Towards More Realistic Probabilistic Models for Data Structures: The External Path Length in Tries under the Markov Model

Transcription

1 Towars More Realistic Probabilistic Moels for Data Structures: The External Path Length in Tries uner the Markov Moel Kevin Leckey an Ralph Neininger Institute for Mathematics J.W. Goethe University Frankfurt Frankfurt am Main Germany {leckey, Wojciech Szpankowski Department of Computer Science Purue University W. Lafayette, IN U.S.A. July, 202 Abstract Tries are among the most versatile an wiely use ata structures on wors. They are pertinent to (internal structure of (store wors an several splitting proceures use in iverse contexts ranging from ocument taxonomy to IP aresses lookup, from ata compression (i.e., Lempel-Ziv 77 scheme to ynamic hashing, from partial-match queries to speech recognition, from leaer election algorithms to istribute hashing tables an graph compression. While the performance of tries uner a realistic probabilistic moel is of significant importance, its analysis, even for simplest memoryless sources, has prove ifficult. Rigorous finings about inherently complex parameters were rarely analyze (with a few notable exceptions uner more realistic moels of string generations. In this paper we meet these challenges: By a novel use of the contraction metho combine with analytic techniques we prove a central limit theorem for the external path length of a trie uner a general Markov source. In particular, our results apply to the Lempel-Ziv 77 coe. We envision that the methos escribe here will have further applications to other trie parameters an ata structures. Introuction We stuy the external path length of a trie built over n binary strings generate by a Markov source. More precisely, we assume that the input is a sequence of n inepenent an ientically istribute ranom strings, each being compose of an infinite sequence of symbols such that the next symbol epens on the previous one an this epenence is governe by a given transition matrix (i.e., Markov moel. Digital trees, in particular, tries have been intensively stuie for the last thirty years [, 8, 9,, 3, 5, 6, 7, 9, 20, 33], mostly uner Bernoulli (memoryless moel assumption. The typical epth uner Markovian moel was analyze in [9, 3], however, to the best of our knowlege, not the external path length. It is well known [33] that the external path length is more challenging ue to stronger epenency. In fact, this is alreay observe for tries uner Bernoulli moel [33]. This author s contribution was mae while visiting J.W. Goethe University Frankfurt a.m. with an Alexaner von Humbolt research awar. This author was also supporte by the NSF STC Grant CCF , NSF Grants DMS an CCF , AFOSR Grant FA , NSA Grant H , an MNSW grant N Also, Visiting Professor at ETI, Gansk University of Technology, Polan.

2 In this paper we establish the central limit theorem for the external path length in a trie built over a Markov moel using a novel use of the contraction metho. Let us first briefly review the contractionmetho. It wasintrouce in 99by Uwe Rösler[27] for the istributional analysis of the complexity of the Quicksort algorithm. Over the last 20 years this approach, which is base on exploiting an unerlying contracting map on a space of probability istributions, has been evelope as a fairly universal tool for the analysis of recursive algorithms an ata structures. Here, ranomness may come from a stochastic moel for the input or from ranomization within the algorithms itself (ranomize algorithms. General evelopments of this metho werepresentein [28, 25, 29, 22, 23, 4, 3, 4, 24]with numerousapplicationsin Theoretical Computer Science. The contractionmetho has been use in the analysisoftries an otherigital trees only uner the symmetric Bernoulli moel (unbiase memoryless source [22, Section 5.3.2], where limit laws for the size an the external path length of tries were re-erive. The application of the metho there was heavily base on the fact that precise expansions of the expectations were available, in particular smoothness properties of perioic functions appearing in the linear terms as well as bouns on error terms which were O( for the size an O(logn for the path lengths. Let us observe that even in the asymmetric Bernoulli moel such error terms seem to be out of reach for classical analytic methos; see the iscussion in Flajolet, Roux, an Vallée [5]. Hence, for the more general Markov source moel consiere in the present paper we evelop a novel use of the contraction metho. Furthermore, the contraction metho applie to Markov sources hits another snag, namely, the Markov moel is not preserve when ecomposing the trie into its left an right subtree of the root. The initial istribution of the Markov source is change when looking at these subtrees. To overcome these problems a couple of new ieas are use for setting up the contraction metho: First of all, we will use a system of istributional recursive equations, one for each subtree. We then apply the contraction metho to this system of recurrences capturing the subtree processes an prove normality for the path lengths conitione on the initial istribution. In fact, our approach avois ealing with multivariate recurrences an instea we reuce the whole analysis to a system of one-imensional equations. We also nee asymptotic expansions of the mean an the variance for applying the contraction metho. However, in contrast to very precise information on perioicities of linear terms for the symmetric Bernoulli moel mentione above our convergence proof oes only require the leaing orer term together with a Lipschitz continuity property for the error term. In this extene abstract we evelop the use of systems of recursive istributional equations in the context of the contraction metho for the external path length of tries uner a general Markov source moel. In particular, we prove the central limit theorem for the external path length, a result that ha been wanting since Lempel-Ziv 77 coe was evise in 977. The methoology use is general enough to cover relate quantities an structures as well. We are confient that our approach also applies with minor ajustments at least to the size of tries, the path lengths of igital search trees an PATRICIA tries uner the Markov source moel as well as other more complex ata structures on wors such as suffix trees. Notations: Throughout this paper we use the Bachmann-Lanau symbols, in particular the big O notation. We eclare xlogx := 0 for x = 0. By B(n,p with n N an p [0,] the binomial istribution is enote, by B(p the Bernoulli istribution with success probability p, by N(0,σ 2 the centere normal istribution with variance σ 2 > 0. We use C as a generic constant that may change from one occurrence to another. 2 Tries an the Markov source moel The Markov source: We assume binary ata strings over the alphabet Σ = {0,} generate by a homogeneous Markov chain. In general, a homogeneous Markov chain is given by its initial istribution µ = µ 0 δ 0 +µ δ on Σ an the transition matrix (p ij i,j Σ. Here, δ x enotes the Dirac 2

3 measure in x R. Hence, the initial state is 0 with probability µ 0 an with probability µ. We have µ 0,µ [0,] an µ 0 +µ =. A transition from state i to j happens with probability p ij, i,j Σ. Now, a ata string is generate as the sequence of states visite by the Markov chain. In the Markov source moel assume subsequently all ata strings are inepenent an ientically istribute accoring to the given Markov chain. We always assume that p ij > 0 for all i,j Σ. Hence, the Markov chain is ergoic an has a stationary istribution, enote by π = π 0 δ 0 +π δ. We have π 0 = p 0 p 0 +p 0, π = p 0 p 0 +p 0. ( Note however, that our Markov source moel oes not require the Markov chain to start in its stationary istribution. The case p ij = /2 for all i,j Σ is essentially the symmetric Bernoulli moel (only the first bit may have a ifferent (initial istribution. The symmetric Bernoulli moel has alreay been stuie thoroughly also with respect to the external path length of tries, see [7, 6, 22]. Hence, we exclue this case subsequently. For later reference, we summarize our conitions as: p ij (0, for all i,j Σ, p ij 2 for some (i,j Σ2. (2 The entropy rate of the Markov chain plays an important role in the asymptotic behavior of tries. In particular, it etermines leaing orer constants of parameters of tries that are relate to epths of leaves an its external path length. The entropy rate for our Markov chain is given by H := i,j Σπ i p ij logp ij = π i H i, (3 i Σ where H i := j Σ p ij logp ij is the entropy of a transition from state i to the next state. Thus, H is obtaine as weighte average of the entropies of all possible transitions with weights accoring to the stationary istribution π. Tries: For a given set of ata strings over the alphabet Σ = {0,} with each ata string a unique infinite path in the infinite complete roote binary tree is associate by ientifying left branches with bit 0 an right branches with bit. Each string is store in the unique noe on its infinite path that is closest to the root an oes not belong to any other ata path: It is the minimal prefix of a string that istinguishes this string from all others; for etails see the monographs of Mahmou [20] or Szpankowski [33]. 3 Recursive Distributional Equations For the Markov source moel a challenge is to set the right framework uner which ata structures to analyze. We formulate in this section a system of istributional recurrences to capture the istribution of the external path length of tries. Our subsequent analysis is entirely base on these equations. We enote by L µ n the external path length of a trie uner the Markov source moel with initial istribution µ holing n ata. We have L µ 0 = Lµ = 0 for all initial istributions µ. The transition matrix is given in avance an suppresse in the notation. We abbreviate L i n := Lδi n for i Σ. Hence, L i n refers to n inepenent strings all starting with bit i an then following the Markov chain. We will stuyl 0 n an L n. Fromthe asymptoticbehaviorofthese twosequenceswecan then irectly obtain corresponing results for L µ n for an arbitrary initial istribution µ = µ 0δ 0 +µ δ as follows: We enote by K n the number of ata among our n which start with bit 0. Then K n has the binomial B(n,µ 0 istribution. Then the contributions of the two subtrees of the trie to its external path length can be represente by the following stochastic recurrence L µ n = L 0 K n +L n K n, n 2, (4 3

4 where (L 0 0,...,L0 n, (L 0,...,L n an K n are inepenent an = enotes that left an right han sie have ientical istributions. We will see later that we can irectly transfer asymptotic results for L 0 n an L n to general Lµ n via (4, see, e.g., the proof of Theorem 6.. For a recursive ecomposition of L 0 n note that we have initial istribution δ 0, thus all ata strings start with bit 0 an are inserte into the left subtree of the root. We enote the root of thisleft subtreebyw. At noew the atastringsaresplitaccoringtotheirseconbit. Weenote by I n the number of ata strings having 0 as their secon bit, i.e., the number of strings being inserte into the left subtree of w. The Markov source moel implies that I n is binomial B(n,p 00 istribute. The right subtree of noe w then hols the remaining n I n ata strings. Consier the left subtreeofwtogetherwith itsrootw. Conitione onits numberi n ofatastringsinserte it is generate by the same Markov source moel as the original trie. However, the right subtree of w together with its root w conitione on its number n I n of ata strings is generate by a Markov source moel with the same transition matrix but another initial istribution, namely δ. Moreover, by the inepenence of ata strings within the Markov source moel, these two subtrees are inepenent conitionally on I n. Phrase in a recursive istributional equation we have L 0 n = L 0 I n +L n I n +n, n 2, (5 with (L 0 0,...,L0 n, (L 0,...,L n an I n inepenent. A similar arguments yiels a recurrence for L n. Denoting by J n a binomial B(n,p istribute ranom variable, we have L n = L 0 n J n +L J n +n, n 2, (6 with (L 0 0,...,L 0 n, (L 0,...,L n an J n inepenent. Our asymptotic analysis of L µ n is base on the istributional recurrence system (5 (6 as well as (4. 4 Analysis of the Mean First we stuy the asymptotic behavior of the expectation of the external path length with a precise error term neee to erive a limit law in Section 6. Theorem 4.. For the external path length L µ n with conitions (2 we have of a binary trie uner the Markov source moel E[L µ n] = H nlogn+o(n, (n, with the entropy rate H of the Markov chain given in (3. The O(n error term is uniform in the initial istribution µ. Our proof of Theorem 4. as well as the corresponing limit law in Theorem 6. epen on refine properties of the O(n error term that are first obtaine for the initial istributions µ = δ 0 an µ = δ an then generalize to arbitrary initial istribution via (4. For µ = δ 0 an µ = δ we enote this error term for all n N 0 an i Σ by f i (n := E[L i n ] nlogn. (7 H The following Lipschitz continuity of f 0 an f is crucial for our further analysis: Proposition 4.2. There exists a constant C > 0 such that for both i Σ an all m,n N 0 f i (m f i (n C m n. The proof of Proposition 4.2 is base of a refine analysis of transfers from growth of toll functions in systems of recursive equations to the growth of the quantities itself. The heart of the proof of Proposition 4.2 an hence Theorem 4. is the following technical transfer result: 4

5 Lemma 4.3. Let (a i (n n 0 an (η i (n n 0 be real sequences an (X i,n n 2 sequences of binomial B(n,p i istribute ranom with p i (0, for i Σ. Assume that for constants c 0,c, 0, (0, with c 0 + = c + 0 = we have for all n 2 an i Σ a i (n = c i E[a i (X i,n ]+ i E[a i (n X i,n ]+η i (n. (8 If furthermore η i (n = O(n α for an α > 0 an both i Σ, then, as n, 5 Analysis of the Variance a i (n = O(, i Σ. To formulate an asymptotic expansion of the variance of the external path length we enote by λ(s the largest eigenvalue of the matrix P(s := (p s ij i,j Σ. Note that λ as a function of s is smooth. We enote its first an secon erivative by λ an λ respectively. Then we have: Theorem 5.. For the external path length L µ n of a binary trie uner the Markov source moel with conitions (2 we have, as n, Var(L µ n = σ 2 nlogn+o(nlogn, (9 where σ 2 > 0 is inepenent of the initial istribution µ an given by σ 2 = λ( λ 2 (. (0 λ 3 ( We start with the analysis of the Poisson variance of the external path length, i.e. ṽ i (λ := Var(L i N λ, i Σ, where N λ has the Poisson(λ istribution an is inepenent of (L i n n 0. In the secon part we use epoissonization techniques of [2] to obtain the asymptotic behavior of Var(L i n. The reason why we consier a Poisson number of strings is that for N λ i.i.. strings with initial istribution δ i the number N λpi0 of strings whose secon bit equals 0 an the number M λpi of strings whose secon bit equals are inepenent an remain Poisson istribute. Hence, in the Poisson case we obtain similarly to (5 an (6 that for i Σ L i N λ = L 0 Nλpi0 +L M λpi +N λpi0 +M λpi {Nλpi0 +M λpi =} ( where (L 0 n n 0, (L n n 0, N λpi0 an M λpi are inepenent, N λpi0 has Poisson(λp i0 istribution an M λpi has Poisson(λp i istribution. Note that {Nλpi0 +M λpi =} is necessary in orer that ( hols when {N λ = }. We enote by ν i (λ := E[L i N λ ], i Σ, the Poisson expectation of the external path length which is Note that ( implies ν i (λ = n=0 e λλn n! E[Li n]. ν i (λ = ν 0 (λp i0 + ν (λp i +λ( e λ. (2 We nee precise information about the mean (secon orer term to erive the leaing term of the variance. We shall use analytic techniques, namely the Mellin transform as surveye in [33] that we iscuss next. A Mellin transform f (s of a real function f(x is efine as f (s = 0 f(xx s x. 5

6 Let ν i (s be the Mellin transform of ν i(λ. Then, by known properties of the Mellin transform [33], the functional equation (2 becomes an algebraic equation for i Σ ν i(s = Γ(s++p s i0 ν 0(s+p s i ν (s. Define the column vector ν (s := (ν0 (s,ν (s an the column vector γ(s := (Γ(s,Γ(s. Then we can write the latter equations as the matrix equation ν (s = γ(s + + P(sν (s that we write as ν (s = (I P(s γ(s+. (3 Then the Mellin transformν (s ofthe mean externalpath length E[L µ N λ ] uner the Poissonmoel satisfies ν (s = Γ(s++µ(sν (s (4 where µ(s := (µ s 0,µ s. To recover the mean external path length uner the Poisson moel we nee to apply the singularity analysis to (4. For matrix P(s, we efine the principal left eigenvector π(s, the principal right eigenvector ψ(s associate with the largest eigenvalue λ(s such that π(s, ψ(s = where we write x,y for the inner prouct of vectors x an y. Then by the spectral representation [33] of P(s we fin ν (s = Γ(sψ(s λ(s +o(/( λ(s that leas to the following asymptotic expansion aroun s = ( ν (s = λ( (s+ 2 + γ s+ λ( + λ( µ( ψ( + +O( (5 2 λ( λ( where ẋ(t an ẍ(t enote the first an secon erivatives of the vector x(t at t. Using (5, inverse Mellin transform, an the resiue theorem of Cauchy, as well as analytic epoissonization of Jacquet an Szpankowski [2] we finally obtain ( E[L µ n ] = H nlogn+n γ λ( + λ( µ( ψ( ++Φ(logn +o(n (6 2 λ( λ( where Φ(x is a perioic function of small amplitue uner certain rationality conition (an zero otherwise; see [3] for etails. The asymptotic analysis of the variance follows the same pattern, however, it is more involve. Our analysis of the Poisson variance ṽ i (λ = Var(L i N λ is base on the following ecomposition: Lemma 5.2. For any λ > 0 an i Σ we have ṽ i (λ = ṽ 0 (λp i0 +ṽ (λp i +2λp i0 µ 0(λp i0 +2λp i µ (λp i (7 +2λe λ ( µ 0 (λp i0 + µ (λp i +λ( e λ +λ 2 e λ (2 e λ where µ i,i Σ, enotes the erivative of µ i, i.e. for z > 0 µ i (z = n= The Mellin transform v i (s of ṽ i(λ is e z z n (n! E[Li n ] µ i(z. vi (s = p s i0 v 0 (s+p s i v (s 2sp s i0 µ 0 (s 2sp s i µ (s Γ(s++F i (s with F i (s the Mellin transform of e λ q i (λ with q i (λ := µ 0 (λp i0+2λp i µ (λp i+λ 2 (2 e λ. Thus, the column vector v (s := (v 0(s,v (s satisfies the following algebraic equation v (s = P(sv (s 2sP(s ν (s γ(s++f(s 6

7 where F (s := (F0 (s,f (s Then, as we i before for the mean analysis, we obtain v(s = 2sΓ(s+ π(s,p(s ψ(s ψ(s ( λ(s 2 +O(/( λ(s. After further computations we fin that the Poisson variance ṽ(λ = Var(L µ N λ is ṽ(λ = ( λ( λ 2 ( λlog2 λ+ 2 λ 3 ( + A λlogλ+o(λ λ 2 ( for some explicitly computable constant A. Finally, with epoissonization, cf. [33], we obtain proving Theorem 5.. Var(L µ n = ṽ(n n[ ν (n] 2 = λ( λ 2 ( nlogn+o(n λ 3 ( 6 Asymptotic Normality Our main result is the asymptotic normality of the external path length: Theorem 6.. For the external path length L µ n with conitions (2 we have of a binary trie uner the Markov source moel L µ n E[L µ n] N(0,σ 2, (n, (8 where σ 2 > 0 is inepenent of the initial istribution µ an given by (0. As in the analysis of the mean, we first erive limit laws for L 0 n an L n an then transfer these to a limit law for L µ n via (4. We abbreviate for i Σ an n N 0 ν i (n := E[L i n ], σ i(n := Var(L i n. Note that we have ν i (0 = ν i ( = σ i (0 = σ i ( = 0 an σ i (n > 0 for all n 2. We efine the stanarize variables by an Y0 i := Y i := 0. Then we have: Yn i := Li n E[L i n], i Σ,n 2, (9 σ i (n Proposition 6.2. For both sequences (Y i n n 0, i Σ, we have convergence in istribution: We now present a brief streamline roa map of the proof. Y i n N(0, (n. (20. Normalization. From the system (5 (6, where we enote there I 0 n := I n an I n := J n, an the normalization (9 we obtain Y i n = σ i(in i σ i (n Y i I + σ i(n In i n i σ i (n Y i n I i n +b i (n, n 2, (2 where b i (n = ( n+νi (I i σ i (n n+ν i (n In ν i i (n, 7

8 an in (2 we have that (Y 0 0,...,Y 0 n, (Y0,...,Y n an (I0 n,i n are inepenent. It can be shown by our expansions of the means ν i (n from section 4 that we have b i (n 0 as n for both i Σ, e.g., in the L 3 -norm which below will be technically sufficient. Furthermore, the asymptotic of the variance from Theorem 5. implies together with the strong law of large numbers that the coefficients in (2 converge: σ i (I i n σ i (n p ii, σ i (n I i n σ i (n p ii, where we recall that σ i (In i is the stanar eviation of Li In i a ranom variable. conitione on In i, hence, in particular 2. System of limit equations. The convergence of the coefficients in (2 suggests, by passing formally with n, that limits Y 0 an Y of Yn 0 an Y n, if they exist, shoul satisfy the system of recursive istributional equations Y 0 = p00 Y 0 + p 00 Y, (22 Y = p Y 0 + p Y, (23 where Y 0 an Y are being inepenent on the right han sies. Clearly, centere normally istribute Y 0 an Y with ientical variances satisfy the system (22 ( The operator of probability istributions. Our approach is base on the system (22 (23 of limit equations together with an associate contracting operator (map on the space of probability istributions as follows: We enote by M s (0, the space of all probability istributions on the real line with mean 0, variance an finite absolute moment of orer s. Later 2 < s 3 will be an appropriate choice for us. With the abbreviation M 2 := M s (0, M s (0, we efine the map T : M 2 M 2 ( p00 (τ 0,τ L( W 0 + p 00 W,L( p W 0 + p W, where W 0, W are inepenent with istributions L(W i = τ i for both i Σ. Note that ranom variables (Y 0,Y solve the system (22 (23 if an only if their pair of istributions (L(Y 0,L(Y is a fixe point of T. Hence the ientification of fixe-points an omains of attraction of such fixe-points plays an important role in the asymptotic behavior of our sequences (Y 0 n n 0 an (Y n n 0 an is a core part of our proof. 4. The Zolotarev metric. In accorance with the general iea of the contraction metho we will enow the space M 2 with a complete metric such that T becomes a contraction with respect to this metric. The issue of fixe-points is then reuce to the application of Banach s fixe-point theorem. Asbuilingblockweusethe Zolotarev metric onm s (0,. It hasbeenstuie in thecontextof the contraction metho systematically in [22]. We only nee the following properties, see Zolotarev [34, 35]: For istributions L(X, L(Y on R the Zolotarev istance ζ s, s > 0, is efine by ζ s (X,Y := ζ s (L(X,L(Y := sup f F s E[f(X f(y] (24 where s = m+α with 0 < α, m N 0, an F s := {f C m (R,R : f (m (x f (m (y x y α }, (25 the space of m times continuously ifferentiable functions from R to R such that the m-th erivative ishölercontinuousoforerαwithhöler-constant. Wehavethatζ s (X,Y <, ifallmoments 8

9 of orers,...,m of X an Y are equal an if the s-th absolute moments of X an Y are finite. Since later on only the case 2 < s 3 is use, for finiteness of ζ s (X,Y it is thus sufficient for these s that mean an variance of X an Y coincie an both have a finite absolute moment of orer s. Convergence in ζ s implies weak convergence on R. Furthermore, ζ s is (s,+ ieal, i.e., we have ζ s (X +Z,Y +Z ζ s (X,Y, ζ s (cx,cy = c s ζ s (X,Y for all Z being inepenent of (X,Y an all c > 0. Now, to measure istances on the prouct space M 2 we efine for (τ 0,τ,( 0, M 2 the istance ζ s ((τ 0,τ,( 0, := ζ s (τ 0, 0 ζ s (τ,. 5. The contraction property. We irectly obtain that T is a contraction in ζ s from the property that ζ s is (s,+ ieal: Denoting the components of T by T 0 an T we have ζ s (T 0 (τ 0,τ,T 0 ( 0, p s/2 00 ζ s(τ 0, 0+( p 00 s/2 ζ s (τ, ( p s/2 00 +( p 00 s/2 ζs ((τ 0,τ,( 0, ζ s (T (τ 0,τ,T ( 0, ( p s/2 ζ s (τ 0, 0+p s/2 ζ s(τ, ( ( p s/2 +p s/2 ζs ((τ 0,τ,( 0,. Hence together with ξ := max i Σ (p s/2 ii +( p ii s/2 we obtain that ζ s (T(τ 0,τ,T( 0, ξζ s ((τ 0,τ,( 0,. Since p ii (0, by assumption (2 we have ξ < for the all s > 2. Furthermore, we only obtain finiteness for ζ s on M s (0, for s 3, thus our choice of s is 2 < s 3. For these s we obtain that T is a contraction in ζ s. 6. Convergence of the Yn. i An intuition why contraction properties of the map T lea to convergence of the Yn i towars the unique fixe-point (N(0,,N(0, of T in M2 is as follows: The map T serves as a limit version of our recurrence system (2. Since in this recurrence system we coul replace the Y i I i n an Y i n I i n on the right han sie by the recurrence (2 itself, iterating these replacements leas approximatively to an iteration of the map T. However, the iteration of T applie to any starting point in M 2 converges to the unique fixe-point of T in the metric ζs. Since convergence in ζ s is strong enough to imply weak convergence an (N(0,,N(0, is the unique fixe point of T this finally yiels Proposition 6.2. A etaile proof is given in the full paper version of this extene abstract. 7. Transfer to arbitrary initial istributions. Finally, we prove Theorem 6.. For this, we have to transfer the convergence of the Yn i from Proposition 6.2 to the convergence of the normalization of L µ n via (4. Proof of Theorem 6.. We write L µ n E[L µ n] = Lµ n ν 0 (K n ν (n K n + ν 0(K n +ν (n K n E[L µ n]. By the Lemma of Slutzky it is sufficient to show, as n, L µ n ν 0(K n ν (n K n ν 0 (K n +ν (n K n E[L µ n ] N(0,σ 2 (26 P 0. (27 9

10 For showing (26 note that by Proposition 6.2 (L i n E[Li n ]/ N(0,σ 2 in istribution for both i Σ. We set A n := [µ 0 n n 2/3,µ 0 n+n 2/3 ] N 0 an A c n := {0,...,n}\A n. Then by Chernoff s boun (or the central limit theorem we have P(K n A n. For all x R we have P ( L µ n ν 0 (K n ν (n K n x ( L 0 = P Kn ν 0 (K n + L n K n ν (n K n nlogn = o(+ j A n P(K n = jp x ( L 0 j ν 0 (j + L n j ν (n j x nlogn nlogn For j A n we have jlogj/ µ 0 an (n jlog(n j/ µ 0. Hence, we have (L 0 j ν 0(j/ N(0,µ 0 σ 2 an (L n j ν (n j/ N(0,( µ 0 σ 2 in istribution an the two summans are inepenent. Together, enoting by N 0,σ 2 an N(0,σ 2 istribute ranom variable we obtain ( L µ P n ν 0 (K n ν (n K n x = o(+ P(K n = j(p ( N 0,σ 2 x +o( nlogn j A n P ( N 0,σ 2 x, where the latter convergence is justifie by ominate convergence. This shows (26. To establish the convergence in probability in (27 note that (4 implies E[L µ n ] = E[ν 0(K n ]+E[ν (n K n ]. Hence, with the notation (7 an g(x := xlogx for x [0,] an enoting the L -norm we have ν 0 (K n +ν (n K n E[L µ n] = ν 0 (K n E[ν 0 (K n ]+ν (n K n E[ν (n K n ]. H g(k n E[g(K n ]+g(n K n E[g(n K n ] + f 0 (K n E[f 0 (K n ] + f (n K n E[f (n K n ]. nlogn nlogn With the concentration of the binomial istribution we obtain g(k n E[g(K n ]+g(n K n E[g(n K n ] ( [ ( ] ( [ ( ] = n g Kn Kn n Kn n Kn E g +g E g n n n n ( = O n /2. The terms f 0 (K n E[f 0 (K n ] an f (n K n E[f (n K n ] are also of the orer O(n /2 by a self-centering argument. Altogether we have ( ν 0 (K n +ν (n K n E[L µ nlogn n ] = O, logn which, by Chebyshev s inequality, implies (27. 0

11 References [] Clément, J., Flajolet, P. an Vallée, B. (200 Dynamical sources in information theory: a general analysis of trie structures. Average-case analysis of algorithms (Princeton, NJ, 998. Algorithmica 29, [2] e la Brianais, R. (959 File searching using variable length keys, in Proceeings of the AFIPS Spring Joint Computer Conference. AFIPS Press, Reston, Va., [3] Drmota, M., Janson, S. an Neininger, R. (2008 A functional limit theorem for the profile of search trees. Ann. Appl. Probab. 8, [4] Fill, J.A. an Kapur, N. (2004 The Space Requirement of m-ary Search Trees: Distributional Asymptotics for m 27. Invite paper, Proceeings of the 7th Iranian Statistical Conference, Available via [5] Flajolet, Ph., Roux, M. an Vallée, B. (200 Digital trees an memoryless sources: from arithmetics to analysis. 2st International Meeting on Probabilistic, Combinatorial, an Asymptotic Methos in the Analysis of Algorithms (AofA 0, Discrete Math. Theor. Comput. Sci. Proc., AM, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, [6] Gusfiel, D. (997 Algorithms on Strings, Trees, an Sequences, Cambrige University Press, Cambrige. [7] Jacquet, Ph. an Régnier, M. (988 Normal limiting istribution of the size an the external path length of tries. Technical Report RR-0827, INRIA-Rocquencourt. [8] Jacquet, Ph. an Régnier, M. (988 Normal limiting istribution of the size of tries. Performance 87 (Brussels, 987, , North-Hollan, Amsteram. [9] Jacquet, Ph. an Szpankowski, W. (989 Analysis of Digital Tries with Markovian Depenency. Computer Science Technical Reports. Report , Purue University. Available via [0] Jacquet, Ph. an Szpankowski, W.(99 Analysis of igital tries with Markovian epenency IEEE Trans. Information Theory, 37, [] Jacquet, Ph. an Szpankowski, W. (995 Asymptotic behavior of the Lempel-Ziv parsing scheme an [in] igital search trees. Special volume on mathematical analysis of algorithms. Theoret. Comput. Sci. 44, [2] Jacquet, Ph. an Szpankowski, W. (998 Analytical Depoissonization an Its Applications, Theoretical Computer Science, 20, 62. [3] Jacquet, P., Szpankowski, W. an Tang, J. (200 Average profile of the Lempel-Ziv parsing scheme for a Markovian source. Mathematical analysis of algorithms. Algorithmica 3, [4] Janson, S. an Neininger, R. (2008 The size of ranom fragmentation trees. Probab. Theory Relate Fiels 42, [5] Kirschenhofer, P. an Proinger, H. (988 Further results on igital search trees. Thirteenth International Colloquium on Automata, Languages an Programming (Rennes, 986. Theoret. Comput. Sci. 58, [6] Kirschenhofer, P., Proinger, H. an Szpankowski, W. (989 On the variance of the external path length in a symmetric igital trie. Combinatorics an complexity (Chicago, IL, 987. Discrete Appl. Math. 25,

12 [7] Kirschenhofer, P., Proinger, H. an Szpankowski, W. (994 Digital search trees again revisite: the internal path length perspective. SIAM J. Comput. 23, [8] Kirschenhofer, P., Proinger, H. an Szpankowski, W. (996 Analysis of a Splitting Process Arising in Probabilistic Counting an Other Relate Algorithms, Ranom Structures & Algorithms, 9, [9] Knuth, D.E. (998 The Art of Computer Programming, Volume III: Sorting an Searching, Secon eition, Aison Wesley, Reaing, MA. [20] Mahmou, H.M. (992 Evolution of Ranom Search Trees, John Wiley & Sons, New York. [2] Neininger, R. (200 On a multivariate contraction metho for ranom recursive structures with applications to Quicksort. Analysis of algorithms (Krynica Morska, Ranom Structures Algorithms 9, [22] Neininger, R. an Rüschenorf, L. (2004 A general limit theorem for recursive algorithms an combinatorial structures. Ann. Appl. Probab. 4, [23] Neininger, R. an Rüschenorf, L. (2004 On the contraction metho with egenerate limit equation. Ann. Probab. 32, [24] Neininger, R. an Sulzbach, H. (202 On a functional contraction metho. Preprint available via [25] Rachev, S.T. an Rüschenorf, L. (995 Probability metrics an recursive algorithms. Av. in Appl. Probab. 27, [26] Rais, B., Jacquet, P. un Szpankowski, W. (993 Limiting istribution for the epth in PATRICIA tries. SIAM J. Discrete Math. 6, [27] Rösler, U. (99 A limit theorem for Quicksort. RAIRO Inform. Théor. Appl. 25, [28] Rösler, U. (992 A fixe point theorem for istributions. Stochastic Process. Appl. 42, [29] Rösler, U. (999 On the analysis of stochastic ivie an conquer algorithms. Average-case analysis of algorithms (Princeton, NJ, 998. Algorithmica 29, [30] Rösler, U. an Rüschenorf, L. (200 The contraction metho for recursive algorithms. Algorithmica 29, [3] Schachinger, W. (995 On the variance of a class of inuctive valuations of ata structures for igital search. Theoret. Comput. Sci. 44, Special volume on mathematical analysis of algorithms. [32] Szpankowski, W. (99 A characterization of igital search trees from the successful search viewpoint. Theoret. Comput. Sci. 85, [33] Szpankowski, W. (200 Average Case Analysis of Algorithms on Sequences, John Wiley, New York. [34] Zolotarev, V. M. (976 Approximation of the istributions of sums of inepenent ranom variables with values in infinite-imensional spaces.(russian. Teor. Veroyatnost. i Primenen. 2, Erratum ibi 22 (977, 90. English transl. Theory Probab. Appl. 2, ; ibi. 22, 88. [35] Zolotarev, V. M. (977 Ieal metrics in the problem of approximating the istributions of sums of inepenent ranom variables. (Russian. Teor. Veroyatnost. i Primenen. 22, English transl. Theory Probab. Appl. 22,