3 research outputs found
A Note on Multiparty Communication Complexity and the Hales-Jewett Theorem
For integers and , the density Hales-Jewett number is
defined as the maximal size of a subset of that contains no
combinatorial line. We show that for the density Hales-Jewett number
is equal to the maximal size of a cylinder intersection in the
problem of testing whether subsets of form a partition.
It follows that the communication complexity, in the Number On the Forehead
(NOF) model, of , is equal to the minimal size of a partition of
into subsets that do not contain a combinatorial line. Thus, the bound
in \cite{chattopadhyay2007languages} on using the Hales-Jewett
theorem is in fact tight, and the density Hales-Jewett number can be thought of
as a quantity in communication complexity. This gives a new angle to this well
studied quantity.
As a simple application we prove a lower bound on , similar to the
lower bound in \cite{polymath2010moser} which is roughly . This lower bound follows from a
protocol for . It is interesting to better understand the
communication complexity of as this will also lead to the better
understanding of the Hales-Jewett number. The main purpose of this note is to
motivate this study
Nondeterministic Communication Complexity with Help and Graph Functions
We define nondeterministic communication complexity in the model of
communication complexity with help of Babai, Hayes and Kimmel. We use it to
prove logarithmic lower bounds on the NOF communication complexity of explicit
graph functions, which are complementary to the bounds proved by Beame, David,
Pitassi and Woelfel
On The Communication Complexity of High-Dimensional Permutations
We study the multiparty communication complexity of high dimensional
permutations, in the Number On the Forehead (NOF) model. This model is due to
Chandra, Furst and Lipton (CFL) who also gave a nontrivial protocol for the
Exactly-n problem where three players receive integer inputs and need to decide
if their inputs sum to a given integer . There is a considerable body of
literature dealing with the same problem, where is replaced by
some other abelian group. Our work can be viewed as a far-reaching extension of
this line of work.
We show that the known lower bounds for that group-theoretic problem apply to
all high dimensional permutations. We introduce new proof techniques that
appeal to recent advances in Additive Combinatorics and Ramsey theory. We
reveal new and unexpected connections between the NOF communication complexity
of high dimensional permutations and a variety of well known and thoroughly
studied problems in combinatorics.
Previous protocols for Exactly-n all rely on the construction of large sets
of integers without a 3-term arithmetic progression. No direct algorithmic
protocol was previously known for the problem, and we provide the first such
algorithm. This suggests new ways to significantly improve the CFL protocol.
Many new open questions are presented throughout