133,608 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
Total Acquisition in Graphs
Let G be a weighted graph in which each vertex initially has weight 1. A total acquisition move transfers all the weight from a vertex u to a neighboring vertex v, under the condition that before the move the weight on v is at least as large as the weight on u. The (total) acquisition number of G, written at(G), is the minimum size of the set of vertices with positive weight after a sequence of total acquisition moves. Among connected n-vertex graphs, at(G) is maximized by trees. The maximum is Θ(√(n lg n) for trees with diameter 4 or 5. It is⌊(n + 1)/3⌋ for trees with diameter between 6 and (2/3)(n + 1), and it is⌈(2n – 1 – D)/4⌉ for trees with diameter D when (2/3)(n + 1) ≤ D ≤ n - 1. We characterize trees with acquisition number 1, which permits testing at(G) ≤ k in time O(nk+2) on trees. If G ≠ C5, then min{at(G), at()} = 1. If G has diameter 2, then at(G) ≤ 32 ln n ln ln n; we conjecture a constant upper bound. Indeed, at(G) = 1 when G has diameter 2 and no 4-cycle, except for four graphs with acquisition number 2. Deleting one edge of an n-vertex graph cannot increase at by more than 6.84√n, but we construct an n-vertex tree with an edge whose deletion increases it by more than (1/2)√n. We also obtain multiplicative upper bounds under products
Three Notes on Distributed Property Testing
In this paper we present distributed property-testing algorithms for graph properties in the CONGEST model, with emphasis on testing subgraph-freeness. Testing a graph property P means distinguishing graphs G = (V,E) having property P from graphs that are epsilon-far from having it, meaning that epsilon|E| edges must be added or removed from G to obtain a graph satisfying P.
We present a series of results, including:
- Testing H-freeness in O(1/epsilon) rounds, for any constant-sized graph H containing an edge (u,v) such that any cycle in H contain either u or v (or both). This includes all connected graphs over five vertices except K_5. For triangles, we can do even better when epsilon is not too small.
- A deterministic CONGEST protocol determining whether a graph contains a given tree as a subgraph in constant time.
- For cliques K_s with s >= 5, we show that K_s-freeness can be tested in O(m^(1/2-1/(s-2)) epsilon^(-1/2-1/(s-2))) rounds, where m is the number of edges in the network graph.
- We describe a general procedure for converting epsilon-testers with f(D) rounds, where D denotes the diameter of the graph, to work in O((log n)/epsilon)+f((log n)/epsilon) rounds, where n is the number of processors of the network. We then apply this procedure to obtain an epsilon-tester for testing whether a graph is bipartite and testing whether a graph is cycle-free. Moreover, for cycle-freeness, we obtain a 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.
These protocols extend and improve previous results of [Censor-Hillel et al. 2016] and [Fraigniaud et al. 2016]
Robustness of Distances and Diameter in a Fragile Network
A property of a graph G is robust if it remains unchanged in all connected spanning subgraphs of G. This form of robustness is motivated by networking contexts where some links eventually fail permanently, and the network keeps being used so long as it is connected. It is then natural to ask how certain properties of the network may be impacted as the network deteriorates. In this paper, we focus on two particular properties, which are the diameter, and pairwise distances among nodes. Surprisingly, the complexities of deciding whether these properties are robust are quite different: deciding the robustness of the diameter is coNP-complete, whereas deciding the robustness of the distance between two given nodes has a linear time complexity. This is counterintuitive, because the diameter consists of the maximum distance over all pairs of nodes, thus one may expect that the robustness of the diameter reduces to testing the robustness of pairwise distances. On the technical side, the difficulty of the diameter is established through a reduction from hamiltonian paths. The linear time algorithm for deciding robustness of the distance relies on a new characterization of two-terminal series-parallel graphs (TTSPs) in terms of excluded rooted minor, which may be of independent interest
Generating realistic scaled complex networks
Research on generative models is a central project in the emerging field of
network science, and it studies how statistical patterns found in real networks
could be generated by formal rules. Output from these generative models is then
the basis for designing and evaluating computational methods on networks, and
for verification and simulation studies. During the last two decades, a variety
of models has been proposed with an ultimate goal of achieving comprehensive
realism for the generated networks. In this study, we (a) introduce a new
generator, termed ReCoN; (b) explore how ReCoN and some existing models can be
fitted to an original network to produce a structurally similar replica, (c)
use ReCoN to produce networks much larger than the original exemplar, and
finally (d) discuss open problems and promising research directions. In a
comparative experimental study, we find that ReCoN is often superior to many
other state-of-the-art network generation methods. We argue that ReCoN is a
scalable and effective tool for modeling a given network while preserving
important properties at both micro- and macroscopic scales, and for scaling the
exemplar data by orders of magnitude in size.Comment: 26 pages, 13 figures, extended version, a preliminary version of the
paper was presented at the 5th International Workshop on Complex Networks and
their Application
A Sublinear Tester for Outerplanarity (and Other Forbidden Minors) With One-Sided Error
We consider one-sided error property testing of -minor freeness
in bounded-degree graphs for any finite family of graphs that
contains a minor of , the -circus graph, or the -grid
for any . This includes, for instance, testing whether a graph
is outerplanar or a cactus graph. The query complexity of our algorithm in
terms of the number of vertices in the graph, , is . Czumaj et~al.\ showed that cycle-freeness and -minor
freeness can be tested with query complexity by using
random walks, and that testing -minor freeness for any that contains a
cycles requires queries. In contrast to these results, we
analyze the structure of the graph and show that either we can find a subgraph
of sublinear size that includes the forbidden minor , or we can find a pair
of disjoint subsets of vertices whose edge-cut is large, which induces an
-minor.Comment: extended to testing outerplanarity, full version of ICALP pape
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