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Optimal Generation of Strictly Increasing Binary Trees and Beyond

Olivier Bodini, Francis Durand, Philippe Marchal

TL;DR

The paper tackles the problem of efficiently and uniformly generating random strictly increasing trees, with emphasis on binary-increasing structures. It introduces two complementary methods: an ad hoc sampler that reduces the problem to generating alternating permutations via a tangent-free Marchal-based scheme, and a recursive Monte Carlo-based sampler that avoids full precomputation of tree counts by exploiting a closed-form tangent generating function. The primary contributions are explicit formulas for the number of trees $t_n$, a constant-rejection sampling strategy for the left-subtree size $M$ with $O(\log n)$ time, and a generalization of the approach to broader families of binary-increasing trees, including binary and binary-unary variants. These results yield entropy-optimal generation with $O(n\log n)$ time and reduced randomness, enabling scalable generation of complex tree families.

Abstract

This article presents two novel algorithms for generating random increasing trees. The first algorithm efficiently generates strictly increasing binary trees using an ad hoc method. The second algorithm improves the recursive method for weighted strictly increasing unary-binary increasing trees, optimizing randomness usage.

Optimal Generation of Strictly Increasing Binary Trees and Beyond

TL;DR

The paper tackles the problem of efficiently and uniformly generating random strictly increasing trees, with emphasis on binary-increasing structures. It introduces two complementary methods: an ad hoc sampler that reduces the problem to generating alternating permutations via a tangent-free Marchal-based scheme, and a recursive Monte Carlo-based sampler that avoids full precomputation of tree counts by exploiting a closed-form tangent generating function. The primary contributions are explicit formulas for the number of trees , a constant-rejection sampling strategy for the left-subtree size with time, and a generalization of the approach to broader families of binary-increasing trees, including binary and binary-unary variants. These results yield entropy-optimal generation with time and reduced randomness, enabling scalable generation of complex tree families.

Abstract

This article presents two novel algorithms for generating random increasing trees. The first algorithm efficiently generates strictly increasing binary trees using an ad hoc method. The second algorithm improves the recursive method for weighted strictly increasing unary-binary increasing trees, optimizing randomness usage.
Paper Structure (8 sections, 3 theorems, 10 equations, 4 algorithms)

This paper contains 8 sections, 3 theorems, 10 equations, 4 algorithms.

Table of Contents

  1. Introduction
  2. An Ad Hoc Approach for Sampling Strictly Increasing Trees
  3. Toward an Efficient Recursive Methods Without Preliminary Calculations
  4. Generating $M$, the size of the left son
  5. Having a direct expression for $t_n$
  6. Adapting the Algorithm for Different Families of Binary-Increasing Trees
  7. Binary Trees
  8. Binary-Unary Trees

Key Result

Lemma 1

For $k\in \{1,2,..,l -1 \}$, $f_1 \geq f_k \geq f_{\lfloor \frac{l}{2} \rfloor}$ and $\frac{f_1}{f_{\lfloor \frac{l}{2} \rfloor}} \leq \frac{4\pi^4}{45} \leq 9$

Theorems & Definitions (3)

  • Lemma 1
  • Lemma 2
  • Theorem 1