Pedagogical value (Gridlandia!)

At the limits of what we can exhaustively enumerate

Sometimes resembles actual states

Goal (6x6 grid → 6 districts, 12 club voters):

evaluate 1,251,677,700 vote distributions over 451,206 districtings

Naively: 564,764,488,306,200 pairings

Decompose districtings into polyominoes.

these are considered to be distinct.

Approach (6x6 → 6):

Decompose ~450,000 districtings into ~2,500 polyominoes

Remove vote distributions that are redundant by symmetry

Compute all polyomino-vote distribution pairings to find best and worst expected seat shares

Solved in ~3 hours over hundreds of cores using Distributed.jl!

code: github.com/pjrule/OptimalVotes.jl

Full enumeration is rather excessive.

(We did it for a very specific reason.)

In practice, the win probability of a polyomino can be estimated via (cheap) uniform sampling of vote distributions.

*Julia notebooks available upon request

not ∞, so connected

read: degree = number of neighbors

Computational challenge: representing the plans alone requires several gigabytes of space (hard to store in memory)

Map/reduce architecture?

Know: 187,497,290,034 plans for 8x8 → 8

(~3 orders of magnitude)

Not outside the realm of possibility if we really wanted to (but why?)

New ZDD methods are thought to speed up enumeration (but may be hard to parallelize)

Conjecture: all balanced NxN → N partitions are ReCom-connected.

We know: the space of balanced NxN → N partitions is ReCom-connected. (N ≤ 6)

Question: is there a polynomial-time algorithm to reconfigure any plan into a canonical configuration

(e.g. vertical strips)?

(This type of question is challenging: for instance, we have tried and failed to produce a non-computational proof for the 6x6 → 3 case.)