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40
A general limit theorem for recursive algorithms and combinatorial structures
 ANN. APPL. PROB
, 2004
"... 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 ..."
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Cited by 53 (25 self)
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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 mary 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.
Second Phase Changes in Random MAry Search Trees and Generalized Quicksort: Convergence Rates
, 2002
"... We study the convergence rate to normal limit law for the space requirement of random mary search trees. While it is known that the random variable is asymptotically normally distributed for 3 m 26 and that the limit law does not exist for m ? 26, we show that the convergence rate is O(n ) for ..."
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Cited by 46 (12 self)
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We study the convergence rate to normal limit law for the space requirement of random mary search trees. While it is known that the random variable is asymptotically normally distributed for 3 m 26 and that the limit law does not exist for m ? 26, we show that the convergence rate is O(n ) for 3 m 19 and is O(n ), where 4=3 ! ff ! 3=2 is a parameter depending on m for 20 m 26. Our approach is based on a refinement to the method of moments and applicable to other recursive random variables; we briefly mention the applications to quicksort proper and the generalized quicksort of Hennequin, where more phase changes are given. These results provide natural, concrete examples for which the BerryEsseen bounds are not necessarily proportional to the reciprocal of the standard deviation. Local limit theorems are also derived. Abbreviated title. Phase changes in search trees.
Phase Change of Limit Laws in the Quicksort Recurrence Under Varying Toll Functions
, 2001
"... We characterize all limit laws of the quicksort type random variables defined recursively by Xn = X In + X # n1In + Tn when the "toll function" Tn varies and satisfies general conditions, where (Xn ), (X # n ), (I n , Tn ) are independent, Xn . . . , n 1}. When the "toll function" Tn ..."
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Cited by 44 (18 self)
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We characterize all limit laws of the quicksort type random variables defined recursively by Xn = X In + X # n1In + Tn when the "toll function" Tn varies and satisfies general conditions, where (Xn ), (X # n ), (I n , Tn ) are independent, Xn . . . , n 1}. When the "toll function" Tn (cost needed to partition the original problem into smaller subproblems) is small (roughly lim sup n## log E(Tn )/ log n 1/2), Xn is asymptotically normally distributed; nonnormal limit laws emerge when Tn becomes larger. We give many new examples ranging from the number of exchanges in quicksort to sorting on broadcast communication model, from an insitu permutation algorithm to tree traversal algorithms, etc.
On a multivariate contraction method for random recursive structures with applications to Quicksort
, 2001
"... The contraction method for recursive algorithms is extended to the multivariate analysis of vectors of parameters of recursive structures and algorithms. We prove a general multivariate limit law which also leads to an approach to asymptotic covariances and correlations of the parameters. As an appl ..."
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Cited by 28 (15 self)
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The contraction method for recursive algorithms is extended to the multivariate analysis of vectors of parameters of recursive structures and algorithms. We prove a general multivariate limit law which also leads to an approach to asymptotic covariances and correlations of the parameters. As an application the asymptotic correlations and a bivariate limit law for the number of key comparisons and exchanges of medianof(2t + 1) Quicksort is given. Moreover, for the Quicksort programs analyzed by Sedgewick the exact order of the standard deviation and a limit law follow, considering all the parameters counted by Sedgewick.
A Characterization Of The Set Of Fixed Points Of The Quicksort Transformation
, 2000
"... The limiting distribution of the normalized number of key comparisons required by the Quicksort sorting algorithm is known to be the unique fixed point of a certain distributional transformation T unique, that is, subject to the constraints of zero mean and finite variance. We show that a dist ..."
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Cited by 18 (10 self)
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The limiting distribution of the normalized number of key comparisons required by the Quicksort sorting algorithm is known to be the unique fixed point of a certain distributional transformation T unique, that is, subject to the constraints of zero mean and finite variance. We show that a distribution is a fixed point of T if and only if it is the convolution of with a Cauchy distribution of arbitrary center and scale. In particular, therefore, is the unique fixed point of T having zero mean. 1
On the internal path length of ddimensional quad trees
, 1999
"... It is proved that the internal path length of a d–dimensional quad tree after normalization converges in distribution. The limiting distribution is characterized as a fixed point of a random affine operator. We obtain convergence of all moments and of the Laplace transforms. The moments of the limit ..."
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Cited by 16 (9 self)
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It is proved that the internal path length of a d–dimensional quad tree after normalization converges in distribution. The limiting distribution is characterized as a fixed point of a random affine operator. We obtain convergence of all moments and of the Laplace transforms. The moments of the limiting distribution can be evaluated from the recursion and lead to first order asymptotics for the moments of the internal path lengths. The analysis is based on the contraction method. In the final part of the paper we state similar results for general split tree models if the expectation of the path length has a similar expansion as in the case of quad trees. This applies in particular to the mary search trees.
Transfer Theorems and Asymptotic Distributional Results for mary Search Trees
, 2004
"... We derive asymptotics of moments and identify limiting distributions, under the random permutation model on mary search trees, for functionals that satisfy recurrence relations of a simple additive form. Many important functionals including the space requirement, internal path length, and the soca ..."
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Cited by 13 (7 self)
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We derive asymptotics of moments and identify limiting distributions, under the random permutation model on mary search trees, for functionals that satisfy recurrence relations of a simple additive form. Many important functionals including the space requirement, internal path length, and the socalled shape functional fall under this framework. The approach is based on establishing transfer theorems that link the order of growth of the input into a particular (deterministic) recurrence to the order of growth of the output. The transfer theorems are used in conjunction with the method of moments to establish limit laws. It is shown that (i) for small toll sequences (tn) [roughly, tn = O(n1/2)] we have asymptotic normality if m ≤ 26 and typically periodic behavior if m ≥ 27; (ii) for moderate toll sequences [roughly, tn = ω(n1/2) but tn = o(n)] we have convergence to nonnormal distributions if m ≤ m0 (where m0 ≥ 26) and typically periodic behavior if m ≥ m0 + 1; and (iii) for large toll sequences [roughly, tn = ω(n)] we have convergence to nonnormal distributions for all values of m.
The Bernoulli sieve
 Bernoulli
, 2004
"... Abstract. Bernoulli sieve is a recursive construction of a random composition (ordered partition) of integer n. This composition can be induced by sampling from a random discrete distribution which has frequencies equal to the sizes of component intervals of a stickbreaking interval partition of [0 ..."
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Cited by 12 (3 self)
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Abstract. Bernoulli sieve is a recursive construction of a random composition (ordered partition) of integer n. This composition can be induced by sampling from a random discrete distribution which has frequencies equal to the sizes of component intervals of a stickbreaking interval partition of [0, 1]. We exploit Markov property of the composition and its renewal representation to derive asymptotics of the moments and to prove a central limit theorem for the number of parts. 1. The Bernoulli sieve can be seen as a generalisation of the ‘game ’ found in [3]. The first round of the game starts with n players and amounts to tossing a coin with probability X1 for tails. Each of the players tosses one time and the players flipping tails must drop out. If all n get heads the trial is disqualified and must be repeated completely with all n players, as many times as necessary until some players do quit. If at least one player remains after the first round, the second round continues with the remaining players, who must toss another coin with probability X2 for tails. The game lasts with probabilities X3, X4,... for tails until all players are sorted out. It is assumed that the probabilities X1, X2,... are independent random variables with a given distribution ω on]0, 1 [ , and that given Xj the individual outcomes at
Limit laws for partial match queries in quadtrees
 ANN. APPL. PROBAB
, 2001
"... It is proved that in an idealized uniform probabilistic model the cost of a partial match query in a multidimensional quadtree after normalization converges in distribution. The limiting distribution is given as a fixed point of a random affine operator. Also a firstorder asymptoticexpansion for th ..."
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Cited by 10 (4 self)
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It is proved that in an idealized uniform probabilistic model the cost of a partial match query in a multidimensional quadtree after normalization converges in distribution. The limiting distribution is given as a fixed point of a random affine operator. Also a firstorder asymptoticexpansion for the variance of the cost is derived and results on exponential moments are given. The analysis is based on the contraction method.