Inner Product and Set Disjointness: Beyond Logarithmically Many Parties

Abstract

A basic goal in complexity theory is to understand the communication complexity of number-on-the-forehead problems $f:(\{0,1{\}}^n{)}^k\to \{0,1\}$ with $k\gg \log n$ parties. We study the problems of inner product and set disjointness and determine their randomized communication complexity for every $k\ge \log n$, showing in both cases that $\Theta (1+\lceil \log n\rceil /\log \lceil 1+k/\log n\rceil )$ bits are necessary and sufficient. In particular, these problems admit constant-cost protocols if and only if the number of parties is $k\ge n^{\epsilon }$ for some constant $\epsilon >0.$