Exponentially Faster Massively Parallel Maximal Matching

01/11/2019
by   Soheil Behnezhad, et al.
0

The study of graph problems in the Massively Parallel Computations (MPC) model has recently seen a burst of breakthroughs. Czumaj et al. [STOC'18], Assadi et al. [SODA'19], and Ghaffari et al. [PODC'18], gave algorithms for finding a 1+ε approximate maximum matching in O( n) rounds using O(n) memory per machine. Despite this progress, we still have a far more limited understanding of the central symmetry-breaking problem of maximal matching. The round complexity of all these algorithms blows up to Ω( n) in this case, which is considered inefficient. In fact, the only known subpolylogarithmic round algorithm remains to be that of Lattanzi et al. [SPAA'11] which undesirably requires a strictly super-linear space of n^1+Ω(1) per machine. We resolve this shortcoming by providing exponentially faster algorithms for maximal matching. Perhaps more importantly, we obtain this by analyzing an extremely simple and natural algorithm. The algorithm edge-samples the graph, partitions the vertices at random, and finds a greedy maximal matching within each partition. We show that this algorithm drastically reduces the vertex degrees. This, among some other results, leads to an O(Δ) round algorithm for maximal matching with O(n) space. The space can be further improved to mildly sublinear in n by standard techniques. As an immediate corollary, we get a 2 approximation for minimum vertex cover in essentially the same rounds and space. This is the best possible approximation factor under standard assumptions, culminating a long line of research. Other corollaries include more efficient algorithms for 1 + ε approximate matching and 2 + ε approximate weighted matching. All these results can also be implemented in the congested clique model within O(Δ) rounds.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
02/22/2018

Improved Massively Parallel Computation Algorithms for MIS, Matching, and Vertex Cover

We present O( n)-round algorithms in the Massively Parallel Computation ...
research
05/05/2021

The Complexity of Symmetry Breaking in Massive Graphs

The goal of this paper is to understand the complexity of symmetry break...
research
04/16/2019

Parallel Balanced Allocations: The Heavily Loaded Case

We study parallel algorithms for the classical balls-into-bins problem, ...
research
11/25/2017

Optimal Gossip Algorithms for Exact and Approximate Quantile Computations

This paper gives drastically faster gossip algorithms to compute exact a...
research
07/12/2019

Space Efficient Approximation to Maximum Matching Size from Uniform Edge Samples

Given a source of iid samples of edges of an input graph G with n vertic...
research
03/16/2020

Beyond Alice and Bob: Improved Inapproximability for Maximum Independent Set in CONGEST

By far the most fruitful technique for showing lower bounds for the CONG...
research
11/08/2017

Coresets Meet EDCS: Algorithms for Matching and Vertex Cover on Massive Graphs

Randomized composable coresets were introduced recently as an effective ...

Please sign up or login with your details

Forgot password? Click here to reset