## A general limit theorem for recursive algorithms and combinatorial structures (2004)

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Venue: | ANN. APPL. PROB |

Citations: | 54 - 25 self |

### BibTeX

@ARTICLE{Neininger04ageneral,

author = {Ralph Neininger and Ludger Rüschendorf},

title = {A general limit theorem for recursive algorithms and combinatorial structures},

journal = {ANN. APPL. PROB},

year = {2004},

pages = {2004}

}

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### Abstract

Limit laws are proven by the contraction method for random vectors of a recursive nature as they arise as parameters of combinatorial structures such as random trees or recursive algorithms, where we use the Zolotarev metric. In comparison to previous applications of this method, a general transfer theorem is derived which allows us to establish a limit law on the basis of the recursive structure and to use the asymptotics of the first and second moments of the sequence. In particular, a general asymptotic normality result is obtained by this theorem which typically cannot be handled by the more common ℓ2 metrics. As applications we derive quite automatically many asymptotic limit results ranging from the size of tries or m-ary search trees and path lengths in digital structures to mergesort and parameters of random recursive trees, which were previously shown by different methods one by one. We also obtain a related local density approximation result as well as a global approximation result. For the proofs of these results we establish that a smoothed density distance as well as a smoothed total variation distance can be estimated from above by the Zolotarev metric, which is the main tool in this article.

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Citation Context ...application of the contraction method in the ℓ2 setting by Geiger [17]. The exponential limit distribution is characterized as the fixed point of X D = U(X + X∗ ), with X, X∗ ,U independent and U unif=-=[0, 1]-=- distributed. The Kn is the number of children of the most recent common ancestor of the population at generation n. 5. Applications: central limit laws. In this section we link expansions of moments ... |

696 |
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Citation Context ...is multinomial M (n \Gammas(m \Gammas1); u). Thus we obtain I (n) n ! (U(1); U(2) \GammasU(1); : : : ; 1 \GammasU(m\Gamma 1)) (50) in L1+". The mean and the variance satisfy for 3 ^ m ^ 26 (see Knuth =-=[31]-=-, Baeza-Yates [2], Mahmoud and Pittel [39], Chern and Hwang [7]) E Yn = 1 2(Hm \Gammas1) n + O(1 + n ff\Gamma 1 ); Var(Yn) = flmn + o(n); with flm ? 0 and ff ! 3=2 depending as well on m. Thus with Co... |

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Citation Context ...Hence we obtain Yn n L \Gamma ! X; X D = U X + 1; (48) where X and U are independent. It is known that this limit distribution is the Dickman distribution also arising in number theory, see Tenenbaum =-=[58]-=- and Hwang and Tsai [24]. This can easily be re-derived by checking that the Dickman distribution satisfies the fixed-point relation (48). For the case where we apply the Quickselect algorithm to sele... |

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Citation Context ...d as shown below for the top-down mergesort. Path lengths in digital structures. The fundamental search trees based on bit comparisons are the digital search tree, the trie and the Patricia trie; see =-=[57]-=-. The cost to build up these trees from n data is measured by their path lengths Yn, that is, the internal path length for digital search trees and the external path lengths for tries and Patricia tri... |

212 |
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Citation Context ... ,...,X (m) , V independent and X (r) ∼ X for r = 1,...,m. 5.3. Normal limit laws. 5.3.1. Linear mean and variance. Size of random m-ary search trees. The size Yn of the random m-ary search tree (see =-=[34]-=-) containing n data satisfies Y0 = 0, Y1 = ···= Ym−1 = 1andthe recursion Yn D m∑ = Y (r) I (n) + 1, n≥m. r r=1 Let V = (U(1),U(2) − U(1),...,1 − U(m−1)) denote the vector of spacings of independent un... |

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Citation Context ...ine ˜ℓr = ˜ℓr,θ by ˜ℓr(X, Y ) = sup |h| h∈R r ℓ(X + hθ, Y + hθ). Smoothing metrics have been used for proving central limit theorems for normalized sums and for martingales in probability theory (cf. =-=[47, 48, 55]-=-). ]AAP ims v.2003/08/05 Prn:1/09/2003; 11:54 F:aap165.tex; (Aiste) p. 15 LIMIT LAWS FOR RECURSIVE ALGORITHMS 15 PROPOSITION 4.2 (Regularity of ˜ℓr). Let r>dand θ satisfy condition (Hr−d). Then ˜ℓr i... |

97 | A Limit Theorem for Quicksort
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Citation Context ... are also strong analytic techniques for the subject, we extend and systematize a more probabilistic approach—the contraction method. This method was first introduced for the analysis of Quicksort in =-=[50]-=- and further developed independently in [51] and [49]; see also the survey article by Rösler and Rüschendorf [53]. The name of the method refers to the fact that the analysis makes use of an underlyin... |

68 | The contraction method for recursive algorithms
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Citation Context ... contraction method. This method was first introduced for the analysis of Quicksort in [50] and further developed independently in [51] and [49]; see also the survey article by Rösler and Rüschendorf =-=[53]-=-. The name of the method refers to the fact that the analysis makes use of an underlying map of measures, which is a contraction with respect to some probability metric. Our article is a continuation ... |

65 |
A survey of recursive trees
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Citation Context ...y successively adjoining children to the existing tree, where for the nth vertex the parent node is chosen uniformly from the vertices labeled 1,...,n− 1. For a survey of recursive trees, we refer to =-=[56]-=-. Mahmoud and Smythe [40] studied the joint distribution of Yn = (Bn,Rn,Gn), whereBn, Rn and Gn are the number of vertices in the tree with out-degree 0, 1 and 2, respectively. Based on a formulation ... |

64 | Asymptotic behavior of the Lempel–Ziv parsing scheme and digital search trees
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Citation Context ...r |k|→∞, we obtain that ϖ3 is infinite differentiable in all cases. Only the C 1 property is needed. This implies asymptotic normality, which for digital search trees and tries was proven in [25] and =-=[27]-=-, respectively; see also [54]: COROLLARY 5.8. The normalized internal (resp., external) path lengths (Yn − EYn)/ √ Var(Yn) of digital search trees, tries and Patricia tries are asymptotically normal i... |

48 | Probability metrics and recursive algorithms
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Citation Context ... we extend and systematize a more probabilistic approach—the contraction method. This method was first introduced for the analysis of Quicksort in [50] and further developed independently in [51] and =-=[49]-=-; see also the survey article by Rösler and Rüschendorf [53]. The name of the method refers to the fact that the analysis makes use of an underlying map of measures, which is a contraction with respec... |

48 | On the analysis of stochastic divide and conquer algorithms
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Citation Context ...ion of the article by Neininger [43], who used the ℓ2 metric approach to establish a general limit theorem for multivariate divide-andconquer recursions, thus extending the one-dimensional results in =-=[52]-=-. Although the ℓ2 approach works well for many problems that lead to nonnormal limit Received October 2001; revised January 2003. 1 Supported by NSERC Grant A3450 and the Deutsche Forschungsgemeinscha... |

46 | Phase changes in random m-ary search trees and generalized quicksort. Random Structures and Algorithms 19
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Citation Context ...al distribution of I (n) given V = u is multinomial M(n− (m − 1), u). Thus we obtain I (n) n → ( ) (51) U(1),U(2) − U(1),...,1 − U(m−1) in L1+ε. The mean and the variance satisfy, for 3 ≤ m ≤ 26 (see =-=[2, 7, 31, 38]-=-), 1 EYn = 2(Hm − 1) n + O(1 + nα−1 ), Var(Yn) = γmn + o(n), with γm > 0andα<3/2 depending as well on m. Thus with Corollary 5.2 we rederive the limit law (see [7, 33, 38]): COROLLARY 5.6. The normali... |

44 | Phase change of limit laws in the quicksort recurrence under varying toll functions
- Hwang, Neininger
(Show Context)
Citation Context ... search trees, secondary cost parameters of Quicksort or the cost of certain tree traversal algorithms, Quicksort, m-ary search tree or generalized Quicksort recursions with small toll functions; see =-=[7, 8, 11, 12, 14, 16, 23]-=-. 5.3.2. Periodic functions in the mean and variance. Size of random tries. The number Yn of internal nodes of a random trie with n keys in the symmetric Bernoulli model (see [34]) satisfies Y0 = 0and... |

36 |
A fixed point theorem for distributions, Stochastic Process
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Citation Context ... subject, we extend and systematize a more probabilistic approach—the contraction method. This method was first introduced for the analysis of Quicksort in [50] and further developed independently in =-=[51]-=- and [49]; see also the survey article by Rösler and Rüschendorf [53]. The name of the method refers to the fact that the analysis makes use of an underlying map of measures, which is a contraction wi... |

35 |
Sorting: A Distribution Theory
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- 2000
(Show Context)
Citation Context ... given as the unique solution of the fixed point equation in (48). Quicksort. The number of key comparisons Yn of the sorting algorithm Quicksort applied to a randomly permuted list of n numbers (see =-=[36]-=-) satisfies Y0 = Y1 = 0 and the recurrence (49) Yn L = YIn + Y ∗ n−In + n − 1, n≥ 2, with (Yn), (Y ∗ n ) and In independent and In unif{0,...,n− 1} distributed. It is well known that EYn = 2n ln n + c... |

32 |
On the distribution of binary search trees under the random permutation model. Random Struct Algor
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- 1996
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Citation Context ... search trees, secondary cost parameters of Quicksort or the cost of certain tree traversal algorithms, Quicksort, m-ary search tree or generalized Quicksort recursions with small toll functions; see =-=[7, 8, 11, 12, 14, 16, 23]-=-. 5.3.2. Periodic functions in the mean and variance. Size of random tries. The number Yn of internal nodes of a random trie with n keys in the symmetric Bernoulli model (see [34]) satisfies Y0 = 0and... |

29 |
Analysis of QUICKSELECT: an algorithm for order statistics
- Mahmoud, Modarres, et al.
- 1995
(Show Context)
Citation Context ...d by ℓ2(µ, ν) = inf { ‖X − Y ‖2 : X ∼ µ, Y ∼ ν } has been used frequently in the analysis of algorithms since its introduction in this context by Rösler [50] for the analysis of Quicksort (see, e.g., =-=[41, 43, 44]-=-); note that ℓ2 is ideal of order 1. This implies that ℓ2 typically cannot be used for fixed-point equations that occur for the normal distribution such as (7) X d = 1 √ 2 X1 + 1 √ 2 X2 d =: TX, where... |

29 | On a multivariate contraction method for random recursive structures with applications to Quicksort. Random Structures Algorithms
- Neininger
- 2001
(Show Context)
Citation Context ...ers to the fact that the analysis makes use of an underlying map of measures, which is a contraction with respect to some probability metric. Our article is a continuation of the article by Neininger =-=[43]-=-, who used the ℓ2 metric approach to establish a general limit theorem for multivariate divide-andconquer recursions, thus extending the one-dimensional results in [52]. Although the ℓ2 approach works... |

27 | Mellin transforms and asymptotics. The mergesort recurrence
- Flajolet, Golin
- 1994
(Show Context)
Citation Context ... in the analysis of the size and path length of bucket digital search trees, see [19]. Mergesort. The number of key comparisons Yn of top-down mergesort (for definition and mathematical analysis, see =-=[15]-=- and [20]), applied to a list of n randomly permuted items, satisfies Y0 = 0and where I (n) 1 Yn D = Y (1) I (n) 1 + Y (2) I (n) + n − Sn, n≥1, 2 (n) (n) =⌈n/2⌉, I 2 = n − I n ) and (Y (2) n ); see [3... |

27 |
Analysis of the space of search trees under the random insertion algorithm
- Mahmoud, Pittel
- 1989
(Show Context)
Citation Context ...al distribution of I (n) given V = u is multinomial M(n− (m − 1), u). Thus we obtain I (n) n → ( ) (51) U(1),U(2) − U(1),...,1 − U(m−1) in L1+ε. The mean and the variance satisfy, for 3 ≤ m ≤ 26 (see =-=[2, 7, 31, 38]-=-), 1 EYn = 2(Hm − 1) n + O(1 + nα−1 ), Var(Yn) = γmn + o(n), with γm > 0andα<3/2 depending as well on m. Thus with Corollary 5.2 we rederive the limit law (see [7, 33, 38]): COROLLARY 5.6. The normali... |

26 | Asymptotic distribution theory for Hoare’s selection algorithm
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(Show Context)
Citation Context ...it laws. LIMIT LAWS FOR RECURSIVE ALGORITHMS 23 Quickselect. The number of key comparisons Yn of the selection algorithm Quickselect, also known as FIND (for definition and mathematical analysis, see =-=[18, 24, 32, 41]-=-) when selecting the smallest order statistic in a set of n data satisfies Y0 = Y1 = 0 and the recurrence Yn D = YIn + n − 1, n≥ 2, where (Yn) and In are independent with In unif{0,...,n− 1} distribut... |

25 |
Digital search trees again revisited: the internal path length perspective
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Citation Context ...log n); Var Yn = n$4(log2 n) + O(log 2 n); where $r are periodic functions (with period 1) varying from one of the structures to the other, see Knuth [31] and Kirschenhofer, Prodinger and Szpankowski =-=[28, 29, 30]-=-. It is known that $4 is continuous and positive in each case. For $3 we have the representations $3(x) = C + 1 ln 2 X k2Znf0g \Gamma (\Gamma !k)e 2kssix ; x 2 R; for the digital search tree and $3(x)... |

24 | Quickselect and the Dickman function
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- 2002
(Show Context)
Citation Context ...it laws. LIMIT LAWS FOR RECURSIVE ALGORITHMS 23 Quickselect. The number of key comparisons Yn of the selection algorithm Quickselect, also known as FIND (for definition and mathematical analysis, see =-=[18, 24, 32, 41]-=-) when selecting the smallest order statistic in a set of n data satisfies Y0 = Y1 = 0 and the recurrence Yn D = YIn + n − 1, n≥ 2, where (Yn) and In are independent with In unif{0,...,n− 1} distribut... |

24 |
On the structure of random plane-oriented recursive trees and their branches, Random Structures Algorithms 4
- Mahmoud, Smythe, et al.
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Citation Context ...Kn ∑ ( (n) ) I s/2 r n r=1 lim sup ≤ 5 + 2s < 1 3 + 3s PROOF. We enumerate the subtrees of the root such that the first subtree in our enumeration is the one with root labeled 2. Then by Theorem 5 in =-=[42]-=-, I (n)/n → V holds almost surely, where V has the beta(1/2, 1) distribution. In particular, EV s/2 = 1/(s + 1) for s>0. An estimate similar to (72) leads to the assertion. □ From this contraction pro... |

22 | An asymptotic theory for Cauchy-Euler differential equations with applications to the analysis of algorithms
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(Show Context)
Citation Context ... search trees, secondary cost parameters of Quicksort, or the cost of certain tree traversal algorithm, Quicksort, m-ary search tree or generalized Quicksort recursions with small toll functions, see =-=[11, 14, 16, 12, 7, 23, 8]-=-. 5.3.2 Periodic functions in the mean and variance Size of random tries: The number Yn of internal nodes of a random trie with n keys in the symmetric Bernoulli model (see Mahmoud [34]) satisfies Y0 ... |

21 |
Limit theorems for the number of maxima in random samples from planar regions, Electron
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Citation Context ...r orders for mean and variance. Maxima in right triangles. We consider the number Yn of maxima of n independent, uniform samples in a right triangle in R2 with vertices (0, 0), (1, 0) and (0, 1); see =-=[3]-=- and [4]. According to Proposition 1 in [3], this number satisfies the recursion Y0 = 0and Yn D = Y (1) I (n) 1 + Y (2) I (n) + 1, 2 where the indices I (n) 1 and I (n) 2 are given as the first two co... |

20 | Limit laws for local counters in random binary search trees, Random Structures Algorithms 2
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(Show Context)
Citation Context ... search trees, secondary cost parameters of Quicksort or the cost of certain tree traversal algorithms, Quicksort, m-ary search tree or generalized Quicksort recursions with small toll functions; see =-=[7, 8, 11, 12, 14, 16, 23]-=-. 5.3.2. Periodic functions in the mean and variance. Size of random tries. The number Yn of internal nodes of a random trie with n keys in the symmetric Bernoulli model (see [34]) satisfies Y0 = 0and... |

20 |
Normal limiting distribution of the size of tries
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- 1987
(Show Context)
Citation Context ...mean and variance satisfy ([22]) (52) EYn = nϖ1(log2 n) + O(1), Var(Yn) = nϖ2(log2 n) + O(1), where ϖ1 and ϖ2 are positive C∞ functions with period 1. We obtain a limit law due to Jacquet and Régnier =-=[26]-=-: COROLLARY 5.7. The normalized size (Yn − EYn)/ √ Var(Yn) of a random trie in the symmetric Bernoulli model converges in distribution to the standard normal distribution. PROOF. For the application o... |

20 |
Ideal metrics in the problem of approximating the distributions of sums of independent random variables
- Zolotarev
- 1977
(Show Context)
Citation Context ...f (m) (x) − f (m) (y) ∥ ∥ ≤‖x − y‖ α} . Convergence in ζs implies weak convergence and moreover, for some c>0, { E(‖X‖ s −‖Y ‖ s ), cζs(X, Y ) ≥ π 1+s (‖X‖, ‖Y ‖), where π is the Prohorov metric (see =-=[60]-=-). There are upper bounds for ζs in terms of difference pseudomoments κs(X, Y ) = sup { |Ef (X) − f(Y)| : ‖f(x)− f(y)‖≤ ∥ ∥ ‖x‖ s−1 x −‖y‖ s−1 y ∥ ∥ } . Note that κs is the minimal metric of the compo... |

19 | Total path length for random recursive trees
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Citation Context ...ion as a sum of independent Bernoulli random variables. Second, we could also subdivide into, for example, the leftist subtree of the root and the rest of the tree (including the root) as was done in =-=[13]-=- for the analysis of the internal path length in random recursive trees. However, the example of Mahmoud and Smythe [40] will, in this decomposition, not be covered by our present approach, since a de... |

18 |
Patterns in random binary search trees, Random Structures Algorithms 11
- Flajolet, Gourdon, et al.
- 1997
(Show Context)
Citation Context ... search trees, secondary cost parameters of Quicksort, or the cost of certain tree traversal algorithm, Quicksort, m-ary search tree or generalized Quicksort recursions with small toll functions, see =-=[11, 14, 16, 12, 7, 23, 8]-=-. 5.3.2 Periodic functions in the mean and variance Size of random tries: The number Yn of internal nodes of a random trie with n keys in the symmetric Bernoulli model (see Mahmoud [34]) satisfies Y0 ... |

16 |
The joint distribution of elastic buckets in multiway search trees
- Lew, Mahmoud
- 1994
(Show Context)
Citation Context ...isfy, for 3 ≤ m ≤ 26 (see [2, 7, 31, 38]), 1 EYn = 2(Hm − 1) n + O(1 + nα−1 ), Var(Yn) = γmn + o(n), with γm > 0andα<3/2 depending as well on m. Thus with Corollary 5.2 we rederive the limit law (see =-=[7, 33, 38]-=-): COROLLARY 5.6. The normalized size (Yn − EYn)/ √ Var(Yn) of a random m-ary search tree with 3 ≤ m ≤ 26 converges in distribution to the standard normal distribution. r=1 r=1AAP ims v.2003/08/05 Pr... |

16 | On the internal path length of d-dimensional quad trees. Random Structures and Algorithms
- Neininger, Rüschendorf
- 1999
(Show Context)
Citation Context ...d by ℓ2(µ, ν) = inf { ‖X − Y ‖2 : X ∼ µ, Y ∼ ν } has been used frequently in the analysis of algorithms since its introduction in this context by Rösler [50] for the analysis of Quicksort (see, e.g., =-=[41, 43, 44]-=-); note that ℓ2 is ideal of order 1. This implies that ℓ2 typically cannot be used for fixed-point equations that occur for the normal distribution such as (7) X d = 1 √ 2 X1 + 1 √ 2 X2 d =: TX, where... |

14 |
On the variance of the external path length in a symmetric digital trie
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Citation Context ...log n); Var Yn = n$4(log2 n) + O(log 2 n); where $r are periodic functions (with period 1) varying from one of the structures to the other, see Knuth [31] and Kirschenhofer, Prodinger and Szpankowski =-=[28, 29, 30]-=-. It is known that $4 is continuous and positive in each case. For $3 we have the representations $3(x) = C + 1 ln 2 X k2Znf0g \Gamma (\Gamma !k)e 2kssix ; x 2 R; for the digital search tree and $3(x)... |

14 |
On the distribution of leaves in rooted subtrees of recursive trees
- Mahmoud, Smythe
- 1991
(Show Context)
Citation Context ...olds. lim sup n→∞ Kn ∑ E r=1 ( (n) ) I s/2 r n ≤ 6 + s < 1 4 + 2sAAP ims v.2003/08/05 Prn:1/09/2003; 11:54 F:aap165.tex; (Aiste) p. 37 LIMIT LAWS FOR RECURSIVE ALGORITHMS 37 PROOF. By Theorem 5.1 in =-=[39]-=- we have, in particular, I (n) 1 /n → U ∼ unif[0, 1] almost surely. Hence it follows that (72) Kn ∑ E r=1 The assertion follows. ( (n) ) s/2 I r n □ ≤ E = E → [(I (n) ) s/2 1 n [(I (n) ) s/2 1 n Kn ∑ ... |

14 |
Approximation of the distributions of sums of independent random variables with values in infinite-dimensional spaces
- Zolotarev
- 1976
(Show Context)
Citation Context ...to be σ 2 > 0 and a finite absolute sth moment is assumed. In our example (7) the fixed point is then unique and, in fact, we can prove the contraction property in the Zolotarev metrics ζs. Zolotarev =-=[59]-=- found the following metric ζs, which is ideal of order s>0, defined for d-dimensional vectors by ( )∣ (8) ζs(X, Y ) = sup ∣E f(X)−f(Y) ∣, f ∈Fs where for s = m + α, 0<α≤ 1, m ∈ N0, Fs := { f ∈ C m (R... |

12 | Limit laws for sums of functions of subtrees of random binary search trees
- Devroye
(Show Context)
Citation Context |

12 | A multivariate view of random bucket digital search trees
- HUBALEK, HWANG, et al.
(Show Context)
Citation Context ...leads to (58) with �n,1 = �n,2 = 1 there; thus we can conclude as in Corollary 5.7. □ For related recursions that arise in the analysis of the size and path length of bucket digital search trees, see =-=[19]-=-. Mergesort. The number of key comparisons Yn of top-down mergesort (for definition and mathematical analysis, see [15] and [20]), applied to a list of n randomly permuted items, satisfies Y0 = 0and w... |

11 |
Asymptotic joint normality of out-degrees of nodes in random recursive trees
- Mahmoud, Smythe
- 1992
(Show Context)
Citation Context ...children to the existing tree, where for the nth vertex the parent node is chosen uniformly from the vertices labeled 1,...,n− 1. For a survey of recursive trees, we refer to [56]. Mahmoud and Smythe =-=[40]-=- studied the joint distribution of Yn = (Bn,Rn,Gn), whereBn, Rn and Gn are the number of vertices in the tree with out-degree 0, 1 and 2, respectively. Based on a formulation as a generalized Pólya– E... |

11 |
Normal convergence problem? Two moments and a recurrence may be the clues
- Pittel
- 1999
(Show Context)
Citation Context ...ormal limit laws, or with s<2, which leads to results where we can weaken the assumption of finite second moments—an assumption that is usually present in the ℓ2 approach. In his 1999 article, Pittel =-=[46]-=- stated as a heuristic principle that various global characteristics of large size combinatorial structures such as graphs and trees are asymptotically normal if the mean and variance are “nearly line... |

9 | The cost distribution of queue-mergesort, optimal mergesorts, and power-of-2 rules
- Chen, Hwang, et al.
- 1999
(Show Context)
Citation Context ...isons (Yn √ − EYn)/ Var(Yn) of top-down mergesort applied to a randomly permuted number of items is asymptotically normal. For other variants of mergesort and a limit law for the queue mergesort, see =-=[6]-=- and the references therein. Note that the limit law for queue mergesort in [6] cannot be obtained by Corollary 5.2 since the corresponding prefactors (g(I (n) r )/g(n)) 1/2 do not converge, r = 1, 2.... |

9 |
On the balance property of Patricia tries: external path length viewpoint. Theoret
- KIRSCHENHOFER, PRODINGER, et al.
- 1989
(Show Context)
Citation Context ...log n); Var Yn = n$4(log2 n) + O(log 2 n); where $r are periodic functions (with period 1) varying from one of the structures to the other, see Knuth [31] and Kirschenhofer, Prodinger and Szpankowski =-=[28, 29, 30]-=-. It is known that $4 is continuous and positive in each case. For $3 we have the representations $3(x) = C + 1 ln 2 X k2Znf0g \Gamma (\Gamma !k)e 2kssix ; x 2 R; for the digital search tree and $3(x)... |

9 | Rates of convergence for Quicksort
- Neininger, Rüschendorf
(Show Context)
Citation Context ...ces (hn) tending to zero sufficiently slowly. The exact order of the rate of convergence of the standardized cost of Quicksort for the ζ3 metric has been identified to be of the order �(ln(n)/n); see =-=[45]-=-. Hence, on the basis of this refinement we obtain as well rates of convergence for global and local convergences by applying Corollaries 4.3 and 4.5. This was made explicit by Neininger and Rüschendo... |

8 |
Limit theorems for mergesort, Random Structures and Algorithms
- Hwang
- 1996
(Show Context)
Citation Context ...d that (Yn \GammasE Yn )= p Var(Yn) satisfies a central limit theorem applying Lyapunov's condition. Hwang [20] found a local limit theorem and large deviations including rates of convergence and, in =-=[21]-=-, gave full (exact) asymptotic expansions for the mean and variance. 26sCramer [9] obtained the (central) limit law applying the contraction method. For methodological reasons we re-derive this limit ... |

7 | Some average measures in m-ary search trees - Baeza-Yates - 1987 |

7 |
An L2 convergence theorem for random affine mappings
- BURTON, RÖSLER
- 1995
(Show Context)
Citation Context ...D L. RÜSCHENDORF where Z (1),...,Z(K),W(1),...,W(K),(A1,...,AK,b) are independent with Z (r) ∼ µ and W (r) ∼ ν. □ REMARK. With respect to the ℓ2 metric, the corresponding contraction property is (see =-=[5, 43]-=-) ℓ2(T µ, T ν) ≤ ∥ E K∑ A r=1 t rAr ℓ2(µ, ν). ∥ op Since ‖E ∑K r=1 (At rAr)‖op ≤ E ∑K r=1 ‖At rAr‖op = E ∑K r=1 ‖Ar‖2 op , the contraction condition for ℓ2 is weaker than that for the comparable ζ2 an... |

7 |
The Art of Computer Programming 3
- Knuth
- 1973
(Show Context)
Citation Context ...al distribution of I (n) given V = u is multinomial M(n− (m − 1), u). Thus we obtain I (n) n → ( ) (51) U(1),U(2) − U(1),...,1 − U(m−1) in L1+ε. The mean and the variance satisfy, for 3 ≤ m ≤ 26 (see =-=[2, 7, 31, 38]-=-), 1 EYn = 2(Hm − 1) n + O(1 + nα−1 ), Var(Yn) = γmn + o(n), with γm > 0andα<3/2 depending as well on m. Thus with Corollary 5.2 we rederive the limit law (see [7, 33, 38]): COROLLARY 5.6. The normali... |

7 | Asymptotic normality of recursive algorithms via martingale difference arrays
- Schachinger
- 2001
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Citation Context ... infinite differentiable in all cases. Only the C 1 property is needed. This implies asymptotic normality, which for digital search trees and tries was proven in [25] and [27], respectively; see also =-=[54]-=-: COROLLARY 5.8. The normalized internal (resp., external) path lengths (Yn − EYn)/ √ Var(Yn) of digital search trees, tries and Patricia tries are asymptotically normal in the symmetric Bernoulli mod... |

6 | Patterns in random binary search trees
- Flajolet, Gourdon, et al.
- 1997
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Citation Context |