The Snake Algorithm: A Rejection-Free Sampler for Binary Matrices with Fixed Margins

Abstract

We study uniform sampling of binary matrices with fixed row and column sums, a recurring problem in ecological null models, Rasch-model testing, network analysis, and combinatorics. We propose the Snake algorithm, a rejection-free Markov chain Monte Carlo sampler that grows an alternating path until its first self-intersection and flips the resulting loop. The chain is reversible and irreducible on the fixed-margin state space, hence has the uniform stationary distribution. We prove that one step flips on the order of $\sqrt{n}$ entries in sparse and balanced square regimes, give upper bounds on the per-step path length, and show that the resulting work per flipped entry is rate optimal in sparse and balanced regimes and near-optimal up to a polylogarithmic factor under a one-sided half-balanced condition. A Markov-chain comparison, combined with the recently established universal spectral-gap bound for the swap chain, proves that the lazy Snake chain is rapidly mixing for every feasible pair of margins; in the permutation-matrix case, the raw chain has the sharp total-variation mixing time $\Theta (n\log n)$. We also describe a directed-graph extension and an equal-margin label-shuffling variant. Numerical experiments against Swap, Rectangle Loop, Curveball, sequential importance sampling, and a directed edge-swap algorithm show consistent gains in move size, wall-clock convergence, and sampling efficiency.