A simple Markov chain for independent Bernoulli variables conditioned on their sum

We consider a vector of $N$ independent binary variables, each with a different probability of success. The distribution of the vector conditional on its sum is known as the conditional Bernoulli distribution. Assuming that $N$ goes to infinity and that the sum is proportional to $N$, exact sampling costs order $N^2$, while a simple Markov chain Monte Carlo algorithm using 'swaps' has constant cost per iteration. We provide conditions under which this Markov chain converges in order $N \log N$ iterations. Our proof relies on couplings and an auxiliary Markov chain defined on a partition of the space into favorable and unfavorable pairs.