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Efficient Sampling of Increasing Trees

Authors Nadja Azzouz , Olivier Bodini , Francis Durand , Bernhard Gittenberger

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LIPIcs.AofA.2026.3.pdf
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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorial algorithms
  • Mathematics of computing → Enumeration
  • Mathematics of computing → Generating functions
Keywords
  • sampling algorithms
  • bit-complexity
  • increasing trees
  • generating functions

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Abstract

We present a new exact-size sampler for increasing trees that outputs a tree of size n uniformly at random while avoiding the global coefficient pre-computation required by the classical recursive method of Flajolet et al. [Philippe Flajolet et al., 1994]. The key idea is a hybrid oracle-driven rejection scheme in which local sampling decisions are made using interval bounds on the coefficients, with a fallback to exact recurrence computation only on rare ambiguous events. In the bit-complexity model this yields an expected running time of O(nlog n) and it consumes a number of random bits within O(n) of the Shannon entropy, which is information-theoretically optimal up to lower-order terms. The sampler proceeds in two phases. We first generate the unlabeled rooted ordered shape by recursively sampling node arities and subtree sizes and then draw a uniform permutation of {1,…,n} and apply a deterministic increasing-labeling procedure.

Cite As Get BibTex

Nadja Azzouz, Olivier Bodini, Francis Durand, and Bernhard Gittenberger. Efficient Sampling of Increasing Trees. In 37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 381, pp. 3:1-3:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.AofA.2026.3

Author Details

Nadja Azzouz
  • Institute of Discrete Mathematics and Geometry, TU Wien, Austria
Olivier Bodini
  • Institut Galilée, Univ. Paris Nord, France
Francis Durand
  • Institut Galilée, Univ. Paris Nord, France
Bernhard Gittenberger
  • Institute of Discrete Mathematics and Geometry, TU Wien, Austria

Funding

  • Azzouz, Nadja: supported by the Austrian Science Foundation FWF, grant I-6744-N.
  • Bodini, Olivier: supported by the Agence Nationale de Recherche, grant ANR-23-CE48-0014
  • Durand, Francis: supported by the Agence Nationale de Recherche, grant ANR-23-CE48-0014
  • Gittenberger, Bernhard: supported by the Austrian Science Foundation FWF, grant I-6744-N.

References

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