Linear-time uniform generation of random sparse contingency tables with specified marginals

TL;DR

This paper presents a linear-time algorithm for uniformly generating sparse contingency tables with fixed marginals, improving efficiency over previous approximate methods based on Markov chain Monte Carlo.

Contribution

The authors introduce a novel linear-time algorithm for exact uniform sampling of sparse contingency tables with specified marginals, under certain sparsity conditions.

Findings

Algorithm runs in expected linear time when ^4< M/5.

Provides exact uniform samples, unlike previous approximate methods.

Outperforms existing algorithms in efficiency for sparse tables.

Abstract

We give an algorithm that generates a uniformly random contingency table with specified marginals, i.e. a matrix with non-negative integer values and specified row and column sums. Such algorithms are useful in statistics and combinatorics. When $\Delta^4< M/5$, where $\Delta$ is the maximum of the row and column sums and $M$ is the sum of all entries of the matrix, our algorithm runs in time linear in $M$ in expectation. Most previously published algorithms for this problem are approximate samplers based on Markov chain Monte Carlo, whose provable bounds on the mixing time are typically polynomials with rather large degrees.