Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Log Diameter Rounds Algorithms for $2$-Vertex and $2$-Edge Connectivity   Many modern parallel systems, such as MapReduce, Hadoop and Spark, can be
modeled well by the MPC model. The MPC model captures well coarse-grained
computation on large data --- data is distributed to processors, each of which
has a sublinear (in the input data) amount of memory and we alternate between
rounds of computation and rounds of communication, where each machine can
communicate an amount of data as large as the size of its memory. This model is
stronger than the classical PRAM model, and it is an intriguing question to
design algorithms whose running time is smaller than in the PRAM model.
  In this paper, we study two fundamental problems, $2$-edge connectivity and
$2$-vertex connectivity (biconnectivity). PRAM algorithms which run in $O(\log
n)$ time have been known for many years. We give algorithms using roughly log
diameter rounds in the MPC model. Our main results are, for an $n$-vertex,
$m$-edge graph of diameter $D$ and bi-diameter $D'$, 1) a $O(\log
D\log\log_{m/n} n)$ parallel time $2$-edge connectivity algorithm, 2) a $O(\log
D\log^2\log_{m/n}n+\log D'\log\log_{m/n}n)$ parallel time biconnectivity
algorithm, where the bi-diameter $D'$ is the largest cycle length over all the
vertex pairs in the same biconnected component. Our results are fully scalable,
meaning that the memory per processor can be $O(n^{\delta})$ for arbitrary
constant $\delta>0$, and the total memory used is linear in the problem size.
Our $2$-edge connectivity algorithm achieves the same parallel time as the
connectivity algorithm of Andoni et al. (FOCS 2018). We also show an
$\Omega(\log D')$ conditional lower bound for the biconnectivity problem.

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The categories of this paper are: gr-qc
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The categories of this paper are: hep-lat
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The categories of this paper are: bayes-an
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The categories of this paper are: eess
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The categories of this paper are: comp-gas
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The categories of this paper are: nlin
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The categories of this paper are: chao-dyn
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The categories of this paper are: q-alg
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The categories of this paper are: math-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: stat
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The categories of this paper are: alg-geom
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The categories of this paper are: chem-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: funct-an
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The categories of this paper are: acc-phys