Equivalence Classes and Conditional Hardness in Massively Parallel Computations

The Massively Parallel Computation (MPC) model serves as a common abstraction of many modern large-scale data processing frameworks, and has been receiving increasingly more attention over the past few years, especially in the context of classical graph problems. So far, the only way to argue lower bounds for this model is to condition on conjectures about the hardness of some specific problems, such as graph connectivity on promise graphs that are either one cycle or two cycles, usually called the one cycle vs. two cycles problem. This is unlike the traditional arguments based on conjectures about complexity classes (e.g., P ? NP), which are often more robust in the sense that refuting them would lead to groundbreaking algorithms for a whole bunch of problems. In this paper we present connections between problems and classes of problems that allow the latter type of arguments. These connections concern the class of problems solvable in a sublogarithmic amount of rounds in the MPC model, denoted by MPC(o(log N)), and some standard classes concerning space complexity, namely L and NL, and suggest conjectures that are robust in the sense that refuting them would lead to many surprisingly fast new algorithms in the MPC model. We also obtain new conditional lower bounds, and prove new reductions and equivalences between problems in the MPC model

On the Distributed Complexity of Large-Scale Graph Computations

Motivated by the increasing need to understand the distributed algorithmic foundations of large-scale graph computations, we study some fundamental graph problems in a message-passing model for distributed computing where $k \geq 2$ machines jointly perform computations on graphs with $n$ nodes (typically, $n \gg k$). The input graph is assumed to be initially randomly partitioned among the $k$ machines, a common implementation in many real-world systems. Communication is point-to-point, and the goal is to minimize the number of communication {\em rounds} of the computation. Our main contribution is the {\em General Lower Bound Theorem}, a theorem that can be used to show non-trivial lower bounds on the round complexity of distributed large-scale data computations. The General Lower Bound Theorem is established via an information-theoretic approach that relates the round complexity to the minimal amount of information required by machines to solve the problem. Our approach is generic and this theorem can be used in a "cookbook" fashion to show distributed lower bounds in the context of several problems, including non-graph problems. We present two applications by showing (almost) tight lower bounds for the round complexity of two fundamental graph problems, namely {\em PageRank computation} and {\em triangle enumeration}. Our approach, as demonstrated in the case of PageRank, can yield tight lower bounds for problems (including, and especially, under a stochastic partition of the input) where communication complexity techniques are not obvious. Our approach, as demonstrated in the case of triangle enumeration, can yield stronger round lower bounds as well as message-round tradeoffs compared to approaches that use communication complexity techniques

A Lower Bound Technique for Communication in BSP

Communication is a major factor determining the performance of algorithms on current computing systems; it is therefore valuable to provide tight lower bounds on the communication complexity of computations. This paper presents a lower bound technique for the communication complexity in the bulk-synchronous parallel (BSP) model of a given class of DAG computations. The derived bound is expressed in terms of the switching potential of a DAG, that is, the number of permutations that the DAG can realize when viewed as a switching network. The proposed technique yields tight lower bounds for the fast Fourier transform (FFT), and for any sorting and permutation network. A stronger bound is also derived for the periodic balanced sorting network, by applying this technique to suitable subnetworks. Finally, we demonstrate that the switching potential captures communication requirements even in computational models different from BSP, such as the I/O model and the LPRAM

Matching on the Line Admits No o(?log n)-Competitive Algorithm

We present a simple proof that the competitive ratio of any randomized online matching algorithm for the line exceeds ?{log?(n +1)}/15 for all n = 2?-1: i ? ?, settling a 25-year-old open question

Communication Lower Bounds for Distributed-Memory Computations

In this paper we propose a new approach to the study of the communication requirements of distributed computations, which advocates for the removal of the restrictive assumptions under which earlier results were derived. We illustrate our approach by giving tight lower bounds on the communication complexity required to solve several computational problems in a distributed-memory parallel machine, namely standard matrix multiplication, stencil computations, comparison sorting, and the Fast Fourier Transform. Our bounds rely only on a mild assumption on work distribution, and significantly strengthen previous results which require either the computation to be balanced among the processors, or specific initial distributions of the input data, or an upper bound on the size of processors\u27 local memories

A time- and message-optimal distributed algorithm for minimum spanning trees

This paper presents a randomized Las Vegas distributed algorithm that constructs a minimum spanning tree (MST) in weighted networks with optimal (up to polylogarithmic factors) time and message complexity. This algorithm runs in $\tilde{O}(D + \sqrt{n})$ time and exchanges $\tilde{O}(m)$ messages (both with high probability), where $n$ is the number of nodes of the network, $D$ is the diameter, and $m$ is the number of edges. This is the first distributed MST algorithm that matches \emph{simultaneously} the time lower bound of $\tilde{\Omega}(D + \sqrt{n})$ [Elkin, SIAM J. Comput. 2006] and the message lower bound of $\Omega(m)$ [Kutten et al., J.ACM 2015] (which both apply to randomized algorithms). The prior time and message lower bounds are derived using two completely different graph constructions; the existing lower bound construction that shows one lower bound {\em does not} work for the other. To complement our algorithm, we present a new lower bound graph construction for which any distributed MST algorithm requires \emph{both} $\tilde{\Omega}(D + \sqrt{n})$ rounds and $\Omega(m)$ messages

Fast Distributed Algorithms for Connectivity and MST in Large Graphs

Motivated by the increasing need to understand the algorithmic foundations of distributed large-scale graph computations, we study a number of fundamental graph problems in a message-passing model for distributed computing where $k \geq 2$ machines jointly perform computations on graphs with $n$ nodes (typically, $n \gg k$). The input graph is assumed to be initially randomly partitioned among the $k$ machines, a common implementation in many real-world systems. Communication is point-to-point, and the goal is to minimize the number of communication rounds of the computation. Our main result is an (almost) optimal distributed randomized algorithm for graph connectivity. Our algorithm runs in $\tilde{O}(n/k^2)$ rounds ($\tilde{O}$ notation hides a \poly\log(n) factor and an additive \poly\log(n) term). This improves over the best previously known bound of $\tilde{O}(n/k)$ [Klauck et al., SODA 2015], and is optimal (up to a polylogarithmic factor) in view of an existing lower bound of $\tilde{\Omega}(n/k^2)$. Our improved algorithm uses a bunch of techniques, including linear graph sketching, that prove useful in the design of efficient distributed graph algorithms. Using the connectivity algorithm as a building block, we then present fast randomized algorithms for computing minimum spanning trees, (approximate) min-cuts, and for many graph verification problems. All these algorithms take $\tilde{O}(n/k^2)$ rounds, and are optimal up to polylogarithmic factors. We also show an almost matching lower bound of $\tilde{\Omega}(n/k^2)$ rounds for many graph verification problems by leveraging lower bounds in random-partition communication complexity