30,536 research outputs found
Round Compression for Parallel Matching Algorithms
For over a decade now we have been witnessing the success of {\em massive
parallel computation} (MPC) frameworks, such as MapReduce, Hadoop, Dryad, or
Spark. One of the reasons for their success is the fact that these frameworks
are able to accurately capture the nature of large-scale computation. In
particular, compared to the classic distributed algorithms or PRAM models,
these frameworks allow for much more local computation. The fundamental
question that arises in this context is though: can we leverage this additional
power to obtain even faster parallel algorithms?
A prominent example here is the {\em maximum matching} problem---one of the
most classic graph problems. It is well known that in the PRAM model one can
compute a 2-approximate maximum matching in rounds. However, the
exact complexity of this problem in the MPC framework is still far from
understood. Lattanzi et al. showed that if each machine has
memory, this problem can also be solved -approximately in a constant number
of rounds. These techniques, as well as the approaches developed in the follow
up work, seem though to get stuck in a fundamental way at roughly
rounds once we enter the near-linear memory regime. It is thus entirely
possible that in this regime, which captures in particular the case of sparse
graph computations, the best MPC round complexity matches what one can already
get in the PRAM model, without the need to take advantage of the extra local
computation power.
In this paper, we finally refute that perplexing possibility. That is, we
break the above round complexity bound even in the case of {\em
slightly sublinear} memory per machine. In fact, our improvement here is {\em
almost exponential}: we are able to deliver a -approximation to
maximum matching, for any fixed constant , in
rounds
An Improved Distributed Algorithm for Maximal Independent Set
The Maximal Independent Set (MIS) problem is one of the basics in the study
of locality in distributed graph algorithms. This paper presents an extremely
simple randomized algorithm providing a near-optimal local complexity for this
problem, which incidentally, when combined with some recent techniques, also
leads to a near-optimal global complexity.
Classical algorithms of Luby [STOC'85] and Alon, Babai and Itai [JALG'86]
provide the global complexity guarantee that, with high probability, all nodes
terminate after rounds. In contrast, our initial focus is on the
local complexity, and our main contribution is to provide a very simple
algorithm guaranteeing that each particular node terminates after rounds, with probability at least
. The guarantee holds even if the randomness outside -hops
neighborhood of is determined adversarially. This degree-dependency is
optimal, due to a lower bound of Kuhn, Moscibroda, and Wattenhofer [PODC'04].
Interestingly, this local complexity smoothly transitions to a global
complexity: by adding techniques of Barenboim, Elkin, Pettie, and Schneider
[FOCS'12, arXiv: 1202.1983v3], we get a randomized MIS algorithm with a high
probability global complexity of ,
where denotes the maximum degree. This improves over the result of Barenboim et al., and gets close
to the lower bound of Kuhn et al.
Corollaries include improved algorithms for MIS in graphs of upper-bounded
arboricity, or lower-bounded girth, for Ruling Sets, for MIS in the Local
Computation Algorithms (LCA) model, and a faster distributed algorithm for the
Lov\'asz Local Lemma
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