The local computation of Linial [FOCS'87] and Naor and Stockmeyer [STOC'93] concerns with the question of whether a locally definable distributed computing problem can be solved locally: for a given local CSP whether a CSP solution can be constructed by a distributed algorithm using local information. In this paper, we consider the problem of sampling a uniform CSP solution by distributed algorithms, and ask whether a locally definable joint distribution can be sampled from locally. More broadly, we consider sampling from Gibbs distributions induced by weighted local CSPs in the LOCAL model. We give two Markov chain based distributed algorithms which we believe to represent two fundamental approaches for sampling from Gibbs distributions via distributed algorithms. The first algorithm generically parallelizes the single-site sequential Markov chain by iteratively updating a random independent set of variables in parallel, and achieves an $O(\Delta\log n)$ time upper bound in the LOCAL model, where $\Delta$ is the maximum degree, when the Dobrushin's condition for the Gibbs distribution is satisfied. The second algorithm is a novel parallel Markov chain which proposes to update all variables simultaneously yet still guarantees to converge correctly with no bias. It surprisingly parallelizes an intrinsically sequential process: stabilizing to a joint distribution with massive local dependencies, and may achieve an optimal $O(\log n)$ time upper bound independent of the maximum degree $\Delta$ under a stronger mixing condition. We also show a strong $\Omega(diam)$ lower bound for sampling independent set in graphs with maximum degree $\Delta\ge 6$. This lower bound holds even when every node is aware of the graph. This gives a strong separation between sampling and constructing locally checkable labelings.
翻译:Linial [FOCS'87] 和 Naor 和 Stockmeyer [STOC'93] 的本地计算对本地可定义的可定义分布计算问题能否在本地解决的问题感到关切:对于特定本地的 CSP 本地的 CSP 解决方案能否使用本地信息的分布算法构建。 在本文中, 我们考虑通过分布算法对统一的 CSP 解决方案进行抽样的问题, 并询问是否可以从本地的可定义联合分配解决方案进行抽样。 更广泛地说, 我们考虑从Gibbs 分布法中加权的本地本地的本地的 CSP 。 我们给两个基于 Markov 链的基于本地可定义的分布算法, 我们认为这代表了通过分布算法的 Gibs 分布法进行抽样的两个基本方法。 第一个算法将单站点的连续的Markov 链条以迭接方式同步地同步更新一个随机独立的变量集, 并在 LOCAL 模式中实现一个$ $ Delta n 的上限。 当Dobrushinal deal deal delist lax lax lax lax lax lady a mail dal dal deal deliver liver liver liver liver list rmalliver ral se seal selevaldal se se se se se se se se, 当我们 rout rlationslations ral ral ral ral raldaldaldaldald 。 当我们算算出一个稳定一个稳定的本地的 raldald ral raldaldaldaldaldaldal raldal ral ral ral ral rl rl ral ral ral ral ral ral ral ral ral ral rl ral rl ral ral ral r r ral r r ral ral ral r), 当在本地的 r r r r ral r