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A survey of maxtype recursive distributional equations
 Annals of Applied Probability 15 (2005
, 2005
"... In certain problems in a variety of applied probability settings (from probabilistic analysis of algorithms to statistical physics), the central requirement is to solve a recursive distributional equation of the form X d = g((ξi,Xi), i ≥ 1). Here(ξi) and g(·) are given and the Xi are independent cop ..."
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Cited by 63 (6 self)
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In certain problems in a variety of applied probability settings (from probabilistic analysis of algorithms to statistical physics), the central requirement is to solve a recursive distributional equation of the form X d = g((ξi,Xi), i ≥ 1). Here(ξi) and g(·) are given and the Xi are independent copies of the unknown distribution X. We survey this area, emphasizing examples where the function g(·) is essentially a “maximum ” or “minimum” function. We draw attention to the theoretical question of endogeny: inthe associated recursive tree process X i,aretheX i measurable functions of the innovations process (ξ i)? 1. Introduction. Write
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.
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.
Smoothness and Decay Properties of the Limiting Quicksort Density Function
, 2000
"... Using Fourier analysis, we prove that the limiting distribution of the standardized random number of comparisons used byQuicksort to sort an array of n numbers has an everywhere positive and infinitely differentiable density f, and that each derivative f (k) enjoys superpolynomial decay at ±∞. In pa ..."
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Cited by 37 (18 self)
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Using Fourier analysis, we prove that the limiting distribution of the standardized random number of comparisons used byQuicksort to sort an array of n numbers has an everywhere positive and infinitely differentiable density f, and that each derivative f (k) enjoys superpolynomial decay at ±∞. In particular, each f (k) is bounded. Our method is sufficiently computational to prove, for example, that f is bounded by 16.
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
Quicksort asymptotics
 Journal of Algorithms
, 2002
"... normality of statistics on permutation tableaux ..."
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.
General combinatorial schemas: Gaussian limit distributions and exponential tails
 Discrete Math
, 1993
"... Under general conditions, the number of components in combinatorial structures defined as sequences, cycles or sets of components admits a Gaussian limit distribution together with an exponential tail. The results are valid, assuming simple analytic conditions on the generating functions of the comp ..."
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Cited by 25 (6 self)
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Under general conditions, the number of components in combinatorial structures defined as sequences, cycles or sets of components admits a Gaussian limit distribution together with an exponential tail. The results are valid, assuming simple analytic conditions on the generating functions of the components. The proofs rely on the continuity theorem for characteristic functions and Laplace transforms as well as techniques of singularity analysis applied to algebraic and logarithmic singularities. Combinatorial applications are in the fields of graphs, permutations, random mappings, ordered partitions and polynomial factorizations. 1.
Approximating the Limiting Quicksort Distribution
, 2001
"... The limiting distribution of the normalized number of comparisons used by Quicksort to sort an array of n numbers is known to be the unique xed point with zero mean of a certain distributional transformation S. We study the convergence to the limiting distribution of the sequence of distributions o ..."
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Cited by 24 (5 self)
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The limiting distribution of the normalized number of comparisons used by Quicksort to sort an array of n numbers is known to be the unique xed point with zero mean of a certain distributional transformation S. We study the convergence to the limiting distribution of the sequence of distributions obtained by iterating the transformation S, beginning with a (nearly) arbitrary starting distribution. We demonstrate geometrically fast convergence for various metrics and discuss some implications for numerical calculations of the limiting Quicksort distribution. Finally, we give companion lower bounds which show that the convergence is not faster than geometric. AMS 2000 subject classications. Primary 68W40; secondary 68P10, 60E05, 60E10, 60F05. Key words and phrases. Quicksort, characteristic function, density, moment generating function, sorting algorithm, coupling, Fourier analysis, Kolmogorv{Smirnov distance, total variation distance, integral equation, numerical analysis, d p metric. Date. January 15, 2001; modied August 22, 2001. 1 Research supported by NSF grant DMS{9803780, and by the Acheson J. Duncan Fund for the Advancement of Research in Statistics. 1 1