Tail Bounds for the Wiener Index of Random Trees

Tail Bounds for the Wiener Index of Random Trees

T¨amur Ali Khan and Ralph Neininger

Department for Mathematics and Computer Science, J.W. Goethe-University Frankfurt, 60054 Frankfurt a. M., Germany

received 17 Feb 2007,revised 23^rdJanuary 2008,accepted.

Upper and lower bounds for the tail probabilities of the Wiener index of random binary search trees are given. For upper bounds the moment generating function of the vector of Wiener index and internal path length is estimated. For the lower bounds a tree class with sufficiently large probability and atypically large Wiener index is constructed. The methods are also applicable to related random search trees.

Contents

1 Introduction and results 279

2 The upper bound 281

3 The lower bound 284

1 Introduction and results

The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices of the graph. The distance between two vertices is defined as the minimum number of edges connecting them. The index was introduced by the chemist Wiener in 1947, in order to study relations between organic compounds and the index of their molecular graphs. For trees the Wiener index has been studied by discrete mathematicians and chemists, cf. the survey of (DEG01).

For random tree models comparatively little is known about the Wiener index. (EMMS94) studied the average Wiener index of simply generated families of trees and showed that the average is asymptotically Kn^5/2, whereKis a constant depending on the simply generated family andn→ ∞denotes the number of nodes. For some of these families (ordinary rooted trees, rooted labeled trees and rooted binary trees) they also gave exact formulæ for the expected Wiener index. (Jan03) proved a limit law for the Wiener index of these tree classes and identified the limit as a functional of the Brownian excursion. (FJ07)

†Supported by an Emmy Noether Fellowship of the Deutsche Forschungsgemeinschaft.

1365–8050 c2007 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France

studied the right tail of this limit. Average Wiener indices of some other tree classes were computed by (Wag06; Wag07).

In this paper we present tail bounds for the Wiener indexWn of random binary search trees with n internal nodes. The average Wiener index of random binary search trees was derived in (HN02),

EWn= 2n²Hn−6n²+ 8nHn−10n+ 6Hn, (1) whereHndenotes the harmonic numberHn=Pn

j=11/j. In (Nei02) the Wiener index of random binary search trees and random recursive trees was studied with respect to limit laws. By setting up a bivariate distributional recurrence for the Wiener index and the internal path length techniques from the contraction method could be used. For the tail bounds of the present paper we also use this recursive description: We denote by(Wn, Pn)the vector of Wiener index and internal path length of the random binary search tree withninternal nodes, and byInandJn =n−1−Inthe cardinalities of the left and right subtree of the root. Then,InandJnare uniformly distributed on{0, . . . , n−1}. We have the recurrence,

Wn

Pn

d

=

1 n−In

0 1

WI_n

PI_n

+

1 n−Jn

0 1

W_J⁰

n

P_J⁰_n

+

2InJn+n−1 n−1

, (2)

where(Wi, Pi),(W_j⁰, P_j⁰),0≤i, j ≤n−1,In are independent andL(W_j⁰, P_j⁰) =L(Wj, Pj). For the rescaled quantitiesY0= (0,0)and

Yn=

Wn− EWn

n² ,Pn−EPn

n

, n≥1,

a bivariate limit law and convergence of the covariance matrix has been shown, see (Nei02).

Here, we present the following tail bounds:

Theorem 1.1 LetL0 .

= 5.0177be the largest root ofe^L = 6L² andc = (L0−1)/(24L²₀) .

= 0.0066.

Then we have for everyt >0and everyn≥0

P

W_n− EW_n n² ≥t

≤











exp(−t²/36), for0≤t≤8.82, exp(−t²/96), for8.82< t≤48L₀, exp(−ct²), for48L0< t≤24L²₀, exp(−t(logt−log(4e)), for24L²₀< t.

The same bound applies to the left tail.

We denote iterated logarithms by log^(k)n, i.e., log⁽¹⁾n := lognandlog^(k+1)n := log(log^(k)n)for k≥1.

Theorem 1.2 For allt >0and alln≥0we have P(|W_n−EW_n| ≥tEW_n)≤exp

−2tlogn

log⁽²⁾n+ logt−log(2e) +o(1) , where theo(1)is with respect ton→ ∞and can also explicitly be bounded.

Furthermore we have a lower bound on the tail probabilities ofW_n:

Theorem 1.3 For all fixedt >0and all sufficiently largenwe have P(W_n− EW_n> tEW_n)≥exp

−8tlogn

log⁽²⁾n+O

log⁽³⁾n .

To derive upper tail bounds in Section 2 we estimate the moment generating function Eexphs,Yni, s ∈ R², from above, see Proposition 2.1, so that tail bounds can be obtained by Chernoff’s bounding technique. The bounds for Eexphs,Y_niare proved by induction on n using recurrence (2) for the induction step. For this, we extend the analysis of the tails of the Quicksort complexity as given in (R¨os91) and refined in (FJ02) to our two-dimensional setting. Note that the second component ofY_n is distributed as the normalized number of key comparisons used by Quicksort.

Another approach to tail bounds is via the method of bounded differences. A Doob martingale onW_n can be defined via an appropriate filtration and its martingale differences can be estimated. We extended earlier analysis of (MH96) for the Quicksort complexity to the Wiener index but do not discuss this here since the resulting bounds we obtained are not tighter than the ones found by the approach presented.

However, details of the application of the method of bounded differences to our problem can be found in the dissertation of (AK06), where also proofs that we omit subsequently are worked out.

In Section 3 we prove Theorem 1.3. For this we construct a class of binary search trees having atypically large Wiener indices and show that the random binary search tree is in that class with sufficiently large probability. This construction also builds upon the analysis of (MH96) for lower tail bounds forPn.

The methods used are applicable to related random search trees such as random (point) quad trees or randomm-ary search trees and depend on a precise expansion of the average Wiener index of the tree.

2 The upper bound

Our tail bounds in Theorem 1.1 are based on the following estimate.

Proposition 2.1 LetL0be as in Theorem 1.1 ands∈R². Then for everyn≥1

Eexphs,Yni ≤







exp 9ksk²

, for0≤ ksk ≤0.49, exp(24ksk²), for0.49<ksk ≤L₀, exp(4e^ksk), forL0<ksk.

To scetch the proof we introduce the following notation: We setwn = EWnandpn = EPn. Further- more, for1≤i≤n−1andj=j(i) =n−i−1we denote

a⁽¹⁾_n (i) =

(i/n)² i(n−i)/n²

0 i/n

, a⁽²⁾_n (i) =a⁽¹⁾_n (j),

C_n⁽¹⁾(i) = 1

n²(w_i+ (n−i)p_i+w_j+ (n−j)p_j−w_n+ 2ij+n−1), C_n⁽²⁾(i) = 1

n(p_i+p_j−p_n+n−1)

andCn(i) = (Cn⁽¹⁾(i), Cn⁽²⁾(i)). With this notation the recurrence forYninduced by recurrence (2) reads Y_n=^d A⁽¹⁾_n Y_In+A⁽²⁾_n Y⁰_J

n+b_n, n≥1, (3)

with

A⁽¹⁾_n , A⁽²⁾_n ,b_n

=

a⁽¹⁾_n (I_n), a⁽²⁾_n (I_n),C_n(I_n) , whereY_i,Y⁰_j,0≤i, j≤n−1,I_nare independent andL(Y_j⁰) =L(Yj).

We collect some useful but technical estimates. We denote byA^T the transpose of a matrixA and set kAk_op:= sup_kxk=1kAxk.

Lemma 2.2 LetU be uniformly distributed on[0,1]and coupleI_n,n≥1, toU by settingI_n =bU nc.

Then we have for alln≥1,

A^(1)T_n A⁽¹⁾_{n op}+

A^(2)T_n A⁽²⁾_n

_op−1<−U(1−U).

Lemma 2.3 We have

sup

n≥0

1≤i≤n−1max kCn(i)k= 1.

Proof of Proposition 2.1: The assertion follows from the next result by choosingL =ksk: For every L >0, denote

K_L=







9, forL≤0.49, 24, for0.49< L≤L₀, 4e^L/L², forL₀< L.

Then

Eexphs,Yni ≤exp KLksk²

, (4)

for everyksk ≤L,n≥0. This will be proved by induction onn. Forn= 0we haveY0= (0,0)and the assertion is true. Assume the assertion is true for someL >0,ksk ≤Land every0≤i≤n−1. Then, conditioning onIn =bU nc=iand using the distributional recurrence (3) we obtain forj =n−i−1 andksk ≤L,

Eexphs,Yni= 1 n

n−1

X

i=0

exphs,Cn(i)iEexpD

s, a⁽¹⁾_n (i)Yi

E

EexpD

s, a⁽²⁾_n (i)Yj

E

≤ 1 n

n−1

X

i=0

exphs,Cn(i)iexp

KL

a⁽¹⁾_n (i)^Ts

2

+KL

a⁽²⁾_n (i)^Ts

2

(5)

≤ 1 n

n−1

X

i=0

exp hs,Cn(i)i+KLksk²

2

X

r=1

a^(r)_n (i)^Ta^(r)_n (i) op

!

=Eexp hs,bni+KLksk²

2

X

r=1

A^(r)T_n A^(r)_n op

!

≤Eexp hs,bni+KLksk²(1−U(1−U))

(6)

=Eexp hs,bni −KLksk²U(1−U)

exp KLksk² . For (5) we applied the induction hypothesis, using

ka^(r)_n (i)^Tsk ≤ ka^(r)_n (i)^Ta^(r)_n (i)k^1/2_op ksk ≤ ksk ≤L,

sinceka^(r)n (i)^Ta^(r)n (i)kop≤1forr= 1,2,0≤i≤n−1, and for (6) we applied Lemma 2.2. Hence the proof is completed by showing that

sup

n≥0Eexp hs,bni −KLksk²U(1−U)

≤1.

We consider the casesL≤0.49andL≥0.49separately.

L≤0.49: The Cauchy-Schwarz inequality yields

Eexp hs,bni −KLksk²U(1−U)

≤Eexp (2hs,bni)^1/2 Eexp −2KLksk²U(1−U)^1/2 , thus it suffices to prove

Eexp (2hs,b_ni)Eexp −2K_Lksk²U(1−U)

≤1.

Withkbnk_∞≤1by Lemma 2.3 and Ehs,bni= 0we obtain Eexp (2hs,b_ni) =E 1 + 2hs,b_ni+

∞

X

k=2

(2hs,b_ni)^k k!

!

= 1 +Ehs,bni²

∞

X

k=2

2^khs,bni^k−2 k!

≤1 +ksk²

∞

X

k=2

2^k(1/2)^k−2 k!

= 1 +ksk²4(e−2). (7)

WithKL= 9we have

Eexp −2KLksk²U(1−U)

≤1−3ksk²+27

5 ksk⁴, (8)

usingexp(−x)≤1−x+x²/2forx≥0. Furthermore, one easily checks that forksk ≤0.49we have 1 +ksk²4(e−2)

1−3ksk²+27 5 ksk⁴

≤1.

Thus (7) and (8) yield that (4) is true forksk ≤L≤0.49withKL= 9.

L >0.49: Again, withkbnk_∞≤1we obtain Eexp hs,bni −KLksk²U(1−U)

≤exp(ksk)Eexp −KLksk²U(1−U) .

It is proved in Section 4 of (FJ01) that the right hand side of this inequality is smaller than1if0.42 ≤ ksk ≤2andK_L= 24, respectively if2≤ ksk ≤LandK_L= 4e^L/L². Thus forK_L= 24L²∨4e^L/L²

we have Eexphs,Y_ni ≤exp(K_Lksk²), for everyksk ≤L,n≥0. Since24L² ≥4e^L/L²forL≤L₀ and24L²≤4e^L/L²forL > L₀, this completes the proof.

Proof of Theorem 1.1: By standard arguments using Markov’s inequality and Proposition 2.1, cf. the proof of Theorem 3.6 in (AKN04).

Proof of Theorem 1.2:Choosetn=twn/n²= 2tlogn+O(1)in Theorem 1.1.

3 The lower bound

In this section we prove Theorem 1.3. The Wiener index of a binary search tree of ordernis rather large, if it has two subtrees which have a large distance from each other and which both have large sizes. Based on this observation we define for every fixedt >0a class of binary search trees of ordern. Every tree in that class has two subtrees with sufficiently large distance from each other and large sizes, such that con- ditioned on the event that the random binary search tree is in that class, the event{Wn−EW_n > tEW_n} has probability tending to1, asn→ ∞. Moreover the probability that the random binary search tree is in that class is at least as large as the bound stated in Theorem 1.3.

Proof of Theorem 1.3:To define the eventAthat the random binary search tree is in the above mentioned class, we denote for fixedt >0

λ:= log⁽³⁾n

log⁽²⁾n, κ:= 8 + 24λ, k:=bκtlognc, s:=

λn tlogn

.

We number nodes in the (complete) binary tree as follows. The root has number1and we count level by level from left to right, cf. figure 1. We denote bySithe size of the subtree rooted at nodeiand setSi= 0 if nodeidoes not belong to the binary search tree. Note that by our count node2^m+ 1is the second leftmost node on levelm.

LetAbe the event thatS2=b(n+ 1)/2cand thatS2^m+1≤s−1, for2≤m≤k, see figure 1. Thus under eventAwe haveS3=d(n−3)/2eandS₂k≥n/2−(k−1)s. Having two large subtrees this far away from each other will yield thatWnis sufficiently large. First note that

P(A)≥ 1 n

s (n+ 1)/2

k−1

≥ 1 n

s n

k−1

= exp(−(k−1) log(n/s)−logn)

≥exp

−8tlogn

log⁽²⁾n+O

log⁽³⁾n

. (9)

From now on, we will assume w.l.o.g. thatnis even. The distance between two nodes in a tree is the number of edges connecting them. From this point of view the Wiener index of a tree can be calculated by counting how often each edge is passed when summing up all node distances. In our notation the incoming edge of nodeiis passedS_i(n−S_i)times. Thus

Wn=X

i∈N

Si(n−Si),

levelk 2^kd

A A A A A A A A A A

S₂k

2^k+ 1 d AA S₂k+1

17d AA S17

level3 9d

AA

S9

level2 5d

AA S₅

2^k−1

levelk−1 d

8d 4d

level1 2d

1d level0

3d

r rr

@

@@

@

@@

@

@@

@

@@

!!!!!!

S3

aa aa aa

A A A A A A A A A A

Fig. 1: Under eventAwe have subtree sizesS3 = d(n−3)/2eandS2^m+1 ≤ s−1, for2 ≤ m ≤ k, thus S₂k≥n/2−(k−1)s.

where exactlyn−1of these summands are nonzero. We set

W_n⁰ =

k

X

m=1

S2^m(n−S2^m).

andW_n⁰⁰ = W_n−W_n⁰ and estimateW_n⁰ andW_n⁰⁰separately under eventA. By construction,W_n⁰ is the number of passings of the edges above the nodes2^m,1≤m≤k. For(s₂, . . . , s_k)∈M ={1, . . . , s}^k−1 letA(s2, . . . , sk)be the event thatS3=d(n+ 1)/2eand thatS2^m+1=sm−1, for2≤m≤k. Thus

A= [

(s₂,...,s_k)∈M

A(s2, . . . , sk).

We denoteσ₁= 0andσ_m=σ_m−1+s_mfor2≤m≤k. Then(m−1)≤σ_m≤(m−1)sand under

eventA(s₂, . . . , s_k)we have W_n⁰ =

k

X

m=1

n 2 +σm

n 2 −σm

=

k

X

m=1

n² 4 −σ²_m

≥ kn² 4 −s²

k

X

m=1

(m−1)²

≥ kn² 4

1−4

3 k²s²

n²

≥

(1 + 3λ)2tlogn−1 4

n²

1−4

3κ²λ²

= 2tn²logn

1 + 3λ− 1

8tlogn 1−4 3κ²λ²

≥2t(1 +λ)n²logn, (10)

for sufficiently largen. For the last inequality in line (10) we use

1 + 3λ− 1

8tlogn 1−4 3κ²λ²

≥(1 + 2λ)

1−4 3κ²λ²

≥1 +λ, for sufficiently largen.

In order to estimateW_n⁰⁰under eventA(s2, . . . , sk)via Chebychev’s inequality, we will use E(W_n⁰⁰|A(s2, . . . , sk))≥w_n/2−1+n

2 + 1

p_n/2−1 (11)

+w_n/2−σk+n 2 +σk

p_n/2−σk (12)

+

k

X

m=2

(w_sm−1+ (n−s_m+ 1)p_sm−1). (13) This inequality is valid, since the right hand side is the expected number of passings of all edges belonging to subtrees rooted at either node3(the summands in line (11)) or node2^k(the summands in line (12)) or node2^m+ 1,2≤m≤k, (the summands in line (13)). WithH`≥log`we get for`≤n

w`+ (n−`)p`≥2`²log`−6`²+o(`<sup>2</sup>) + (n−`) (2`log`−4`)

≥n(2`log`−6`+o(`)).

Thus

E(W_n⁰⁰|A(s₂, . . . , s_k))≥2nn 2 −1

logn 2 −1

+ 2nn 2 −σ_k

logn 2 −σ_k +

k

X

m=2

2n(sm−1) log(sm−1)−6n²+o(n²)

≥2n(n−σk−1) logn 2 −σk

+ 2n(k−1)(ˆs−1) log(ˆs−1)−6n²+o(n²), by convexity ofx7→xlogx, wheresˆ= 1/(k−1)Pk

m=2sm. Withσk = (k−1)ˆs≤(k−1)swe have (n−σ_k−1) logn

2 −σ_k

≥(n−(k−1)ˆs−1)

logn+ log

1−2(k−1)s n

−log 2

=nlogn−(log 2)n−(k−1)ˆslogn+o(n).

Together this yields

E(W_n⁰⁰|A(s2, . . . , sk))

≥2n²logn−2n(k−1)(ˆs−1) log n

ˆ s−1

−(6 + 2 log 2)n²−2n(k−1) logn+o(n²)

≥2n²logn−2n(k−1)(s−1) log n

s−1

−(6 + 2 log 2)n²+o(n²)

= 2n²logn−2κλn²log

tlogn λ

−(6 + 2 log 2)n²+o(n²)

≥2n²logn−(16 +o(1))n²log⁽³⁾n,

for all sufficiently largen, where we use thatx7→xlog(n/x)is increasing for0 < x < n/e. Similarly to (13) we have

Var(W_n⁰⁰|A(s₂, . . . , s_k)) = Var

W_n/2−1+n 2 + 1

P_n/2−1 + Var

W_n/2−σk+n 2 +σk

P_n/2−σk +

k

X

m=2

Var (W_sm−1+ (n−s_m+ 1)P_sm−1).

For`≤n,

Var (W`+ (n−`)P`) = Var(W`) + (n−`)<sup>2</sup>Var(P`) + 2(n−`)Cov(W`, P`)

≤O(`<sup>4</sup>) +n<sup>2</sup>O(`²) + 2nO(`³),

sinceVar(Wn) = O(n⁴)andCov(Wn, Pn) = O(n³), as shown in (Nei02), andVar(Pn) = O(n²).

Thus

Var(W_n⁰⁰|A(s₂, . . . , s_k)) =O(n⁴) and hence by Chebychev’s inequality

P

W_n⁰⁰≥2n²logn−17n²log⁽³⁾n|A(s2, . . . , sk)

→1 asn→ ∞. (14) This convergence is uniform over all(s2, . . . , sk)∈M. For sufficiently largen,

2t(1 +λ)n²logn+ 2n²logn−17n²log⁽³⁾n >(1 +t)EW_n. (15)

Using estimates (9), (10), (14) and (15) we get P(Wn >(1 +t)EWn)

≥P(Wn >(1 +t)EWn|A)P(A)

= X

(s2,...,sk)∈M

P(Wn>(1 +t)EWn|A(s2, . . . , sk))P(A(s2, . . . , sk))

≥ X

(s₂,...,s_k)∈M

P(W_n⁰⁰>2n²logn−17n²log⁽³⁾n|A(s₂. . . , s_k))P(A(s₂, . . . , s_k))

= (1 +o(1))P(A)

= exp

−8tlogn

log⁽²⁾n+O

log⁽³⁾n . This completes the proof.

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