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

We give an algorithm which generates a uniformly random contingency table with specified marginals, that is, a matrix with nonnegative integer values and specified row and column sums. Such algorithms are useful in statistics and combinatorics. When Δ4<M/5, where Δ 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.