A general limit theorem for recursive algorithms and combinatorial structures

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 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 2 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.

Generating Random Permutations by Coin Tossing

“…Asymptotic Normality. By applying either the contraction method (see Neininger and Rüschendorf [2004]) or the refined method of moments (see Hwang [2003]), we can establish the convergence in distribution of the centered and normalized random variables (X n − μ n )/σ n to the standard normal distribution, where μ n := E(X n ) and σ 2 n := V(X n ). The latter is also useful in providing stronger results such as the following.…”

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“…The fixed-point problem in Theorem 1.1 mirrors the execution of a message passing algorithm on the Galton-Watson tree. The study of this fixed-point problem, for which we use the contraction method [41], is the key technical ingredient of our proof. We believe that this strategy provides an elegant framework for tackling many other problems in the theory of random graphs as well.…”

On rates of convergence in the probabilistic analysis of algorithms

“…As already mentioned, recurrences of this form come up in various fields, see Rösler and Rüschendorf [83] and Neininger and Rüschendorf [70] for many concrete examples. Just to name a few, possible applications range from complexity measures of recursive algorithms (e.g., the number of key comparisons used by Quicksort, Mergesort or Quickselect) to parameters of random trees (e.g., the size of tries and m-ary search trees, path lengths in digital search trees, (PATRICIA) tries and m-ary search trees or the number of leaves in quadtrees) to quantities of stochastic geometry (e.g., the number of maxima in right triangles).…”

“…Up to the relaxed independence assumption (1.4), this is the framework of Neininger and Rüschendorf [70], where some general convergence results are shown for appropriate normalizations of the Y n . The content of this thesis is to additionally study the rates of convergence in such general limit theorems.…”

“…The name of the method refers to the fact that the limiting distribution is characterized as the fixed-point of a contracting map of measures. A general limit theorem for recursive algorithms and combinatorial structures based on the contraction method as well as numerous applications can be found in Neininger and Rüschendorf [70].…”