# Broadcast Congested Clique: Planted Cliques and Pseudorandom Generators

## TL;DR

This paper introduces new techniques for proving lower bounds in the Broadcast Congested Clique model, applies them to the planted clique problem, and constructs an efficient pseudo-random generator to reduce randomness in algorithms.

## Contribution

It develops a novel method for lower bounds in natural input distributions and constructs an optimal seed pseudo-random generator for the model.

## Key findings

- Lower bounds for directed planted clique when k = n^(1/4 - eps)
- A simple, efficient pseudo-random generator that fools the model
- First average-case lower bound and time hierarchy for the model

## Abstract

We develop techniques to prove lower bounds for the BCAST(log n) Broadcast Congested Clique model (a distributed message passing model where in each round, each processor can broadcast an O(log n)-sized message to all other processors). Our techniques are built to prove bounds for natural input distributions. So far, all lower bounds for problems in the model relied on constructing specifically tailored graph families for the specific problem at hand, resulting in lower bounds for artificially constructed inputs, instead of natural input distributions. One of our results is a lower bound for the directed planted clique problem. In this problem, an input graph is either a random directed graph (each directed edge is included with probability 1/2), or a random graph with a planted clique of size k. That is, k randomly chosen vertices have all of the edges between them included, and all other edges in the graph appear with probability 1/2. The goal is to determine whether a clique exists. We show that when k = n^(1/4 - eps), this problem requires a number of rounds polynomial in n. Additionally, we construct a pseudo-random generator which fools the Broadcast Congested Clique. This allows us to show that every k round randomized algorithm in which each processor uses up to n random bits can be efficiently transformed into an O(k)-round randomized algorithm in which each processor uses only up to O(k log n) random bits, while maintaining a high success probability. The pseudo-random generator is simple to describe, computationally very cheap, and its seed size is optimal up to constant factors. However, the analysis is quite involved, and is based on the new technique for proving lower bounds in the model. The technique also allows us to prove the first average case lower bound for the Broadcast Congested Clique, as well as an average-case time hierarchy.