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

Andrzej Lingas

TL;DR

This work investigates whether problems with substantially super-linear sequential time can be solved in $N^{o(1)}$ rounds on a congested clique with about $N^{1/2}$ nodes. It introduces the exponents $\mathrm{opt}(P)$ for sequential time and $\mathrm{ave}(B)$ for average local computation, and derives a lower bound $\mathrm{ave}(B) \ge \frac{\mathrm{opt}(P)-\tfrac{1}{2}-o(1)}{\tfrac{1}{2}+\log_N t_B(N)}$, with the corollary that $t_B(N)\le N^{o(1)}$ implies $\mathrm{ave}(B) \ge 2\,\mathrm{opt}(P)-1-o(1)$. Applying the framework to matrix multiplication and APSP, and using known bounds for $\mathrm{opt}$, the paper shows that for $N^{o(1)}$ rounds the average local computation must exceed the problem’s sequential exponent, signaling that such ultra-fast protocols are highly non-trivial to design. The results help explain why the fastest congested-clique protocols for MM and APSP are near-threshold and motivate strategies like data compression or shared computation, while also extending the analysis to generalized CONGEST models.

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.

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

TL;DR

This work investigates whether problems with substantially super-linear sequential time can be solved in rounds on a congested clique with about nodes. It introduces the exponents for sequential time and for average local computation, and derives a lower bound , with the corollary that implies . Applying the framework to matrix multiplication and APSP, and using known bounds for , the paper shows that for rounds the average local computation must exceed the problem’s sequential exponent, signaling that such ultra-fast protocols are highly non-trivial to design. The results help explain why the fastest congested-clique protocols for MM and APSP are near-threshold and motivate strategies like data compression or shared computation, while also extending the analysis to generalized CONGEST models.

Abstract

We study the possibility of designing -round protocols for problems of substantially super-linear polynomial-time (sequential) complexity on the congested clique with about nodes, where 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.
Paper Structure (3 sections, 3 theorems, 3 equations)

This paper contains 3 sections, 3 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. The Argument
  3. Discussion

Key Result

theorem 1

Consider a problem $P$ solvable in polynomial time and a protocol $B$ that solves an input instance of $P$ of size $N$ using $t_B(N)$ rounds on the congested clique. If $ave(B)$ is well defined then the following inequality holds:

Theorems & Definitions (5)

  • theorem 1
  • proof
  • corollary 1
  • corollary 2
  • proof