We study a distributed sampling problem where a set of processors want to output (approximately) independent and identically distributed samples from a joint distribution with the help of a common message from a coordinator. Each processor has access to a subset of sources from a set of independent sources of "shared" randomness. We consider two cases -- in the "omniscient coordinator setting", the coordinator has access to all these sources of shared randomness, while in the "oblivious coordinator setting", it has access to none. All processors and the coordinator may privately randomize. In the omniscient coordinator setting, when the subsets at the processors are disjoint (individually shared randomness model), we characterize the rate of communication required from the coordinator to the processors over a multicast link. For the two-processor case, the optimal rate matches a special case of relaxed Wyner's common information proposed by Gastpar and Sula [IEEE Information Theory Workshop, 2019] thereby providing an operational meaning to the latter. We also give an upper bound on the communication rate for the "randomness-on-the-forehead" model where each processor observes all but one source of randomness and we give an achievable strategy in the omniscient coordinator setting for the general case where the processors have access to arbitrary subsets of sources of randomness. Also, we consider a more general model where the processors observe components of correlated sources (with the coordinator observing all the components), where we characterize the communication rate when all the processors wish to output the same random sequence. In the oblivious coordinator setting, we completely characterize the trade-off region between the communication and shared randomness rates for the general case where the processors have access to arbitrary subsets of sources of randomness.

翻译：我们研究一个分布式抽样问题, 一组处理器想要在协调员的共同信息帮助下从联合分发中输出( 约) 独立和同样分布的样本。 每个处理器都可以从一组独立的“ 共享” 随机性来源获得一组来源的通信。 我们考虑两个案例—— 在“ 未知的协调器设置” 中, 协调员可以获取所有这些共享随机性来源, 在“ 明显的协调器设置” 中, 协调员可以访问所有这些来源, 而“ 明显的协调器设置 ”, 所有处理器和协调员都可以私下随机地进行。 在无意识的协调员设置中, 当处理器的子组分分脱节性( 单独共享随机性模式) 时, 每个处理器都可以从协调员到多个处理器的独立源的通信速度。 在两种处理器中, 最佳的通信率符合一个特例, 即Wyner的共享随机性源, 而在“ 共享- 共享- 共享- 随机性流程” 模型中, 我们可以将所有通性交易源的通信速性数据连接到总操作器 。