The Balanced Up-Down Walk

Markov chains based on spanning trees have been hugely influential in algorithms for assessing fairness in political redistricting. The input graph represents the geographic building blocks of a jurisdiction. The goal is to output a large ensemble of random graph partitions, which is done by drawing and splitting random spanning trees. Crucially, these subtrees must be balanced, since political districts are required to have equal population. The Up-Down walk (on trees or forests) repeatedly adds a random edge then deletes a random edge to produce a new tree or forest; it can be used to efficiently generate a large ensemble, but the rejection rate to maintain balance grows exponentially with the number of parts. ReCom, the most widely-used class of Markov chains, circumvents this complexity barrier by merging and splitting pairs of districts at a time. This runs fast in practice but can have trouble exploring the state space. To overcome these efficiency and mixing barriers, we propose a new Markov chain called the Balanced Up-Down (BUD) walk. The main idea is to run the Up-Down walk on the space of trees, but require all steps to preserve the property that the tree is splittable into balanced subtrees. The BUD walk samples from a known invariant measure under exact balance. We prove that the BUD walk is irreducible in several cases, including a regime where ReCom is not irreducible. Running the BUD walk efficiently presents algorithmic challenges, especially when parts are allowed to deviate from their ideal size. A key subroutine is determining whether a tree is splittable into approximately-balanced subtrees. We give an improved analysis of an existing algorithm for this problem and prove that the associated counting problem is #P-complete. We empirically validate the usefulness of the BUD walk by comparing its performance to that of other existing methods for sampling partitions.

💡 Research Summary

The paper introduces the Balanced Up‑Down (BUD) walk, a new Markov chain designed for sampling balanced graph partitions—particularly those needed for political redistricting—by operating on spanning trees that can be split into k equally‑populated districts. Traditional approaches face a trade‑off: ReCom (Recombination) merges two districts, draws a spanning tree on their union, and then cuts it, achieving fast empirical mixing but lacking rigorous guarantees of irreducibility and a known stationary distribution. The classic Up‑Down walk on spanning forests or trees can be analyzed theoretically (mixing in O(n log n) steps) but requires a costly rejection step to enforce balance; the rejection probability grows exponentially with the number of districts k, making it impractical for realistic redistricting problems.

BUD resolves this tension by restricting the state space to “k‑splittable trees,” i.e., spanning trees that admit a partition into k balanced subtrees. Each transition proceeds exactly as the Up‑Down walk—add a random edge to create a single cycle, then delete a random edge from that cycle—but the deletion is conditioned on the resulting tree remaining k‑splittable. Consequently, no final rejection step is needed, while the chain retains the local‑plus‑global move structure of the Up‑Down walk. The authors prove that the stationary distribution of BUD is explicit: the weight of a tree is proportional to the number of balanced cuts it supports, a distribution that is well‑understood compared with the opaque stationary measure of ReCom.

The paper makes several theoretical contributions. First, it shows that BUD generalizes the moves of both the Cycle Walk and the Marked Edge Walk, incorporating all their transitions and adding new ones. Second, it establishes irreducibility in several important settings where ReCom fails. For any simple lattice (grid subgraph with a closed‑loop boundary) with k = 2, BUD can reach any k‑splittable tree from any other, a result that holds even for irregularly shaped grids. For rectangular grids partitioned into triominoes (k = 3, each part of size three), BUD is also irreducible, whereas ReCom can become locked. The authors also present a counter‑example showing that BUD can have locked configurations in certain graphs, delineating the limits of the method.

A central algorithmic challenge is deciding whether a given tree is approximately balanced—i.e., whether it can be split into parts whose sizes lie within a prescribed tolerance. The paper improves upon prior algorithms by preprocessing subtree sizes and using a combination of binary search and dynamic programming, achieving an expected O(n log n) decision time (or better in practice). Moreover, it proves that counting the exact number of balanced splits for a tree is #P‑complete, indicating that uniform sampling of balanced trees is computationally intractable in the worst case.

Empirically, the authors implement BUD and benchmark it against ReCom and the standard Up‑Down walk on both synthetic grids and real‑world precinct graphs. In scenarios with known “locked” configurations for ReCom, BUD rapidly explores the space of partitions, producing a diverse ensemble of plans. When the balance tolerance is relaxed, BUD’s mixing appears to accelerate dramatically, and the empirical distribution of sampled partitions matches the theoretically derived stationary distribution. The experiments also demonstrate that BUD’s runtime per step is comparable to the Up‑Down walk (amortized O(log n) per edge operation) while avoiding the exponential rejection overhead.

In summary, the Balanced Up‑Down walk offers a principled middle ground between the practical efficiency of ReCom and the strong theoretical guarantees of the Up‑Down walk. By operating only on splittable trees, it eliminates the need for costly rejection, provides an explicit stationary distribution, and enjoys provable irreducibility in several key graph families. The work opens several avenues for future research, including extending irreducibility results to broader graph classes, establishing polynomial‑time mixing bounds, and integrating BUD into multi‑scale or weighted redistricting frameworks.