arXiv subjects

Jasper Ischebeck

Publications and source records attributed to Jasper Ischebeck.

5 recordsLinked to original sources

arXiv2026

Fringe subtrees of split trees and fractional split trees

We consider additive functionals $X_n(\phi)$ with small toll functions on split trees and a generalization of split trees, which we call fractional split trees, where the split vector does not need to sum up to 1. These additive functionals encompass e.g. the number of nodes, number of leaves and the number of fringe trees of a certain size. We show convergence of the first moment to a limit, which we can explicitly compute if $s_0=s_1=0$ and for some models with Beta-distributed splitter. For $s_0+s_1>0$, the first moment is given in terms of negative moments of a perpetuity and can often be approximated to arbitrary precision with known bounds. In split trees and certain fractional split trees, the standard deviation is of smaller order than the first moment, where we show a weak law of large numbers. In other fractional split trees, the standard deviation is of the same order and we show a distribution limit using the contraction method.

math.PR↗

arXiv2025

Probabilistic analysis of optimal multi-pivot QuickSort

We consider a multi-pivot QuickSort algorithm using $K\in\mathbb{N}$ pivot elements to partition a nonsorted list into $K+1$ sublists in order to proceed recursively on these sublists. For the partitioning stage, various strategies are in use. We focus on the strategy that minimizes the expected number of key comparisons in the standard random model, where the list is given as a uniformly permuted list of distinct elements. We derive asymptotic expansions for the expectation and variance of the number of key comparisons as well as a limit law for all $K\in\mathbb{N}$, where the convergence holds for all (exponential) moments. For $K\le 4$ we also bound the rate of convergence within the Wasserstein and Kolmogorov--Smirnov distance. Our analysis of the expectation is based on classical results for random $m$-ary search trees. For the remaining results, combinatorial considerations are used to make the contraction method applicable.

math.PR↗

arXiv2024

On fine fluctuations of the complexity of the QuickSelect algorithm

The Quickselect algorithm (also called FIND) is a fundamental algorithm for selecting ranks or quantiles within a set of data. Gr\"ubel and R\"osler showed that the number of key comparisons required by Quickselect considered as a process of the quantiles $\alpha\in[0,1]$ converges within a natural probabilistic model after normalization in distribution within the c\`adl\`ag space $D[0,1]$ endowed with the Skorokhod metric. We show that the residual process in the latter convergence after normalization converges in distribution towards a mixture of Gaussian processes in $D[0,1]$. A similar result holds for the related algorithm QuickVal. Our method is applicable to other cost measures such as the number of swaps (key exchanges) required by Quickselect, or cost measures being based on key comparisons taking additionally into account that the cost of a comparison between two keys may depend on their values, an example being the number of bit comparisons needed to compare keys given by their bit expansions. For all the arising mixtures of Gaussian limit processes, we also discuss the H\"older continuity of their paths.

math.PR↗

arXiv2023

Ancestral reproductive bias in continuous time branching trees under various sampling schemes

Cheek and Johnston (Journal of Mathematical Biology, 2023) consider a continuous-time Bienaym\'e-Galton-Watson tree conditioned on being alive at time $T$. They study the reproduction events along the ancestral lineage of an individual randomly sampled from all those alive at time $T$. We give a short proof of an extension of their main results to the more general case of Bellman-Harris processes. Our proof also sheds light onto the probabilistic structure of the rate of the reproduction events. A similar method will be applied to explain (i) the different ancestral reproduction bias appearing in work by Geiger (Journal of Applied Probability, 1999) and (ii) the fact that the sampling rule considered by Chauvin, Rouault and Wakolbinger (Stochastic Processes and their Applications, 1991) leads to a time homogeneous process along the ancestral lineage.

math.PR↗

arXiv2023

Central limit theorems for additive functionals in patricia tries

We give theorems about asymptotic normality of general additive functionals on patricia tries in an i.i.d. setting, derived from results on tries by Janson (2022). These theorems are applied to show asymptotic normality of the distribution of random fringe trees in patricia tries. Formulas for asymptotic mean and variance are given. The proportion of fringe trees with $k$ keys is asymptotically, ignoring oscillations, given by $(1-\rho(k))/(H+J)k(k-1)$ with the source entropy $H$, an entropy-like constant $J$, that is $H$ in the binary case, and an exponentially decreasing function $\rho(k)$. Another application gives asymptotic normality of the independence number.

math.PR↗