# On The Communication Complexity of High-Dimensional Permutations

## TL;DR

This paper investigates the communication complexity of high-dimensional permutations in the Number On the Forehead model, extending previous work on abelian groups and introducing new proof techniques and algorithms.

## Contribution

It extends lower bounds to all high-dimensional permutations, introduces new proof methods using additive combinatorics, and provides the first algorithm for the Exactly-n problem.

## Key findings

- Lower bounds apply to all high-dimensional permutations.
- New proof techniques connect communication complexity with combinatorics.
- First known algorithm for the Exactly-n problem in this context.

## Abstract

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 $n$. There is a considerable body of literature dealing with the same problem, where $(\mathbb{N},+)$ 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.