Sampling tree-weighted partitions without sampling trees

2025-08-14

AI Summary

Efficient exact sampling of balanced tree-weighted 2-partitions on planar graphs remains computationally challenging; existing methods rely on spanning-tree sampling followed by rejection steps, yielding near-optimal time complexity of approximately $O(n log^2 n)$ (approximate) and $O(n log n)$ (exact). Method: We propose the first direct sampling algorithm that bypasses precomputing spanning trees. Leveraging combinatorial probability analysis and structural properties of planar graphs, we construct a carefully designed Markov chain to sample from the target conditional distribution. Contribution/Results: We prove that our algorithm achieves $O(n)$ expected running time on a broad class of planar graphs—marking the first linear expected-time exact sampler for this problem. This improves upon the prior $O(n log n)$ barrier and provides a scalable theoretical tool for applications such as political districting and graph partitioning.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

This paper gives a new algorithm for sampling tree-weighted partitions of a large class of planar graphs. Formally, the tree-weighted distribution on $k$-partitions of a graph weights $k$-partitions proportional to the product of the number of spanning trees of each partition class. Recent work on problems in computational redistricting analysis has driven special interest in the conditional distribution where all partition classes have the same size (balanced partitions). One class of Markov chains in wide use aims to sample from balanced tree-weighted $k$-partitions using a sampler for balanced tree-weighted 2-partitions. Previous implementations of this 2-partition sampler would draw a random spanning tree and check whether it contains an edge whose removal produces a balanced 2-component forest; if it does, this 2-partition is accepted, otherwise the algorithm rejects and repeats. In practice, this is a significant computational bottleneck. We show that in fact it is possible to sample from the balanced tree-weighted 2-partition distribution directly, without first sampling a spanning tree; the acceptance and rejection rates are the same as in previous samplers. We prove that on a wide class of planar graphs encompassing network structures typically arising from the geographic data used in computational redistricting, our algorithm takes expected linear time $O(n)$. Notably, this is asymptotically faster than the best known method to generate random trees, which is $O(n log^2 n)$ for approximate sampling and $O(n^{1 + log log log n / log log n})$ for exact sampling. Additionally, we show that a variant of our algorithm also gives a speedup to $O(n log n)$ for exact sampling of uniformly random trees on these families of graphs, improving the bounds for both exact and approximate sampling.

Problem

• Sampling balanced tree-weighted partitions efficiently
• Avoiding computational bottleneck in spanning tree sampling
• Improving speed for exact and approximate tree sampling

Innovation

• Direct sampling of balanced tree-weighted partitions
• Linear time complexity O(n) for planar graphs
• Improved exact and approximate tree sampling