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Sarah Cannon

Publications and source records attributed to Sarah Cannon.

At least 19 recordsLinked to original sources
arXiv2026

Statistical Assessments of Representational Reforms: A Case Study from Los Angeles

Electoral reforms -- including changes to district boundaries, electoral rules, and the size of elected bodies -- are a recurrent feature of American state and municipal politics. The representational impact of such reforms is challenging to assess before they are implemented, and even well-intentioned civic reformers are often surprised by the unintended consequences of changes to the political system. We develop a statistical framework for the prospective evaluation of electoral reforms by integrating several modes of analysis: descriptive statistics of the voter file, probabilistic race and ethnicity imputation, ecological inference, and redistricting simulations via ensembles. Together, these tools provide a unified assessment of how proposed reforms affect whose voices are ultimately heard in government. We apply this framework to reforms currently being considered for the Los Angeles City Council, including increasing the number of single-member districts, adopting ranked-choice voting, and introducing multimember districts elected by proportional ranked choice voting. We find that expanding the Council alone is unlikely to substantially improve representation for underrepresented communities. In contrast, eliminating low-turnout primary elections and adopting multimember districts produce larger improvements in representational equity. Our analyses were shared with L.A.'s Charter Reform Commission via public testimony in late 2025. Although motivated by Los Angeles, the framework is broadly applicable to evaluating prospective electoral reforms across American jurisdictions.

stat.AP↗
arXiv2026

Census Dual Graphs: Properties and Random Graph Models

In the computational study of political redistricting, feasibility necessitates the use of a discretization of regions such as states, counties, and towns. In nearly all cases, researchers use a dual graph, whose vertices represent small geographic units (such as census blocks or voting precincts) with edges for geographic adjacency. A political districting plan is a partition of this graph into connected subgraphs that satisfy certain additional properties, such as connectedness, compactness, and equal population. Though dual graphs underlie nearly all computational studies of political redistricting, little is known about their properties. This is a unique graph class that has been described colloquially as `nearly planar, nearly triangulated,' but thus far there has been a lack of evidence to support this description. In this paper we study dual graphs for counties, census tracts, and census block groups across the United States in order to understand and characterize this graph class. We also consider several random graph models (most based on randomly perturbing grids or Delauney triangulations of random point sets), and determine which most closely resemble dual graphs under key metrics. This work lays an initial foundation for understanding and modeling the properties of dual graphs; this will provide invaluable insight to researchers developing algorithms using them to understand, assess, and quantify the properties of political districting plans.

cs.DM↗
arXiv2026

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.

cs.DM↗
arXiv2025

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 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, rejecting if not. 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. We implement our algorithm and benchmark it on grid graphs, finding that it outperforms the standard bipartitioning method in the widely-used GerryChain library.

cs.DS↗
arXiv2024

Pirogov--Sinai Theory Beyond Lattices

Pirogov--Sinai theory is a well-developed method for understanding the low-temperature phase diagram of statistical mechanics models on lattices. Motivated by physical and algorithmic questions beyond the setting of lattices, we develop a combinatorially flexible version of Pirogov--Sinai theory for the hard-core model of independent sets. Our results illustrate that the main conclusions of Pirogov--Sinai theory can be obtained in significantly greater generality than that of $\mathbb Z^{d}$. The main ingredients in our generalization are combinatorial and involve developing appropriate definitions of contours based on the notion of cycle basis connectivity. This is inspired by works of Tim\'{a}r and Georgakopoulos--Panagiotis.

math.PR↗
arXiv2023

Sampling Balanced Forests of Grids in Polynomial Time

We prove that a polynomial fraction of the set of $k$-component forests in the $m \times n$ grid graph have equal numbers of vertices in each component, for any constant $k$. This resolves a conjecture of Charikar, Liu, Liu, and Vuong, and establishes the first provably polynomial-time algorithm for (exactly or approximately) sampling balanced grid graph partitions according to the spanning tree distribution, which weights each $k$-partition according to the product, across its $k$ pieces, of the number of spanning trees of each piece. Our result follows from a careful analysis of the probability a uniformly random spanning tree of the grid can be cut into balanced pieces. Beyond grids, we show that for a broad family of lattice-like graphs, we achieve balance up to any multiplicative $(1 \pm \varepsilon)$ constant with constant probability, and up to an additive constant with polynomial probability. More generally, we show that, with constant probability, components derived from uniform spanning trees can approximate any given partition of a planar region specified by Jordan curves. These results imply polynomial time algorithms for sampling approximately balanced tree-weighted partitions for lattice-like graphs. Our results have applications to understanding political districtings, where there is an underlying graph of indivisible geographic units that must be partitioned into $k$ population-balanced connected subgraphs. In this setting, tree-weighted partitions have interesting geometric properties, and this has stimulated significant effort to develop methods to sample them.

cs.DM↗
arXiv2023

Irreducibility of Recombination Markov Chains in the Triangular Lattice

In the United States, regions are frequently divided into districts for the purpose of electing representatives. How the districts are drawn can affect who's elected, and drawing districts to give an advantage to a certain group is known as gerrymandering. It can be surprisingly difficult to detect gerrymandering, but one algorithmic method is to compare a current districting plan to a large number of randomly sampled plans to see whether it is an outlier. Recombination Markov chains are often used for this random sampling: randomly choose two districts, consider their union, and split this union in a new way. This works well in practice, but the theory behind it remains underdeveloped. For example, it's not known if recombination Markov chains are irreducible, that is, if recombination moves suffice to move from any districting plan to any other. Irreducibility of recombination Markov chains can be formulated as a graph problem: for a graph $G$, is the space of all partitions of $G$ into $k$ connected subgraphs ($k$ districts) connected by recombination moves? We consider three simply connected districts and district sizes $k_1\pm 1$ vertices, $k_2\pm 1$ vertices, and $k3\pm 1$ vertices. We prove for arbitrarily large triangular regions in the triangular lattice, recombination Markov chains are irreducible. This is the first proof of irreducibility under tight district size constraints for recombination Markov chains beyond small or trivial examples.

cs.DM↗
arXiv2022

Spanning Trees and Redistricting: New Methods for Sampling and Validation

Deciding whether a political districting plan was distorted by a hidden agenda, or whether it dilutes the voting power of some group, requires a neutral baseline for comparison. Remarkably, all nine U.S. Supreme Court justices have now signed on to decisions that find that computational methods can provide key evidence. Today, the leading approaches for benchmarking districting plans are based on the use of spanning trees for sampling graph partitions. We present a new *reversible recombination* algorithm and rigorously prove its fundamental properties. Furthermore, we argue for a canonical sampling distribution called the *spanning tree distribution* that is well adapted to redistricting and provides a principled foundation for comparing and validating methods. Together with a highly efficient (and open-source) implementation that can generate and handle large datasets, this work provides the most powerful null model to date for the gerrymandering problem, meeting an urgent democratic challenge with sound scientific methodology.

physics.soc-ph↗
arXiv2022

Voting Rights, Markov Chains, and Optimization by Short Bursts

Finding outlying elements in probability distributions can be a hard problem. Taking a real example from Voting Rights Act enforcement, we consider the problem of maximizing the number of simultaneous majority-minority districts in a political districting plan. An unbiased random walk on districting plans is unlikely to find plans that approach this maximum. A common search approach is to use a biased random walk: preferentially select districting plans with more majority-minority districts. Here, we present a third option, called short bursts, in which an unbiased random walk is performed for a small number of steps (called the burst length), then re-started from the most extreme plan that was encountered in the last burst. We give empirical evidence that short-burst runs outperform biased random walks for the problem of maximizing the number of majority-minority districts, and that there are many values of burst length for which we see this improvement. Abstracting from our use case, we also consider short bursts where the underlying state space is a line with various probability distributions, and then explore some features of more complicated state spaces and how these impact the effectiveness of short bursts.

physics.soc-ph↗
arXiv2022

Fast and perfect sampling of subgraphs and polymer systems

We give an efficient perfect sampling algorithm for weighted, connected induced subgraphs (or graphlets) of rooted, bounded degree graphs. Our algorithm utilizes a vertex-percolation process with a carefully chosen rejection filter and works under a percolation subcriticality condition. We show that this condition is optimal in the sense that the task of (approximately) sampling weighted rooted graphlets becomes impossible in finite expected time for infinite graphs and intractable for finite graphs when the condition does not hold. We apply our sampling algorithm as a subroutine to give near linear-time perfect sampling algorithms for polymer models and weighted non-rooted graphlets in finite graphs, two widely studied yet very different problems. This new perfect sampling algorithm for polymer models gives improved sampling algorithms for spin systems at low temperatures on expander graphs and unbalanced bipartite graphs, among other applications.

cs.DS↗
arXiv2020

Programming Active Cohesive Granular Matter with Mechanically Induced Phase Changes

Active matter physics and swarm robotics have provided powerful tools for the study and control of ensembles driven by internal sources. At the macroscale, controlling swarms typically utilizes significant memory, processing power, and coordination unavailable at the microscale, e.g., for colloidal robots, which could be useful for fighting disease, fabricating intelligent textiles, and designing nanocomputers. To develop principles that that can leverage physics of interactions and thus can be utilized across scales, we take a two-pronged approach: a theoretical abstraction of self-organizing particle systems and an experimental robot system of active cohesive granular matter that intentionally lacks digital electronic computation and communication, using minimal (or no) sensing and control, to test theoretical predictions. We consider the problems of aggregation, dispersion, and collective transport. As predicted by the theory, as a parameter representing interparticle attraction increases, the robots transition from a dispersed phase to an aggregated one, forming a dense, compact collective. When aggregated, the collective can transport non-robot "impurities" in their environment, thus performing an emergent task driven by the physics underlying the transition. These results point to a fruitful interplay between algorithm design and active matter robophysics that can result in new nonequilibrium physics and principles for programming collectives without the need for complex algorithms or capabilities.

cond-mat.soft↗
arXiv2019

Counting independent sets in unbalanced bipartite graphs

We give an FPTAS for approximating the partition function of the hard-core model for bipartite graphs when there is sufficient imbalance in the degrees or fugacities between the sides $(L,R)$ of the bipartition. This includes, among others, the biregular case when $λ=1$ (approximating the number of independent sets of $G$) and $Δ_R \geq 7Δ_L \log(Δ_L)$. Our approximation algorithm is based on truncating the cluster expansion of a polymer model partition function that expresses the hard-core partition function in terms of deviations from independent sets that are empty on one side of the bipartition. As a consequence of the method, we also prove that the hard-core model on such graphs exhibits exponential decay of correlations by utilizing connections between the cluster expansion and joint cumulants.

cs.DS↗
arXiv2019

A Markov Chain Algorithm for Compression in Self-Organizing Particle Systems

In systems of programmable matter, we are given a collection of simple computation elements (or particles) with limited (constant-size) memory. We are interested in when they can self-organize to solve system-wide problems of movement, configuration and coordination. Here, we initiate a stochastic approach to developing robust distributed algorithms for programmable matter systems using Markov chains. We are able to leverage the wealth of prior work in Markov chains and related areas to design and rigorously analyze our distributed algorithms and show that they have several desirable properties. We study the compression problem, in which a particle system must gather as tightly together as possible, as in a sphere or its equivalent in the presence of some underlying geometry. More specifically, we seek fully distributed, local, and asynchronous algorithms that lead the system to converge to a configuration with small boundary. We present a Markov chain-based algorithm that solves the compression problem under the geometric amoebot model, for particle systems that begin in a connected configuration. The algorithm takes as input a bias parameter $λ$, where $λ> 1$ corresponds to particles favoring having more neighbors. We show that for all $λ> 2+\sqrt{2}$, there is a constant $α> 1$ such that eventually with all but exponentially small probability the particles are $α$-compressed, meaning the perimeter of the system configuration is at most $α\cdot p_{min}$, where $p_{min}$ is the minimum possible perimeter of the particle system. Surprisingly, the same algorithm can also be used for expansion when $0 < λ< 2.17$, and we prove similar results about expansion for values of $λ$ in this range. This is counterintuitive as it shows that particles preferring to be next to each other ($λ> 1$) is not sufficient to guarantee compression.

cs.DC↗
arXiv2018

A Stochastic Approach to Shortcut Bridging in Programmable Matter

In a self-organizing particle system, an abstraction of programmable matter, simple computational elements called particles with limited memory and communication self-organize to solve system-wide problems of movement, coordination, and configuration. In this paper, we consider a stochastic, distributed, local, asynchronous algorithm for "shortcut bridging", in which particles self-assemble bridges over gaps that simultaneously balance minimizing the length and cost of the bridge. Army ants of the genus Eciton have been observed exhibiting a similar behavior in their foraging trails, dynamically adjusting their bridges to satisfy an efficiency trade-off using local interactions. Using techniques from Markov chain analysis, we rigorously analyze our algorithm, show it achieves a near-optimal balance between the competing factors of path length and bridge cost, and prove that it exhibits a dependence on the angle of the gap being "shortcut" similar to that of the ant bridges. We also present simulation results that qualitatively compare our algorithm with the army ant bridging behavior. Our work gives a plausible explanation of how convergence to globally optimal configurations can be achieved via local interactions by simple organisms (e.g., ants) with some limited computational power and access to random bits. The proposed algorithm also demonstrates the robustness of the stochastic approach to algorithms for programmable matter, as it is a surprisingly simple extension of our previous stochastic algorithm for compression.

cs.DC↗
arXiv2018

A Local Stochastic Algorithm for Separation in Heterogeneous Self-Organizing Particle Systems

We present and rigorously analyze the behavior of a distributed, stochastic algorithm for separation and integration in self-organizing particle systems, an abstraction of programmable matter. Such systems are composed of individual computational particles with limited memory, strictly local communication abilities, and modest computational power. We consider heterogeneous particle systems of two different colors and prove that these systems can collectively separate into different color classes or integrate, indifferent to color. We accomplish both behaviors with the same fully distributed, local, stochastic algorithm. Achieving separation or integration depends only on a single global parameter determining whether particles prefer to be next to other particles of the same color or not; this parameter is meant to represent external, environmental influences on the particle system. The algorithm is a generalization of a previous distributed, stochastic algorithm for compression (PODC '16), which can be viewed as a special case of separation where all particles have the same color. It is significantly more challenging to prove that the desired behavior is achieved in the heterogeneous setting, however, even in the bichromatic case we focus on. This requires combining several new techniques, including the cluster expansion from statistical physics, a new variant of the bridging argument of Miracle, Pascoe and Randall (RANDOM '11), the high-temperature expansion of the Ising model, and careful probabilistic arguments.

cs.DC↗
arXiv2017

Phototactic Supersmarticles

Smarticles, or smart active particles, are small robots equipped with only basic movement and sensing abilities that are incapable of rotating or displacing individually. We study the ensemble behavior of smarticles, i.e., the behavior a collective of these very simple computational elements can achieve, and how such behavior can be implemented using minimal programming. We show that an ensemble of smarticles constrained to remain close to one another (which we call a supersmarticle), achieves directed locomotion toward or away from a light source, a phenomenon known as phototaxing. We present experimental and theoretical models of phototactic supersmarticles that collectively move with a directed displacement in response to light. The motion of the supersmarticle is approximately Brownian, and is a result of chaotic interactions among smarticles. The system can be directed by introducing asymmetries among the individual smarticle's behavior, in our case by varying activity levels in response to light, resulting in supersmarticle biased motion.

cs.RO↗
arXiv2016

Polynomial mixing of the edge-flip Markov chain for unbiased dyadic tilings

We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall, and Spencer in 2002. A dyadic tiling of size n is a tiling of the unit square by n non-overlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the form [a2^{-s}, (a+1)2^{-s}] \times [b2^{-t}, (b+1)2^{-t}] for non-negative integers a,b,s,t. The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling. Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n^{4.09}), which implies that the mixing time is at most O(n^{5.09}). We complement this by showing that the relaxation time is at least Ω(n^{1.38}), improving upon the previously best lower bound of Ω(n\log n) coming from the diameter of the chain.

math.PR↗