11 research outputs found
Faster and Simpler Distributed Algorithms for Testing and Correcting Graph Properties in the CONGEST-Model
In this paper we present distributed testing algorithms of graph properties
in the CONGEST-model [Censor-Hillel et al. 2016]. We present one-sided error
testing algorithms in the general graph model.
We first describe a general procedure for converting -testers with
a number of rounds , where denotes the diameter of the graph, to
rounds, where is the number of
processors of the network. We then apply this procedure to obtain an optimal
tester, in terms of , for testing bipartiteness, whose round complexity is
, which improves over the -round algorithm by Censor-Hillel et al. (DISC 2016). Moreover, for
cycle-freeness, we obtain a \emph{corrector} of the graph that locally corrects
the graph so that the corrected graph is acyclic. Note that, unlike a tester, a
corrector needs to mend the graph in many places in the case that the graph is
far from having the property.
In the second part of the paper we design algorithms for testing whether the
network is -free for any connected of size up to four with round
complexity of . This improves over the
-round algorithms for testing triangle freeness by
Censor-Hillel et al. (DISC 2016) and for testing excluded graphs of size by
Fraigniaud et al. (DISC 2016).
In the last part we generalize the global tester by Iwama and Yoshida (ITCS
2014) of testing -path freeness to testing the exclusion of any tree of
order . We then show how to simulate this algorithm in the CONGEST-model in
rounds
Distributed Detection of Cycles
Distributed property testing in networks has been introduced by Brakerski and
Patt-Shamir (2011), with the objective of detecting the presence of large dense
sub-networks in a distributed manner. Recently, Censor-Hillel et al. (2016)
have shown how to detect 3-cycles in a constant number of rounds by a
distributed algorithm. In a follow up work, Fraigniaud et al. (2016) have shown
how to detect 4-cycles in a constant number of rounds as well. However, the
techniques in these latter works were shown not to generalize to larger cycles
with . In this paper, we completely settle the problem of cycle
detection, by establishing the following result. For every , there
exists a distributed property testing algorithm for -freeness, performing
in a constant number of rounds. All these results hold in the classical CONGEST
model for distributed network computing. Our algorithm is 1-sided error. Its
round-complexity is where is the property
testing parameter measuring the gap between legal and illegal instances