Boltzmann Sampling for Powersets without an Oracle

TL;DR

The paper tackles sampling from powerset combinatorial classes under the Boltzmann model without evaluating the generating function. It introduces an occupancy-model view with a Poisson-number of elements and applies thinning to recover the Boltzmann distribution, enabling an oracle-free sampler for bounded counting sequences. The method is implemented and tested in Python, showing runtimes on par with existing Boltzmann samplers and validating against limit shapes for strict partitions. This yields a practical, oracle-free approach to Boltzmann sampling of powersets in settings with bounded counting sequences, broadening applicability and simplifying implementation.

Abstract

We show that powersets over structures with a bounded counting sequence can be sampled efficiently without evaluating the generating function. An algorithm is provided, implemented, and tested. Runtimes are comparable to existing Boltzmann samplers reported in the literature.

Figures (3)

• Figure 1: The orange hard line represents a sampled partition, and the blue dotted line represents the scaling limit. On the left panel the partition is of size $100$ and on the right panel it is of size $1\,000\,000$.
• Figure 2: Sampling times (in $ms$) for the free sampler (left panel) and the free sampler (right panel). The shaded region represents the interval between the centiles $10$ and $90$.
• Figure 3: The orange hard line represents a sampled partition, and the blue dotted line represents the scaling limit. On the left panel the partition has been sampled with $\mathbb{E}(M)=50$, $\mathbb{E}(N)=10^9$ and on the right panel it is has been sampled with $\mathbb{E}(M)=100$, $\mathbb{E}(N)=10^{12}$.

Theorems & Definitions (13)

• Definition 1
• Definition 2
• Definition 3
• Proposition 4
• proof
• Proposition 5
• proof
• Proposition 6
• proof
• Proposition 7