# A Parallel Algorithm for Sampling Matchings from an Almost Uniform Distribution

## Abstract

In this paper we present a randomized parallel algorithm to sample matchings from an almost uniform distribution on the set of matchings of all sizes in a graph. First we prove that the direct NC simulation of the sequential Markov chain technique for this problem is P-complete. Afterwards we present a randomized parallel algorithm for the problem. The technique used is based on the definition of a genetic system that converges to the uniform distribution. The system evolves according to a non-linear equation. Little is known about the convergence of these systems. We can define a non-linear system which converges to a stationary distribution under quite natural conditions. We prove convergence for the system corresponding to the almost uniform sampling of matchings in a graph (up to know the only known convergence for non-linear systems for matchings was matchings on a tree 5). We give empirical evidence that the system converges faster, in polylogarithmic parallel time.

## Keywords

Markov Chain Initial Distribution Parallel Algorithm Genetic System Markov Chain Monte Carlo Method## Preview

Unable to display preview. Download preview PDF.

## References

- 1.M. Jerrum and A. Sinclair.
*The Markov Chain Monte Carlo method: An approach to approximate counting and integration*, pages482–520PWS, Boston1995.Google Scholar - 2.R. Kannan. Markov chains and polynomial time algorithms. In
*35th IEEE Symposium on Foundations of Computer Science*, pages656–6711994.Google Scholar - 3.R. Motwani and P. Raghavan.
*Randomized Algorithms*. Cambridge University Press, 1995.Google Scholar - 4.Y. Rabani, Y. Rabinovich, and A. Sinclair. A computational view of population genetics. In
*27th ACM Symposium on Theory of Computing*, pages 83–921995.Google Scholar - 5.Y. Rabinovich, A. Sinclair, and A. Wigderson. Quadratic dynamical systems. In
*33th IEEE Symposium on Foundations of Computer Science*, pages 304–3131992.Google Scholar - 6.A. Renyi.
*Probability Theory*North-Holland, Amsterdam1970.Google Scholar - 7.A. Sinclair.
*Algorithm for random generation and counting: A Markov chain approach*. Birkhäuser, Boston1993.Google Scholar - 8.Shang-Hua Teng. Independent sets versus perfect matchings.
*Theoretical Computer Science*, pages1–10 1995. Amrican Mathematical societyGoogle Scholar - 9.V. Vazirani. Rapidly mixing Markov chainsIn B. Bollobas editor
*Probabilistic combinatorics and its applications*, pages 99–121. 1991.Google Scholar