Sampling tree-weighted partitions without sampling trees

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.