We study the distributed simulation problem where @math players aim to generate sequences of random coin flips where some subsets of the players share an independent common coin which can be tossed multiple times, and there is a publicly seen blackboard through which the players communicate with each other. We provide a tight representation of the optimal communication rates via linear programming, and more importantly, propose explicit algorithms for the optimal distributed simulation for a wide class of hypergraphs. In particular, the optimal communication rate in complete hypergraphs is still achievable in sparser hypergraphs containing a path-connected cycle-free cluster of topologically connected components. Some key steps in analyzing the upper bounds rely on two different definitions of connectivity in hypergraphs, which may be of independent interest.
A related work deserving special attention is @cite_18 , which studied the also on hypergraphs. Specifically, it showed that if the @math -uniform hypergraph is (a notion introduced in @cite_18 ), then there is a strategy achieving the optimal communication rate @math and outputting each hyperedge . Our work differs from @cite_18 in the following aspects. First, instead of outputting each hyperedge exactly once, we may output a subset of hyperedges or some hyperedges with different repetitions (cf. Section ). Consequently, our lower bound in Theorem is stronger. Second, the achievability scheme in @cite_18 requires large blocklengths to achieve some small probability of error, while our achievability scheme is more combinatorial and achieves zero error. Third, their type @math condition is an information-theoretic condition, with an unclear relationship to combinatorial hypergraph theory. It is also unknown whether it can be verified in polynomial time that some subset of (possibly repeated) hyperedges in a given hypergraph forms a hypergraph of type @math . In contrast, our conditions in Theorems and are more combinatorial in nature, and the topological connectivity can be efficiently checked using matrix ranks. Hence, our work presents an alternative approach which sheds more lights on the combinatorial perspective.
@cite_18
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Let h>w>0 be two fixed integers. Let H be a random hypergraph whose hyperedges are all of cardinality h. To w-orient a hyperedge, we assign exactly w of its vertices positive signs with respect to the hyperedge, and the rest negative. A (w,k)-orientation of H consists of a w-orientation of all hyperedges of H, such that each vertex receives at most k positive signs from its incident hyperedges. When k is large enough, we determine the threshold of the existence of a (w,k)-orientation of a random hypergraph. The (w,k)-orientation of hypergraphs is strongly related to a general version of the off-line load balancing problem. The graph case, when h=2 and w=1, was solved recently by Cain, Sanders and Wormald and independently by Fernholz and Ramachandran, thereby settling a conjecture made by Karp and Saks. Motivated by a problem of cuckoo hashing, the special hypergraph case with w=k=1, was solved in three separate preprints dating from October 2009, by Frieze and Melsted, by Fountoulakis and Panagiotou, and by Dietzfelbinger, Goerdt, Mitzenmacher, Montanari, Pagh and Rink.