The space requirements of an $m$-ary search tree satisfies a well-known phase
transition: when $m\leq 26$, the second order asymptotics is Gaussian. When
$m\geq 27$, it is not Gaussian any longer and a limit $W$ of a complex-valued
martingale arises. We show that the distribution of $W$ has a square integrable
density on the complex plane, that its support is the whole complex plane, and
that it has finite exponential moments. The proofs are based on the study of
the distributional equation $ W\egalLoi\sum_{k=1}^mV_k^{\lambda}W_k$, where
$V_1, ..., V_m$ are the spacings of $(m-1)$ independent random variables
uniformly distributed on $[0,1]$, $W_1, ..., W_m$ are independent copies of W
which are also independent of $(V_1, ..., V_m)$ and $\lambda$ is a complex
number

Chauvin, Brigitte

Liu, Quansheng

Pouyanne, Nicolas

English

arXiv

International audienceThe space requirements of an $m$-ary search tree satisfies a well-known phase transition: when $m\leq 26$, the second order asymptotics is Gaussian. When $m\geq 27$, it is not Gaussian any longer and a limit $W$ of a complex-valued martingale arises. We show that the distribution of $W$ has a square integrable density on the complex plane, that its support is the whole complex plane, and that it has finite exponential moments. The proofs are based on the study of the distributional equation $ W \overset{\mathcal{L}}{=} \sum_{k=1}^mV_k^{\lambda}W_k$, where $V_1, ..., V_m$ are the spacings of $(m-1)$ independent random variables uniformly distributed on $[0,1]$, $W_1, ..., W_m$ are independent copies of W which are also independent of $(V_1, ..., V_m)$ and $\lambda$ is a complex number

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Support and density of the limit $m$-ary search trees distribution

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Brigitte Chauvin

Quansheng Liu

Nicolas Pouyanne

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Support and density of the limit m-ary search trees distribution

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The space requirements of an $m$-ary search tree satisfies a well-known phase transition: when $m\leq 26$, the second order asymptotics is Gaussian. When $m\geq 27$, it is not Gaussian any longer and a limit $W$ of a complex-valued martingale arises. We show that the distribution of $W$ has a square integrable density on the complex plane, that its support is the whole complex plane, and that it has finite exponential moments. The proofs are based on the study of the distributional equation $ W \overset{\mathcal{L}}{=} \sum_{k=1}^mV_k^{\lambda}W_k$, where $V_1, ..., V_m$ are the spacings of $(m-1)$ independent random variables uniformly distributed on $[0,1]$, $W_1, ..., W_m$ are independent copies of W which are also independent of $(V_1, ..., V_m)$ and $\lambda$ is a complex number

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AofA’12 DMTCS proc. AQ, 2012, 191–200Support and density of the limit m-ary searchtrees distributionBrigitte Chauvin1†, Quansheng Liu2‡ and Nicolas Pouyanne1§1Université de Versailles-St-Quentin, Laboratoire de Mathématiques de Versailles, CNRS, UMR 81002Université de Bretagne Sud, Laboratoire de Mathématiques de Bretagne Atlantique, CNRS, UMR 6205The space requirements of an m-ary search tree satisfy a well-known phase transition: when m ≤ 26, the secondorder asymptotics is Gaussian. When m ≥ 27, it is not Gaussian any longer and a limit W of a complex-valuedmartingale arises. We show that the distribution of W has a square integrable density on the complex plane, that itssupport is the whole complex plane, and that it has finite exponential moments. The proofs are based on the study ofthe distributional equation W L=∑mk=1 Vλk Wk, where V1, ..., Vm are the spacings of (m − 1) independent randomvariables uniformly distributed on [0, 1], W1, ...,Wm are independent copies of W which are also independent of(V1, ..., Vm) and λ is a complex number.Keywords: m-ary search trees. Characteristic function. Smoothing transformation. Absolute continuity. Support.Exponential moments. Mandelbrot cascades.1 IntroductionSearch trees are fundamental data structures in computer science used in searching and sorting. Forintegers m ≥ 2, m-ary search trees generalize the binary search tree. The quantity m is called thebranching factor.A random m-ary search tree is an m-ary tree in which each node has the capacity to contain (m − 1)elements called the data or the keys. The keys can be considered as i.i.d. random variables xi, i ≥ 1, withany absolutely continuous distribution on the interval [0, 1].The tree Tn, n ≥ 0, is recursively defined as follows: T0 has an empty node-root; T1 has a node-root which contains x1; T2 has a node-root which contains x1 and x2; . . .Tm−1 still has one node-root,containing x1, . . . xm−1. As soon as the (m−1)-th key is inserted in the root,m empty subtrees of the rootare created, corresponding from left to right to them ordered intervals I1 =]0, x(1)[, . . . , Im =]x(m−1), 1[,where 0 < x(1) < · · · < x(m−1) < 1 are the ordered first (m−1) keys. Each following key xm, xm+1, . . .is recursively inserted in the subtree corresponding to the unique interval Ij to which it belongs. As soonas a node is saturated, m empty subtrees of this node are created.†Email: brigitte.chauvin@uvsq.fr.‡Email: quansheng.liu@univ-ubs.fr.§Email: nicolas.pouyanne@uvsq.fr.1365–8050 c© 2012 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France192 Brigitte Chauvin, Quansheng Liu and Nicolas PouyanneFor each i = {1, . . . ,m− 1} and n ≥ 1, X(i)n is the number of nodes in Tn which contain (i− 1) keys(and i gaps or free places) after insertion of the n-th key; such nodes are named nodes of type i. We onlytake into consideration the external nodes and not the internal nodes which are the saturated nodes. Thevector Xn is called the composition vector of the m-ary search tree. It provides a model for the spacerequirement of the algorithm. By spreading the input data in m directions instead of only 2, as is the casefor a binary search tree, one seeks to have shorter path lengths and thus quicker searches. One can referto Mahmoud’s book [Mah92] for further details on search trees.The following figure is an example of 4-ary search tree obtained by insertion of the successive numbers0.3 ,0.1, 0.4, 0.15, 0.9, 0.2, 0.6, 0.5, 0.35, 0.8, 0.97, 0.93, 0.23, 0.84, 0.62, 0.64, 0.33, 0.83. Thecorresponding composition vector is X18 = t(9, 2, 2).0.1 0.3 0.4   0.15 0.2 0.23  0.33 0.35  0.5 0.6 0.9        0.62 0.8 0.84 0.93 0.97  0.64 0.83  A vast literature is devoted to the asymptotic behavior of this composition vector. A famous phasetransition appears. When m ≤ 26, the random vector admits a central limit theorem with convergencein distribution to a Gaussian vector: see also Mahmoud’s book [Mah92] or Janson [Jan04] for a vectorialtreatment.When m ≥ 27, an almost sure asymptotics for the composition vector has been obtained in [CP04]:Xn = nv1 + <(nλ2Wv2) + o(nσ2) a.s. (1)(throughout the paper we write <(z) or <z for the real part of z, and =(z) or =z for the imaginal partof z), where λ2 = σ2 + iτ2 is the root of the polynomialm−1∏k=1(z + k)−m!, (2)Support and density of the limit m-ary search trees distribution 193having the second largest real part σ2 and a positive imaginary part τ2, v1 and v2 are two deterministicvectors, and W is the limit of a complex-valued martingale that admits moments of all positive orders.Heated conjectures about the second order complex-valued limit distribution W remain open (see[CP04], [Pou05], Chern and Hwang [CH01], Janson [Jan04]). Namely, what is the true order of mag-nitude of o(nσ2) in (1)? Can theW distribution be expressed by means of usual distributions? How heavyare the tails of W ? Can one identify the distributions of |W | and Arg(W )?A significant step is achieved by Fill and Kapur in [FK04], who establish that W satisfies the followingdistributional equation called the smoothing equation:WL=m∑k=1V λ2k Wk, (3)where V1, ..., Vm are the spacings of (m − 1) independent random variables uniformly distributed on[0, 1], W1, ...,Wm are independent copies of W which are also independent of (V1, ..., Vm). The precisedefinition of Vj will be given hereunder. By a contraction method, Fill and Kapur prove that W is theunique solution of Eq. (3) in the space M2(C) of square integrable probability measures having C =E(W ) as expectation. The present paper is based on this characterization of W .It has been recently proved [CLP11] that the continuous-time embedding of the process (Xn)n hasan analogous asymptotic behavior, with a second-order term which is a solution of some distributionalequation (not the same one). Inspired by this study of the continuous-time case we prove the followingtheorem.Theorem 1 Let W be the second order limit distribution of an m-ary search tree for m ≥ 27, definedby (1).(i) The support of W is the whole complex plane.(ii) The law of W admits a continuous square integrable density on C.(iii) Eeδ|W | < ∞ for some δ > 0. The exponential moment generating series of W (thus) has apositive radius of convergence.Thanks to Fill and Kapur results [FK04], these results are immediate corollaries of Theorems 3 and 6proved in the next two sections.In the sequel, let V1, . . . , Vm be the spacings of (m − 1) independent random variables uniformlydistributed on [0, 1]. In other words, let U1, . . . , Um−1 be independent random variables uniformly dis-tributed on [0, 1] and let U(1) ≤ · · · ≤ U(m−1) be their order statistics. Denote also U(0) := 0, U(m) := 1.For any k ∈ {1, . . . ,m}, the random variable Vk is defined byVk := U(k) − U(k−1).The variables Vk are Beta(1,m− 1)-distributed and satisfy∑mk=1 Vk = 1 almost surely.Remark 2 Details about roots of (2) can be found in Hennequin [Hen91] and Mahmoud [Mah92]. Notethat for m = 2, the polynomial (2) has the unique root λ = 1. For m ≥ 3, it is known that if λ2 is a root194 Brigitte Chauvin, Quansheng Liu and Nicolas Pouyanneof the polynomial (2) having the second largest real part, then λ2 is nonreal, <λ2 < 1 for any m ≥ 3,<λ2 > 0 if and only if m ≥ 14 (relate this to Hwang [Hwa03]), and<λ2 >12⇐⇒ m ≥ 27.2 SupportThe limit distribution W satisfies (3). From now on, we consider the solutions of the distributional equa-tionZL=m∑k=1V λk Zk, (4)where V1, ..., Vm are the spacings of (m − 1) independent random variables uniformly distributed on[0, 1], Z1, ..., Zm are independent copies of Z which are also independent of (V1, ..., Vm) and λ is anonreal complex number.We assume thatλ is a nonreal root of (2) having a positive real part σ. (5)Indeed, V1, . . . , Vm are Beta(1,m− 1)-distributed and <(λ) > 0 guarantees that E|V λ1 | <∞. More-over, we are interested in solutions of (4) having a nonzero expectation and the existence of such solutionsimplies that λ is a root of (2). Note that when m ≤ 13, no λ satisfies (5).The following theorem implies Theorem 1(i) because W is integrable with EW = 1Γ(1+λ2) 6= 0(see [Pou05] for instance).Theorem 3 Let λ be a nonreal complex number having a positive real part. If Z is a solution of (4)having a nonzero expectation, then the support of Z is the whole complex plane.The proofs of Theorem 3 and Theorem 6 make use of the complex-valued random variableA :=m∑k=1V λk . (6)Notice that the existence of an integrable solution Z of (4) such that E(Z) 6= 0 implies thatE(A) = 1, (7)which just means that λ is a root of the polynomial (2).PROOF OF THEOREM 3. For a complex valued random variable X , we denote its support bySupp(X) = {x ∈ C, ∀ε > 0, P(|X − x| < ε) > 0}.Let Z be a solution of (4) having a nonzero expectation. We first prove that ∀a ∈ C, ∀z ∈ C,[a ∈ Supp(A) and z ∈ Supp(Z)]=⇒ az ∈ Supp(Z). (8)Support and density of the limit m-ary search trees distribution 195Indeed, let ε > 0, a ∈ Supp(A) and z ∈ Supp(Z). Let also Z1, . . . , Zm be i.i.d. copies of Z. Then, withpositive probability, |A− a| ≤ ε and |Zk − z| ≤ ε for any k. Therefore, with positive probability,∣∣∣∣∣m∑k=1V λk Zk − az∣∣∣∣∣ =∣∣∣∣∣m∑k=1V λk (Zk − z) + z(A− a)∣∣∣∣∣ ≤ (m+ ε)ε+ |z|ε.Since ε is arbitrary, this inequality shows that az ∈ Supp(V λ1 Z1 + · · ·+ V λmZm)which implies thataz ∈ Supp(Z) because of (4).Let z ∈ Supp(Z)\{0}. Such z exist because E(Z) 6= 0. Iterating (8), any complex number of the forma1 . . . anz where a1, . . . , an ∈ Supp(A) belongs to Supp(Z). Therefore, Lemmas 4 and 5 below implythat Supp(Z) contains C \ {0} which suffices to get the conclusion since the support of a probabilitymeasure is a closed set. Lemma 4 There exist c, c′ ∈ C\{0} and respective open neighbourhoods V and V ′ of c and c′ such that|c| > 1, |c′| < 1 and V ∪ V ′ ⊆ Supp(A).PROOF. Obviously,Supp(A) ={m∑k=1tλk , 0 ≤ tk ≤ 1,m∑k=1tk = 1}.In particular, Supp(A) contains the set f([0, 1]2)(the image of [0, 1]2 by f ), where f is defined byf : [0, 1]2 → C(s, t) 7→ (st)λ + (s(1− t))λ + (1− s)λ.We show that there exist (sc, tc) and (sc′ , tc′) in ]0, 1[2 such that c := f(sc, tc) and c′ := f(sc′ , tc′) satisfy|c| > 1, 0 < |c′| < 1 and f is a local diffeomorphism in some respective neighbourhoods of (sc, tc) and(sc′ , tc′), which implies the result.Let σ and τ be respectively the real part and the imaginary part of λ. By assumption, 0 < σ < 1. Weassume that τ > 0; if not, replace Z and λ by their conjugates. For any integer k ≥ 1, denoteuk = exp(−2kπτ)and u′k = exp(π − 2kπτ).Then, uk and u′k are real numbers in ]0, 1[ that tend to 0 as k tends to infinity, and they satisfyuλk = uσk ∈]0, 1[ and u′kλ= −u′kσ ∈]− 1, 0[.Denote moreover sk := uk + u2k, tk :=11 + uk,s′k := u′k + u′k2, t′k :=11 + u′k.As 0 < σ < 1, we have|f(sk, tk)| =∣∣∣uλk + uk2λ + (1− uk − uk2)λ∣∣∣ = 1 + uσk +O (uk)196 Brigitte Chauvin, Quansheng Liu and Nicolas Pouyanneand|f(s′k, t′k)| =∣∣∣∣u′kλ + u′k2λ + (1− u′k − u′k2)λ∣∣∣∣ = 1− u′kσ +O (u′k)when k tends to infinity, so that |f(sk, tk)| > 1 and 0 < |f(s′k, t′k)| < 1 when k is large enough. It remainsto show that f is a local diffeomorphism in neighbourhoods of (sk, tk) and (s′k, t′k). Let Φ : [0, 1]2 → R2be defined asΦ(s, t) =(<f(s, t),=f(s, t)).It suffices to show that the Jacobian of Φ at suitable (sc, tc) and (sc′ , tc′) does not vanish to show that fis a local diffeomorphism at these points. This sufficient condition is equivalent to requiring that∂f∂s× ∂f∂tis nonreal (the overline denotes the complex conjugacy).For any k ≥ 1, after computation, one gets∂f∂s(sk, tk)×∂f∂t(sk, tk) = |λ|21 + ukskuk[uλk(1 + uλk)− sk(1− sk)λ−1][uλ+1k − u2λk]= |λ|2 1 + ukskuk[uσk + u2σk − sk(1− sk)λ−1][uσ+1k − u2σk].The above number is real if and only if (1 − sk)λ ∈ R, i.e. if and only if τ log(1 − sk) ∈ πZ. Sincesk 6= 0 and sk tends to zero when k tends to infinity, τ log(1 − sk) /∈ πZ as soon as k is large enough.Therefore, taking (sc, tc) = (sk, tk) for k large enough suffices to get the result. An argument of the samekind applied to the sequence (s′k, t′k)k leads to the result on the existence of (sc′ , tc′). Lemma 5 Let V and V ′ be respectively open neighbourhoods of c ∈ C and c′ ∈ C with |c| > 1 and0 < |c′| < 1, which do not contain 0. LetM := {v1v2 . . . vn, n ≥ 1, v1, v2, . . . , vn ∈ V ∪ V ′} .Then M = C \ {0}.PROOF. Let ` and `′ be complex numbers such that <` > 0 and <`′ < 0. Let U and U ′ be respectivelyopen neighbourhoods of ` and `′. Denote byM the additive submonoid of C/2iπZ generated by U ∪ U ′mod 2iπ; it is the set of classesM := {u1 + u2 + · · ·+ un mod 2iπ, n ≥ 1, u1, u2, . . . , un ∈ U ∪ U ′}.We prove hereunder thatM = C/2iπZ. Taking the exponential, this suffices to prove the lemma.Support and density of the limit m-ary search trees distribution 197PPPPPPPPPPPPPPPPPPPPPPPPP0PPPPPP dU`&%'$p`pUPPPPPPiU ′`′Szz − 2iπqTake an integer p ≥ 1 large enough so that pU contains a whole mesh of the lattice generated by `and `′, i.e. such that pU ⊇ p` + [0, 1]` + [0, 1]`′. Then,M contains the classes mod 2iπ of the sectorS := p`+ R≥0`+ R≥0`′. Since <`× <`′ < 0, when z is any complex number, there exists q ∈ Z suchthat z − 2iπq ∈ S, which proves the result. 3 Density and exponential momentsAs in the beginning of Section 2, Theorem 1(ii) and (iii) are straightforward corollaries of the followingtheorem.Theorem 6 Let λ be a nonreal complex number and Z a solution of (4) having a nonzero expectation.(i) If <(λ) > 0, then Z admits a continuous square integrable density on C.(ii) If <(λ) > 12 , then Eeδ|Z| < ∞ for some δ > 0. The exponential moment generating series of Z(thus) has a positive radius of convergence.PROOF. It runs along the same lines as in [CLP11] and uses the Fourier transform ϕ of Z, namelyϕ(t) := E exp{i〈t, Z〉} = E exp{i<(tZ)}, t ∈ C,where 〈x, y〉 = <(xy) = <(x)<(y) + =(x)=(y). In terms of Fourier transforms, Eq. (4) readsϕ(t) = E(m∏k=1ϕ(tV λk)). (9)• To get (i), we prove that ϕ is in L2(C) because it is dominated by |t|−δ for some δ > 1 so thatthe inverse Fourier-Plancherel transform provides a square integrable density for Z. The guiding ideaconsists in adapting methods (developed in [Liu99] and [Liu01]) usually applied to positive real-valued198 Brigitte Chauvin, Quansheng Liu and Nicolas Pouyannerandom variables to the present complex-valued case. For any r ≥ 0, denoteψ(r) := max|t|=r|ϕ(t)|.Using Theorem 3, one can step by step mimick the proof of Theorem 7.17 in [CLP11] to get the result.We just give hereunder an overview of this proof, written as a succession of hints.Show first that Theorem 3 implies that ψ(r) < 1 for any r > 0. Then, notice thatψ(r) ≤ E(m∏k=1ψ (r|Vk|σ)). (10)By Fatou’s lemma, (10) implies that lim sup+∞ ψ(r) ∈ {0, 1}. Iterating suitably inequality (10) leads tolim+∞ ψ(r) = 0. Finally, applying (10) again we can show that ψ(r) = O(r−δ) for some δ > 1, so thatϕ is square integrable on C, which leads to the result.• To get (ii), like in [CLP11], we use Mandelbrot’s cascades. Denote V := (V1, V2, . . . , Vm). Let U bethe set of finite sequences of positive integers between 1 and m, namelyU :=⋃n≥1{1, 2, . . . ,m}n.Elements of U are denoted by concatenation. Let Vu := (Vu1, Vu2, . . . , Vum), u ∈ U be independentcopies of V , indexed by all finite sequences of integers u = u1 . . . un ∈ U .Introduce the martingale (Yn)n≥1 defined byYn :=∑u1...un∈{1,...,m}nV λu1Vλu1u2 . . . Vλu1...un .By (7), E(Yn) = E(A) = 1. It can be easily seen thatYn+1 =m∑k=1V λk Yn,k, (11)where Yn,k for 1 ≤ k ≤ m are independent of each other and independent of the Vk and each has thesame distribution as Yn. Besides, since σ > 12 , mEV2σ1 < 1 and, by Cauchy-Schwarz inequality,E|A|2 ≤ E(m∑k=1|V λk |)2= E(m∑k=1V σk)2≤ 2Em∑k=1V 2σk = 2mEV 2σ1 < 2.Therefore for n ≥ 1, Yn is square integrable andVarYn+1 = (E|A|2 − 1) +mEV 2σ1 VarYn,where VarX = E(|X − EX|2)denotes the variance of X . Thus, the martingale (Yn)n is bounded inL2, so that when n→ +∞,Yn → Y∞ a.s. and in L2,Support and density of the limit m-ary search trees distribution 199where Y∞ is a (complex-valued) random variable with varianceVar(Y∞) =E|A|2 − 11−mEV 2σ1.Passing to the limit in Eq. (11) shows that Y∞ is a solution of Eq. (4) and by unicity, (ii) in Theorem 6holds as soon as it holds for Y∞.This last fact comes from an adaptation of Lemma 8.29 in [CLP11], giving some constants C > 0 andε > 0 such that for all t ∈ C with |t| ≤ ε, we haveEe〈t,Y∞〉 ≤ e<(t)+C|t|2.The adaptation relies on∑mk=1 V2σk < 1 a.s. for σ >12 . The last assertion implies that Eet|Y∞| < ∞for t > 0 small enough, so that the exponential moment generating series of Y∞ has a positive radius ofconvergence. References[CH01] H.-H. Chern and H.-K. Hwang. Phase changes in random m-ary search trees and generalizedquicksort. Random Structures and Algorithms, 19:316–358, 2001.[CLP11] B. Chauvin, Q. Liu, and N. Pouyanne. Limit distributions for multitype branching processes ofm-ary search trees. arXiv:1112.0256v1, math.PR, 2011.[CP04] B. Chauvin and N. Pouyanne. m-ary search trees when m > 26: a strong asymptotics for thespace requirements. Random Structures and Algorithms, 24(2):133–154, 2004.[FK04] J.A. Fill and N. Kapur. The space requirement of m-ary search trees: distributional asymptoticsfor m ≥ 27. Proceedings of the 7th Iranian Conference, page arXiv:math.PR/0405144, 2004.[Hen91] P. Hennequin. Analyse en moyenne d’algorithme, tri rapide et arbres de recherche. PhD Thesis,Ecole Polytechnique, 1991.[Hwa03] H.-K. Hwang. Second phase changes in random m-ary search trees and generalized quicksort:convergence rates. Ann. Probab., 31, 2:609–629, 2003.[Jan04] S. Janson. Functional limit theorem for multitype branching processes and generalized Pólyaurns. Stochastic Processes and their Applications, 110:177–245, 2004.[Liu99] Q. Liu. Asymptotic properties of supercritical age-dependent branching processes and homo-geneous branching random walks. Stochastic Processes and their Applications, 82(1):61–87,1999.[Liu01] Q. Liu. Asymptotic properties and absolute continuity of laws stable by random weighted mean.Stochastic Processes and their Applications, 95:83–107, 2001.[Mah92] H.M. Mahmoud. Evolution of Random Search Trees. John Wiley & Sons Inc., New York, 1992.[Pou05] N. Pouyanne. Classification of large Pólya- Eggenberger urns with regard to their asymptotics.Discrete Mathematics and Theoretical Computer Science, AD, pages 275–286, 2005.200 Brigitte Chauvin, Quansheng Liu and Nicolas Pouyanne

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Support and density of the limit $m$ -ary search trees distribution