696 research outputs found
Coresets Meet EDCS: Algorithms for Matching and Vertex Cover on Massive Graphs
As massive graphs become more prevalent, there is a rapidly growing need for
scalable algorithms that solve classical graph problems, such as maximum
matching and minimum vertex cover, on large datasets. For massive inputs,
several different computational models have been introduced, including the
streaming model, the distributed communication model, and the massively
parallel computation (MPC) model that is a common abstraction of
MapReduce-style computation. In each model, algorithms are analyzed in terms of
resources such as space used or rounds of communication needed, in addition to
the more traditional approximation ratio.
In this paper, we give a single unified approach that yields better
approximation algorithms for matching and vertex cover in all these models. The
highlights include:
* The first one pass, significantly-better-than-2-approximation for matching
in random arrival streams that uses subquadratic space, namely a
-approximation streaming algorithm that uses space
for constant .
* The first 2-round, better-than-2-approximation for matching in the MPC
model that uses subquadratic space per machine, namely a
-approximation algorithm with memory per
machine for constant .
By building on our unified approach, we further develop parallel algorithms
in the MPC model that give a -approximation to matching and an
-approximation to vertex cover in only MPC rounds and
memory per machine. These results settle multiple open
questions posed in the recent paper of Czumaj~et.al. [STOC 2018]
Training Gaussian Mixture Models at Scale via Coresets
How can we train a statistical mixture model on a massive data set? In this
work we show how to construct coresets for mixtures of Gaussians. A coreset is
a weighted subset of the data, which guarantees that models fitting the coreset
also provide a good fit for the original data set. We show that, perhaps
surprisingly, Gaussian mixtures admit coresets of size polynomial in dimension
and the number of mixture components, while being independent of the data set
size. Hence, one can harness computationally intensive algorithms to compute a
good approximation on a significantly smaller data set. More importantly, such
coresets can be efficiently constructed both in distributed and streaming
settings and do not impose restrictions on the data generating process. Our
results rely on a novel reduction of statistical estimation to problems in
computational geometry and new combinatorial complexity results for mixtures of
Gaussians. Empirical evaluation on several real-world datasets suggests that
our coreset-based approach enables significant reduction in training-time with
negligible approximation error
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