Local Rejection Sampling with Soft Filters
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We study a new LLL-like framework for rejection sampling, which naturally generalizes the variable-framework for the Lovász local lemma (LLL) and the LLL-based Partial Rejection Sampling of Guo, Jerrum and Liu [STOC'17]. In this new framework, the occurrence of a bad event is biased by the current evaluation of variables, rather than being determined by them as in the standard setting of LLL. Such generalization allows us to model rejection samplings with soft filters. We give a generic algorithm, called Local Rejection Sampling, for sampling from the correct distribution arising from the rejection rules. For rejection sampling with soft filters, this algorithm only updates variables which are local to the violated events in the dependency graph, making it suitable for sampling with local computation or on dynamic input. As a nontrivial consequence, for any distribution defined by local constraints, if a decay of correlation property known as the strong spatial mixing holds, there is a distributed zero-error Las Vegas algorithm for sampling precisely from this distribution within O(^3 n) rounds.
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