On Solving Problems of Substantially Super-linear Complexity in $N^{o(1)}$ Rounds in the MPC Model

TL;DR

This paper investigates the limitations of designing sublinear-round protocols for super-linear problems in the MPC model, showing inherent complexity constraints without large local memory.

Contribution

It establishes that without large local memory, the local computation complexity must match the problem's inherent complexity in sublinear rounds.

Findings

Protocols with fewer than N^{o(1)} rounds require high local computation complexity.

Local memory constraints prevent solving super-linear problems efficiently in few rounds.

Inherent complexity of problems cannot be bypassed without large local memory.

Abstract

We study the possibility of designing $N^{o(1)}$-round protocols for problems of substantially super-linear polynomial-time (sequential) complexity in the model of Massively Parallel Computation, where $N$ is the input size. We show that if the machines are not equipped with relatively large local memory and their number does not exceed $N$, then the exponent of the average time complexity of the local computation performed by a machine in a round (in terms of local memory size) in such protocols must be larger than the exponent of the time complexity of the given problem.