On the Hardness of Massively Parallel Computation

TL;DR

This paper demonstrates inherent limits of parallelization in the MPC model by constructing functions that are efficiently computable sequentially but require many rounds in MPC with small local memory, highlighting fundamental parallelization barriers.

Contribution

It introduces a method to prove lower bounds on MPC round complexity using cryptographic and data structure techniques, showing certain functions are inherently sequential in the MPC setting.

Findings

Existence of functions hard to parallelize in MPC with small memory

Almost optimal separation between RAM and MPC complexities

Adaptation of compression arguments to MPC lower bounds

Abstract

We investigate whether there are inherent limits of parallelization in the (randomized) massively parallel computation (MPC) model by comparing it with the (sequential) RAM model. As our main result, we show the existence of hard functions that are essentially not parallelizable in the MPC model. Based on the widely-used random oracle methodology in cryptography with a cryptographic hash function $h:\{0,1\}^n \rightarrow \{0,1\}^n$ computable in time $t_h$, we show that there exists a function that can be computed in time $O(T\cdot t_h)$ and space $S$ by a RAM algorithm, but any MPC algorithm with local memory size $s < S/c$ for some $c>1$ requires at least $\tilde{\Omega}(T)$ rounds to compute the function, even in the average case, for a wide range of parameters $n \leq S \leq T \leq 2^{n^{1/4}}$. Our result is almost optimal in the sense that by taking $T$ to be much larger than $t_h$, \textit{e.g.}, $T$ to be sub-exponential in $t_h$, to compute the function, the round complexity of any MPC algorithm with small local memory size is asymptotically the same (up to a polylogarithmic factor) as the time complexity of the RAM algorithm. Our result is obtained by adapting the so-called compression argument from the data structure lower bounds and cryptography literature to the context of massively parallel computation.