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29
On the Analysis of Stochastic Divide and Conquer Algorithms.
, 1999
"... This paper develops general tools for the analysis of stochastic divide and conquer algorithms. We concentrate on the average performance and the distribution of the duration of the algorithm. In particular we analyse the average performance and the running time distribution of the 2k + 1median ..."
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Cited by 48 (1 self)
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This paper develops general tools for the analysis of stochastic divide and conquer algorithms. We concentrate on the average performance and the distribution of the duration of the algorithm. In particular we analyse the average performance and the running time distribution of the 2k + 1median version of Quicksort.
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.
Quicksort Algorithm Again Revisited
 Discrete Math. Theor. Comput. Sci
, 1999
"... this paper, we establish an integral equation for the probability density of the number of comparisons L n . Then, we investigate the large deviations of L n . We shall show that the left tail of the limiting distribution is much "thinner" (i.e., double exponential) than the right tail (which is onl ..."
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Cited by 37 (5 self)
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this paper, we establish an integral equation for the probability density of the number of comparisons L n . Then, we investigate the large deviations of L n . We shall show that the left tail of the limiting distribution is much "thinner" (i.e., double exponential) than the right tail (which is only exponential). Our results contain some constants that must be determined numerically. We use formal asymptotic methods of applied mathematics such as the WKB method and matched asymptotics. Keywords: Algorithms, Analysis of algorithms, Asymptotic analysis, Binary search tree, Quicksort, Sorting. 1 Introduction
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.
Asymptotic laws for nonconservative selfsimilar fragmentations
 Electronic J. Probab
, 2004
"... Abstract We consider a selfsimilar fragmentation process in which the generic particle of size x is replaced at probability rate x α by its offspring made of smaller particles, where α is some positive parameter. The total of offspring sizes may be both larger or smaller than x with positive probab ..."
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Cited by 14 (2 self)
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Abstract We consider a selfsimilar fragmentation process in which the generic particle of size x is replaced at probability rate x α by its offspring made of smaller particles, where α is some positive parameter. The total of offspring sizes may be both larger or smaller than x with positive probability. We show that under certain conditions the typical size in the ensemble is of the order t −1/α and that the empirical distribution of sizes converges to a random limit which we characterise in terms of the reproduction law. 1
A Stochastic Fixed Point Equation Related to Weighted Branching
"... For real numbers C, T1, T2,... we find all solutions µ to the stochastic fixed point equation W � j≥1 TjWj + C, where W, W1, W2,... are independent realvalued random variables with distribution µ and means equality in distribution. All solutions are infinitely divisible. The set of solutions depend ..."
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Cited by 13 (3 self)
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For real numbers C, T1, T2,... we find all solutions µ to the stochastic fixed point equation W � j≥1 TjWj + C, where W, W1, W2,... are independent realvalued random variables with distribution µ and means equality in distribution. All solutions are infinitely divisible. The set of solutions depends on the closed multiplicative subgroup of R ∗ = R\{0} generated by the Tj. If this group is continuous, i.e. R ∗ itself or the positive halfline R>, then all nontrivial fixed points are stable laws. In the remaining (discrete) cases further periodic solutions arise. A key observation is that the Lévy measure of any fixed point is harmonic with respect to Λ = � j≥1 δT, i.e. Γ = Γ ⋆ Λ, where ⋆ means multiplicative j convolution. This will enable us to apply the powerful ChoquetDeny theorem.
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.
Convergence Conditions for Weighted Branching Processes
, 2000
"... We consider some aspects of the weighted branching processes and, in particular, consider convergence conditions for the processes being a certain analogue of the wellknown X ln X condition for ordinary branching processes and branching random walks on R. ..."
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Cited by 9 (3 self)
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We consider some aspects of the weighted branching processes and, in particular, consider convergence conditions for the processes being a certain analogue of the wellknown X ln X condition for ordinary branching processes and branching random walks on R.
Weighted height of random trees
 Manuscript
"... We consider a model of random trees similar to the split trees of Devroye [30] in which a set of items is recursively partitioned. Our model allows for more flexibility in the choice of the partitioning procedure, and has weighted edges. We prove that for this model, the height H n of a random tree ..."
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Cited by 5 (4 self)
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We consider a model of random trees similar to the split trees of Devroye [30] in which a set of items is recursively partitioned. Our model allows for more flexibility in the choice of the partitioning procedure, and has weighted edges. We prove that for this model, the height H n of a random tree is asymptotic to c log n in probability for a constant c that is uniquely characterized in terms of multivariate large deviations rate functions. This extension permits us to obtain the height of pebbled tries, pebbled ternary search tries, dary pyramids, and to study geometric properties of partitions generated by kd trees. The model also includes all polynomial families of increasing trees recently studied by Broutin, Devroye, McLeish, and de la Salle [17].