A Note on Solving Problems of Substantially Super-linear Complexity in $N^{o(1)}$ Rounds of the Congested Clique

TL;DR

This paper investigates the feasibility of designing extremely fast distributed protocols for complex problems on the congested clique, revealing that local computation must be significantly more intensive than the problem's sequential complexity.

Contribution

It establishes a fundamental lower bound on local computation time for sub-polynomial round protocols solving super-linear problems in the congested clique model.

Findings

Local computation time must be substantially larger than problem complexity.

Protocols with $N^{o(1)}$ rounds cannot have efficient local computation.

Super-linear problems require intensive local processing in this model.

Abstract

We study the possibility of designing $N^{o(1)}$-round protocols for problems of substantially super-linear polynomial-time (sequential) complexity on the congested clique with about $N^{1/2}$ nodes, where $N$ is the input size. We show that the average time complexity of the local computation performed at a clique node (in terms of the size of the data received by the node) in such protocols has to be substantially larger than the time complexity of the given problem.