PROFILES OF TRIES 1. Introduction. Tries are prototype data

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SIAM J. COMPUT. Vol. 38, No. 5, pp. 1821–1880

c 2009 Society for Industrial and Applied Mathematics

PROFILES OF TRIES∗ § , AND ` GAHYUN PARK† , HSIEN-KUEI HWANG‡ , PIERRE NICODEME

WOJCIECH SZPANKOWSKI¶ Abstract. Tries (from retrieval) are one of the most popular data structures on words. They are pertinent to the (internal) structure of stored words and several splitting procedures used in diverse contexts. The proﬁle of a trie is a parameter that represents the number of nodes (either internal or external) with the same distance from the root. It is a function of the number of strings stored in a trie and the distance from the root. Several, if not all, trie parameters such as height, size, depth, shortest path, and ﬁll-up level can be uniformly analyzed through the (external and internal) proﬁles. Although proﬁles represent one of the most fundamental parameters of tries, they have hardly been studied in the past. The analysis of proﬁles is surprisingly arduous, but once it is carried out it reveals unusually intriguing and interesting behavior. We present a detailed study of the distribution of the proﬁles in a trie built over random strings generated by a memoryless source. We ﬁrst derive recurrences satisﬁed by the expected proﬁles and solve them asymptotically for all possible ranges of the distance from the root. It appears that proﬁles of tries exhibit several fascinating phenomena. When moving from the root to the leaves of a trie, the growth of the expected proﬁles varies. Near the root, the external proﬁles tend to zero at an exponential rate, and then the rate gradually rises to being logarithmic; the external proﬁles then abruptly tend to inﬁnity, ﬁrst logarithmically and then polynomially; they then tend polynomially to zero again. Furthermore, the expected proﬁles of asymmetric tries are oscillating in a range where proﬁles grow polynomially, while symmetric tries are nonoscillating, in contrast to most shape parameters of random tries studied previously. Such a periodic behavior for asymmetric tries implies that the depth satisﬁes a central limit theorem but not a local limit theorem of the usual form. Also √the widest levels in symmetric tries contain a linear number of nodes, diﬀering from the order n/ log n for asymmetric tries, n being the size of the trees. Finally, it is observed that proﬁles satisfy central limit theorems when the variance goes unbounded, while near the height they are distributed according to Poisson laws. As a consequence of these results we ﬁnd typical behaviors of the height, shortest path, ﬁll-up level, and depth. These results are derived here by methods of analytic algorithmics such as generating functions, Mellin transform, Poissonization and de-Poissonization, the saddle-point method, singularity analysis, and uniform asymptotic analysis. Key words. digital trees, tries, proﬁle, depth, height, shortest path, ﬁll-up level, analytic Poissonization, Mellin transform, saddle-point method, singularity analysis AMS subject classifications. Primary, 68W40; Secondary, 68P5, 68P10, 05C05, 60F05 DOI. 10.1137/070685531

1. Introduction. Tries are prototype data structures useful for many indexing and retrieval purposes. They were ﬁrst proposed by de la Briandais [9] in the late 1950s for information processing; Fredkin [27] suggested the current name as it is ∗ Received by the editors March 19, 2007; accepted for publication (in revised form) August 19, 2008; published electronically January 9, 2009. http://www.siam.org/journals/sicomp/38-5/68553.html † Department of Computer Science, State University of New York at Geneseo, Geneseo, NY 14454 ([email protected]). ‡ Institute of Statistical Science, Academia Sinica, 11529 Taipei, Taiwan ([email protected]. edu.tw). This author’s work was partially supported by a grant from the National Science Council of Taiwan. § Laboratory LIX, Ecole ´ Polytechnique, 91128 Palaiseau Cedex, France (nicodeme@lix. polytechnique.fr). ¶ Department of Computer Sciences, Purdue University, 250 N. University Street, West Lafayette, IN 47907-2066 ([email protected]). This author’s work was supported in part by NSF grants CCF0513636, DMS-0503742, DMS-0800568, and CCF-0830140; NIH grant R01 GM068959-01; NSA grant H98230-08-1-0092; and AFOSR grant FA8655-08-1-3018.

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part of retrieval. Tries are multiway trees whose nodes are vectors of characters or digits. Due to their simplicity and eﬃciency, tries found widespread use in diverse applications including document taxonomy, IP address lookup, data compression, dynamic hashing, partial-match queries, speech recognition, leader election algorithms, and distributed hashing tables (see [29, 50, 53, 79]). In this paper, we are concerned with probabilistic properties of the proﬁles of tries, where the proﬁle of a tree is the sequence of numbers each corresponding to the number of nodes with the same distance from the root. We discover several new phenomena in the proﬁles of tries built over strings generated by a random memoryless source, and develop asymptotic tools to describe them. Structure and usefulness of tries. Tries are a natural choice of data structure when the input records involve a notion of alphabets or digits. They are often used to store such data so that future retrieval can be made eﬃcient. Given a sequence of n words over the alphabet {a1 , . . . , am }, m ≥ 2, we can construct a trie as follows. If n = 0, then the trie is empty. If n = 1, then a single (external) node holding the word is allocated. If n ≥ 1, then the trie consists of a root (internal) node directing words to the m subtrees according to the ﬁrst alphabet of each word, and words directed to the same subtree are themselves tries (see [50, 53, 79] for more details). For simplicity, we deal only with binary tries in this paper. Unlike other search trees such as digital search trees and binary search trees where records or keys are stored at the internal nodes, the internal nodes in tries are branching nodes used merely to direct records to each subtrie, with all records stored in external nodes that are leaves of such tries. A trie has more internal nodes than external nodes (ﬁxed to be n throughout this paper), diﬀering from almost all other search trees. In Figure 1 we plot a binary trie of ﬁve strings. The simple organizing procedure used to construct tries and the general eﬃciency they achieve make tries one of the most popular digital search trees. Since their invention, tries have found frequent use in many computer science applications. For example, tries are widely used in algorithms for automatically correcting words in texts (see [51]) and in algorithms for taxonomies and toolkits of regular language (see the Ph.D. thesis [80]); they are also used to represent the event history in datarace detection for multithreaded object-oriented programs (see [6]); another example is the Internet IP address lookup problem (see [60, 74]), where the search time for the IP address problem is directly related to the distribution of the ﬁll-up level (see below for a more precise deﬁnition) and other trie parameters. For applications to other problems in searching, sorting, dynamic hashing, coding, polynomial factorization, Lempel–Ziv compression schemes, and molecular biology, see [29, 79]. The structure of tries also has a close connection to several splitting procedures using coin-ﬂipping; these include algorithms for resolving collisions in multiaccess (or broadcast) communication models, algorithms for loser selection or leader election, etc.; see [45]. Thus most shape parameters in tries have direct interpretations in terms of other related objects. Random tries under the Bernoulli model. Throughout the paper, we write Bn,k to denote the number of external nodes (leaves) at distance k from the root; the number of internal nodes at distance k from the root is denoted by In,k . For simplicity, we will refer to Bn,k as the external proﬁle and In,k as the internal proﬁle. Figure 1 shows a trie and its proﬁles. In this paper we study the proﬁles of a trie built over n binary strings generated by a memoryless source. More precisely, we assume that the input is a sequence of n independent and identically distributed random variables, each being composed of an

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Bn,0 = 0, In,0 = 1 1

0

Bn,1 = 0, In,1 = 2 0

0

1 Bn,2 = 1, In,2 = 2

10 0

0

110 0 0000

1 111

Bn,3 = 2, In,3 = 1

1 0001

Bn,4 = 2, In,4 = 0

Fig. 1. A trie of n = 5 records and its proﬁles: the circles represent internal nodes, and rectangles holding the records are external nodes.

inﬁnite sequence of Bernoulli random variables with mean p, where 0 < p < 1 is the probability of a “1” and q := 1 − p is the probability of a “0.” The corresponding trie constructed from these n binary strings is called a random trie. This simple model may seem too idealized for practical purposes; however, the typical behaviors under such a model often hold under more general models such as Markovian or dynamical sources, although the technicalities are usually more involved (see, for example, [8, 12, 15, 35]). The motivation of studying the proﬁles is multifold. First, they are ﬁne shape measures closely connected to many other cost measures on tries; some of them are indicated below. Second, they are also asymptotically close to the proﬁles of suﬃx trees, which in turn have a direct combinatorial interpretation in terms of words; see [36, 59, 78, 79] for more information and another interpretation in terms of urn models. Third, not only are the analytic problems mathematically challenging, but the diverse new phenomena they exhibit are highly interesting and unusual. Fourth, our ﬁndings imply several new results on other shape parameters (see section 8). Finally, most properties of random tries have also a prototype character and are expected to hold for other varieties of digital search trees (and under more general random models), although the proofs are generally more complicated. Major cost measures on random tries. Due to the usefulness of tries, many cost measures, discussed below, on random tries have been studied in the literature since the early 1970s, and most of these measures can be expressed and analyzed through the proﬁles studied in this paper: • depth: the distance from the root to a randomly selected node; its distribution is given by the expected external proﬁle divided by n; see [10, 12, 13, 20, 24, 33, 36, 42, 46, 52, 65, 69, 71, 75, 76]; • total path length: the sum of distances between nodes and the root, or, equivalently, j jIn,j ; see [8, 11, 24, 44, 58, 57, 70, 71, 72, 73, 75]; • size: the total number of internal nodes, or j In,j ; see [8, 24, 34, 36, 40, 41, 43, 47, 50, 57, 67, 68, 69, 70, 71, 72, 73]; • height : the length of the longest path from the root, or max{j : Bn,j > 0}; see [8, 11, 12, 13, 14, 22, 26, 33, 48, 64, 65, 77]; • shortest path: the length of the shortest path from the root to an external node, or min{j : Bn,j > 0}; see [64, 65];

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• ﬁll-up (or saturation) level : the largest full level, or max{j : In,j = 2j }, where the levels of a tree denote the sets of nodes with the same distance from the root; see [49]; • Horton–Strahler number and stack-size: certain notions of heights related to the traversal of tries; see [4, 17, 54, 55, 56]; • distance of two randomly chosen nodes; see [1, 7]; • pattern occurrences in tries (including page usage or b-tries); see [22, 42, 43, 58, 71, 76]; • one-sided height (or leader election or loser selection); see [21, 39, 66, 81, 82]. The reader is referred to the book [79] and the papers [15, 37, 71] for a systematic treatment of several of these quantities. The general analytic context. The major diﬀerence between most previous study and the current paper is that we are dealing with the asymptotics of bivariate recurrence, in contrast to univariate recurrences (with or without maximization or minimization) addressed in the literature. To be more precise, we observe that, by assumption of the model, the probability generating function Pn,k (y) := E(y Bn,k ) of the external proﬁle satisﬁes the recurrence n Pn,k (y) = (1) (n ≥ 2; k ≥ 1) pj q n−j Pj,k−1 (y)Pn−j,k−1 (y) j 0≤j≤n

with the initial conditions Pn,k (y) = 1 + δn,1 δk,0 (y − 1) when either n ≤ 1 and k ≥ 0 or k = 0 and n ≥ 0, where δa,b is the Kronecker symbol. Observe that this recurrence depends on two parameters n and k, which makes the analysis quite challenging, as we will demonstrate in this paper. The probability generating functions of the internal proﬁle satisfy the same recurrence (1) but with diﬀerent initial conditions; see section 6. From (1), the moments of Bn,k and In,k (centered or not) are seen to satisfy a recurrence of the form n pj q n−j (xj,k−1 + xn−j,k−1 ) xn,k = an,k + j 0≤j≤n

with suitable initial conditions, where an,k are known (either explicitly or inductively). approach is to consider the Poisson generating function f˜k (z) := A standard −z n e n xn,k z /n!, which in turn satisﬁes the functional equation f˜k (z) = g˜k (z) + f˜k−1 (pz) + f˜k−1 (qz) with a suitable g˜k (z). This equation can be solved explicitly by a simple iteration argument and asymptotically by using the Mellin transform (see [23, 79]). The ﬁnal step is to invert from the asymptotics of the Poisson generating function f˜k (z) to recover the asymptotics of xn,k . This last step is guided by the Poisson heuristic, which roughly states that (2) if a sequence {xn }n is “smooth enough,” then xn ∼ e−n j≥0 xj nj /j!, where xn ∼ yn if limn→∞ xn /yn = 1. Such a Poisson heuristic has appeared in diverse contexts under diﬀerent forms such as Borel summability and Tauberian theorems; it dates back at least to Ramanujan’s Notebooks; see the book by Berndt [3, pp. 57–66] for more details. It is known as analytic de-Poissonization when justiﬁed

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by complex analysis and the saddle-point method, and was the subject of intensive analysis, resulting in a robust solution presented in [37]. By means of the Poisson heuristic (2), we expect that μn,k ∼ e−n j≥0 μj,k nj /j!. However, as we will see, such a heuristic holds in our case when q 2k n → 0 but fails otherwise. The reason is that μn,k is too small in this range. Also it should be mentioned that the asymptotic analysis of the above functional equation is in general more intricate because we have an additional parameter k to be taken into account and we need uniformity for our asymptotic approximations in k (varying with n) and in z (in some region in the complex plane) in order to invert the results to obtain xn,k by suitable complex analysis. Known results for proﬁles. As far as probabilistic properties of the proﬁles of random tries are concerned, very little is known in the literature. Since the distribution of the depth Dn in random tries is given by P(Dn = k) = μn,k /n, where μn,k := E(Bn,k ), the asymptotics of the expected proﬁle μn,k for n → ∞ and varying k = k(n) can be regarded as local limit theorems for Dn . Although many papers have addressed the limiting behaviors of the depth, none has dealt with the local limit theorem of Dn and the asymptotics of μn,k for varying k. We will see in section 8 that our result implies an unusual type of local limit theorem for Dn . However, it should be mentioned that the central limit theorem for the depth was developed in [13, 34, 35]. On the other hand, Pittel [65] showed that the distribution of the number of pairs of input-strings having a common preﬁx of length at least k is asymptotically Poisson when k is close to the height. Devroye [14] showed that if

E(Bn,k ) √ → ∞, n

if E(In,k ) → ∞,

then then

Bn,k → 1 in probability; E(Bn,k ) In,k → 1 in probability, E(In,k )

under very general assumptions on the underlying models (see also [15] for further reﬁnements). References [65] and [14] represent known results concerning proﬁles. We will see that convergence in probability in the two “if statements” holds as long as the variance tends to inﬁnity. Sketch of the major phenomena. In the next section we present an in-depth discussion of our results. Here, we brieﬂy summarize our main ﬁndings. We focus mostly on the proﬁles of asymmetric tries (when p = q) since the symmetric tries (when p = q = 1/2) are comparatively easier. We will ﬁrst derive asymptotic approximations to the average external proﬁle μn,k for all ranges of k. Our results show inter alia that for k ≤ (1 − ε) log n/ log(1/q) the average proﬁle μn,k is exponentially small where ε > 0 is small. When k increases and lies in the range (log n − log log log n + O(1))/ log(1/q), then μn,k decays to zero logarithmically until k > k ∗ for a speciﬁc threshold k ∗ in this range beyond which μn,k suddenly grows unbounded at a logarithmic rate. The rate becomes polynomial Θ(nυ ) for some 0 < υ ≤ 1 when 2 1 (1 + ε) log n ≤ k ≤ (1 − ε) log n. log(1/q) log(1/(p2 + q 2 )) Surprisingly enough, for this range of k an oscillating √ factor emerges in the expected proﬁle behavior; that is, E(Bn,k ) ≈ G(logp/q pk n)nv / log n, where G is a bounded periodic function. Such behavior is a consequence of an inﬁnite number of saddlepoints appearing in the integrand of the associated Mellin integral transform. This

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was ﬁrst observed by Nicod`eme [59]. For larger values of k, these oscillations disappear since the behavior of the expected proﬁle is dominated by a polar singularity. Analogous results also hold for the internal proﬁle. In addition, we prove that the variances of both proﬁles are asymptotically of the same order as their expected values. This suggests a central limit theorem for both external and internal proﬁles for a wide range of k. We show that this is indeed true; furthermore, we also show that for k near the height the limiting distribution of the proﬁles becomes Poisson. Some of these results were already anticipated in [63] and constitute the Ph.D. thesis of the ﬁrst author [62]. Proﬁles of digital and nondigital log trees. In passing, we observe that most random trees in the discrete probability literature fall into two major categories according √ to their expected height being of order n (referred to as square-root trees for brevity) or of order log n (referred to as log trees), where n is the tree size. While most random square-root trees were introduced in combinatorics and probability, the majority of log trees arise from data structures and computer algorithms. We can further classify log trees into “digital-type” and “nondigital-type” log trees, according to the nature of construction (or search) of the tree. Proﬁles of nondigital-type search trees of logarithmic height for which binary search trees are representative have received much recent attention and are shown to exhibit several interesting phenomena such as bimodality of the variance and multifaceted behaviors of the limiting distributions; see [5, 18, 19, 28, 31] for more information. In contrast, proﬁles of digital-type search trees have not been addressed as much, and most properties remain unknown; see [14, 15, 65] for tries and [2, 38] for digital search trees. We will show that the limiting behaviors of the proﬁles are very diﬀerent from those of nondigital search trees. In particular, while in no range will the normalized proﬁles in random binary search trees lead to asymptotic normality (in the sense of convergence in distribution), proﬁles of random tries, when properly centered and normalized, all converge to the standard normal law when the variance goes unbounded in the limit. As is often the case for proving asymptotic normality, we need more precise asymptotic approximation to the variance, rendering our analysis more complicated. Organization of the paper. The paper is organized as follows. In the next section, we present (rather informally) a more detailed summary of our main ﬁndings. This section is to help the reader to comprehend the richness of our results in their fullness but without resorting to rather abstruse mathematical formulations. Sections 3–8 are devoted to precise formulations of our results. This paper contains two major parts: The ﬁrst, section 3, develops the asymptotic tools we need for deriving the diverse asymptotic approximations to the expected external proﬁle μn,k . Most proofs of the second part (sections 4–8) are then sketched because they extend the same methods of proof as in the ﬁrst part. Except for sections 7 and 8, we assume p = q throughout this paper. Among these sections, section 4 derives the asymptotics of the variance of Bn,k , the corresponding results of convergence in distribution being given in section 5. The internal proﬁles are addressed in section 6, and results for symmetric tries are given in section 7. Consequences of our ﬁndings are discussed in section 8, where we establish typical behaviors of the height, the width, the shortest path, the ﬁll-up level, and the right-proﬁle, as well as a rather atypical local limit theorem for the depth. 2. Summary of main results. In this section we discuss informally our main results. We focus here on describing the major phenomena arising in the analysis of

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proﬁles rather than presenting the precise and complicated results to which we devote the remaining sections of this paper. Crucial to our analysis of the proﬁles is the asymptotics of the expected proﬁles. Not only are the results fundamental and highly interesting, but the analytic methods we used are also of certain generality. From (1), we see that the expected external proﬁle μn,k := E(Bn,k ) satisﬁes the following recurrence: n (3) μn,k = pj q n−j (μj,k−1 + μn−j,k−1 ) j 0≤j≤n

for n ≥ 2 and k ≥ 1 with the initial values μn,0 = 0 for all n = 1 and 1 for n = 1. Furthermore, μ0,k = 0, k ≥ 0, and μ1,k = 0 for k ≥ 1 and is equal to 1 when k = 0. Throughout we assume that p > q = 1 − p unless stated otherwise. The polynomial growth of μn,k . In section 3, we solve asymptotically (3) for various ranges of k when p = q; a crude description of the asymptotics of μn,k is as follows: ⎧ 0 if α ≤ α1 , ⎪ ⎪ ⎨ log μn,k −ρ + α log(p−ρ + q −ρ ) if α1 ≤ α ≤ α2 , → (4) if α2 ≤ α ≤ α3 , 2 + α log(p2 + q 2 ) ⎪ log n ⎪ ⎩ 0 if α ≥ α3 , where (5) α1 :=

1 , log(1/q)

α2 :=

p2 + q 2 , p2 log(1/p) + q 2 log(1/q)

and α3 :=

2 log(1/(p2 + q 2 ))

are delimiters of α := limn k/ log n (k = k(n)), and ρ :=

1 log log(p/q)

1 − α log(1/p) α log(1/q) − 1

.

Note that α1 ≤ α2 ; see Figure 2. The limiting estimate (4) gives a rough picture of μn,k as follows: μn,k is of polynomial growth rate when α1 + ε ≤ α ≤ α3 − ε and is smaller than any polynomial powers when 0 ≤ α ≤ α1 − ε and α ≥ α3 + ε. Near the two boundaries α1 and α3 , the behaviors of μn,k will undergo phase changes from being subpolynomial to being polynomial or the other way around. More reﬁned asymptotics. To derive more precise asymptotics of μn,k than the phase transitions (4) of the polynomial order of μn,k , we divide all possible values of k into four overlapping ranges. (I) Elementary range: 1 ≤ k ≤ α1 (log n − log log log n + O(1)). √ (II) Saddle-point range: α1 (log n−log log log n+Kn ) ≤ k ≤ α2 (log n−Kn log n). (III) Gaussian transitional range: k = α2 log n +√o((log n)2/3 ). (IV) Polar singularity range: k ≥ α2 log n + Kn log n, where, throughout this paper, Kn ≥ 1 represents a (generic) sequence tending to inﬁnity. More precisely, in Theorem 1 we prove that for k lying in range (I) the expected external proﬁle μn,k ﬁrst decays exponentially fast (asymptotic to q k n(1 − q k )n−1 ).

` G. PARK, H.-K. HWANG, P. NICODEME, AND W. SZPANKOWSKI

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10

α3

1

8

α2

0.8 p = 0.85

0.6

6

p = 0.9

4

0.4

2

0.2

0.5

0.6

0.7

0.8

α1 p 0.9 1

α2

0

2

4

6

8

10

Fig. 2. Left: a plot of α1 , α2 , and α3 (deﬁned in (5)) as functions of p. Right: the (nonzero) limiting order of log μn,k / log n plotted against α = limn k/ log n for p = 0.55, 0.6, . . . , 0.9 (the spans of the curves increase as p grows). The vertical lines represent the positions of α2 (to the right of which the curves are straight lines); see (4).

Then, when k is around α1 (log n − log log log n + log(p/q − 1) + m log(p/q)) for some integer m ≥ 0, μn,k ∼

k m m k−m −npm qk−m p q ne , m!

which is of order μn,k = O

log log n

logξ−m n

for some ξ. Thus, for m < ξ the expected external proﬁle decays only logarithmically, but for m ≥ ξ it increases logarithmically. The behavior of μn,k in range (II) is described in Theorem 2. The situation becomes highly nontrivial and interesting. More precisely, for α1 (1 + ε) log n ≤ k ≤ α2 (1 − ε) log n, we ﬁnd that

pρ q ρ (p−ρ + q −ρ ) nυ1 , μn,k ∼ G1 ρ; logp/q pk n ·√ log n 2παn,k log(p/q) where (αn,k := k/ log n) υ1 = −ρ + αn,k log(p−ρ + q −ρ ), −1 − αn,k log q 1 ρ=− log , log(p/q) 1 + αn,k log p and G1 (ρ; x) is a periodic function. We plot in Figures 3 and 4 the periodic parts of G1 (−1, x) for a few values of p and ρ, respectively. Analytically, these oscillations are consequences of an inﬁnite number of saddle-points appearing in the integrand of the associated Mellin transform of the expected proﬁle, but visually they look like certain sine waves due to the fact that the corresponding Fourier expansions involve a

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3 × 10−7

6 × 10−3 p = 0.95

p = 0.65 1 p = 0.55 −6 × 10−23

p = 0.85 1

1

−3 × 10−7

p = 0.75

−6 × 10−3

Fig. 3. The ﬂuctuating part around the mean of the periodic function G1 (−1; x) for p = 0.55, 0.65, . . . , 0.95 and for x in the unit interval; its amplitude tends to zero when p → 0.5+ . 1.5 × 10−11

ρ = −1.5

3 × 10−6

ρ = 3.5

1

1.5 × 10−11

10

1

1

3 × 10−6

ρ = 8.5

−10

Fig. 4. The ﬂuctuating part around the mean of the periodic function G1 (ρ; x) for ρ ∈ {−1.5, 3.5, 8.5} and x ∈ [0, 1]. The amplitude increases as ρ grows.

Gamma function with increasing parameters, which decreases very fast along a ﬁxed vertical line for an increasing imaginary part, so that only a few terms dominate. Finally, in Theorem 3 we prove that for k in range (IV) μn,k ∼ 2pqn2 (p2 + q 2 )k−1 =

2pq nυ2 , p2 + q 2

where υ2 = 2 + αn,k log(p2 + q 2 ), and the periodic function disappears. In this region, the asymptotic behavior of the expected proﬁle is dictated by the expected number of pairs (of input-strings) having common preﬁxes of length at least k. This property is analytically reﬂected by a polar singularity in the associated Mellin transform. The asymptotics of μn,k in range (III) for k = α2 log n + o(log2/3 n) are presented in Theorem 4. In this transitional range, the saddle-point coalesces with the polar singularity, so we use the Gaussian integral to describe the behavior of μn,k . In summary, our results roughly state that μn,k → 0 when 1 ≤ k ≤ k ∗ for some ∗ k close to α1 (log n − log log log n + O(1)), and then μn,k tends abruptly to inﬁnity at a logarithmic rate when k > k ∗ . Such an abrupt change has already been observed in the literature for the shortest path and the ﬁll-up level (see [49, 65]), but not much is known for μn,k beyond that. Then we show that μn,k grows polynomially when k lies in the range α1 (1 + ε) log n ≤ k ≤ α3 (1 − ε) log n, reaching the peak where it is √ of order n/ log n; it decays at a slower rate afterwards until it tends to zero again when k ≥ α3 (log n + Kn ). A salient feature here is the presence of an oscillating function in the asymptotic approximation when p = q.1 In Figure 5, a plot of the rough silhouettes of μn,k is presented. 1 The expected values of many shape characteristics of random tries often exhibit the asymptotic pattern: ∼ F (logc n)n if log p/ log q is rational for some periodic function F and constant c expressible in terms of p, and ∼ Cn if log p/ log q is irrational; see [37, 71, 79].

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α1 log

n log log n

+ O(1)

α1 log

n log log n

+ O(1)

α0 log n log n+O(1) p log(1/p)+q log(1/q)

α2 log n

α3 log n + O(1)

log n+O(1) p log(1/p)+q log(1/q)

α2 log n

α3 log n + O(1)

Fig. 5. The silhouettes of the expected external (left) and internal (right) proﬁles of an asymmetric trie (p = 0.75). Note that the right subtrees of the asymmetric trie have more nodes than their left siblings since p > 1/2. Also, the ﬁrst few levels contain almost no external nodes but are almost full of internal nodes.

Asymptotics of the expected internal proﬁle. The expected value of the internal proﬁle E(In,k ) is discussed in section 6. In particular, √ the expected internal proﬁle is asymptotically equivalent to 2k for k ≤ α (log n−K log n), where α0 := 2/(log(1/p) 0 n √ + log(1/q)). When k ≥ α2 (log n + Kn log n), then E(In,k ) ∼ (p2 + q 2 )E(Bn,k )/pq. Between these two ranges, it is again the inﬁnite number of saddle-points that yield the dominant asymptotic approximation. Unlike μn,k , an additional √ phase transition appears in the asymptotics of the E(In,k ) when k = α0 log n + O( log n), reﬂecting the structural change of the internal nodes from being asymptotically full to being of the same order as the number of external nodes. The silhouettes of the expected internal proﬁles for a symmetric trie and an asymmetric (p = 0.75) trie are presented in Figure 6. Variance and limiting distributions. In section 4 we deal with the variance of the proﬁle. In particular, in Theorem 7 we derive asymptotic approximations to the variance of the proﬁle, which asymptotically turns out to be of the same order as the expected value for all ranges of k ≥ 1; namely, V(Bn,k ) = Θ(E(Bn,k )). In fact, we show that V(Bn,k ) ∼ E(Bn,k ) in range (I), for range (IV) V(Bn,k ) ∼ 2E(Bn,k ), and in range (II) (polynomial growth) the variance and the expected proﬁle diﬀer only by the oscillating functions. The variance of the internal proﬁle behaves almost identically to the variance of the external proﬁle; roughly, V(In,k ) = Θ(V(Bn,k )) for all k. The methods used to derive these results are the same as those used in section 3. We then prove, in section 5, that both internal and external proﬁles, after proper normalization, are asymptotically normally distributed iﬀ the variance tends to inﬁnity (see Theorems 8 and 9). The limiting distribution is Poisson when the variance

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log2

n log n

+ O(1) log2

n log n

+ O(1)

log2 n + O(1)

log2 n + O(1)

2 log2 n + O(1)

2 log2 n + O(1)

Fig. 6. The silhouettes of the expected external and internal proﬁles of a symmetric; compare Figure 5.

remains bounded away from zero and inﬁnity. In particular, we will prove that when V(Bn,k ) = Θ(1), P (Bn,k = 2m) =

λm 0 e−λ0 + o(1) and P (Bn,k = 2m + 1) = o(1), m!

where λ0 := pqn2 (p2 + q 2 )k−1 , while for V(In,k ) = Θ(1), we ﬁnd P(In,k = m) =

λm 1 e−λ1 + o(1) m!

(m = 0, 1, . . . ),

where λ1 := n2 (p2 + q 2 )k /2. These results hold for both symmetric and asymmetric tries, but the ranges where the variances become unbounded are diﬀerent. Symmetric tries. For the symmetric case, we have α1 = α2 = 1/ log 2. This means that the two ranges separated by α2 coalesce into one for symmetric tries; see Figure 2. The analysis then becomes simpler as shown in section 7. An interesting property is that unlike for asymmetric tries, the fattest levels of proﬁles of symmetric tries contain a linear number of nodes. The global picture of a random symmetric trie is roughly as follows (α1 = 1/ log 2): • When 1 ≤ k ≤ α1 (log n − log log n + O((log n)−1 )), each level is almost full of internal nodes (In,k ≈ 2k ), the number of external nodes tending to zero; in particular, the variances of both proﬁles tend to zero. • When α1 (log n − log log n + Kn / log n) ≤ k ≤ 2α1 (log n − Kn ), where Kn is any sequence tending to inﬁnity, the variances of both proﬁles tend to inﬁnity, and we prove the asymptotic normality of both proﬁles. • When k = 2α1 (log n + O(1)), both proﬁles are asymptotically Poisson distributed, but Bn,k assumes only even values. • When k ≥ α1 (log n + Kn ), then nodes appear very unlikely. Section 8 describes some consequences of our main results. In particular, we point out a rather unusual form of the local limit theorem for the depth due to the oscillating factor in the expected proﬁle. Then we apply our results to rederive typical behavior for the height, the shortest path, and the ﬁll-up level. Also, the width and

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` G. PARK, H.-K. HWANG, P. NICODEME, AND W. SZPANKOWSKI

the right-proﬁle (counting only right branches and neglecting the left ones) are brieﬂy discussed. This completes the summary of our main results. Precise formulations and proofs are presented in the next ﬁve sections. Enjoy the reading! 3. Expected external profile. We derive asymptotic approximations to the expected external proﬁle μn,k in this section, starting with a few useful expressions for μn,k . Notation. Throughout this paper, p ∈ [1/2, 1) is ﬁxed and q = 1−p. Let k = k(n) and α := limn k/ log n, whenever the limit exists. The constants α1 , α2 , and α3 are deﬁned in (5). For convenience, we also write Ln := log n,

LLn := log log n,

LLLn := log log log n.

The generic symbol ε is always used to represent a suitably small constant whose value may vary from one occurrence to another, and Kn denotes any sequence tending to inﬁnity. The symbol f (n) = Θ(g(n)) means that there are positive constants C and C such that C|g(n)| ≤ |f (n)| ≤ C |g(n)|. 3.1. Exact expressions and integral representations. ˜Denote by Mk (z) the probability generating function n≥0 μn,k z n /n! of μn,k and by M k (z) the correspond−z ˜ ing Poisson generating function Mk (z) := e Mk (z). ˜ k (z) satisﬁes the integral represenLemma 1. The Poisson generating function M tation

k ˜ k (z) = 1 (6) z −s Γ(s + 1)g(s) p−s + q −s ds M 2πi (ρ) for k ≥ 1 and (z) > 0, where Γ denotes the Gamma function, g(s) := 1 − 1/(p−s + ρ+i∞ q −s ), and (ρ) stands for the integral ρ−i∞ . The integral with ρ > −2 is absolutely convergent for (z) > 0. Proof. By taking the derivative with respect to y on both sides of (1) and then substituting y = 1, we see that μn,k satisﬁes the recurrence (3) with the initial conditions μn,k = δn,1 δk,0 when either n ≤ 1 and k ≥ 0 or k = 0 and n ≥ 0. Note that μn,1 = n pq n−1 + qpn−1 (n ≥ 2). It follows that Mk (z) = eqz Mk−1 (pz) + epz Mk−1 (qz)

(k ≥ 2),

˜ k (z) satisﬁes with M1 (z) = z(peqz + qepz − 1). Thus M (7)

˜ k (z) = M ˜ k−1 (pz) + M ˜ k−1 (qz). M

Iterating this equation yields (8)

˜ k (z) = M

k − 1 ˜ 1 (pj q k−1−j z), M j

0≤j −2, and the Mellin transform of M1 (pj q k−1−j z) is p−sj q −s(k−1−j) M1∗ (s) (see [23, 79]). To justify the absolute convergence of the integral, we apply the Stirling formula for the Gamma function (with complex parameter)

s s √ 1 + O |s|−1 , Γ(s + 1) = 2πs e uniformly as |s| → ∞ and | arg s| ≤ π − ε, which implies that |Γ(ρ + it)| = Θ(|t|ρ−1/2 e−π|t|/2 ),

(9)

uniformly for |t| → ∞ and ρ = o(|t|2/3 ). The integrand in (6) is analytic for (s) > −2 and bounded above by k z −ρ−it Γ(ρ + 1 + it)g(ρ + it) p−ρ−it + q −ρ−it

= O |z|−ρ |t|ρ+1/2 e−π|t|/2+arg(z)t (p−ρ + q −ρ )k for large |t|. This completes the proof of the lemma. Corollary 1. The expected external proﬁle μn,k satisﬁes, for n, k ≥ 1, k n−1 pj q k−j n 1 − pj q k−j μn,k = j 0≤j≤k k − 1 n−1 (10) − pj q k−1−j n 1 − pj q k−1−j j 0≤j −2),

where g(s) = 1 − 1/(p−s + q −s ). Proof. By deﬁnition and (8)

k − 1 j+1 k−1−j j k−j j k−1−j )z )z pj q k−1−j z pe(1−p q + qe(1−p q )z − e(1−p q . Mk (z)= j 0≤j 0. The proof of (26) follows directly from the next proposition in view of (8) and [z n ]M1 (z) ≥ 0. Proposition 2. Let f (z) be an entire function and let z = reiθ , where r ≥ 0 and |θ| ≤ π. If |ez f (z)| ≤ er f (r) (r ≥ 0; |θ| ≤ π), where f (r) ≥ 0, then the sum fk (z) := 0≤j≤k kj f (pj q k−j z) satisﬁes (27)

2

|ez fk (z)| ≤ er fk (r)e−crθ ,

(28)

uniformly for k ≥ 0, r ≥ 0, and |θ| ≤ π, where c > 0 is independent of z and k. Proof. By (27) and the elementary inequality 1 − cos θ ≥

(29) we obtain

2 2 θ π2

(|θ| ≤ π),

k i k−j j k−j |e fk (z)| ≤ e(1−p q )r cos θ ep q r f (pj q k−j r) j 0≤j≤k k i k−j 2 2 j k−j ≤ e(1−p q )r(1−2θ /π ) ep q r f (pj q k−j r) j z

0≤j≤k

≤ e−2rθ

2

(1−pk )/π 2 r

e fk (r).

This proves (28) with, say, c = 2(1 − p)/π 2 .

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` G. PARK, H.-K. HWANG, P. NICODEME, AND W. SZPANKOWSKI

˜ k (z) more precisely in the folProof of (22) in Theorem 1. We next evaluate M lowing lemma whose proof is presented in Appendix A. Let k − 1 m k−m −pm qk−m z p q ze . Sk,m (z) := m Lemma 3. (i) (m = 0). If 1 ≤ k ≤ k0 − α1 Kn /LLn , then ˜ k (z) = q k ze−qk z 1 + O(e−Kn ) , M (30) −1/2

uniformly for |z| = n and arg(z) = o(LLn (ii) (m ≥ 1). If (31)

).

k = α1 (Ln − LLLn + log(p/q − 1) + m log(p/q) − η) ,

where m ≥ 1 and Kn Kn ≤ η ≤ log(p/q) − , LLn LLn then ˜ k (z) = Sk,m (z) 1 + O(me−Kn ) , M

(32)

−1/2

uniformly for |z| = n and arg(z) = o(LLn ). Using the above lemma, we now prove Theorem 1. It remains to evaluate the integral in (25). We ﬁrst consider the case m = 0. By substituting (30) into the integral in (25), and by completing the arc | arg(z)| ≤ θ0 to a full circle, we see that

k k n! ˜ k (z)dz = q n! z −n−1 ez M z −n e(1−q )z dz + O(E1 ) |z|=n |z|=n 2πi 2πi | arg(z)|≤θ0

| arg(z)|≤θ0

k

= q n![z where E1 := e

−Kn

n!n

−n k

θ0

]e(1−q

e(1−q

q n

E2 := q k n!n1−n

n−1

k

k

)z

+ O(E2 ) + O(E1 ),

)n cos θ

dθ,

−θ0 π

e(1−q

k

)n cos θ

dθ.

θ0

By inequality (29), we have

∞ −Kn 1/2 k −qk n −2n(1−qk )θ 2 /π 2 E1 = O e n q ne e dθ −∞

k = O e−Kn q k ne−q n . Similarly,

k 1/5 2 E2 = O q k ne−q n n−1/10 e−2n /π .

This completes the proof of (22) when m = 0. For m ≥ 1, we proceed in a similar manner but using part (ii) of Lemma 3.

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Proof of (23) in Theorem 1. We now consider the remaining gaps when k is of √ the form (31) with η = x/LLn , where x = o( LLn ). In this case, the same analysis as above shows that both terms Sk,m (z) and Sk,m+1 (z) are asymptotically close so that ˜ k (z) = (Sk,m (z) + Sk,m+1 (z)) (1 + O(E3 )) , M

(33)

where the error E3 introduced is bounded above by ⎞ ⎛ Sk,j (z) Sk,j (z) ⎠ E3 = O ⎝ Sk,m (z) + Sk,m (z) 0≤j<m

m+2≤j≤k

η

= O (m + 1)L−(1−qe n

cos θ/p)

⎛

+ O ⎝m!

(pα1 /q)j j≥2

(j + m)!

j

⎞

j− (p/q) −1 eη cos θ ⎠ Ln p/q−1

= O (m + 1)L−(1−q/p) + (m + 1)−1 L−(p/q−1) n n

, = O (m + 1)L−(1−q/p) n

since 1 − q/p ≤ p/q − 1, where we used the inequality tj − 1 t+1 ≥ j t−1 2

(t > 1; j ≥ 2),

−1/2

). Thus the same analysis as above gives

η

k m m k−m −pm qk−m n pL1−e n , p q 1 + O (m + 1)L−(1−q/p) = ne 1+ n m! q(m + 1) log(1/q)

and θ = o(LLn μn,k

which implies (23). 3.4. Range (II): A saddle-point analysis. We now assume that α1 (Ln − LLLn + Kn ) ≤ k ≤ α2 (Ln − Kn Ln ), (34) and proceed by the saddle-point method (see [79, 83]) to derive the following main result of this subsection. Theorem 2 (asymptotics of μn,k in range (II)). If k satisﬁes (34), then

n−ρ (p−ρ + q −ρ )k 1 1 + 1+O (35) μn,k = G1 ρ; logp/q pk n , k(p/q)ρ k(ρ + 2)2 2πβ2 (ρ)k where ρ = ρ(n, k) > −2 is chosen to satisfy the saddle-point equation ⎧ d ρ −ρ −ρ −ρ ⎪ ⎪ ρ e n (p + q −ρ )k = 0 if ρ ≥ 1, ⎨ dρ (36) ⎪ d −ρ −ρ ⎪ ⎩ n (p + q −ρ )k = 0 if ρ ≤ 1, dρ and (37)

p−ρ q −ρ log(p/q)2 β2 (ρ) := , (p−ρ + q −ρ )2 G1 (ρ; x) = g(ρ + itj )Γ(ρ + 1 + itj )e−2jπix j∈Z

(tj := 2jπ/ log(p/q))

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` G. PARK, H.-K. HWANG, P. NICODEME, AND W. SZPANKOWSKI

with g(s) = 1 − 1/(p−s + q −s ), and G1 (ρ, x) is a 1-periodic function (see Figures 3 and 4). We devote the rest of this subsection to the proof of Theorem 2. 3.4.1. Two-step saddle-point method. Here we outline the main steps of the proof of Theorem 2. The approach may be called a two-step saddle-point method since the saddle-point method will be applied twice. First, we start from the Mellin integral ˜ k (reiθ ) for (6) and apply the saddle-point method to obtain precise asymptotics of M small θ (i.e., around the real axis) and large r. The proof here is complicated by the fact that −ρ−it p (38) + q −ρ−it = p−ρ + q −ρ when t = tj = 2πj/ log(p/q), j ∈ Z, which implies that the number of saddle-points with the same real part is inﬁnite, yielding the 1-periodic function G1 (ρ; x). This ﬁrst application of the saddle-point method yields a good approximation to ˜ k (z) by another ˜ k (z) for z large and near the real axis; then we de-Poissonize M M ˜ application of the saddle-point method and establish that μn,k ∼ Mk (n). Ultimately, we will use the de-Poissonization result of Proposition 1; however, in the ﬁrst approximation we do de-Poissonization by “bare hands” by applying the argument already used in the proof of Proposition 1, namely, (17) and (18). Thus we focus on the evaluation of the Cauchy integral (13) but with |θ| ≤ n−2/5 (the ﬁrst integral of (25)). k

3.4.2. Location of saddle-points. The integrand z −s Γ(s+1)g(s) (p−s + q −s ) of the integral in (6) has simple poles at s = −j, j = 2, 3, . . . , the rightmost (dominant) one being at s = −2; it also has saddle-points, which are the zeros of the equation (39)

d Γ(s + 1)n−s (p−s + q −s )k = 0 ds

(note that g(s) is uniformly bounded for all s). In view of (38), there are inﬁnitely many saddle-points of the form ρ + itj / log(p/q) (j = 0, ±1, . . .), where the real part ρ satisﬁes (39). Also it is easy to see that 1 , ρ → +∞ if Lkn ↓ log(1/q) k 1 ρ → −∞ if Ln ↑ log(1/p) . We distinguish between two cases ρ ≥ 1 and −2 < ρ < 1. In the former, the saddle-points are asymptotically determined, by Stirling’s formula for the Gamma function, by the ﬁrst equation in (36), which is simpler than (39), while in the latter case they are asymptotically determined by the second equation of (36) since Γ(ρ + 1) is uniformly bounded and thus does not contribute signiﬁcantly to the saddle-point location. More precisely, ﬁrst consider the case when ρ ≥ 1 (the choice of 1 being arbitrary). In this case, by (36), we obtain p−ρ + q −ρ k = −ρ , Ln − log ρ p log(1/p) + q −ρ log(1/q) which can be written in the form 1 log ρ= log(p/q)

Ln − log ρ − k log(1/p) k log(1/q) − Ln + log ρ

,

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PROFILE OF TRIES

whenever Ln − log ρ < k log(1/q), which will be seen to be the case when k satisﬁes (34). On the other hand, when ρ ≤ 1, we consider the second equation in (36) or k p−ρ + q −ρ , = −ρ Ln p log(1/p) + q −ρ log(1/q) which is solved to be (40)

ρ=

1 log log(p/q)

Ln − k log(1/p) k log(1/q) − Ln

.

It follows that if k satisﬁes (34), then log(p/q) 1 (41) + o(1) , ρ≤ LLn − log Kn + log log(p/q) log(1/q) implying, in particular, that ρ = O(LLn ). Also, if k = α1 (Ln − LLLn + log log(p/q) + Kn ), then 1 log(p/q) −1 ρ= + O(Kn ) . LLn − log Kn + log log(p/q) log(1/q) However, if k is close to the right boundary of (34), more precisely, k = α2 (1 − εn )Ln , where εn = o(1), then ρ = −2 +

εn + O(ε2n ). α2 β2 (−2)

Thus ρ = O(1). From (41), we see that if ρ ≥ 1 and k satisﬁes (34), then kβ2 (ρ) = Θ(k(p/q)ρ ) and k(p/q)ρ ≥

Kn + o(1); log(p/q)

−1/2

on the other hand, if ρ ≥ −2 + Kn Ln , then k(ρ + 2)2 ≥ Kn2 . Thus the O-term in (35) is small if we choose Kn suﬃciently large. 3.4.3. More transparent behaviors of μn,k . Before we present a formal proof of Theorem 2, we ﬁrst discuss more transparent behaviors of μn,k in some speciﬁed ranges. The central range: α ∈ [α1 + ε, α2 − ε]. In this case, G1 is bounded and G1 (ρ; x) ∼ G1 (ρ ; x), where 1 1 − α log(1/p) (42) ρ := log ; log(p/q) α log(1/q) − 1 also β2 (ρ) ∼ β2 (ρ ). Note that g(ρ + itj ) = 1 − pitj /(p−ρ + q −ρ ) and

G1 ρ; logp/q pk n = G1 ρ; logp/q q k n .

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` G. PARK, H.-K. HWANG, P. NICODEME, AND W. SZPANKOWSKI

More precisely, if k = α(Ln + x αβ2 (ρ )Ln ), where α ∈ [α1 + ε, α2 − ε] and 1/6 x = o(Ln ), then

μn,k = G1

k n−ρ p−ρ + q −ρ 1 + |x|3 k −x2 /2 √ ρ ; logp/q p n e 1+O , Ln 2παβ2 (ρ )Ln

uniformly in x. In particular, when α = 1/h, where h := p log(1/p) + q log(1/q) is the entropy of the Bernoulli variate, then ρ = −1, and it follows that

(43)

μn,k

√ h G1 −1; logp/q pk n 1 + |x|3 n −x2 /2 √ √ 1+O , = e ·√ log(p/q) 2πpq Ln Ln 1/6

1/6

uniformly for x = o(Ln ). Other approximations can be derived for Ln x = √ o( Ln ). Thus μn,k reaches the maximum for k near Ln /h + O(1); also, μn,k increases with k when α < 1/h and decreases with k when α > 1/h; see Figure 2. See also Figure 3 for a plot of G1 (−1; x) for a few p’s. −1/2

The left boundary: ρ → −2+ and ρ + 2 Ln periodicity vanishes because G1 (ρ; x) ∼

. In this case, the dominant

2pq |g(−2)| = 2 ; ρ+2 (p + q 2 )(ρ + 2)

thus (44)

μn,k ∼ √

k 2 k −1/2 n−ρ p−ρ + q −ρ . 2π log(p/q)(ρ + 2)

The right boundary: k/Ln → 1/ log(1/q)+ . In this case, ρ → ∞ and ρ = O(LLn ). The periodicity in the leading term of (35) does not vanish because we have G1 (ρ; x) ∼

Γ(ρ + 1 + itj )e−2jπix ,

j∈Z

and G1 is not bounded. Indeed, the periodicity becomes more pronounced for increasing ρ since

Γ(ρ + 1 + it) = O e−t2 /(2ρ)+O(t4 /ρ3 ) Γ(ρ + 1) for large ρ and t = o(ρ); see Figure 4. This estimate also implies that ⎛ G1 (ρ; x) = O ⎝

⎞

|Γ(ρ + 1 + itj )|⎠ = O e−ρ ρρ+1 = O ρ1/2 Γ(ρ + 1) .

j∈Z

The order is tight. This means that even if we normalize G1 (ρ; x) by Γ(ρ+1), |G1 (ρ; x)| still goes to inﬁnity with ρ.

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3.4.4. Proof of Theorem 2. In view of (25) (more generally, de-Poissonization ˜ k (n) and obtain precise local expansions for Proposition 1), we need only evaluate M iθ ˜ Mk (ne ) when |θ| ≤ θ0 in order to estimate the ﬁrst integral of (25). We ﬁrst focus ˜ k (n) and then extend the same approach to derive the asymptotics on estimating M iθ ˜ ˜ k (n). Later in subsection 3.8 we of Mk (ne ). This suﬃces to prove that μn,k ∼ M reﬁne this analysis to obtain a better error term. ˜ k (n) by the inverse Mellin transform, ﬁrst we move the line In order to evaluate M of integration of (6) to (s) = ρ so that

∞ ˜ k (n) = 1 M (45) Jk (n; ρ + it)dt, 2π −∞ −s where ρ > −2 is the saddle-point chosen according to (36) and Jk (n; √ s) := n Γ(s + −s −s k 1)g(s)(p + q ) . We now show that the above integral with |t| ≥ Ln is asymptotically smaller than the dominant term in (35) √ and then assess the main contribution of saddle-points falling into the range |t|√≤ Ln . Estimate of the integral when |t| ≥ Ln . Assume from now on that ρ is chosen as described above in (36). Since our ρ > −2 satisﬁes (40), we have, by (9),

∞ 1 −ρ −ρ −ρ k Jk (n; ρ + it)dt = O n (p + q ) √ |Γ(ρ + 1 + it)|dt 2π |t|≥√Ln Ln

∞ −ρ −ρ −ρ k ρ+1/2 −πt/2 = O n (p + q ) √ t e dt Ln

√ = O Lρ/2+1/4 e−π Ln /2 n−ρ (p−ρ + q −ρ )k . n −1/2

On the other hand, since ρ = O(LLn ) and ρ ≥ −2 + Kn Ln , we then obtain

√ √ √ 2 Lρ/2+1/4 e−π Ln /2 = O e−π Ln /2+O(LLn ) = O Γ(ρ + 2)e− Ln n −1/2

and ρ for large enough n; the last O-term holds uniformly for ρ ≥ −2 + Kn Ln satisfying (41). integer j for which Contribution√from each saddle-point. Let j0 be the largest 2jπ/ log(p/q) ≤ Ln . Then we can split the integral over |t|≤√Ln as follows:

Jk (n; ρ + it)dt = Jk (n; ρ + it)dt √ |t|≤ Ln

|j|<j0

+

|t−tj |≤π/ log(p/q)

√ tj0 ≤|t|≤ Ln

Jk (n; ρ + it)dt.

The last integral is bounded above by

√ O Γ(ρ + 2)n−ρ (p−ρ + q −ρ )k e− Ln , by the same argument used above. It remains to evaluate the integrals

1 Tj := Jk (n; ρ + it)dt 2π |t−tj |≤π/ log(p/q) for |j| < j0 .

` G. PARK, H.-K. HWANG, P. NICODEME, AND W. SZPANKOWSKI

1844

We ﬁrst derive a uniform bound for |p−ρ−it + q −ρ−it |. By the elementary inequalities (29) and √ x 1−x≤1− 2

(x ∈ [0, 1]),

we have

p 1 − cos t log q p p−ρ q −ρ ) log 1 − cos (t − t ≤ p−ρ + q −ρ 1 − −ρ j −ρ 2 (p + q ) q 2 −ρ −ρ p 2p q ≤ p−ρ + q −ρ 1 − 2 −ρ (t − tj )2 log −ρ 2 π (p + q ) q −ρ 2 ≤ p + q −ρ e−c0 (t−tj ) ,

−ρ−it p + q −ρ−it = p−ρ + q −ρ

(46)

2p−ρ q −ρ 1 − −ρ (p + q −ρ )2

uniformly for |t − tj | ≤ π/ log(p/q), where c0 = c0 (ρ) :=

2p−ρ q −ρ log(p/q)2 2 = 2 β2 (ρ). π 2 (p−ρ + q −ρ )2 π

We now take v0 :=

k −2/5 (c0 k)−2/5

if − 2 < ρ ≤ 1, if ρ ≥ 1,

and split the integration range into two parts: |t − tj | ≤ v0 and v0 < |t − tj | ≤ π/ log(p/q). (We assume that k is so large that v0 < π/ log(p/q).) First consider the case when −2 < ρ ≤ 1. From the inequality (46), it follows that

1 (47) Tj := Jk (n; ρ + it)dt 2π v0 ≤|t−tj |≤π/ log(p/q)

k ∞ 2 e−c0 kv dv = O |Γ(ρ + 2 + itj )|n−ρ p−ρ + q −ρ k−2/5 1/5 |Γ(ρ + 1 + itj )| if j = 0 k = O n−ρ p−ρ + q −ρ k −3/5 e−c0 k × 1 if j = 0 for each |j| ≤ j0 . When ρ ≥ 1 and satisﬁes (34), we have Tj

= O |Γ(ρ + 1 + itj )|n

−ρ

−ρ k p + q −ρ

∞

e (c0 k)−2/5

−c0 kv 2

dv

k 1/5 = O |Γ(ρ + 1 + itj )|n−ρ p−ρ + q −ρ (c0 k)−3/5 e−(c0 k) for |j| ≤ j0 .

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PROFILE OF TRIES

The dominant terms. It remains to evaluate the integrals Tj for t in the range |t − tj | ≤ v0 . Note that, by our choice of tj , p−ρ−itj + q −ρ−itj = p−itj p−ρ + q −ρ = q −itj p−ρ + q −ρ , so that i (t − tj ) p−ρ−itj log(1/p) + q −ρ−itj log(1/q) p−ρ−it + q −ρ−it · = 1 + −ρ−it −ρ−it j + q j p ! p−ρ−itj + q −ρ−itj ≥1

=1+

i (t − tj ) ≥1

!

·

p−ρ log(1/p) + q −ρ log(1/q) , p−ρ + q −ρ

where we recall that tj = 2πj/ log(p/q). It follows that β (ρ) i (t − tj ) , log p−ρ−it + q −ρ−it = log p−ρ−itj + q −ρ−itj + ! ≥1

where, in particular, β1 (ρ) =

p−ρ log(1/p) + q −ρ log(1/q) . p−ρ + q −ρ

The remaining manipulation by using the saddle-point method is then straightforward. We use the local expansions ⎛ ⎞ k −ρ−it β (ρ) p + q −ρ−it i (t − tj ) + O(k|β4 (ρ)||t − tj |4 )⎠ = exp ⎝k p−ρ−itj + q −ρ−itj ! 1≤≤3

and Γ(ρ + 1 + it)g(ρ + it) ⎧ (t − tj )2 ⎪ ⎪ C + C i(t − t ) + O if − 2 < ρ ≤ 1, 1 j ⎪ ⎨ 0 (ρ + 2)2 = Γ(ρ + 1 + itj )e(log ρ)i(t−tj ) 1 + C2 i(t − tj ) + O(|C2 |3 |t − tj |2 ) ⎪ ⎪ ⎪ ⎩ if ρ ≥ 1, × g(ρ + itj ) + g (ρ + itj )i(t − tj ) + O |t − tj |2 where

C0 := Γ(ρ + 1 + itj )g(ρ + itj ), C1 := g(ρ + itj )Γ(ρ + 1 + itj )ψ(ρ + 1 + itj ) + g (ρ + itj )Γ(ρ + 1 + itj ),

ψ(s) = Γ (s)/Γ(s) being the logarithmic derivative of the Gamma function, and C2 := ψ(ρ + 1 + itj ) − log ρ

(ρ ≥ 1).

Here C0 and C1 are deﬁned to be their limits when ρ = −1 and j = 0, namely, C0 := p log(1/p) + q log(1/q), C1 := − 2p−1 p log(p)2 − q log(q)2 − C0 γ − 2pq log(p) log(q). 2

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` G. PARK, H.-K. HWANG, P. NICODEME, AND W. SZPANKOWSKI

Note that ψ(ρ + 1 + itj ) − log ρ = O(log(1 + |tj |)). It follows that for |j| < j0 k g(ρ + itj ) Γ(ρ + 1 + itj )n−ρ−itj p−ρ + q −ρ p−iktj Tj = 2πβ2 (ρ)k 1 1 + × 1+O . kβ2 (ρ) k(ρ + 2)2 Summing over all |j| < j0 and collecting all estimates, we obtain −ρ −ρ −ρ k ˜ k (n) = n (p + q ) M g(ρ + itj )Γ(ρ + 1 + itj )(pk n)−itj 2πβ2 (ρ)k |j|<j0 1 1 × 1+O + . k(p/q)ρ k(ρ + 2)2

˜ k (z). To complete the de-Poissonization, we An asymptotic approximation to M ˜ need a more precise expansion of Mk (neiθ ) for small θ. The above proof by the saddle˜ k (z) for complex values point method can be easily extended mutatis mutandis to M of z lying in the right half-plane since we can write (7) as

k ˜ k (neiθ ) = 1 M n−s e−iθs Γ(s + 1)g(s) p−s + q −s ds, 2πi (ρ) where ρ > −2 and |θ| ≤ π/2 − ε. The result is −σ −σ k ˜ k (neiθ ) = (p + q ) M g(σ + itj )Γ(σ + 1 + itj )(neiθ )−ρ−itj p−iktj 2πβ2 (ρ)k |j|<j 0 1 1 × 1+O + (48) , k(p/q)ρ k(ρ + 2)2

uniformly for |θ| ≤ π/2 − ε and k lying in the range (34). Note that the index of the sum can be extended to inﬁnity, but it is easier to manipulate a ﬁnite sum than an inﬁnite series since we substitute the right-hand side into the Cauchy integral (13) and then integrate term by term. This completes the proof of (35). 3.5. Range (IV): A singularity analysis. We consider range (IV) ﬁrst, leaving to the next subsection the analysis in the transitional range when k = α2 Ln + 2/3 o(Ln ). √ We show that, for k ≥ α2 Ln + Kn Ln , the asymptotics of the expected proﬁle ˜ k (n) are dictated by the simple pole at s = −2 in (6) or, structurally, by the number M of pairs of input-strings sharing the same preﬁxes of length at least k. Theorem 3. If

(49) k ≥ α2 Ln + Kn α2 β2 (−2)Ln , where β2 is deﬁned in (37), then (50)

2 3 √ μn,k = 2pqn2 (p2 + q 2 )k−1 1 + O Kn−1 e−Kn /2+O(Kn / Ln ) ,

√ uniformly for 1 Kn = o( Ln ).

PROFILE OF TRIES

1847

Proof. To prove (50), we move the line of integration (by absolute convergence of the integral) of the integral in (6) to (s) = , where Kn := −2 − . α2 β2 (−2)Ln ˜ k (neiθ ) equals the residue of the integrand at s = −2 (the dominant term in Thus M (50)) plus the integral along (s) = :

∞ ˜ k (neiθ ) = |g(−2)|n2 e2iθ (p2 + q 2 )k + 1 M Jk (neiθ ; + it)dt, 2π −∞ where |g(−2)| = 2pq/(p2 + q 2 ). It remains only to estimate the last integral. By the same analysis used for Tj (see (47)) and the inequality (46), we have ⎛ ⎞

1 ⎝ ⎠ Jk (neiθ ; + it)dt + 2π |t|≤π/ log(p/q) |t−t |≤π/ log(p/q) j |j|≥1

− − − k −c0 kt2 = O |Γ( + 1)|n p +q e dt |t|≤π/ log(p/q)

⎛

k Γ + 1 + 2|j| − 1 πi e(2|j|+1)π|θ|/ log(p/q) + O ⎝n− p− + q − log(p/q) |j|≥1

2

e−c0 k(t−tj ) dt

×

|t−tj |≤π/ log(p/q)

k k =O n− p− + q − | + 2|

k , = O Kn−1 n− p− + q − −1/2

where we used (9) to bound the sum Γ + 1 + 2|j| − 1 πi e(2|j|+1)π|θ|/ log(p/q) log(p/q) |j|≥1 ⎛ ⎞ 2 π (2|j| − 1) (2|j| + 1)π|θ| ⎠ + = O⎝ (2|j| − 1)+1/2 exp − 2 log(p/q) log(p/q) |j|≥1

= O(1), uniformly for |θ| ≤ π/2 − ε. By our choice of and by straightforward expansion, we have k

Kn−1 n− (p− + q − ) k

n2 (p2 + q 2 )

Thus (51)

2 3 k k = O Kn−1 e−Ln (+2)+ α2 (+2)+ 2 β2 (−2)(+2) +O(k|+2| )

2 3 √ = O Kn−1 e−Kn /2+O(Kn / Ln ) .

√ ˜ k (neiθ ) = |g(−2)|(neiθ )2 (p2 + q 2 )k 1 + O K −1 e−Kn2 /2+O(Kn3 / Ln ) , M n

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` G. PARK, H.-K. HWANG, P. NICODEME, AND W. SZPANKOWSKI

uniformly for |θ| ≤ π/2 − ε. Substituting this into (25), we deduce the desired result (50). √ √ Remarks. (i) When Kn ≥ ε Ln , we can either take Kn = ε Ln or reﬁne the analysis to give a better error term. (ii) The asymptotic approximation (50) can also be derived from the exact expression (10) by using only elementary arguments. (iii) Also √ note that the range (49) implies that the saddle-point ρ satisﬁes ρ ≤ −2−Kn / Ln , but the contribution from this saddle-point is asymptotically negligible (compared to the polar singularity). 3.6. Range (III): A uniform analysis. We consider in this subsection the 2/3 transitional range k = α2 Ln + o(Ln ) and show that the transitional behavior of μn,k in this range is well described by a Gaussian distribution function. Theorem 4. If

(52) k = α2 Ln + ξ α2 β2 (−2)Ln , 1/6

where ξ = ξn,k = o(Ln ), then

k 1 + |ξ|3 √ μn,k = |g(−2)|Φ(ξ)n2 p2 + q 2 1+O , Ln ξ 2 uniformly in ξ, where Φ(ξ) = (2π)−1/2 −∞ e−t /2 dt. Proof. We assume ﬁrst that k satisﬁes (52) and k < α2 L (or ξ < 0). We move the line of integration of the integral in (11) to (s) = ρ, where ρ is taken to be of the same form as in (40); asymptotically (53)

ξ ρ = −2 − + O ξ 2 L−1 . n α2 β2 (−2)Ln

(54)

By a similar analysis as the proof of Theorem 3, we obtain

˜ k (neiθ ) = 1 M Jk (neiθ ; ρ + it)dt 2π |t|≤L−2/5 n

k 1/5 )|n−ρ p−ρ + q −ρ e−c0 Ln + O |Γ(ρ + 1 + iL−2/5 n

k , + O k −1/2 n−ρ p−ρ + q −ρ where |θ| < π/2. By (54), we have |Γ(ρ + 1 + It follows that ˜ k (neiθ ) = 1 M 2π

iL−2/5 )| n

−2/5

|t|≤Ln

=O

1 −1/2 |ξ|Ln

+

−2/5 Ln

= O(L2/5 n ).

k . Jk (neiθ ; ρ + it)dt + O k −1/2 n−ρ p−ρ + q −ρ

Note that since s → Γ(s + 1) + 1/(s + 2) is analytic for |s + 2| ≤ 1 − ε, we have

k n−ρ−it e−iθ(ρ+it) −ρ−it ˜ k (neiθ ) = |g(−2)| M p + q −ρ−it dt −2/5 2π ρ + 2 + it |t|≤Ln

−ρ k −1/2 −ρ . p + q −ρ n +O k

PROFILE OF TRIES

1849

The integral on the right-hand side is evaluated as follows:

k n−ρ−it e−iθ(ρ+it) −ρ−it + q −ρ−it dt p −2/5 ρ + 2 + it |t|≤Ln

2 3 k |g(−2)| −ρ −iθρ −ρ eθt−β2 (ρ)kt /2+O(k|t| ) = n e dt p + q −ρ −2/5 2π ρ + 2 + it |t|≤Ln

∞ −t2 /2 |g(−2)| −ρ −iθρ −ρ |t| + |t|3 e −ρ k √ = n e p +q dt, 1+O 2π Ln −∞ ξ0 + it

|g(−2)| 2π

(55) where

ξ0 := (ρ + 2) β2 (ρ)k > 0.

−1/2 by (52) and (54). Since ξ0 > 0, we have Note that ξ0 = −ξ + O ξ 2 Ln 1 2π

∞

−∞

∞

∞ 2 1 e−t /2 −t2 /2 dt = e e−v(ξ0 +it) dvdt ξ0 + it 2π −∞ 0

∞

∞ 2 1 e−vξ0 e−t /2−itv dtdv = 2π 0 −∞

∞ 2 1 e−v /2−vξ0 dv = √ 2π 0 2

= eξ0 /2 Φ(−ξ0 ). The error term in (55) is estimated similarly and satisﬁes L−1/2 n

∞ 2 (|t| + |t|3 )e−t /2 −1/2 3 −v 2 /2−vξ0 dt = O Ln (v + v )e dv . |(ρ + 2) β2 (ρ)k + it| 0

∞

−∞

Observe that e

(56)

x2 /2

Φ(−x) =

O x−1 if x → ∞, x2 /2 O e if x → −∞.

Also

∞

(v + v 3 )e−v

2

/2−vx

0

dv =

O x−2 if x → ∞, 3 x2 /2 O |x| e if x → −∞,

so that

∞

v 3 e−v

2

/2−vx

2 dv = O ex /2 Φ(−x)(1 + |x|3 ) .

0

Thus + |ξ|3 ˜ k (neiθ ) = |g(−2)|(neiθ )−ρ p−ρ + q −ρ k eξ02 /2 Φ(−ξ0 ) 1 + O 1 √ M Ln

−ρ −1/2 −ρ −ρ k (57) , +O k p +q n

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` G. PARK, H.-K. HWANG, P. NICODEME, AND W. SZPANKOWSKI

uniformly for |θ| ≤ π/2 − ε. Substituting this in (25) and using the expansions k k 2 3 −1/2 n−ρ p−ρ + q −ρ = n2 p2 + q 2 e−ξ /2+O(|ξ| Ln ) ,

2 2 , eξ0 /2 Φ(ξ0 ) = eξ /2 Φ(ξ) 1 + O |ξ|3 L−1/2 n we deduce (53) when ξ < 0. The restriction that ξ < 0 can now be removed by continuity (when ξ0 = 0 the integral path has to be properly indented) or by a similar analysis. This proves (53). One can easily check, by (56), that the asymptotic estimate (53) coincides with the two estimates (44) and (50) when ξ → −∞ and ξ → ∞, respectively. Remark. The appearance of the normal distribution function is typical when a saddle-point coalesces with a simple pole; see [83]. Also, the polynomial order (4) of μn,k now follows from (35), (50), and (53). 3.7. The range where the expected profile grows unbounded. An important consequence of the preceding results is the following characterization of the range where μn,k → ∞, which also will be seen to be the range where Bn,k is asymptotically normally distributed. Theorem 5. Deﬁne 1 2 . m0 := and α3 := 1 p/q − 1 log p2 +q 2 Then μn,k → ∞ iﬀ p LLLn − Kn α1 Ln − LLLn − log m0 + m0 log ≤ k ≤ α3 (Ln − Kn ) − q m0 LLn as n → ∞. Proof. Consider ﬁrst the upper bound. If k ≤ α3 Ln − x, then n2 (p2 + q 2 )k ≥ (p2 + q 2 )−x , which tends to inﬁnity if x → ∞; on the other hand, if k ≥ α3 Ln − x, then the reverse inequality holds and the right-hand side remains bounded if x is less than a positive constant. For the lower bound, we use the estimate (24). First, if k ≤ k0 = α − 1 (Ln − LLLn + log(p/q − 1)) (see (20)), then μn,k = Θ(q k ne−q

k

n

) = o(1).

Next, if km−1 ≤ k ≤ km , m ≥ 1, then by (24) η

μn,k = Θ(Sn,k,m ) = Θ(Lm−e n

/(p/q−1)

LLn ),

where k is written in the form (31). Since η ∈ [0, log(p/q)], we have m−

p eη q ≤m− ≤m− . p−q p/q − 1 p−q

Also, by the deﬁnition of m0 , we have the inequalities m0 − 1
m − m0 − 1 ≥ 0, p/q − 1 p−q

when m ≥ m0 + 1. Therefore, μn,k → ∞ if k ≥ km0 (and remains in the range k ≤ α1 (Ln − LLLn + Kn )). m −eη /(p/q−1) The remaining range is km0 −1 ≤ k ≤ km0 in which μn,k = Θ(Ln 0 LLn ), where k = α1 (Ln − LLLn + log(p/q − 1) + m0 log(p/q) − η). We distinguish three cases: (i) if η ≥ log m0 + log(p/q − 1) + m−eη /(p/q−1)

then μn,k = Θ(Ln

LLLn + Kn , m0 LLn

LLn ) and η

Lm−e n

/(p/q−1)

LLn ≤ e−Kn → 0;

(ii) if η = log m0 + log(p/q − 1) +

LLLn + x , m0 LLn

then μn,k ∼ Sn,k,m0 ∼

0 αm 1 e−x , (m0 − 1)!

uniformly for x = O(1); and (iii) if η ≤ log m0 + log(p/q − 1) + m−eη /(p/q−1)

then μn,k = Θ(Ln

LLLn − Kn , m0 LLn

LLn ) and η

Lm−e n

/(p/q−1)

LLn ≥ eKn → ∞.

Thus μn,k is bounded away from zero and inﬁnity in the second case. This proves the theorem when k lies in ranges (I) and (IV). The remaining cases follow easily from (35) and (53). Let {x} denote the fractional part of x. The lower bound can be further reﬁned as follows.

` G. PARK, H.-K. HWANG, P. NICODEME, AND W. SZPANKOWSKI

1852

Corollary 3. Let LLLn (58) , kˆ := α1 Ln − LLLn − log m0 + m0 log(p/q) − m0 LLn ˆ ≤ k ≤ α3 (Ln − Kn ); (ii) where m0 = 1/(p/q − 1). Then (i) μn,k → ∞ for k ˆ − 2; and (iii) μn,k → 0 for k ≤ k ˆ = O(LL−1 ), = Θ(1) if {k} n μn,k −1 ˆ →0 otherwise. Proof. The proof is similar to that of Theorem 5. We consider only the last case. First write ˆ − 1 = kˆ − {k} ˆ = α1 (Ln − LLLn + log(p/q − 1) + m0 log(p/q) − η ) , k where η = log m0 + log

p LLLn ˆ − 1 + {k}/α . 1+ q m0 LLn

(We assume that kˆ is not an integer.) Then we follow the same proof as above by distinguishing three cases. In particular, the case when kˆ is an integer is also covered by the bounded case. The result is to be compared with Pittel’s result in [65], which says that the probability that the shortest path equals either κn or κn + 1 tends to 1, where x denotes the nearest integer to x and j κn = α1 Ln − LLLn − log max j(q/p) . j≥1

Note that − log max j(q/p)j = − log m0 + m0 log(p/q). j≥1

Our result is slightly more precise; see section 8. 3.8. Refinement of μn,k by de-Poissonization. All expansions for μn,k that −1 we have derived so far are in terms of slowly decreasing powers of L−1 n or LLn , which will turn out to be insuﬃcient for the asymptotics of the variance because of cancellation of dominant terms. Thus in this section we derive a more eﬀective ˜ k (n) and its higher derivatives; namely, we derive an expansion for μn,k in terms of M expression of the form (19). The major diﬀerence here is that we do not substitute ˜ k (n) into the Cauchy integral representation for μn,k , the asymptotic expansions for M resulting in a less explicit asymptotic approximation to μn,k but with a much better error term. We start with a lemma in which we again use k0 = α1 (Ln − LLLn + log(p/q − 1)). Lemma 4. Deﬁne ⎧ k ⎨ q n if 1 ≤ k ≤ k0 , ρ0 := ρ if ρ ≥ 1 and k ≥ k0 , (59) ⎩ 1 if ρ ≤ 1,

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PROFILE OF TRIES

where ρ is given by (36). Then

˜ () (neiθ ) = O ρ0 n− M ˜ k (n) , M k

(60)

−1/2

uniformly for θ = o(LLn ). Proof. If ≥ 1, then, by (8), k − 1 ˜ () (z) ˜ () (z) = (pj q k−1−j ) M M 1 k j 0≤j 0 satisﬁes the saddle-point equation (36), β2 (ρ) is the same as in (37), and (ρ + 1 + itj )Γ(ρ + itj )e−2jπix , G3 (ρ; x) = j∈Z

where tj := 2jπ/ log(p/q).

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PROFILE OF TRIES 2/3

Asymptotics of E(In,k ) when k = α0 (Ln + o(Ln )). In this range, we write k = α0 (Ln + ξ α0 β2 (0)Ln ), 1/6

where α0 β2 (0) = 2(log(1/p)+log(1/q))/ log(p/q)2 and ξ = o(Ln ). The same uniform asymptotic analysis we used for proving (53) gives 1 + |ξ|3 √ E(In,k ) = 2k Φ(−ξ) 1 + O , Ln uniformly in ξ, where Φ(x) denotes the standard √ normal distribution function. √ Asymptotics of E(In,k ) when α0 (Ln + Kn Ln ) ≤ k ≤ α2 (Ln − Kn Ln ). The same saddle-point method and de-Poissonization procedure yield (87) E(In,k ) = G3

n−ρ (p−ρ + q −ρ )k 1 1 ρ; logp/q p n + 1+O , k(p/q)ρ k(ρ + 2)2 2πβ2 (ρ)k k

with ρ, β2 (ρ), and G3 as deﬁned above. 2/3 Asymptotics of E(In,k ) when k = α2 (Ln + o(Ln )). In this case, we write k = α0 (Ln + ξ α2 β2 (−2)Ln ), and we have E(In,k ) =

1 + |ξ|3 1 √ Φ(ξ)n2 (p2 + q 2 )k 1 + O , 2 Ln

1/6

uniformly for ξ = o(Ln ). √ Asymptotics of E(In,k ) when k ≥ α2 (Ln + Kn Ln ). In this case, the simple pole at s = −2 in the integrand of (85) dominates, and we have

2 3 −1/2 1 E(In,k ) = n2 (p2 + q 2 )k 1 + O Kn−1 e−Kn /2+O(Kn Ln ) 2 as n → ∞. 6.2. Asymptotics of V(In,k ). Since V(In,k ) = V(I¯n,k ), we can apply the same analysis used for proving Theorem 7 to derive asymptotic approximations to V(In,k ). The auxiliary function we need is ⎞2 ⎛ 2 E(I¯n,k E(I¯n,k ) ) [I] V˜k (z) := e−z z n − ⎝e−z zn⎠ , n! n! n≥0

which satisﬁes (88)

[I] V˜k (z) =

n≥0

k [I] V˜ (pj q k−j z) j 0

(k ≥ 0),

0≤j≤k

[I] where V˜0 (z) = (1 + z)e−z (1 − (1 + z)e−z ). Thus we have

1 [I] ˜ Vk (z) = z −s (s + 1)Γ(s) 1 − 2−s − s2−s−2 (p−s + q −s )k ds, 2πi (ρ)

where ρ > −2.

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` G. PARK, H.-K. HWANG, P. NICODEME, AND W. SZPANKOWSKI

Asymptotics of V(In,k ) when 1 ≤ k ≤ α1 (1 + o(1))Ln . In this range, we have V(In,k ) ∼ V(Bn,k ) ∼ E(Bn,k ).

√ Asymptotics of V(In,k ) when α1 (Ln − LLLn + Kn ) ≤ k ≤ α2 (Ln − Kn Ln ). We have

n−ρ (p−ρ + q −ρ )k 1 1 k + 1+O , V(In,k ) = G4 ρ; logp/q p n k(p/q)ρ k(ρ + 2)2 2πβ2 (ρ)k where ρ = ρ(n, k) > −2 satisﬁes the saddle-point equation (36) and G4 (ρ; x) = (ρ + 1 + itj )Γ(ρ + itj ) 1 − 2−ρ−itj − (ρ + itj )2−ρ−2−itj e−2jπix . j∈Z

√ Asymptotics of V(In,k ) when k ≥ α2 (Ln + Kn Ln ). In this case, the simple pole at s = −2 again dominates, and we have V(In,k ) ∼ E(In,k ). Observe that, unlike for the external proﬁle, the variance of the internal proﬁle is asymptotically equivalent to the mean of the internal proﬁle near the height of a trie. From these asymptotic estimates and from Chebyshev’s inequality, we see that In,k /E(In,k ) → 1 in probability if E(In,k ) → ∞; see [15]. 6.3. Limiting distributions. The same limiting Gaussian–Poisson behavior for Bn,k holds for In,k . We state formally our main result for the internal proﬁle in the following theorem. The proof is indeed simpler than that for Theorem 8 since the [I] base function P0 (z, y) has a simpler form than P0 (z, y). Theorem 9. (i) If V(In,k ) → ∞, then In,k − E(In,k ) d −→ N (0, 1). V(In,k ) (ii) If V(In,k ) = Θ(1), then, with λ1 := n2 (p2 + q 2 )/2, (89)

P(In,k = m) =

λm 1 e−λ1 + o(1) m!

for all m ≥ 0. The theorem states that asymptotic normality (in the sense of convergence in distribution) holds as long as ˆ ≤ k ≤ α3 Ln − Kn k for any sequence Kn → ∞, where kˆ is deﬁned in (58). On the other hand, In,k is asymptotically Poisson distributed when k = α3 Ln + O(1). A result related to (89) was given in [65] by a method of moments, as a key step in deriving the asymptotic distribution of the height. 7. Profiles under the unbiased Bernoulli model. All exact expressions we have derived up to now, as well as most asymptotic approximations, also hold when p = q = 1/2. The major diﬀerence is reﬂected by the fact that α1 = α2 (see Figure 2), so that the saddle-point range between α1 and α2 does not exist, and most of the analysis we give above becomes much simpler. For simplicity of presentation, we omit all error terms in our asymptotic estimates.

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Expected external proﬁle. By (8), the Poisson generating function of E(Bn,k ) is given exactly by

˜ k (z) = z e−z/2k − e−z/2k−1 (90) (k ≥ 1). M From this we deduce, by our de-Poissonization procedures, that n−1 if 2−k n → ∞, n 1 − 2−k E(Bn,k ) ∼ (91) ˜ Mk (n) if 4−k n → 0, where the condition 4−k n → 0 is due to the properties that

˜ () (z) = O 2−k |M ˜ k (z)| M (| arg(z)| ≤ π/2 − ε) k and 2−k = o(n−1/2 ); see Proposition 1 and compare with (61). In particular, if 2−k n → t ∈ (0, ∞), ne−t 1 − e−t E(Bn,k ) ∼ −k 2 2 n if 2−k n → 0. Note that these approximations can also be easily derived by the exact formula n−1 n−1 E(Bn,k ) = n 1 − 2−k (92) − n 1 − 21−k , by (90) or (10). But such an elementary approach becomes messier for the calculation of the variance. Also n max E(Bn,k ) ∼ , k 4 which is reached when k ∼ log2 n − 1. Expected internal proﬁle. In a similar manner, we have, by (84), ˜ [I] (z) = 2k − (2k + z)e−z/2k M k Therefore, the expected internal proﬁles satisfy n−1 2k − n 1 − 2−k E(In,k ) ∼ ˜ [I] (n) M k

(k ≥ 0).

if 2−k n → ∞, if 4−k n → 0.

Consequently, E(In,k ) ∼ Note that

if 2−k n → t ∈ (0, ∞), 2k 1 − (1 + t)e−t −k−1 2 2 n if 2−k n → 0.

n n−1 − n 1 − 2−k E(In,k ) = 2k 1 − 1 − 2−k

and (93)

max E(In,k ) ∼ c3 n, k

where c3 ≈ 0.298 denotes the maximum value achieved by the function (1 − (1 + x)e−x )/x for x ∈ R+ .

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Asymptotics of the variances. Similarly, by (63) and (88), we have

k k−1 k−1 k k−1 2 + 2−k z 2 e−z/2 V˜k (z) = z e−z/2 − e−z/2 − 21−k z 2 e−z/2 − e−z/2 , k k−1 2 [I] V˜k (z) = (2k + z)e−z/2 − 2k 1 + 2−k e−z/2 ; accordingly, if n/2k → ∞, then n−1 V(Bn,k ) ∼ V(In,k ) ∼ E(Bn,k ) ∼ n 1 − 2−k , and if n/4k → 0, then V(Bn,k ) ∼ V˜k (n)

[I] and V(In,k ) ∼ V˜k (n),

uniformly in k. These approximations imply that ne−t 1 − (1 + t)e−t + 2te−2t (2 − e−t ) if 2−k n → t ∈ (0, ∞), (94) V(Bn,k ) ∼ if 2−k n → 0 2E(Bn,k ) ∼ 21−k n2 and

V(In,k ) ∼

2k (1 + t)e−t 1 − (1 + t)e−t if 2−k n → t ∈ (0, ∞), −k−1 2 n if 2−k n → 0. E(In,k ) ∼ 2

Limiting distributions. Both Theorems 8 and 9 (asymptotic normality of Bn,k and In,k , respectively) hold when p = q = 1/2 by the same method of proof. Note that both bivariate generating functions become simpler (see (72) and (83)): k−1 2 2 z z/2k z/2k−1 2z Pk (z, y) = e + (y − 1) k−1 e − 1 + (y − 1) k , 2 4 k

k z 2 [I] Pk (z, y) = yez/2 + (1 − y) 1 + k . 2 Observe that, as n → ∞, E(Bn,k ) → ∞ iﬀ V(Bn,k ) → ∞ iﬀ V(In,k ) → ∞ iﬀ 1 Kn 2 (Ln − Kn ) (95) ≤k≤ Ln − LLn + log 2 Ln log 2 for any sequence Kn → ∞ with n; compare Theorem 5 for the asymmetric case. Theorem 10. (i) If k lies in the range (95), then Bn,k − E(Bn,k ) d −→ N (0, 1), V(Bn,k )

In,k − E(In,k ) d −→ N (0, 1). V(In,k )

(ii) If k = 2(Ln + O(1))/ log 2, then, with λ2 := 2−k−1 n2 , ⎧ λm ⎪ 2 ⎪ e−λ2 + o(1), P(Bn,k = 2m + 1) = o(1), ⎨ P(Bn,k = 2m) = m! ⎪ λm ⎪ ⎩ P(In,k = m) = 2 e−λ2 + o(1), m! uniformly for m ≥ 0. Note that when p = q, λ0 = λ1 = λ2 .

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8. Applications of results. In this section, we brieﬂy discuss a few properties of some shape characteristics of random tries, as implied either by our results or by our approaches. We consider only depth, height, shortest path, ﬁll-up level, width, and right-proﬁle. Depth. The distribution of the depth Dn is given by P(Dn = k) = μn,k /n. Our asymptotic approximations for μn,k give very precise results for the distribution of Dn . First consider the case when p = q. By deﬁnition, we see that the result (4) for the limiting behaviors of log μn,k / log n also describes those of −1 + log P(Dn = k)/ log n, or essentially the large deviations of the distribution of Dn . Furthermore, (43) can be regarded as a local limit theorem for Dn . Indeed, we have, for k = h−1 (Ln + x h−1 β2 (−1)Ln ), where h := p log(1/p) + q log(1/q) is the entropy rate, (96)

P(Dn = k) = G1

1 + |x|3 √ −1; logp/q p n 1+O , Ln 2πV(Dn ) k

e−x

2

/2

1/6

uniformly for x = o(Ln ), where V(Dn ) ∼ (h2 − h2 )/h3 log n, with h2 := p log2 p + q log2 q (see [33, 76]), is also rederived below in (97). Because of the appearance of the uncommon periodic function G1 , we see that Dn satisﬁes a central limit theorem but not a local limit theorem (of the usual form). It can be shown that the right-hand side indeed sums (over all k) asymptotically to 1. The result (96) is new. If p = q, then, by the exact formula (92), we have n−1 n−1 − 1 − 21−k , P(Dn = k) = 1 − 2−k which implies that

−+{log n} 1−+{log2 n} 2 P(Dn = log2 n + ) = e−2 − e−2 1 + O n−1 2− , uniformly for ∈ Z, where {x} denotes the fractional part of x. On the other hand, if one is interested in the cumulative distribution functions or tail probabilities, then, by (6) and by partial summation, ez P(Dn ≤ k) = (n − 1)![z ] 2πi

n

k z −s Γ(s + 1) p−s + q −s ds

(ρ)

for k ≥ 1, where ρ > −1. Equivalently, by (11), we have (see [35]) P(Dn ≤ k) =

1 2πi

(ρ)

k Γ(n)Γ(s + 1) −s p + q −s ds, Γ(n + 1 + s)

where ρ > −1. Asymptotics of such integrals can be treated by our approaches, which give not only the central limit theorem of Dn with convergence rate (since there is a simple pole at s = −1) but also precise estimates for tail probabilities. Indeed, we have

1 + |x|3 −1 −1 √ P Dn ≤ h (Ln + x h β2 (−1)Ln ) = Φ(x) 1 + O , Ln

` G. PARK, H.-K. HWANG, P. NICODEME, AND W. SZPANKOWSKI

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uniformly for x = o(Ln ), as already shown in [33, 35] (but without rate). Furthermore, log P(Dn ≤ αLn ) → ρ + 1 − α log(p−ρ + q −ρ ) (α1 ≤ α ≤ h−1 ), log n log P(Dn ≥ αLn ) ρ + 1 − α log(p−ρ + q −ρ ) if h−1 ≤ α ≤ α2 , − → −1 − α log(p2 + q 2 ) if α2 ≤ α ≤ α3 , log n

−

both tails being asymptotic to −1 for smaller and larger α, respectively, where ρ is given in (42). These results imply, in particular, that E(Dn ) ∼ Ln /h and (97)

V(Dn ) ∼ β2 (−1)h−3 Ln =

pq log2 (p/q) h2 − h2 L = Ln , n (p log(1/p) + q log(1/q))3 h3

where h2 := p log2 p + q log2 q; see [13, 76]. Note that the constant on the right-hand side becomes zero when p = q. Width. The width of tries Wn is deﬁned to be Wn := maxk In,k , or the size of the most abundant level(s). As a natural lower bound for E(Wn ), we consider maxk E(In,k ). By (87) and a similar analysis for (43), we have, when p = q,

√ h G3 −1; logp/q pk n n

√ 1 + O(L−1/2 E(In,k ) = ) , ×√ n log(p/q) 2πpq Ln uniformly for k = Ln /h + O(1). This approximation, together with the estimates for E(In,k ) in other ranges given in section 6.1, yields E(Wn ) ≥ max E(In,k ) = Θ(nL−1/2 ), n k

when p = q. Indeed, we have E(Wn ) = Θ(nL−1/2 ). n The upper bound can be proved by applying the arguments used in [16], which start from the inequality E(Wn ) ≤ Mn +

2/3 |k−Ln /h|≤εLn

V(In,k ) + Mn − E(In,k )

E(In,k ),

2/3 |k−Ln /h|>εLn

where Mn : maxk E(In,k ), and then using the asymptotics of E(In,k ) and V(In,k ) given in section 6.1. Details are omitted here. Finer results for E(Wn ) can be derived, but the proof is more involved due to the presence of the periodic function G3 (whose parameter involves k). For symmetric tries, we easily have E(Wn ) = Θ(n), by (93) and the trivial bound E(Wn ) ≤ n. Thus random symmetric tries are “fatter,” and most nodes lie near the most abundant levels k = log2 n + O(1). Height. We next derive an estimate for the height of random tries, as a consequence of our estimates for the external proﬁles together with the use of the ﬁrst and second moment methods (see [79]). Corollary 6 (height of a trie). Let Hn := max{k : Bn,k > 0} be the height of a random trie. Then Hn / log n → α3 in probability.

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Proof. Let kH := α3 Ln . First we derive an upper bound for Hn as follows: P(Hn > (1 + ε)kH ) ≤ P(Bn,k ≥ 1) (for some k ≥ (1 + ε)kH ) ≤ E(Bn,k ) → 0, where the last inequality follows from Theorem 3 when p = q and (91) when p = q. For the lower bound, we use the second moment method (see [79]) to ﬁnd P(Hn < (1 − ε)kH ) ≤ P(Bn,(1−ε)kH = 0) V(Bn,(1−ε)kH ) (E(Bn,(1−ε)kH ))2 1 =O →0 E(Bn,(1−ε)kH )

≤

by Theorems 3 and 7 and (94). Combining the two estimates, we obtain the required result. Corollary 6 is not new and has already been derived in Devroye [12], Pittel [64, 65], and Szpankowski [77]. Shortest path. The shortest path Sn := min{j : Bn,j > 0} of a random trie, discussed next, has attracted much less attention than the height (see [79]) in the literature. It is closely related to the behaviors of the external proﬁle in range (I) near k = α1 (Ln − LLLn + O(1)) as discussed in Theorem 1 and its reﬁnement in Corollary 3. Deﬁne ⎧ LLLn ⎨ α − LLL − log m + m log(p/q) − L if p = q, 1 n n 0 0 kˆ := m0 LLn ⎩ if p = q, α1 (Ln − LLn ) where m0 := 1/(p/q − 1), and kS :=

ˆ k ˆ k

if p = q, if p = q.

Corollary 7 (shortest path length of tries). If p = q, then ˆ kS if {k}LL n → ∞, Sn = ˆ kS or kS − 1 if {k}LLn = O(1) with high probability;2 if p = q = 1/2, then kS + 1 Sn = kS or kS + 1

ˆ n → ∞, if {k}L ˆ n = O(1) if {k}L

with high probability. ˆ Proof. Assume p = q. Consider ﬁrst the case {k}LL n → ∞. In this case we have, by Corollary 3, μn,kS → ∞, μn,k → 0 for k ≤ kS − 1. 2 We say that an event holds with high probability if it holds with probability tending to 1 as n → ∞.

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Thus, again by the second moment method, P(Sn > kS ) ≤ P(Bn,kS

V(Bn,kS ) = 0) ≤ =O (E(Bn,kS ))2

1

μn,kS

→ 0.

On the other hand, by using the ﬁrst moment method, we have P(Sn < kS ) ≤ P(Bn,k ≥ 1) (for some k < kS ) ≤ μn,k → 0. These two estimates imply that P(Sn = kS ) → 1. ˆ Now if {k}LL n = O(1), then, again by Corollary 3, ⎧ ⎨ μn,kS → ∞, μn,kS −1 = Θ(1), ⎩ μn,k → 0 for k ≤ kS − 2. Thus applying mutatis mutandis the same proof gives P(Sn = kS ) + P(Sn = kS − 1) → 1. The proof for the symmetric case is similar, because μn,k → ∞ when k lies in the range (95), and from this result we deduce that μn,kS +1 → ∞, μn,kS −1 → 0, and μn,kS

ˆ n → ∞, →0 if {k}L ˆ n = O(1). = Θ(1) if {k}L

This completes the proof. Fill-up level. We now consider the ﬁll-up level Fn = max{k : In,k = 2k } of a random trie, which was also analyzed previously by Devroye [12], Pittel [64, 65], and Knessl and Szpankowski [49]. Corollary 8 (ﬁll-up level of a trie). If p = q, then ˆ kS − 1 if {k}LL n → ∞, Fn = ˆ kS − 2 or kS − 1 if {k}LLn = O(1) with high probability; if p = q = 1/2, then kS Fn = kS or kS − 1

if if

ˆ n → ∞, {k}L ˆ {k}Ln = O(1).

Proof. Observe that Fn = max{k : I¯n,k = 0} = min{k : I¯n,k > 0} − 1. By (86), we have E(I¯n,k ) ∼ μn,k when k ≤ α1 (1 + o(1))Ln . Thus the proof of Corollary 7 applies with little modiﬁcation. Proﬁle enumerating only right branches. We consider the random variable Rn,k , which denotes the number of external nodes in random tries that are away from the root by k right branches. Since a right branch means a “1” in the input-string, Rn,k enumerates the number of strings with exactly k 1’s; it also has other concrete interpretations in splitting processes and conﬂict resolution algorithms. All of our

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tools can be extended to Rn,k , although Rn,k exhibits very diﬀerent behaviors. For example, unlike Bn,k or In,k , there is no need to distinguish between symmetric and asymmetric tries, all results being uniform in p; also, the Poisson heuristic holds for all k ≥ 0. This example further reveals the power of our approaches. The probability generating function Fn,k (y) := E(y Rn,k ) of Rn,k satisﬁes the recurrence n pj q n−j Fj,k−1 (y)Fn−j,k (y) (n ≥ 2; k ≥ 0), Fn,k (y) = j 0≤j≤n

with the initial conditions Fn,k (y) = 1 forn ≤ 1 or k < 0 and F2,1 (y) = y. Thus the bivariate generating function Fk (z, y) := n Fn,k (y)z n /n! satisﬁes k+j−1 F0 (pk q j z, y)( j ) , Fk (z, y) = Fk (qz, y)Fk−1 (pz, y) = j≥0

where F0 (z, y) = epz F0 (qz, y) + p(1 − p/2)(y − 1)z 2 , which is further solved to be (98)

F0 (z, y) = ez + p(1 − p/2)(y − 1)

(q j z)2 e(1−q

j

)z

.

j≥0

From this we deduce that the expected right-proﬁle is given by

ez p−ks E(Rn,k ) = p(1 − p/2)n![z n] z −s Γ(s + 2) ds, 2πi (ρ) (1 − q −s )k+1 where −2 < ρ < 0. The integral is not of the same type as (6) but is similar, and our methods of proof easily extend. It has simple poles at s = −2, −3, . . . and poles of order k + 1 at s = 2jπi/ log(1/q), j ∈ Z. Thus the asymptotics of E(Rn,k ) are divided into four overlapping ranges. • If 0 ≤ k = o(log n), then the residues of the poles on the imaginary lines are dominant, and we have ⎛ ⎞ (log pk n)k ⎝1 + E(Rn,k ) ∼ p(1 − p/2) Γ(1 + χj )(pk n)−χj ⎠ , k!(log(1/q))k+1 j=0

uniformly in k, where χj := 2jπi/ log(1/q). √ • If k → ∞ and k ≤ α∗ (Ln − Kn Ln ), where Kn → ∞ and α∗ :=

1 − q2 , (1 − q 2 ) log(1/p) − q 2 log(1/q)

then by the saddle-point method p(1 − p/2)q ρ/2 k −ρ (p n) (1 − q −ρ )−k E(Rn,k ) ∼ √ Γ(ρ + 1 + χj )(pk n)−χj , 2πk log(1/q) j∈Z uniformly in k, where ρ = log1/q

log(pk n) . log(pk n/q k )

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` G. PARK, H.-K. HWANG, P. NICODEME, AND W. SZPANKOWSKI

• If k = α∗ Ln + x α∗ (1 + α∗ log(p/q))(1 + α∗ log p)Ln , then E(Rn,k ) ∼

1 Φ(x)(pk n)2 (1 − q 2 )−k , 2

1/6

uniformly for x = o(Ln ). • If k ≥ α∗ Ln + Kn α∗ (1 + α∗ log(p/q))(1 + α log p)Ln , then E(Rn,k ) ∼

1 k 2 (p n) (1 − q 2 )−k . 2

These results imply that, as n → ∞, E(Rn,k ) → ∞ iﬀ 1≤k≤

2 Ln − Kn , log 2−p p

where Kn → ∞ with n. Note that log e−z F0 (z, y) = log(1 + (y − 1)τ (z)), j where τ (z) := p(1 − p/2) j≥0 (q j z)2 e−q z satisﬁes τ (z) = O(|z|2 ) as z → 0, and, by Mellin transform, τ (z) = O(1) as |z| → ∞ in a small sector containing the real axis. This yields, by a straightforward modiﬁcation of our approaches, that V(Rn,k ) = Θ(E(Rn,k )) for all k = k(n) ≥ 0 and that Rn,k − E(Rn,k ) d −→ N (0, 1) V(Rn,k ) whenever E(Rn,k ) or V(Rn,k ) → ∞. Two remaining cases are k = 0 and k = 2Ln / log 2−p p + O(1). In the ﬁrst case, Rn,0 by (98) is Bernoulli distributed with mean equal to τ (n), which is asymptotic to the periodic function ⎛ ⎞ 1 ⎝1 + Γ(2 − χj )n−χj ⎠ , log(1/q) j=0

and in the second case, P(Rn,k = m) =

λm 3 e−λ3 + o(1), m!

where λ3 := (pk n)2 (1 − q 2 )−k /2. Appendix A: Proof of Lemma 3. In this appendix, we prove Lemma 3. For −1/2 part (i) let z = neiθ , where θ = o(LLn ). By (8)

k − 1 j k−j j k−1−j n cos θ ˜ Mk (z) = pj q k−j ze−p q z 1 + O e−(p−q)p q j 0≤j 0, ∗

ε ei¯ ∞

xρ+it−1 f (x)dx

∞ i¯ ε(ρ+it) =e xρ+it−1 f (xei¯ε )dx 0

1

∞ −¯ εt ρ+1 −¯ εt ρ −qx cos ε¯ = O(e x dx) + O e x e dx 0 1

= O e−¯εt ρ−1 + e−¯εt q −ρ ρ1/2 (ρ/e)ρ ,

f (ρ + it) =

0

uniformly in ρ and t. If t < 0, then changing ei¯ε to e−i¯ε gives

f ∗ (ρ + it) = O e−¯ε|t| ρ−1 + e−¯ε|t| q −ρ ρ1/2 (ρ/e)ρ . When −2 < ρ ≤ 1, f ∗ (ρ + it) = O(e−¯ε|t| ) for large |t| by the same argument. On the other hand, by the ﬁrst estimate in (101), we also have f ∗ (s) = O |s + 2|−1 (s → 2). With these estimates and the Mellin inversion integral

1 fk (z) = z −s f ∗ (s)(p−s + q −s )k−1 ds, 2πi (ρ) ˜ k (z) and prove that fk (z) = Θ(|M ˜ k (z)|) for we can apply the arguments used for M | arg(z)| ≤ ε¯. REFERENCES [1] R. Aguech, N. Lasmar, and H. Mahmoud, Distribution of inter-node distances in digital trees, in Proceedings of the 2005 International Conference on Analysis of Algorithms, C. Mart´ınez, ed., Discrete Mathematics and Theoretical Computer Science, Nancy, France, 2005, pp. 1–10. [2] D. Aldous and P. Shields, A diﬀusion limit for a class of randomly-growing binary trees, Probab. Theory Related Fields, 79 (1988), pp. 509–542. [3] B. C. Berndt, Ramanujan’s Notebooks. Part I, Springer-Verlag, New York, 1985. [4] J. Bourdon, M. Nebel, and B. Valle´ e, On the stack-size of general tries, Theor. Inform. Appl., 35 (2001), pp. 163–185. [5] B. Chauvin, M. Drmota, and J. Jabbour-Hattab, The proﬁle of binary search trees, Ann. Appl. Probab., 11 (2001), pp. 1042–1062. [6] J.-D. Choi, K. Lee, A. Loginov, R. O’Callahan, V. Sarkar, and M. Sridharan, Eﬃcient and precise datarace detection for multithreaded object-oriented programs, in Proceedings of the 2002 ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI), 2002, pp. 258–269.

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