Common words between two random strings

HAL Id: inria-00071368

https://hal.inria.fr/inria-00071368

Submitted on 23 May 2006

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Common words between two random strings

Philippe Jacquet

To cite this version:

Philippe Jacquet. Common words between two random strings. [Research Report] RR-5899, INRIA.

2006. �inria-00071368�

ISSN 0249-6399 ISRN INRIA/RR--5899--FR+ENG

a p p o r t

d e r e c h e r c h e

Thème COM

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Common words between two random strings

Philippe Jacquet

N° 5899

Avril 2006

Unité de recherche INRIA Rocquencourt

Domaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex (France)

Téléphone : +33 1 39 63 55 11 — Télécopie : +33 1 39 63 53 30

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Œ7d=UQ

z2 ∈ C/ ’mpa:a:rbFGR†Q:zlT]z3TVR†Q eI=<aO3RbH[HLOVQ:R!<T]JLRbOVQ r†=<;‹;‹T OVQ

Cn(z2) ^{TVQde OVQ} g(n, z2)^{”Œ=}zV= J} Cn,m=Cn(m) +O(_mⁿ + 1) =C(n, m) +O(_mⁿ +^m_n)^

(' )* +,+- /.0 21345 6 68791:; <+=>+= ?@

… =TVH[HLP:;‹=7:=UY[=BJL7dT]J

p1=p2 =p^TVQde q1=q2 =q5ZJL7:=HLOVPdY[NU=B;‹OIeI=<rbH`TVYL=wRSeI=UQqJ[RbN<T]r'²Q’JL7:RSH N<TVHL=‡Œ=NUTVQe:= 4dQ:=

c(z) =C(z, z)Œ7:RSNW7’HLT]JLRSH 4d=MH75

c(z) =c(pz) +c(qz) + (1−e^−z)^{2 j}

… =`ŒRbr†r1a:Y[O‚oV=JL7T‚J

Cn,n=n^{2 log 2}_h +Q(logn)n+o(n)^”VŒR†JL7 h=−plogp−qlogqJ[7:=`=UQqJ[YLO3aqF YWT‚J[=VBA‡PdTVQqJLR†J^F Š˜P:QdN JLRbOVQ

Q() ^RSH™T HL;ŽT]rbra=<YLRbOIeIRbNŠ˜PdQdNšJ[R†O3Q T];‹a:rbR†JLPdeI= HL;ŽT]rbr†=<Y‹JL7dTVQ

10⁻⁶

Œ7dRbNW7RbH`Q:OVQ !U=<YLO‹OVQdr†F™Œ7d=UQ

logp logq

RbY[Y[T]JLRbOVQdTVr = •:N =<aIJL=MeŒ7:=UQ

p=q= ¹₂^

… =>“qP:RSNGr†F’adT3HLH‡O‚oV=<Y`J[7:=>N<TVHL=

p=q= ¹₂ »JL7d=ˆ:RbQdTVYLF’P:Q:R†Š˜OVY[;DNUT3HK=/Œ7:=<YL=JL7:=

c(z) ^7dTVH

J[7:=HLR†;‹a:rb=w= •Ia:Y[=<H[HKRbOVQ

c(z) = 2z−1 +e^−2z ²QJ[7:RbHpNUTVHL=BŒ=w7dT–o3=

Cn,n = 2n+O(1)^

… =‹Q:O‚Œ N O3QdHKRSeI=<YpJ[7:=‹zV=UQd=UYWT]rNUTVHL=VEGRbQdNU=

c(z)^RbH O(z²)^Œ7:=UQ z →0^{T]Qde RbH} O(z) Cn, n

RSHHL;‹TVr†rb=UYBJ[7dT]QlJL7:=ŽQGP:;ˆ=<YO]ŠQ:OIeI=<HwRbQTJ[YLRb=‹ŒRJ[7

2nR†QeI=Ua1=UQde:=UQqJHK=M“qP:=UQdNU=<Hš” J[7:=™A-=<r†rbR†Q J[Y[TVQdHKŠ˜OVY[;O]Š

c(z)^” c^∗(s)^RSH`eI= 4Q:=<eŠ˜OVY

<(s)∈]−2,−1[T]QeH[T‚J[RbH(4d=MH

c^∗(s) = (2^−s−2)Γ(s) 1−p^−s−q^−s

k

… =’PdHL=}J[7:=’R†QGoV=<Y[HL=’A-=Urbr†RbQ

c(z) = _2iπ¹ Rc+i∞

c−i∞ c^∗(s)z^−sds ^Š˜O3Y c ^{RbQ J[7:=-eI= 4Q:RJ[R†O3Q} e:OV;ŽT]RbQ OVŠ

c^∗(s)^

‡HKRbQ:z‹JL7d=YL=MHKRSeIPdHJL7d=UOVY[=U;‰OVQJL7:=HLRbQ:zVP:rST]Y[R†JLRb=<HO]ŠtJL7:=RbQqJL=Uz3Y[TVQdeŒ7:RbNW7’T]Y[=+5

¹t¹©8:9;<<

% T‚J

s=−1ŒR†JL7’Y[=<HLRbe:PdH

−^{2zlog 2}_h ^

%

T‚J s=sk

” k6= 0JL7:=eI=<QGP:;‹=UYWT]ˆ:rb=HK=UJ`O]Š%JL7:=OVJL7:=<YYLOGO]JWH8O]Š

1−p^−s−q^−sŒRJ[7-YL=MHKRSeIPdH

z^−sk_p−sklog^2−sk_p+q⁻²_−sklogqΓ(sk)^

687d=UY[= Š˜OVY[= ˜RbzVQ:O3YLRbQ:z>YL=MHKRSeIPdHT]J

<(s)≥0 c(z) =2zlog 2

h −X

k6=0

z^−sk 2^−sk−2

p^−sklogp+q^−sklogqΓ(sk) +O(1) ^{^xMu}

O]J[RbNU=iJL7dT]JP:Q:rb=<H[H J[7:=+rbOVzqT]Y[RJ[7:;ŽH O]ŠGJ[7:=Za:Y[OVˆdTVˆ:Rbr†R†JLRb=<HtT]Y[=iR†QqJL=<zV=<Y;P:r†JLRba:rb=

logp logq

RSHYWT‚J[R†O3QdT]r

RbQlJL7:=Žˆ:RbQdTVYLF N<TVHL=%‡Œ=>7T–oV=

∀k 6= 0 : <(sk)> −1 ^ˆ:PIJ lim inf<(sk) =−1Ž687:=<YL=UŠ˜OVY[=J[7:=

rb=<T3eIRbQ:zJL=UY[; RSH

2 log 2

h n^{ … 7:=UQ log}_log^p_q RSHY[T]JLRbOVQdTVrJL7:=<Q JL7:=<YL=ŽRSHT’HLP:ˆdHL=<“qP:=UQN =ŽO]ŠJL7d=

sk

J[7dT‚J

TVYL=wY[=Uz3P:rbTVYLRbrbF™HKadT3N =Me}O3QJ[7:=o3=UYLJLRSNUT]r T]•GRSH

<(s) =−1Z687:RSHRb;‹adTVN J[H8J[7:=r†=MTVeIRbQ:z‹J[=UY[;‰ŒR†JL7-T HL;ŽT]rbr a=<YLRbOIeIRSNBNUOVY[YL=MNšJ[R†Q:zJL=<YL;

zQ(logz)^

)* +,+- /.0 21345!" .0@ 4 1$ " 1 >+= ?@

… =ŒRbr†rdHL7:O‚Œ RbQŽTBrbTVYLz3=HL= JZOVŠ1NUT3HK=MHiT]Qde>N O3Q‚\^=<N JLP:Y[=8R†Q‹OVJL7:=<YZNUTVHL=<H%J[7dT‚J

C(z, z) =O(z^κ)^Œ7d=UY[=

κ≤1^T]Qde κRbHpJL7d=HL;‹TVr†rb=<HKJpY[=<TVrQGP:;ˆ1=UYwHLPdNW7’JL7:=<YL== •IRSH^JWH

s1, s2 >0 HKPNW7’JL7dT]J

s1+s2=κ

TVQde

p^s₁¹p^s₂²+q₁^s1q^s₂² = 1^

… =ŒTVQqJtJ[OBPHK=8O]ŠJL7:=`e:OVP:ˆ:rb=A-=UrbrbR†Q>JLYWT]QH^Š˜O3YL;

R RC(z1, z2)z^s₁¹⁻¹z^s₂²⁻¹dz1dz2

”]RbQqJL=<zVYWT‚JL=Me

Š˜O3Y

<(s1) =c1

T]Qe <(s2) =c2

ŒR†JL7

(c1, c2) R†Q’J[7:=eI= 4Q:RJ[R†O3Q eIOV;ŽTVR†Q’O]Š

C^∗(s1, s2)PIJ`JL7dRbH RSH

Q:OHLPdNW7}eIO3;ŽT]RbQ™HKRbQdN =‡Œ=‡7dT–oV=

C(z1, z2) =O(zi)Š˜OVY8T]rbrNUT3HK=MH75

zi→0^TVQde zi→ ∞

i ∈ {1,2} ^{ 6tO} YL=<;‹O‚oV=™J[7:= eI=MTVeIrbOIN Œ=-eI=74dQ:=

C(z˜ 1, z2) = C(z1, z2)−D(z1, z2) ^Œ7d=UY[=

D(z1, z2) =z1e^−z1D1(z2) +z2e^−z2D2(z1)−C1,1z1z2e^{−z1−z2”IŒ7:=UY[=} D1(z) = _∂z^∂

1C(0, z)^T]Qde D2(z) = _∂z^∂

2C(z,0)^

687dT]J>Œ8T–F

C(z˜ 1, z2) = O(z²_i) ^Œ7d=UQ zi → 0 ^T]Qde eIOVP:ˆdr†=A-=Urbr†RbQ JLYWT]QdHKŠ˜OVY[;

C^∗(s1, s2) ^OVŠ

C(z˜ 1, z2) = •IRbHKJ[HŠ˜O3Y

−2<<(si)<−1^

€  &tI€ ž €  ~ &

687d=ZA-=Urbr†RbQwJLYWT]QH^Š˜O3YL; OVŠ

Di(z)= •IRbHKJ[HT]QdeBRSH =<“qPdTVr‚JLO

D^∗_i(s) =−_1−pip^Γ(s)−s

j −qiq^−s_j

”–HKRbQdN =

∂

∂ziC(z1, z2) = pi ∂

∂ziC(p1z1, p2z2) +qi ∂

∂ziC(q1z1, q2z2) +e^−zi(1−e^−zj)^

687:=<YL=UŠ˜OVY[=ˆGFadrbT–FGRbQ:zwŒRJ[7‹A =Urbr†RbQJ[Y[TVQdHKŠ˜OVY[;

Di(z) =^z_h^logz

ij +O(z)^ŒRJ[7 hij =−pilogpj− qilogqj



 ¢ ¡ | & % && <€–t~ ¡ €

… =7dT–o3=pJ[7:=;‹OGe:R.4d=MeŠ˜P:QdNšJ[R†O3QdT]r=<“qPdT‚J[R†O3Q 5

C(z˜ 1, z2) = C(p˜ 1z1, p2z2) + ˜C(q1z1, q2z2) + (1−e^−z1)(1−e^−z2) + +D(p1z1, p2z2) +D(q1z1, q2z2)−D(z1, z2).

·˜¸%¹·»º

687d=UY[= Š˜OVY[=

C^∗(s1, s2) =

1

1−R(s1, s2)+ s1

1−R(−1, s2)+ s2

1−R(s1,−1)+ s1s2

1−R(−1,−1)

×

×Γ(s1)Γ(s2)

ŒR†JL7

R(s1, s2) = p^−s₁ p^−s₂ ² +q₁^−s1q^−s₂ ^2”ZRbQ adTVH[HKRbQ:zlŒ=7dT–oV=

C1,1 = _{1−R(−1,−1)}¹ 9‡HKRbQ:zJL7:=

RbQGoV=<Y[HL=BA =Urbr†RbQJLYWT]QdHKŠ˜OVY[;Œ=w7dT–o3=

C(z, z) =˜ 1

2iπ 2Z

<(s1)=c1

Z

<(s2)=c2

C^∗(s1, s2)z^−s1−s2ds1ds2

KxVx%

… =ŒRbr†r;‹O‚o3=‡J[7:=RbQ3J[=Uz3Y[T]JLRbOVQ’eIOV;ŽT]RbQO]Š

s1

JLO

<(s1) =c1

T]Qde}JL7d=R†QqJ[=UzVYWT‚J[R†O3QeIO3;ŽT]RbQOVŠ

s2

JLO

<(s2) =c2

Š˜OVY

c1, c2 =−1 + ^ŒRJ[7 >0 ^{HKPdNW7 JL7dT]J} R(s1, s2)HKJ[T–FIHpŠT]YwT–ŒT–FŠ˜Y[OV; xV

687d=‡O3Q:r†FŽa1O]J[=UQqJLRST]r/a1OVrb=<H=UQN OVPdQ3J[=UY[=<eT]Y[=‡T‚J

s1=−1^T]Qde s2=−1T]QdeT]Y[=‡N<T]QdNU=<T]rb=<e™ˆqF™JL7:=

T3e:eIR†JLRbOVQdTVr/JL=<YL;ŽH<

687:=<YL=UŠ˜OVY[=’Œ=-7dT–o3=’=<“qPdT‚J[R†O3QKxVx/‹H^J[R†rbr`o‚T]rbRbe Š˜O3Y

<(s1) = c1

TVQde

<(s2) = c2

Š˜O3Y}HKO3;‹=

c1, c2>−1^

@‡QlJL7:=ŽQd=UŒRbQqJL=<zVYWT‚JLRbOVQeIOV;ŽTVR†QŒ=ŽN<T]QlŠ˜OGNUPdHL=>O3Q:rbF OVQ JL=UY[;

R R Γ(s1)Γ(s2)

1−R(s1,s2)z^{−s1−s2”JL7:=}

OVJL7:=<Yrb=<T3eIRbQ:zJ[O=U•IaO3Q:=UQqJLRST]rbrbF>Qd=UzVrbRbzVRbˆ:r†=‡JL=<YL;ŽH< ²QdeI=U=MeŽJ[7:=BJL=UY[;

s1Γ(s1)Γ(s2) 1−R(−1,s2)

7dTVHQ:OaO3r†=‡Š˜OVY

<(s1)>−1”T]QeJL=UY[;

s2Γ(s1)Γ(s2) 1−R(s1,−1)

7dT3HQ:O™a1OVrb=Š˜O3Y

<(s2)>−18687:=J[=UY[;

s1s2Γ(s1)Γ(s2) 1−R(−1,−1)

7dTVH

QdO>a1OVrb=T‚J`TVr†r'

687:=<YL=UŠ˜OVY[=wŒ=w7dT–o3=

C(z, z) = 1

2iπ 2Z

<(s1)=c1

Z

<(s2)=c2

Γ(s1)γ(s2)

1−R(s1, s2)z^−s1−s2ds1ds2+O(z^−M) ^KxMs+

dOVY`TVYLˆ:R†JLYWT]Y[F

M >0^

… =‹ŒR†rbrr†OGO+-T]J‡J[7:=>HLRb;>adr†R-4d=<eNUT3HK=Œ7:=UY[=

p1 =q1 = ¹₂ ^T]Qde (p2, q2) = (p, q)z3=UQ:=<Y[TVr) ²Q J[7:RSH`NUT3HK=

R(s1, s2) = 2^s1R(s2)^ŒRJ[7 R(s) =p^−s+q^−s J[7:=c=<FqQdR†=<Y=UQqJLY[OVaGFV

O‚Œ Œ=‡ŒR†rbr/;>O‚o3=JL7d=‡RbQqJL=Uz3Y[T]JLRbOVQŽrbR†Qd=pO]Š

s1

JLO‚Œ8T]YWe‹aOqHKR†JLRboV=‡YL=MT]r/T‚•IRSH+eIRbYL=MNšJ[R†O3Q

=B;‹=U= J

O3Q:=pa1OVrb=‡T]J

s1= 0 ˜Y[=<HLRbeIP

1ši687:=‡aO3r†=MH+OVŠ Š˜P:QdN JLRbOVQ

(1−2^s1R(s2))^{−1 T‚J} s1=−log₂R(s2) +

2ikπ log 2

”IŠ˜O3Y

kRbQqJL=<zV=UY YL=MHKRSeIPdH

−log 2¹

JL7:=<YL=UŠ˜OVY[=‡Œ=7dT–o3=+5

C(z, z) =˜ − 1 2iπ

Z Γ(s2)

1−R(s2)z^−s2ds2+

+X

k

1 2iπlog 2

Z Γ

−L(s2) +2ikπ log 2

Γ(s2)z^{L(s2)−s2−2ikπlog 2}ds2+ +O(z^−M)

¹t¹©8:9;<<

ŒR†JL7

L(s) = log₂R(s) ^T]Qde M >0ˆ1=URbQ:z}T]Y[ˆ:R†JLYWT]Y[F ˜ˆ:P:J 4:•I=<eJ[O™7dT–oV=wP:QdRŠ˜O3YL;J^F}O]Š

O(.) ^{ J}

J[P:Y[QdHOVP:JJL7dT]J`Š˜P:QdN JLRbOVQ

−1−R(s)¹

TVQdeŠ˜P:QdNšJ[R†O3Q

P

k

Γ(^{−L(s)+2ikπ}log 2)

log 2 z^{L(s)−2ikπlog 2} 7dT–o3=HLTV;>=wa1OVrb=<H TVQde JL7:=<R†YŽY[=<HLRbe:PdHŽNUTVQdN =MT]r' ²Qde:=U=<e J[7:=-YLOGOVJ>OVŠ

logR(s)−2ikπ TVYL=’= •:T3NšJLrbF JL7d=-YLOGO]J

sk

OVŠ

1−R(s)T]QeJL7:=Y[=<HLRSeIPdH8O]Š

Γ

L(s)−^2ikπ_{log 2} RSHJ[7:=H[T];‹=TVHJL7:=Y[=<HLRSeIP’O]Š

1 1−R(s)

8687:=<YL=UŠ˜OVY[=

J[7:=>R†QqJL=<zVYWT‚J[R†O3Q-adT‚J[7 NUTVQ ˆ=>;‹O‚oV=MeJLO‚Œ8T]YWe-T]Y[ˆ:RJ[Y[TVYLFaOqHKR†JLRbOVQwmpQGFq7dO‚Œ Œ=N<T]Q:QdO]J‡RbQdNUYL=MTVHL=

J[OGOp;PNW7JL7:=o‚T]rbP:=+OVŠ

c2

ˆ=MNUTVPdHK=+JL7:=Š˜P:QdN JLRbOVQ

z^{L(s)−s−2ikπlog 2} RbQdNUYL=MTVHL=<HŒ7d=UQ

s→ ∞%687:=UY[= Š˜O3YL=

Œ=ŽNUTVQ J[T+V=>Š˜OVY

c2

J[7:=‹o‚T]rbP:=‹O]ŠŒ7dRbNW7;‹R†Q:Rb;‹R-!<=<H

L(s)−s ˆ=UJ^Œ=<=UQ

−1,0^{ = JHNUTVr†r} κ ^J[7:RSH

;‹RbQ:Rb;P:; o–TVr†Pd=V

=NUT]QRbzVQ:O3YL=‡JL7:=wJ[=UY[; RbQ

z^−s 1−R(s)

HKRbQdN =

L(s)>0^Š˜OVY s ˆ=UJ^Œ=<=UQ

−1,0^

687:=<YL=UŠ˜OVY[=wŒ=w7dT–o3=

C(z, z) = 1 2iπlog 2

Z

<(s)=c₂

X

k

Γ

−L(s) +2ikπ log 2

Γ(s)z^{L(s)−s−2ikπlog 2}ds+O(z^κ−) ^Kx

@‡Qw=<T3NW7BJL=UY[; J[7:=UY[=iRSHTH[TVe:eIrb=aO3R†QqJtT]J

s=c2

JL7d=UY[= Š˜OVY[=i=<T3NW7‡OVŠGJL7:=J[=UY[;‹H

Γ

−L(s) +^2ikπ_{log 2}

Γ(s)z^{L(s)−s−2ikπlog 2}

ŒRbrbrdN O3Q3J[YLRbˆ:PIJ[=TVH

z^κ

√2πα2logzΓ

c1+^2ikπ_{log 2}

Γ(c2)z^{−2ikπlog 2 ŒRJ[7} c1=−c2−κ687:RbHRSHi=M“3PdR†o‚T]rb=UQqJ

J[O √ z^κ

2πα2logzΓ(c1)Γ(c2)^ŒR†JL7 dPNšJLPT‚JLRbQ:z>JL=<YL;ŽH8OVŠ%a1=UY[R†OIe

log 2^

7:=<Q

logp logq

RbHYWT‚JLRbOVQdQdT]r'”]J[7:=UQ}JL7:=‡Š˜P:QdN JLRbOVQ

R(s)RbHa1=UY[R†OIeIRSN`O3QŽJL7:=BRb;‹TVzVRbQdT]Y[F‹T‚•IRbHrb=<TVe:R†Q:z J[OT]QŽRbQ"4dQdRJ^F‹O]ŠH[TVedeIr†=paO3R†QqJ[H<687:=UY[= Š˜O3YL=8J[7:=p;ŽT]RbQŽJ[=UY[;

z^κ

√2πα2logzΓ(c1)Γ(c2)z3O]JTVrbHLOO]J[7:=UY PdNšJ[PdT‚J[R†Q:z JL=<YL;ŽHO]Š`a1=UY[R†OIe Q:OVJQ:=MN =MHLH[T]Y[F ;P:rJ[R†adr†=O]Š

log 2^3²Q NUOVQdNUr†PdHLRbOVQ JL7d=dPNšJLPT‚JLRbQ:z J[=UY[;

z^κQ(logz)”IŒ7:RbNW7’NUTVQˆ1=N O3;‹a:r†=UJL=<r†FNW7dT]YWTVN JL=UY[R!U=<e”3;ŽT–F™Q:OVJ`ˆ=a1=UY[RbOGe:RbNwT‚J`TVr†r'

687:=;ŽTVR†QJL=<YL;NUT]Qˆ1=YL=<ŒYLR†JL=<Q}J[7:=wŠ˜OVrbr†O‚ŒRbQ:z>ŒT–F

C(z, z) = z^κ

√π∆R∇RlogzΓ(c1)Γ(c2) +z^κQ(logz) +o(z^κ) ^Kx7*

ŒR†JL7

∆R=_∂s^∂22

1R+_∂s^∂22

2R^TVQde ∇R= ¹₂(_∂s^∂₁R+_∂s^∂₂R)^T‚J (c1, c2

Y[OGO]J[HOVŠ

1−R(s1, s2)^Œ7:RSNW7

;‹RbQ:Rb;>R!U=MH

s1+s2



?rSTVH[HKRSNwT]QdTVr†FIHLRbHz3R†o3=<HŠ˜OVY8J[7:=ˆ:RbQdT]Y[F}T]rba:7dTVˆ=UJ 5

κ=log^q_q²₁log log^q_q²₁ + log^p_p¹₂log log^p_p¹₂−log^q_q²₁^p_p¹₂ log log^q_q²₁^p_p¹₂ logp1logq2−logp2logq1

KxMh+

%RbzVP:Y[=}xe:RbHLa:rST–FGH`JL7:=>o‚T]Y[R†O3PdHo‚T]rbP:=O]Š

κ^ OVJLRSN =JL7dT]JBN O3QqJLYWT]Y[F™JLO 4dzVPdYL=

κeIY[OVadHpŒ=<r†rtJLO

!<=UY[OŒ7d=UQ

p2

OVY

q2

TVYL=wN rbO3HL=‡JLO !U=<YLO>Œ7:=UQ

p1=q1=¹₂”Iˆ:PIJJL7d=wNUOVQGoV=<YLz3=UQdNU=pRSHoV=UY[F™HLr†O‚Œ R†Q

1 logp2



·˜¸%¹·»º