## A functional limit theorem for the profile of search trees (2008)

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Venue: | Annals of Applied Probability |

Citations: | 21 - 11 self |

### BibTeX

@ARTICLE{Drmota08afunctional,

author = {Michael Drmota and Svante Janson and Ralph Neininger and Tu Wien},

title = {A functional limit theorem for the profile of search trees},

journal = {Annals of Applied Probability},

year = {2008},

pages = {288--333}

}

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### Abstract

We study the profile Xn,k of random search trees including binary search trees and m-ary search trees. Our main result is a functional limit theorem of the normalized profile Xn,k/EXn,k for k =⌊α log n ⌋ in a certain range of α. A central feature of the proof is the use of the contraction method to prove convergence in distribution of certain random analytic functions in a complex domain. This is based on a general theorem concerning the contraction method for random variables in an infinite-dimensional Hilbert space. As part of the proof, we show that the Zolotarev metric is complete for a Hilbert space. 1. Introduction. Search

### Citations

3016 |
Convergence of Probability Measures
- Billingsley
- 1968
(Show Context)
Citation Context ... there exists a sequence λn of strictly increasing continuous functions that map I onto itself such that λn(x) → x and fn(λn(x)) → f(x), uniformly on every compact subinterval of I; see, for example, =-=[2]-=-, Chapter 3, (I = [0,1]), [24], [18], Chapter VI, [21], Appendix A2 ([0, ∞)), [19], Section 2. It is of technical importance that this topology can be induced by a complete, separable metric [2], Chap... |

523 |
Limit Theorems for Stochastic Processes
- Jacod, Shiryaev
- 2003
(Show Context)
Citation Context ...ctly increasing continuous functions that map I onto itself such that λn(x) → x and fn(λn(x)) → f(x), uniformly on every compact subinterval of I; see, for example, [2], Chapter 3, (I = [0,1]), [24], =-=[18]-=-, Chapter VI, [21], Appendix A2 ([0, ∞)), [19], Section 2. It is of technical importance that this topology can be induced by a complete, separable metric [2], Chapter 14, [18], Theorem VI.1.14, [21],... |

381 | Analytic Combinatorics - Flajolet, Sedgewick - 2009 |

337 |
Foundations of Modern Probability
- Kallenberg
- 2002
(Show Context)
Citation Context ... a closed subspace of L 2 (R 2 ), and is thus a separable Hilbert space, see e.g. [22, §1.4]. Since these spaces are metric spaces, we can use the general theory in e.g. Billingsley [2] or Kallenberg =-=[21]-=- for convergence in distribution of random func9stions in these spaces (equipped with their Borel σ-fields). In particular, recall that “convergence in distribution” really means convergence of the co... |

317 |
Singularity analysis of generating functions
- Flajolet, Odlyzko
- 1990
(Show Context)
Citation Context ... R. NEININGER Fortunately, we are in the situation where the relevant generating functions can be analytically continued outside the unit disc so that the singularity analysis of Flajolet and Odlyzko =-=[13]-=- (see also Flajolet and Sedgewick [14], Chapter 6) applies. As pointed out in [9], page 197, this simplifies the arguments considerably, so we consider only this case. We introduce the generating func... |

212 |
Evolution of random search trees
- Mahmoud
- 1992
(Show Context)
Citation Context ...sets Ii of keys (ignoring Ii if it is empty) and attach the roots of these trees as children of the root in the search tree. In the case m = 2, t = 0, we thus have the well-studied binary search tree =-=[4, 6, 7, 11, 12, 15, 26]-=-. In the case t ≥ 1, the only difference is that the pivots are selected in a different way, which affects the probability distribution of the set of pivots and thus of the trees. We now select mt+m−1... |

212 |
Topological Vector Spaces, Distributions and Kernels, volume 25 of Pure and Applied Mathematics
- Treves
- 1967
(Show Context)
Citation Context ...th a topology that can be defined by a complete metric, and it has (by Montel’s theorem on normal families) the property that every closed bounded subset is compact (see, e.g., [28], Chapter 1.45, or =-=[29]-=-, Example 10.II and Theorem 14.6). It is easily seen that the topology is separable [e.g., by regarding H(D) as a subspace of C ∞ 0 (D)]. • B(D) is the Bergman space of all square-integrable analytic ... |

165 |
Probability in Banach Spaces: isoperimetry and processes
- Ledoux, Talagrand
- 1991
(Show Context)
Citation Context ...tic functions ∫ ei〈x,y〉 dμ(x) and ∫ ei〈x,y〉 dν(x) are equal, which implies that all finite-dimensional projections coincide for μ and ν, andμ = ν then follows by a monotone class argument (see, e.g., =-=[23]-=-, Section 2.1). We continue by constructing some other functions in F ∗ s . Taking small positive multiples of them, we thus obtain functions in Fs.A FUNCTIONAL LIMIT THEOREM FOR THE PROFILE OF SEARC... |

163 |
Function Theory of Several Complex Variables
- Krantz
- 2001
(Show Context)
Citation Context ...e norm given by �f�2 � B(D) = D |f(z)|2 dm(z), where m is the two-dimensional Lebesgue measure. B(D) can be regarded as a closed subspace of L2 (R2 ) and is thus a separable Hilbert space (see, e.g., =-=[22]-=-, Chapter 1.4).sA FUNCTIONAL LIMIT THEOREM FOR THE PROFILE OF SEARCH TREES11 Since these spaces are metric spaces, we can use the general theory in, for example, Billingsley [2] or Kallenberg [21] for... |

144 |
Urn Models and Their Applications
- Johnson, Kotz
- 1977
(Show Context)
Citation Context ...th balls of m colors, initially containing t + 1 balls of each color, where we draw balls at random and replace each drawn ball together with a new ball of the same color (see, e.g., Johnson and Kotz =-=[20]-=-, Section 4.5.1). This distribution can be obtained by first taking a random vector V with the Dirichlet distribution above and then a multinomial variable with parameters n − (mt + m − 1) and V ([20]... |

70 | Functional Analysis, 2nd ed - Rudin - 1991 |

65 |
Foundations of Modern Probability. 2nd ed
- Kallenberg
- 2002
(Show Context)
Citation Context ...ntinuous functions that map I onto itself such that λn(x) → x and fn(λn(x)) → f(x), uniformly on every compact subinterval of I; see, for example, [2], Chapter 3, (I = [0,1]), [24], [18], Chapter VI, =-=[21]-=-, Appendix A2 ([0, ∞)), [19], Section 2. It is of technical importance that this topology can be induced by a complete, separable metric [2], Chapter 14, [18], Theorem VI.1.14, [21], Theorem A2.2. Not... |

54 | A general limit theorem for recursive algorithms and combinatorial structures
- Neininger, Rï£¡schendorf
- 2004
(Show Context)
Citation Context ... FUNCTIONAL LIMIT THEOREM FOR THE PROFILE OF SEARCH TREES19 6. Contraction method for Hilbert spaces. In this section, we extend the contraction method as developed for the Zolotarev metric on R d in =-=[27]-=- to random variables in a separable Hilbert space H. We denote by P(H) the set of all probability distributions on H. The limit distributions occurring subsequently are characterized as fixed points o... |

38 | Martingales and Profile of Binary Search Trees
- Chauvin, Klein, et al.
- 2005
(Show Context)
Citation Context ...ctor Xn = (Xn,k)k≥0, where Xn,k is the number of keys that are stored in nodes with depth k. The profile of binary search trees (and related structures) has been intensively studied in the literature =-=[4, 6, 7, 8, 10, 11, 12, 15, 17, 25]-=-. Most results concern 1st and 2nd moments. However, there are also distributional results, particularly for binary search trees and recursive trees [4, 6, 15] that are of the form X n,⌊αlogn⌋ EX n,⌊α... |

30 |
Weak convergence of probability measures and random functions in the function space D(0
- Lindvall
- 1973
(Show Context)
Citation Context ...f strictly increasing continuous functions that map I onto itself such that λn(x) → x and fn(λn(x)) → f(x), uniformly on every compact subinterval of I; see, for example, [2], Chapter 3, (I = [0,1]), =-=[24]-=-, [18], Chapter VI, [21], Appendix A2 ([0, ∞)), [19], Section 2. It is of technical importance that this topology can be induced by a complete, separable metric [2], Chapter 14, [18], Theorem VI.1.14,... |

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- Chern, Hwang, et al.
(Show Context)
Citation Context ...e. This splitting procedure was first introduced by Hennequin for the study of variants of the Quicksort algorithm and is referred to as the generalized Hennequin Quicksort (cf. Chern, Hwang and Tsai =-=[9]-=-). In particular, in the case m = 2, we let the pivot be the median of 2t + 1 randomly selected keys (when n ≥ 2t + 1). We describe the splitting of the keys by the random vector Vn = (Vn,1,Vn,2, ...,... |

20 |
Ideal metrics in the problem of approximating the distributions of sums of independent random variables
- Zolotarev
- 1977
(Show Context)
Citation Context ... strict contraction. Proof. This is similar to the proof of Lemma 3.1 in [27]. Note that for a linear operator A in H and L(X), L(Y ) ∈ Ps,z, we have ζs(A(X),A(Y )) ≤ �A� s op ζs(X,Y ) (cf. Zolotarev =-=[31]-=-, Theorem 3). � Lemma 6.2 and Theorem 5.1 imply that the restrictions of T in Lemma 6.2 have unique fixed points in Ps and Ps,0, respectively. We consider a sequence (Xn)n≥0 of random variables in H s... |

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- Drmota, Hwang
(Show Context)
Citation Context ...ctor Xn = (Xn,k)k≥0, where Xn,k is the number of keys that are stored in nodes with depth k. The profile of binary search trees (and related structures) has been intensively studied in the literature =-=[4, 6, 7, 8, 10, 11, 12, 15, 17, 25]-=-. Most results concern 1st and 2nd moments. However, there are also distributional results, particularly for binary search trees and recursive trees [4, 6, 15] that are of the form X n,⌊αlogn⌋ EX n,⌊α... |

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- Hwang
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Citation Context ...ctor Xn = (Xn,k)k≥0, where Xn,k is the number of keys that are stored in nodes with depth k. The profile of binary search trees (and related structures) has been intensively studied in the literature =-=[4, 6, 7, 8, 10, 11, 12, 15, 17, 25]-=-. Most results concern 1st and 2nd moments. However, there are also distributional results, particularly for binary search trees and recursive trees [4, 6, 15] that are of the form X n,⌊αlogn⌋ EX n,⌊α... |

16 | The density of the ISE and local limit laws for embedded trees - Bousquet-Mélou, Janson - 2006 |

16 | Profiles of random trees: limit theorems for random recursive trees and binary search trees. Algorithmica
- Fuchs, Hwang, et al.
(Show Context)
Citation Context |

15 |
Orthogonal decompositions and functional limit theorems for random graph statistics
- JANSON
- 1994
(Show Context)
Citation Context ...I onto itself such that λn(x) → x and fn(λn(x)) → f(x), uniformly on every compact subinterval of I; see, for example, [2], Chapter 3, (I = [0,1]), [24], [18], Chapter VI, [21], Appendix A2 ([0, ∞)), =-=[19]-=-, Section 2. It is of technical importance that this topology can be induced by a complete, separable metric [2], Chapter 14, [18], Theorem VI.1.14, [21], Theorem A2.2. Note that it matters significan... |

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- Drmota, Hwang
- 2005
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14 |
Approximation of the distributions of sums of independent random variables with values in infinite-dimensional spaces
- Zolotarev
- 1976
(Show Context)
Citation Context ... LIMIT THEOREM FOR THE PROFILE OF SEARCH TREES13 5. The Zolotarev metric on a Hilbert space. We recall the definition of the Zolotarev metric for probability measures in a Banach space; see Zolotarev =-=[30]-=-. If B and B1 are Banach spaces and f :U → B1 is a function defined on an open subset U ⊆ B, then f is said to be (Fréchet) differentiable at a point x ∈ U if there exists a linear operator Df(x):B → ... |

12 |
Analytic combinatorics. Cambridge Univ
- Flajolet, Sedgewick
- 2009
(Show Context)
Citation Context ...he situation where the relevant generating functions can be analytically continued outside the unit disc so that the singularity analysis of Flajolet and Odlyzko [13] (see also Flajolet and Sedgewick =-=[14]-=-, Chapter 6) applies. As pointed out in [9], page 197, this simplifies the arguments considerably, so we consider only this case. We introduce the generating function Ψ(ζ;z) := � n≥0 EWn(z)ζ n . Let Λ... |

12 |
More combinatorial problems on certain trees
- Lynch
- 1965
(Show Context)
Citation Context |

11 | Transitional behaviors of the average cost of quicksort with medianof-(2t+1
- Chern, Hwang
(Show Context)
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9 | Connecting Yule process, Bisection and Binary Search Tree via Martingales
- Chauvin, Rouault
- 2004
(Show Context)
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7 | The random multisection problem, travelling waves, and the distribution of the height of m-ary search trees, Algorithmica (2006
- Chauvin, Drmota
(Show Context)
Citation Context ...2 + = F(λ1(β 2 + ))]. First, by definition, mt+m−2 � 1 logF(λ1(β+)) = (λ1(β+) − 1) λ1(β+) + i i=t . Moreover, with S+ := �mt+m−2 i=t (λ1(β+) + i) −1 , it follows by a convexity argument (compare with =-=[5]-=-, Lemma 3.2) that, for every λ ≥ 1, logF(λ) ≤ (λ − 1)S+, with equality only for λ = λ1(β+). Consequently, log F(2λ1(β+) − 1) < (2λ1(β+) − 2)S+ = 2logF(λ1(β+)). Thus, we have F(2λ1(β+) − 1) < F(λ1(β+))... |

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An asymptotic theory for Cauchy Euler differential equations with applications to the analysis of algorithms
- Chern, Hwang, et al.
- 2002
(Show Context)
Citation Context ...e. This splitting procedure was first introduced by Hennequin for the study of variants of the Quicksort algorithm and is referred to as the generalized Hennequin Quicksort (cf. Chern, Hwang and Tsai =-=[9]-=-). In particular, in the case m = 2, we let the pivot be the median of 2t + 1randomly selected keys (when n ≥ 2t + 1). We describe the splitting of the keys by the random vector Vn = (Vn,1,Vn,2, ...,V... |

3 |
Estimates of distances between sums of independent random elements in Banach spaces. Teor. Veroyatnost. i Primenen
- Rachkauskas
- 1984
(Show Context)
Citation Context ...nce λn of strictly increasing continuous functions that map I onto itself such that λn(x) → x and fn(λn(x)) → f(x), uniformly on every compact subinterval of I; see, for example, [2], Chapter 3, (I = =-=[0,1]-=-), [24], [18], Chapter VI, [21], Appendix A2 ([0, ∞)), [19], Section 2. It is of technical importance that this topology can be induced by a complete, separable metric [2], Chapter 14, [18], Theorem V... |

3 |
On the central limit theorem in Hilbert space
- Giné, León
- 1980
(Show Context)
Citation Context ...ent and given kth moments zk, 1 ≤ k < s. Moreover, if Xn,X are H-valued random variables with distributions in Ps,z(H) and ζs(Xn,X) → 0, then Xn d −→ X. The final assertion is proved by Giné and Léon =-=[16]-=-. For completeness, we include a short proof using lemmas needed for the first part. Proof of Theorem 5.1. First, note that ζs is a metric on Ps,z(H) [30]; the fact that ζs(µ,ν) = 0 implies µ = ν for ... |

3 |
Estimates for the distance between sums of independent random elements in Banach spaces
- Bentkus, Rachkauskas
- 1984
(Show Context)
Citation Context ...ts a sequence λn of strictly increasing continuous functions that map I onto itself such that λn(x) → x and fn(λn(x)) → f(x), uniformly on every compact subinterval of I; see e.g. [2, Chapter 3] (I = =-=[0, 1]-=-), [24], [18, Chapter VI], [21, Appendix A2] ([0, ∞)), [19, §2]. It is of technical importance that this topology can be induced by a complete, separable metric [2, §14], [18, Theorem VI.1.14], [21, T... |

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The profile of binary search
- Chauvin, Drmota, et al.
- 2001
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