Balanced Labeled Trees: Density, Complexity and Mechanicity

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HAL Id: inria-00159564

https://hal.inria.fr/inria-00159564v2

Submitted on 4 Jul 2007

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Balanced Labeled Trees: Density, Complexity and Mechanicity

Nicolas Gast, Bruno Gaujal

To cite this version:

Nicolas Gast, Bruno Gaujal. Balanced Labeled Trees: Density, Complexity and Mechanicity. [Re- search Report] RR-6240, INRIA. 2007, pp.25. �inria-00159564v2�

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Thème NUM

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Balanced Labeled Trees: Density, Complexity and Mechanicity

Nicolas Gast — Bruno Gaujal

N° 6240

Juillet 2007

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n+ 1^amibtika P(A, n+ 1) =P(A, n)R&S1eUa b0SUXWhXikaCmydUXDw<dUzmydf`OmydUXXcz>Xmyidf²myefbCm<¨tXwy|S¸dUm1zfXR&S w<bCVDXw<dUa/b0S w<b/i£¨tqXa0bw<h0b{¨hmyV w

XWh0bXM¦m<¨b0SUX/hw<xfSIªqXCi¬¨mykm³ wzfXWbXWh0Vidfikab0ik|x w<b0SOamb0SUXWhXCikaKXM¦fwy|Wb0`

P(A, n) ¨»wy|WbmyhaKm<¨

kXWdfb0S

k ≤n

ÁÁ ' '@Á % * ,f,H8

A ⁵⁸⁶^{3 f8.} ^L⁼ A ^CU@0Bg ⁶^8.X α^c ^8DCE α ⁶⁵⁸⁶^{3 JL}

{3^3

= L akXWb|Sts[a

A ikathw<b0ikmyd w<¶ªfb0SUXWhX/XM¦>ika0ba

k a0eU|Sb0S w<bb0SUX[b0hXcX{ika|cmyVxfkXWbXW`zfXM UdUXcz«Q`iba a0ef«>¢¶b0hXcXca

(A₁, . . . , A_k) m<¨SUXWiS1b

kKsKa+b0SUXb0hXcX6ikahw<b0ikmyd w<¶ªµXwy|SOaef«>¢¶b0hXcX

A_i S wyaKb±qm|Sfikz>hXWd

(A_i₁, A_i₂)¸X|cmydUa0ikzfXWhKw\¹w<hkm|S w<id

(X_n) myd²b0SUX/aXWb

{A₁, . . . , A_k}^¤ P(X_n+1=A_i₁|X_n=A_i) =P(X_n+1 =A_i₂|X_n=A_i) = 1

2.

s aXcjQeUXWdU|cX

X₀, . . . , X_n, . . . zfXM UdUXcawefdfikjQeUXx w<b0S¹idb0SUXb0hXcX|cmyh0hXca0xmydUz>idf²bmwqmyhz

w

SUXWhX

w_i ika&b0SUX[w<«XWm<¨b0SUXKhm$mybm<¨

A_i¸X{|w<µzfXWdUa0ib§`m<¨b0Sfikatx w<b0Sb0SUX{zfXWdUa0ib§`m<¨¬b0SUX{tmyhz

w

i£¨¥ibXM¦$ika0ba³

´±¨¬b0SUX{\Ow<hkm³|S w<idikaqih0hXcz>eU|Wi«fkXCw<dUzw<xXWh0ikm1z>ik|ª>b0SUXWhX{XM¦>ika0ba

α a0eU|Sb0S w<bw<x w<b0SUa

w^{S wX}

w<VDma0b{a0efhXW`wzfXWdUa0ib±`

α |cmyh0hXcax©mydUz>idfbmb0SUX6a0bw<b0ikmyd w<h0`pz>ika0b0h0i«fefb0ikmydm<¨b0SUX6|S w<idIªSfik|S¹ika wamyefb0ikmydm<¨whw<b0ikmyd w<IidUXw<hKa0`$abXWV w<dUzikab0SUXWhXM¨»myhX/hw<b0ikmyd w<¶

kXWb{eUa[xfik|k¹waef«>¢¶b0hXcXwy|c|cmyhz>idfbm

Π^/uIXWb w «©Xwhw<dUzfmyV x w<b0S¸m<¨°kXWdfb0S

n «XWidfdfidfw<b b0SUXhm$mybqm<¨µb0Sfika°b0hXcX´±bikatw<kamz>ikab0h0i«fefbXczwy|c|cmyhz>idfbm/b0SUX+abw<b0ikmyd w<h0`¡z>ika0b0h0i«fefb0ikmydIY[m)kXWbqeUa

xfik|kb§qm¡hw<dUzfmyV x w<b0SUaKm<¨¥kXWdfb0S

n−1 id²b0SUXhXca0b+m<¨b0SUXaef«>¢¶b0hXcX z>ikan0myidQb+¨@hmyV b0SUXxfhXW1ikmyeUa x w<b0SMª

sKKm<¨/b0SUXcaXx w<b0S w<hXpz>ika0b0h0i«fefbXcz wy|c|cmyhz>idfºb0SUXOa0bw<b0ikmyd w<h0` z>ika0b0h0i«fefb0ikmydI u¬XWb

C_i ^«Xb0SUX

z>ika0b0h0i«fefb0ikmydOm<¨¥b0SUXCdQefV«XWh{m<¨

1 idOwx w<b0Sm<¨kXWdfb0S

n−it´±¨b0SUX/d1efV«XWh[m<¨°wb0hXcX6m<¨¥kXWdfb0S

n

ikaz>ika0b0h0i«fefbXczpwya

Bn

ªftX/S wcX/b0SUXChXWw<b0ikmydI¤

h(B_n) ∼ C_n+B_n−1+B_n−2+B_n−3+· · ·+B₁

∼ C_n+C_n−1+ 2C_n−2+ 2²C_n−3+· · ·+ 2ⁿ⁻²C₁,

(10)

iefhX¡U¤o¦fw<VxfkXm<¨+zfXc|cmyVxma0ib0ikmyd m<¨Kw²b0hXcXm<¨SUXWiSQb

5 idºx w<b0SUam<¨+kXWdfb0S>ªUª>ª¹w<dUz

o°wy|Sp|cmykmyh+|cmyh0hXca0xmydUz>idfDbm¡wx w<b0SI

Bn

ikaKwa0efV m<¨

Cn

b0SQeUa+idwcXWhw<Xª b0SUXCzfXWdUa0ib±`m<¨

An

ika α

R&SUX&d1efV«XWhm<¨

1m<¨b0SUX&a0ef«>¢¶b0hXcXm<¨SUXWiS1b

2nika¥b0SUXtd1efV«XWh°m<¨

1idb0SUX&a0ef«>¢¶b0hXcX

A_nm<¨ SUXWiS1b

n^xfeUa 2ⁿ b0iVDXca&b0SUX{d1efV«XWh+m<¨

1 idb0SUX/a0ef«>¢¶b0hXcXca

(a₁, . . . , a₂ⁿ) m<¨SUXWiS1b

nb0S w<b+w<hX/|Sfikz>hXWdm<¨

An

R&SQeUa+b0SUX/dQefV6«©XWh[m<¨

1 ^id An

ikadUXWii«fkX|cmyVx w<hXczbmb0SUX/dQefV6«©XWhKid

(a₁, . . . , a₂ⁿ)

´±¨b0SUX\Ow<hkm³p|S w<id¹ika/w<xXWh0ikm1z>ik|ª©b0SUX

a_ia/w<hX6z>ikab0h0i«fefbXcz¸wy|c|cmyhz>idfbmb0SUXa0bw<b0ikmyd w<h0`w

w<dUzb0SUXWdb0SUX[dQefV«XWh+m<¨

1^ika α´¶¨b0SUX[|S w<idika&x©XWh0ikm$z>ik|[b0SUX{|S w<idViSQb+S wcXCwzfXWdUa0ib±` »w<dUzib ika α&«fefbKw<kamDViSQb+dUmybKS wcXwzfXWdUa0ib±`ªid²b0S w<b+|wyaXªUqXaFwc`b0S w<b

α ikab0SUX/xXWh0ikm$z>ik|M¢±zfXWdUa0ib±`²m<¨

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´±¨b0SUX6|S w<id¹ikaKhXcz>eU|Wi«fkXªb0SUXWdOb0SUXw<d w<`$a0ika{|w<dO«©X|w<h0h0ikXczOeUa0idfb0SUXWmydUXa0bXWxbXc|Sfdfikj1eUX

¨»myhb0hw<dUa0ikXWdQb[\Ow<hkm³|S w<idUa

´±¨µb0SUX|S w<idikahXcz>eU|Wi«fkXª$b0SUXaXWbtm<¨µiba°a0bw<bXcaq|w<d¡«Xz>i1ikzfXcz¡id¡a0ef«UaXWba

S₁, . . . , Sm

myd¡Sfik|S

b0SUX|S w<id¸ika[ih0hXcz>eU|Wi«fkXxfeUaCwaXWb

O m<¨°b0SUXmyb0SUXWh/a0bw<bXca{b0S w<b/«XWkmydfbmdUm

S_i^Cu¬XWb α_i ^«©X6b0SUX

zfXWdUa0ib§`m<¨¥b0SUX/x w<b0SUaidb0SUX/a0ef«UaXWb

Si

´±¨qtXDa0bw<h0b{idUa0ikzfX¡w<d

Si

ªµqXbk1dUm³ b0S w<bCqXia0bw`¨»myhXWXWh/idb0Sfika

Si

S w<b/tXtw<d1bbm

a0SUm³ikaCb0S w<bCi£¨tXDa0bw<h0bmyefba0ikzfX¡b0SUXDaXWbm<¨&b0SUX

S_iªItXDS wX¨»myh6w<

i whw<b0ikmyd w<qxfhmy« w<«fiib±`¸m<¨

XWdUz>idfDid

Si

´±¨/b0SUXWhX²ika

l a0bw<bXca¡id

OªtqXpzfXM UdUXpwV¡w<b0h0i£¦

X m<¨Ca0i¯X

l×l^¤ Xij

ikaDb0SUXxfhmy« w<«fiib±`)m<¨

b0hw<dUa0ib0ikmyd¨@hmyV

i ^bm j @Sfik|SpXcj1e w<kaK>ª /myh

1/2MUmyh+Xwy|Spih0hXcz>eU|Wi«fkXa0ef«UaXWb

S_iªfkXWb+eUazfXM UdUX

ai

ww<h0iw<«fkXw<dUzkXWbeUa+zfXM UdUX/b0SUX/Xc|Wbmyh m<¨a0i¯X

l R SUXWhX{¨»myh+w<

o∈ O^¤ R_o =X

i

P(X_n+1∈S_i|X_n=o)a_i.

o°wy|S

Ro

ikawidUXw<h|cmyV6«fid w<b0ikmydDmydb0SUXtaXWb

{ai}uIXWbeUa¥zfXM UdUXwKXc|Wbmyh

P =P

iP(∃n/Xn∈ S_i|X₀ =o)a_i ^SUXWhX P(∃n/X_n ∈S_i|X₀ =o) b0SUXxfhmy« w<«fiib±`pm<¨XWdUz>idfidOw

S_i a0bw<h0b0idf¡id

o^{ª ib}

aw<b0ika" Xcab0SUXCXcjQe w<ib§`

P =XP +R.

´§d²myhzfXWhbmxfhmXmyefhhXca0efbª>qXCtw<dQbKbmxfhmX/b0S w<b+b0Sfikaa0`>a0bXWV S wya+wefdfikjQeUXamyefb0ikmydw<dUz

b0S w<bb0Sfikaamyefb0ikmyd²ikahw<b0ikmyd w<¶

(11)

S₁, α₁ S₂, α₂ o₁

o₂

o₃

1 2

1 2 1 2

1 2

R&SUXC|cmyh0hXca0xmydUz>idf¡V¡w<b0h0ik|cXca

X^ªR ^w<dUz P ^w<hX¤

X = 1 2





0 0 0 1 0 0 0 1 0



 R=



 a₁

1 2a₂



 P =





a₁

1

2(a₁+a₂)

1

4(a₁+ 3a₂)



.

iefhX[>¤¥o¥¦Uw<VxfkX{m<¨w/hXcz>eU|Wi«fkX{\¹w<hkm¡|S w<idI

Si

w<dUz

S₂ w<hXKb0SUX+b§qmih0hXcz>eU|Wi«fkX{a0ef«UaXWb¸X wyaa0efVDXb0S w<b[Xwy|S

S_i ika+xXWh0ikm1z>ik|w<dUz²S wya[zfXWdUa0ib§`

α_iR&SUXWd¹wDb0hXcX«©XWidfdfidfib0S

o₁ iS wX zfXWdUa0ib§`

α1

ªfib0S

o2 α1+α2

2

w<dUzib0S

o3

¤ α1+3α2

4

s[a°dUmXWkXWVDXWdQbm<¨

O «©XWkmydfatbmw<dih0hXcz>eU|Wi«fkX/aef«UaXWbª¨»myhw<©a0ef«UaXWba

O ⊂ OªQb0SUXWhX[XM¦>ika0baw<b

kXwya0b+mydUX6a0bw<bX/id

O a0eU|Spb0S w<bb0SUXCxfhmy« w<«fiib§`m<¨¥kXwc$idf

O ikahXw<bXWh+b0S w<d

0¤kXWbeUaK|w<¬ib

j₀

¸X/S wX

P

i∈OX_j_O_i<1 »w<dUz²m<¨|cmyefhaX{¨»myh+w<

i∈O^¤ P

i∈OX_ij ≤1^M

´±¨©b0SUXWhX+XM¦$ika0baw/Xc|Wbmyh

xa0eU|SDb0S w<b

X.x=xªkXWb°eUa|w<

O b0SUXaef«UaXWb

O={i/x_i 6= 0}R&SUXWd b0SUXWhXXM¦>ika0ba

jO

a0eU|SDb0S w<b

P

j∈OXij<1w<dUzb0SUXWd

P

ixi=P

i

P

jXijxj ≤P

i

P

jXijP

jxj <

P

jx_j Sfik|Sp|cmydQb0hwyz>ik|Wba

X.x=x°rtmydUaXcj1eUXWdQb0`ª b0SUX/V¡w<b0h0i£¦

Id−X ikaih0hXcz>eU|Wi«fkX

R&SUXa0`>a0bXWV

(Id−X)P =R S wyaKwDefdfikjQeUXamyefb0ikmyd¹a0idU|cX

Id−X ika+ih0hXcz>eU|Wi«fkXw<dUz¨ myh{w<

oª>b0SUXWhXCXM¦$ikaba

Poi

a0eU|S²b0S w<b¤

R_o=X

i

P_oia_i, P_oi ikab0SUXxfhmy« w<«fiib§` b0S w<bDa0bw<h0b0idf¨hmyV2b0SUXa0bw<bX

oª°b0SUX\¹w<hkm |S w<id)iXWdUzid b0SUX a0ef«UaXWb

Si

\OmyhXcmXWh{w<

Poi

w<hX{hw<b0ikmyd w<aidU|cXCw<IdQefV6«©XWha

Xoj

w<dUz

Ro

w<hX{hw<b0ikmyd w<¶

R&S1eUai£¨¥b0SUX/hm1mybm<¨¥b0SUX/b0hXcX/ika

r^ªUwya k bXWdUzfa+bmid> Udfib±`ªwa0ef«>¢¶b0hXcX/S wyaidOwcXWhw<X6wzfXWdUaib±`

α ^SUXWhX

α=X

i

Poiαi.

R&S1eUab0SUX{xXWh0ikm1z>ik|/zfXWdUa0ib§`m<¨b0SUX/b0hXcX{ika

α°R&S w<bikaSQ`i£¨b0SUX{b0hXcX/S wyawzfXWdUaib±`ªfibiµ«X

α∈Q

(12)

R&SUXb0hXcXika¡hw<b0ikmyd w<+a0idU|cXibDS wyaDb±qmºz>i£¨@¢

¨XWhXWd1bOaef«>¢¶b0hXcXcaOm<¨DSUXWiS1b¸ a0efh0hmyefdUzfXcz

S wya+wzfXWdUa0ib§`

R&Sfika¹mydUX S wyab§qm a0ef«>¢¶b0hXcXca¹m<¨DSUXWiSQb¸>

m$z>ik|

iefhXD>¤{R&Sfika{ UefhXhXWxfhXcaXWdQbab±tmhw<b0ikmyd w<b0hXcXcaR&SUX\Ow<hkm³O|S w<id |cmyh0hXca0xmydUz>idf²bmb0SUX

1/3{R&SUX\Ow<hkm³|S w<id|cmyh0hXca0xmydUz>idfbm¡b0SUX

kXM¨bmydUX/ikaxXWh0ikm1z>ik|Cw<dUzb0Sfika+b0hXcX/zfm1XcadUmybS wcXwDzfXWdUa0ib±`«fefbibaxXWh0ikm1z>ik|M¢±zfXWdUaib±`ika

1 2

+6 !©

sKdid> UdfibX6qmyhz

n >0ª b§qma0ef«UaXcj1eUXWdU|cXcaKm<¨

w m<¨kXWdfb0S

n

iµS wcXw<VDma0bb0SUX/aw<VDXCzfXWdUaib±`m<¨

1\OmyhX/xfhXc|WikaXW`ªfi£¨qXC|w<

1 ^idb0SUX

qmyhz

x^ª w ika« w<w<dU|cXcz 6

¨»myh+w<

n >0 w<dUz¨»myh+w<

x, y m<¨kXWdfb0S

n^¤

|h(x)−h(y)| ≤1.

s « w<w<dU|cXczºqmyhz S wya6V¡w<dQ`¸xfhmyxXWh0b0ikXcaO´±d¸x w<h0b0ik|Wefw<hªqw²« w<w<dU|cXcz qmyhz w<tw`$awyz>Vibaw

zfXWdUa0ib§` @Sfik|SVDXw<dUa

limn→∞h(w1...wn) n

XM¦>ika0batw<dUzw<dpw<xXWh0ikm1z>ik|C« w<w<dU|cXczqmyhzikaK_1b0efh0Viw<dI

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mydUXw<dUz¹xfhmy« w<«f`pb0SUXVDma0b{d w<b0efhw<mydUXªika{S w<b/tX|w<°w\)]8.¤/wb0hXcXika/aw<ikzObm«X

« w<w<dU|cXcz²i£¨w<Im<¨¥ibahm1mybXcza0ef«>¢¶b0hXcXca+m<¨¥SUXWiSQb

1^¤

"½$#&%(' '@Á)% i,.* .hw<w<dU|cXcz¸b0hXcX,R/

6 A 6

8a8. A ⁶¡« w<w<dU|cXcz 6= =

3 S&)3^

n ⁼ ^3S

k≥0^T

|h(A_[n,k])−h(A_[n0,k])| ≤1.

uIXWb¥eUaw<kam[idQb0hm$z>eU|cXwKa0b0hmydfXWhzfXM Udfib0ikmydI¤wKb0hXcXtika_8.3:[YXRJ i£¨w<>m<¨ iba¥dUmyd>¢¶hm$mybXcz

a0ef«>¢¶b0hXcXCm<¨¥SUXWiSQb

k w<dUzikzfX

1^¤

"½$#&%(' '@Á)% , ¶_1b0hmydf`« w<w<dU|cXcz¹b0hXcX,/ 8.

A ⁶^0H ⁶^8Z3 a0b0hmydf`p« w<w<dU|cXcz 6=4=

3S

k, q≥0 ⁼ ^{30Di 3^} n, n⁰^T

|h(A_[n,k])−h(A_[n,q])| − |h(A_[n0,k])−h(A_[n0,q])|

≤1.

(13)

R&SfikaKw<bXca0bKzfXM Udfib0ikmydOika+|WkXw<h0`²ab0hmydfXWh+b0S w<db0SUX6a0bw<dUzUw<hzOmydUXa0idU|cX/bw@k$idf

q = 0iVxfikXca

h(A_[n,k])−h(A_[n0,k])

≤1¸Xi©aXcXid¡b0SUX+dUXM¦$bqx w<h0b°b0S w<bqb0SfikaqzfXM Udfib0ikmydika°a0b0h0ik|Wb0`Dab0hmydfXWh a0idU|cX{b0SUXWhX6XM¦$ika0bb0hXcXcab0S w<bKw<hX/« w<w<dU|cXcz«fefbdUmybKa0b0hmydf`« w<w<dU|cXczµ

sKb0SUmyefSb0SUX6zfXM Udfib0ikmydOm<¨°w« w<w<dU|cXczpb0hXcX/ikaKqXw@kXWh{w<dUzaXw<VDaKVDmyhXCd w<b0efhw<¬¨myhKwDXWdUXWh"¢

w<i¯³w<b0ikmyd¨ hmyV tmyhzfaª>qX{iµaXcX[b0S w<ba0b0hmydf`« w<w<dU|cXczb0hXcX[S wcXCw<VDma0bb0SUX{aw<VDX[xfhmyxXWh0b0ikXca

b0S w<d²iba|cmyefd1bXWh0x w<h0b[mydtmyhzfa

}pao q\r-t_yvw x

t Er w

rr

´§db0Sfikax w<h0bª

Umyhw<

n^ª |h(A_[n,k])−h(A_[n0,k]| ≤1 b0SQeUaq¨»myh&w<

k tX{|w<dzfXM UdUX

m_k Sfik|S²wyatb0SUX+VidfiV6efV m<¨

h(A_[n,k]) m³XWhKw<

n

m_k≤h(A_[n,k])≤m_k+ 1.

s a0ef«>¢¶b0hXcXKm<¨SUXWiS1b

q+k ika°|cmyVxmaXczm<¨Iw/b0hXcX+m<¨Ia0i¯X

mq

w<dUz

mq+ 1

mydUXcaib0S

2^q |Sfikz>hXWdpm<¨a0i¯X

k Sfik|S²S wXC«XWb±tXcXWd

m_k ^w<dUz m_k+ 1 mydUXca³R&SQeUa¤

2^qmk+mq≤m_q+k≤2^q(mk+ 1) +mq+ 1.

ÁÁ ' '@Á % i,J {XWdUa0ib±`m<¨« w<w<dU|cXczb0hXcX,?M8

A ^{VJ 8.^L}

CEHW

6

8DR9U

6

GM9g α^c DJB8DCgzfXWdUa0ib±` 3 =

A ^9gCe8DCU8 ⁼ ^3VS n^T

k→∞lim Π(A_[n,k]) =α.

]33>@

∀n, k:|h(A_[n,k])− b(2^k−1)αc| ≤1.

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2^k−1 ≤Π(A_[n,k+q])−Π(A_[n,k])≤ m_q+k+ 1

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2^k−1 ≤Π(A_[n,k+q])−Π(A_[n,k])≤ 2^q(mk+ 1) +mq+ 1

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