Communication complexity and combinatorial lattice theory

AbstractIn a recent paper, Hajnal, Maass, and Turán analyzed the communication complexity of graph connectivity. Building on this work, we develop a general framework for the study of a broad class of communication problems which has several interesting special cases including the graph connectivity problem. The approach is based on the combinatorial theory of alignments and lattices

Lattices, MEbius Functions and Communication Complexity

Abstract. In a recent paper, Hajnal, Maass and Tura'n analyzed the communication complexity of graph connectivity. Building on this work, we develop a general framework for the study of a broad class of communication problems which has several interesting special cases including the graph connectivity problem. The approach is based on combinatorial lattice theory. LINTRODUCTION Communication complexity is concerned with the question: how much information do two processors need to exchange to compute a specified function fthat depends on both of their inputs? The minimum number of bits that must be communicated is the deterministic communicaiion complexity off. Set Disjointness. Each processor is given a set (from a finite universe X of size n ) and they must determine whether the sets are disjoint. Trivially n bits of communication suffice: one processor tells the other the incidence vector of its subset. There is also a simple information theoretic lower bound of logn. It has in fact been shown that the upper bound is tight. Natural variants of this problem arise in a number of settings: Convex Sei Disjointness. The universe X is a finite subset of Euclidean d-space. Each processor is given a finite subset of X and they must determine whether the intersection of their convex hulls contains a point from X. Vector Disjoinmess. The universe X is a finite set of vectors from some vector space. Each processor is given a subset of X and they must determine whether the intersection of the subspaces spanned by each set contains a common vector from X. Tree Disjointness. Each processor is given a subtree of a fixed tree on vertex set X and they must determine whether the trees have a vertex in common. Other problems with a similar flavor include: Each processor is given a graph on vertex set V and they must determine if the union of the graphs is connected. Graph (s.t)-Connectivity. Each processor is given a graph on vertex set V and they must determine if the union has a path from s to t. Vector Space Span. Each processor is given a set of vectors (from a finite set of vectors X ) and they must determine whether the union of their sets spans the whole space. For each of these problems one can establish trivial upper and lower bounds on their communication complexity. Recently, Hajnal et al. [HMT] showed that the trivial upper bound on the graph connectivity problem is tight. As we will see, the trivial upper bound is also tight for the Znd, 5* and 6* problems, while for the other two there are protocols that come very close to the lower bound. In this paper, we define and investigate a broad class of communication problems that include all of the above as special cases. The natural setting for studying these problems is the combinatorial theory of alignments. A family L of subsets of X is an alignment if whenever A,B are contained in L then so is their intersection. For an alignment L, we consider: Disjointness Problem for L. Each processor gets a set from L and they must determine whether the sets are disjoint. It is not hard to see that the three variants of set 0272-5428/88/0000/0081.$01.00 0 1988 IEEE 8