZ\_^t²¦jiYZ_*rxz_*] \§^CD¡µn − λ
√n ≤ Zn ≤ µn + λ
√ncZ_*z_
§ £=§ r§ §._wxkkns]7_λ = o(
√n)§]_\zrx]/mZ\uvkcB_wx£ koraZwX©x_x¡j®rDzc_^wDqZµ\«l_pu tTmo_*D_*z
j > 0¡
(µn)j(1 − λ√
nµn
)j ≤ (Zn)j ≤ (µn)j(1 + λ√
nµn
)j §
YZ_zo_pkns\£mt\r%±®rx£ £r%k ?Ç\zr%©Tuvl_pK
uvkqZ\rTkn_^t¦\u -_*trxs\DZPAknu tq*_x¡lmZ\_(wxkkos\]lmou rxtkrDtλwt
µn_*tTmwxu£9mZw%m
(1 + λ√
nµn
)j u kcwm.]rDknm 1 + j2λ√
nµn
¡lcZ\_*z_^wDk(1− λ
√n
µn)j u kcwm£ _^wDkm 1− 2jλ
√n
µn2
+^! 4c¬ !
YZ_~w£ xrxzumoZ\] zo_pkn_^tDm_^ ut mZ\uvk·kn_pq¤murDt uvk·w©D_*zi knu ]\£ _ µzknmw%mnm_*]lm Lkorx£ slmou rxt ®rDz-moZ\_zorD¦\£_^]³wmZwt¬§ =¨£mZ\rxs\DZLutL]-wxtTiq*wDkn_pkumK\rT_pkt\rxm£_pwxmorw©x_*ziko]-w£ £?lrD]7u twmou t\·kn_*m^¡umz_*\z_^ko_*tTmkw7twmos\zw£¢¦N_*tqZ]wxzo¶®rDz¨wtji-]rxz_)zo_*µt\_pZ\_^s\zu knmouvq§
( ½ Ç ._®rDzo_moZ\_¨µzknm.knmo_^·r¢mZ\_)w£ xrxzumoZ\]moZ\_Kxzw\Z·q*rxtkouvkmkQr?waknu t\D£_©D_*zomo_*«v0wxt S = v0
§*=.meknmo_*tuGmoZ_(t\_^c£i"x_^t\_*zw%m_^"©D_*zomo_«
vtlrj_^kKt\rxm¨ZwX©x_awtjit_*u xZj¦rDs\zku t
S ?Çu;§ _x§ Γ(vt) ∩ S = ∅ AQmZ\_*t vtu k¨wx\l_p·mr S §
tVmoZ_®rxzomoZq*rx]ut"\u kqskkourDtm
uvkwµ\«l_^²rTknumou ©x_u tTmo_*D_*zp§ ¬_mXt
\_*t\rxmo_7moZ\_·kou *_r.moZ\_\rx]utw%mou t\kn_*m S q*rx]\slm_^¦ji =£ xrDzoumoZ\] Ce¦N_®rDzo_
vtuvkcwxl_^mor7moZ_qs\zzo_^tTm.Dzwx\Zwxt£ _m
µt = E (Xt)§[_\rDzxzw\ZkKx_^t\_*zw%m_^Vwxq*q*rxzlu t\·mrmZ\_
GR,mt
]rj\_*£=¡¢moZ\_\zrx¦w¦\u £umi"moZw%mvt]uvkoko_^k.mZ\_lrD]7u twmou t\-kn_*muvk
(1 − Xt
t )m §]U_*tq_._)q^wtczoumo_
Õ ÕÚk **¤á
b ]c324 ^3 gS
µt+1 = µt + E [(1 − Xt
t )m]§
?_mx = x(m)
¦N_mZ\_(st\uTs\_7knrD£s\mou rxt~rxmZ\_7_Tsw%murDtx = (1 − x)m u t
(0, 1)§¨Yw¦\£ _7Du©D_^k
mZ\_©%w£ s\_^krxx®rDzmoZ\_)µzknm®_*¯©Xwx£s_^kr
m§
B¹ K' 12 < ρ < 1
M3HSKH 041' R IOfR- S2H<%< M3H3LPC3HM3
4KP!"K t > 0
|µt − xt| ≤ Ctρ ^
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|µt+1 − x(t+ 1)| ≤ |µt − xt| +O
(
√
log tt
) §
YZu kBq*wt-¦N_\zr%©x_p7¦Tiautlsq¤murDt·rxtt§ .i7\_µt\umou rxt
X1 = 1¡TZ\_*tq*_ |µ1 −x| = 1−x §°V_Kw£vknrZwX©x_ |µt+1 −x(t+1)| = |µt −xt+E [(1− Xt
t )m]− (1−x)m| §GYZ\_Klu9_*z_*tq*_ E [(1− Xt
t )m]−(1−x)m q*wt-¦N_¨z_*czumnmo_^t·wxk −m
t (µt −xt)+ E [(1− Xt
t )m]− 1+mµt
t − (1−x)m +1−mx §U¨_*tq*_
|µt+1 − x(t + 1)| = |(1 − mt )(µt − xt) + E [(1 − Xt
t )m] − 1 +mµt
t − (1 − x)m + 1 −mx| §
Y r-qrD]7£_*mo_emoZ\_zorjrt\rmu q*_)moZwm^¡\¦ji ?_*]]-w Cx¡
E [(1 − Xt
t )m] − 1 +mµt
t = (1 − µt
t )m − 1 +mµt
t +O(√
log tt )
§
wxt mZ\_~®s\tq¤mou rxtf(z) = (1 − z)m − 1 + mz
kowmouvkµ_pk |f(z1) − f(z2)| ≤ m|z1 − z2|¡c®rxz
z1, z2 ∈ [0, 1]§
2
Yw¦\£ _ Q!s]7_^zouvq*wx£¢©%w£ s\_^k\_µt\_pu t¬_^]7]-w·\§
m xC § § 5RbD5 § 5HC X § D § KXT4C § D Cp § xN5T
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¹j½ ¹ Xt ∼ xt
^
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u t-Y wx¦\£ _*CwxtaekoZ\r%k^§ YZ\uvku k_pknN_^q*u wx££ i)mozs\_.®rxz
m = 1cZ\_^zo_.t\r)u ]7zor%©D_*]_*tTmq*rxs\£va¦_rx¦\mwu t\_p¢§V_rxz
£vwzx_^zQ©%w£ s\_^krm¡Tw(¦_*mnm_*z..wXiar¢µt\ut\7kn]-wx££Nlrx]u tw%mut\7kn_*mku kBrx¦lmwu t\_^¦jirlq*q^wxkourDtw£ £i
wx££ r%cu t\ ©x_^znmu q*_^kmr ¦_²lzrx\N_^ ®zrx] S § mu kqrDtT©D_*t\u _*tTm-mr q£vwxkknu®i moZ\_V©x_*zomouvq_pk-u t moZ\_\rx]utw%mou t\kn_*mwxk24OH]?;kn_*m P ABwxt 1S2(^NM33N-^?Çko_m R A¤§QYZjsk S = P ∪R § ?_m
Ptwt
Rt\_*t\rxmo_emoZ\_kou *_^krxkosqZ"kn_*mkcwmcmou ]7_
t?Çko_m
P1 = 0wxt
R1 = 1A¤§
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§]=Émo_*zvtuvkcqz_^wmo_^wtq*rxt\t_^q¤m_^mor
mt_*u xZj¦rDs\zk*¡ju
Γ(vt) ∩ P 6= ∅ moZ\_^t vtuvk]r%©x_p-morV \ S §Bb¨mZ\_*zcu ko_ vt
uvkcwxl_^mor R uΓ(vt) ∩R = ∅ ¡lrmoZ_*zcu ko_ vt
uvkcwx\\_^mor Pwxtw£ £¢©x_*zomouvq_pkutΓ(vt) ∩ R wxzo_e]r%©x_^mor
V \ S §
YZ_e_*«l_pq¤mwmou rxtkπt = E (Pt)
wtρt = E (Rt)
kw%mouvkn®iQ
πt+1 = πt + E [(1 − Pt
t )m] −E [(1 − Pt
t − Rt
t )m],
ρt+1 = ρt + E [(1 − Rt
t − Pt
t )m] −mE [Rt
t (1 − Pt
t )m−1].
_µt\_αR
up
wDkp+ r
¡cZ_*z_p = p(m)
wtr = r(m)
kowmouvk®i
r = (1−p)m−p1+m(1−p)m−1 ,
p = (1 − p)m − (1 − p− r)m.
B¹ K[K 12 < ρ < 1
M3H3LP 041' N $R- S2H<%<9M34LPIC1
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t > 0 |πt − pt| ≤ C1t
ρ ( |ρt − rt| ≤ C2tρ ^
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πt§]=.mcmoZ_ut\sq¤mu©D_)knmo_^JQ
Õ ÕÚk **¤á
Cp ]c324 ^3 gS
|πt+1 − p(t+1)| = |πt − pt+E [(1− Pt
t )m]− (1− p)m−E [(1− Pt
t + Rt
t )m]− (1− p− r)m| §
YZ_\zrTrx¢u kBqrx]\£ _m_^-¦jil_^q*rx]NrDkout\)mZ\_K\u 9_*z_*tq*_^kE [(1− Pt
t )m]− (1−p)m wxtE [(1−
Pt
t − Rt
t )m] − (1 − p− r)m u twzomk.mZw%m¨wxzo_)\zrxNrxzomou rxtwx£mr-_*umoZ\_^zPt − πt
rDzπt − pt
§YZ_ez_^kos\£m¨w¦Nrxslm
ρtuvk\zr%©x_^knu ]u£vwz£ iw%Ém_*zct\rxmouvqu t\7moZwm
rkw%mu knµ_^kcQ
r = (1 − p− r)m −mr(1 − p)m−1 §
2
?_mX2
t
l_*t\rxmo_BmoZ\_ckou^_BrxmZ\_.\rx]utw%mou t\)kn_*mz_ms\zt\_^(¦TiT=¨£DrxzumZ\] ¨cZ\_*t7zost(rxt7w¨zwtlrx]Dzwx\Zzorlq_pkokstDmu£¬mou ]7_
t§
¹j½ ¹ X2
t ∼ (p+ r)t0 ^
½¬½ YZ_*rxz_*]l§WCu ]7£u _^kmoZwmmoZ\_7kns]p+ r
uvk¨w\§ w\§ k^§B©D_*ziq£ rDko_mr |S|t = Pt
t + Rt
t
§YZ\_z_^kos\£m®rx£ £r%k^§
2
YZ_e©%wx£s\_pkcrp+ r
®rxzm ≤ 7
wz_Kz_*Nrxzomo_pu tmoZ\_q*rx£ s\]t£vw¦N_*£ £_pαR
up
rGY wx¦\£_[CD§
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§ twzomouvqs\£vwz.moZ_)®rx£ £r%cu t\zo_^£ wmou rxtkoZ\u k.Z\rx£v
E (Xt+1) = E (Xt) + E [(1 − Dt
2mt )m],
E (Dt+1) = (1 + 12t )E (Dt) +mE [(1 − Dt
2mt )m].
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k > 0§
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mt_*u xZj¦rDs\zk*¡D£_*m
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t = |Vm+i \ S|u tGC,m
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t =∑
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vtu k
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©D_^qmorxz(Y 0
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m§O=¨£ kor¡
Y nt +
∑(k−1)mi=0 (m + i)Y i
t = 2mt¡?wxt¢¡?wm)_pwxqZkm_*
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mmozu wx£ kQwX©%wxu£vw¦\£ _.morumuvkw\zorX«lu ]-w%mo_^£i9?®rx]umnmut\
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t
2mt
?ÇcZ\_*z_δi,n = 1
ui = n
wxt²*_^zorrxmoZ\_^zocuvko_.A¤§_rxz
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t
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rx _^©x_*tTmk(Ed)d∈0,...,n
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0 )m
?Çt\rmw%murDtSb
a
knmwt\k¢®rDzPa+. . .+Pb
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a ≡ 0ua > b
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t+1 − Y it | Ed)
¦ji·mZ\__«l\z_^kknu rxtgQ
∑
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∏di=0 P
hi
i1
χd
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Zwxkn + 1
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h0, . . . , hdknsqZ moZwm ∑d
i=0 hi = mwxt
hd > 0¡wt
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w7mo_^zo]ψi,s
®rxzcmZ\_(qZwxt\x_emrY i
t
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tutΓ(vt)
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§
YZ_Kµzknmr moZ\_pkn_uvkcknu ]\£ iδi,0
§G°²_w£vkor7ZwX©x_
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?®u=§ _D§ uvt
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t
¡BrxmoZ\_^zocuvkn_Y d
t
u k\_^qz_^wDkn_pwxt
m+ d+ 1s\t\umkwxzo_)wxl_^mor
Y nt
A¤¡wt
ψi,s = δs,n × δi,s + (1 − δs,n) × ((m+ s)δs,n−1 + 1)δi,s+1 − δi,s
?Çus = n
moZ_*tY n
t
u kutq*zo_pwxko_^7¦jiarDt\_x¡Dus = n−1
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2
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B¹ (41 R I e!3 M)IKf(x, y) : IR → IR
HM3 H H!"T3NM3 2H<%<HLSRm
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Srx9kou^_
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(S dom. S) ≤ (S∗ dom. S∗)cZ\_*z_
S∗ = [s] = 1, ..., s §=Kk
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vt) =1
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k¡NmoZ\_a£_*Ém
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k¥=moZ~©D_*zomo_*«9¡Nwtknr
v = xk+1§)gjs\rTkn_(moZwm
xkmoZ_
k¥=moZ~zu xZTmK_^tlNrxu tDm
Zwxk£vw¦N_*£2k + s+ a1 + 1
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(v, u)£ _m(mZ\_
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ut±moZ_Vm.r q*wxko_^kwxk®rD££ r%kQ
(G(a1, a2)) = FLFR,L/Φ(2n)cZ\_*z_
FLu k(moZ\_tjs\](¦N_*z7rK£_*Ém7wxznmu wx£wxuzu t\Dk(r
2(k +1) + s+ a1 + a2
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wu zou t\7rxt2n− (2(k + 1) + s+ a1 + a2)
£ wx¦_^£ k^§ t_*umoZ\_^z¨q*wDkn_
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2n− (2(k + 1) + s+ a1 + a2)
s+ a1 + a2
)
(s+ a1 + a2)!Φ(2(n− ((k+ 1) + s+ a1 + a2))).? C.A
moZ_*z_euvkct\r(v, u)
_^\x_emoZ\_^t
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(
2k + s− 4
s
)
Φ(2(k − 2)),?;RA
wxtumoZ\_^zo_uvkcw(v, u)
_^lD_KmZ\_*t
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(
2k + s− 4
s
)
Φ(2(k − 2)).?I5A
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(v, u)u tGmZ\_*tL_*«jzo_pkokourDtka? C<A¤¡
?=NA.wz_)s\tqZwxt\x_prDtkow\\u t\wt (G) =
(φ(G))
§ moZ\_^zo_uvkcwt_^\x_(v, u)
moZ\_^t
(G(a1, a2)) = c(a1 + 1)?IXA
(φ(G(a1, a2))) =
(G(a2, a1)) = c(a2 + 1)
?=NAcZ_*z_
cuvkxu ©x_*tªwx¦r%©D_x§°V_kn_^_~mZw%m"u
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S?SlrD]utwmo_pk
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CcZu qZZwX©x_emZ\_\zrxN_*zomi (C) ≤ (φ(C))
§ tzo_^wzw%murDt·®rxzmZ\uvkcB_)t\rxmo_RQ?_m
σ = dn/ logne ¡\£_*m
BI = G : ∃i ≥ σkosqZmZw%m
GZwxkcwmc£_pwxknm7_plx_pk
(xi+1, xi)BII = G : ∃j > i ≥ σ
knsqZmZw%mGZwxkcwxt_plx_
(xi+1, xi)wxt"w%mc£ _^wxknm¨a_^lD_^k
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xw%mknmo_*
nrx
(n/x)1/2c log3 n?;kn_^_K_^§
:Ñ.;´¡: b<;AK®rxzw£ £x ≥ σ
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wtmoZwm (BII) = O(log6 n/n)
§Y r7mwx¶x_wxl©%wxtDmwD_Krx?moZu k^¡\._etrmo_)mZw%m
(S dom. S) ≤ (S ∪ [σ] dom. S \ [σ]).
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wDk[σ]
uvkwx£wXilku tmoZ\_lrD]utwmou t\-kn_*m^§?_m
Au = G ∈ : ∃(v, u)_plx_)u t
G ∩ (BI ∪ BII),
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∩ (BI ∪ BII),wxt£_*m
Cu = G ∈ :tr-_^lD_
(v, u).YZ_\zrTrx®rDz
C®rD££ r%kQlu zo_pq¤m£i(®zorD] mZ\_¨swxzq*wDkn_wDk
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BIwDkmoZ\_^zo_¨u kG_*«\wxq¤m£iarxt\_
(v, u)_plx_
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G(a1, a2) ∈ AB_)l_*µt\_emoZ\_]u t\u ]wx£¢xzw\Z
G(α1, α2)rGwxkB®rx£ £ r%kQ!¬_*m
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Sut
G§³YZjsku
e ∈ α2mZ\_*t
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S\rx]utw%mo_pk
S¡9¦\slm
G(a1, α2)lrj_^k^§ mKuvk Ts\umo_arTkokou¦£_mZw%m
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G(a1, a2)Zwxkcwm£_pwxknmrDt\_)]ut\u ]-w£¬Dzwx\Z¬§°²_)®rxz]/moZ\_ko_m
G(a1, a2) = G(α1 − I, α2 + I) : I ⊆ α1.YZjsk
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?®uA +eu©D_*tG(a1, a2) ∈ A
moZ\_^tG(α1, α2)
u ks\t\u Ds_x§?ÇuuA
G(a1, a2) ⊆ A§
?®u uuA G(a, b), G(a′, b′)
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A§
?_mG(a1, a2)
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A¡
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x > vknrmoZw%m
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u tq*£s\_wxt_^lD_(®zrx]x§'=£vknr
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xwxkemZ\_*z_-wz_
tr-wzw£ £_^£¬_^\x_^k^§BYZjskmoZ\_^zo_uvktr·_^lD_)®zorD]xutα1 = a1 + β2
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q^wt\trm\rx]utw%mo_
xwxt
G(α1, α2)u k¦wD®rxz
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(G(α1, α2)) ≤
(φ(G(α1, α2))).
?;RA
Õ ÕÚk **¤á
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(G(b2, b1))
wxkmoZ\_7rxslmo¥=_plx_aknwk¨ru, v
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(v, u)_^\x_LmZ\_*t
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(G(b2, b1))
§ moZ\_^zo_)uvkcw(v, u)
_^lD_KmZ\_*t®zrx] ?)5RA¡g?XA¤¡J?;RA
(G(b1, b2)) = c(b1 + 1),
(G(b2, b1)) = c(b2 + 1).=Kk
G(α1, α2) = G(α1, α2), ..., G(α1 − I, α2 + I), ..., G(∅, α1 + α2)¡\B_)ZwX©D_
(G(α1, α2)) = c
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j=0
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α1
j
)
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)
,cZ_*z_^wxk
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α2, ∅)¡\wtmoZjsk
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α1∑
j=0
(
α1
j
)
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= c(
(α1 + α2 + 1)2α1 − α12α1−1
)
.YZjsk
(φ(G(α1, α2))) −
(G(α1, α2)) = cα22
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m ≥ 2§
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(x(1), ..., x(m))rxkos\¦l¥´©x_^znmu q*_^krxl_^xz_*_
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1, ..., 2mn§¬_*mmoZ\_
(v, u)_plx_cu t Ds_^knmou rxt¦_
ei = (v(i), u(j))§°²_knw7moZ_c_^lD_^kGrx
v(i)wt
u(j)_«\q*_*lm
eiwt
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u(j + k)®rxzkow\\u t\§
?_mGS = Gm
t ¦N_amoZ\_-kowxq*_(rkoq^w£ _¥=®zo_^_xzwZk¨rxrxs\mn¥Øl_*Dzo_^_ m §aYZ\_7kowDq_GS
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GC = GC,mt l_^kqzu ¦_pu t·mZ\uvkBwN_*zp§ t_*umoZ\_^zB]rll_^£;¡jrxtwx\\umurDtr¢mZ\_erxslmo¥=_plx_pk
ej , j = 1, ...,mrx?©x_^znm_«
vt¡jmoZ\_
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v1, ..., vt−1wz_)utq*£sl_^¢§QYZjsku tmoZ\_kq*wx£_*¥;®z_*_e]rll_*£
(ej
qZrTrTkn_pkB®zorD]rxslm(v1, ..., vt−1)) =
2m(t− 1)
2m(t− 1) + 2(j − 1) + 1j = 1, ...,m.
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SrGknu *_ |S| ≤ dt) = O(ct),
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w£ ]rDknmkns\z_*£ i-®rDzd > 0
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(S∗ dom. S∗) = exp
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Unité de recherche INRIA Sophia Antipolis2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France)
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ÉditeurINRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)
ISSN 0249-6399