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Bonsai: A class of effective methods for independent sampling of graph partitions

Jeanne Clelland, Kristopher Tapp

Abstract

We develop effective methods for constructing an ensemble of district plans via independent sampling from a reasonable probability distribution on the space of graph partitions. We compare the performance of our algorithms to that of standard Markov Chain based algorithms in the context of grid graphs and state congressional and legislative maps. For the case of perfect population balance between districts, we provide an explicit description of the distribution from which our method samples.

Bonsai: A class of effective methods for independent sampling of graph partitions

Abstract

We develop effective methods for constructing an ensemble of district plans via independent sampling from a reasonable probability distribution on the space of graph partitions. We compare the performance of our algorithms to that of standard Markov Chain based algorithms in the context of grid graphs and state congressional and legislative maps. For the case of perfect population balance between districts, we provide an explicit description of the distribution from which our method samples.
Paper Structure (13 sections, 3 theorems, 17 equations, 12 figures, 1 table, 4 algorithms)

This paper contains 13 sections, 3 theorems, 17 equations, 12 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Set-up
  3. Complete Cut
  4. Bonsai: A practical algorithm for independent sampling
  5. Bonsai for partitions with perfect population balance
  6. Bonsai for partitions with imperfect population balance
  7. Variations
  8. Empirical results
  9. $7 \times 7$ grid into $7$ districts
  10. The $50 \times 50$ grid
  11. Pennsylvania and North Carolina VTD ensembles
  12. Conclusion
  13. The distribution from which Algorithm \ref{['alg:perfect-balance-1']} samples

Key Result

Proposition 1

The probability that Complete Cut selects the partition $\mathcal{P} = \{D_1,...,D_k\}$ is where $\text{ST}(\mathcal{G}/\mathcal{P})$ denotes the number of spanning trees on the quotient multi-graph $\mathcal{G}/\mathcal{P}$, in which, for each $i$, all vertices of $D_i$ are identified.

Figures (12)

  • Figure 1: Bonsai algorithm for partitioning a $6 \times 6$ grid graph into $6$ equal-size districts: (a) uniformly sampled spanning tree; (b) valid cut edges on spanning tree; (c) partial partition of graph; (d) uniformly sampled spanning trees on double-district-sized-pieces; (e) valid cut edges on spanning trees; (f) completed partition into districts.
  • Figure 2: When Bonsai gets stuck: (a) a partial partition that contains an uncuttable piece; (b) uncuttable piece merged with piece from prior cut; (c) new uniformly sampled spanning tree on merged piece; (d) valid cut edges on new spanning tree; (e) completed partition into districts.
  • Figure 3: A toy example for which Algorithm \ref{['alg:imperfect-balance-3']} fails when "best" is defined as "most balanced".
  • Figure 4: Ensemble statistics for plan-wide cut edges for Bonsai vs. Complete Cut, (a) using minimum spanning trees, (b) using uniform spanning trees, for $7$-district plans on a $7 \times 7$ grid
  • Figure 5: Ensemble statistics for plan-wide cut edges for Bonsai vs. ReCom variants, (a) using minimum spanning trees, (b) using uniform spanning trees, for $7$-district plans on a $7 \times 7$ grid
  • ...and 7 more figures

Theorems & Definitions (9)

  • Proposition 1
  • proof
  • Example 1
  • Example 2
  • Example 3
  • Proposition 2
  • proof
  • Proposition 3
  • Example 4