Fast distributed algorithms for testing graph properties

Censor-Hillel, Keren; Fischer, Eldar; Schwartzman, Gregory; Vasudev, Yadu

doi:10.1007/s00446-018-0324-8

Cited by 45 publications

(91 citation statements)

References 51 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the property testing setting, an algorithm has to decide, with probability at least 2 3 , if the input graph is (a) H-free (i.e., does not contain a subgraph isomorphic to H) or (b) ε-far from being H-free (that is, the goal is to distinguish whether the input graph G is H-free or one needs to modify more than ε|E(G)| edges of G to obtain a graph that is H-free); in the intermediate case, the algorithm can perform arbitrarily (see e.g., [3,7] for more details). Property testing of H-freeness in the CONGEST model has received a lot of attention lately (see, e.g., [1,2,7,8,9]). In particular, it has been shown [7] that testing H-freeness can be done in O(1/ε) round in the CONGEST model for any constant-size graph H containing an edge (x, y) such that any cycle in H contains at least one of x, y.…”

Section: Property Testing Of H-freenessmentioning

confidence: 99%

See 1 more Smart Citation

Detecting cliques in CONGEST networks

Czumaj

Konrad

2019

Distrib. Comput.

View full text Add to dashboard Cite

The problem of detecting network structures plays a central role in distributed computing. One of the fundamental problems studied in this area is to determine whether for a given graph H, the input network contains a subgraph isomorphic to H or not. We investigate this problem for H being a clique K in the classical distributed CONGEST model, where the communication topology is the same as the topology of the underlying network, and with limited communication bandwidth on the links.Our first and main result is a lower bound, showing that detecting K requires Ω( √ n/b) communication rounds, for every 4 ≤ ≤ √ n, and Ω(n/( b)) rounds for every ≥ √ n, where b is the bandwidth of the communication links. This result is obtained by using a reduction to the set disjointness problem in the framework of two-party communication complexity. We complement our lower bound with a two-party communication protocol for listing all cliques in the input graph, which up to constant factors communicates the same number of bits as our lower bound for K4 detection. This demonstrates that our lower bound cannot be improved using the two-party communication framework. detecting K 4 and K in Section 2, to ensure full generality of presentation, we will make the analysis parametrized by the message size b, in which case we will refer to such model of distributed computation as CONGEST b , the CONGEST model with messages of size b.)Our goal is, for a given network G = (V, E) and ≥ 4, to solve the subgraph detection problem for a clique K , that is, to design an algorithm in the CONGEST model such that (i) if G contains a copy of K , then with probability at least 2 3 at least one node outputs 1, and (ii) if G does not contain any copy of K , then with probability at least 2 3 no node outputs 1. The subgraph detection problem is a local problem: it can be solved efficiently solely on the basis of local information. In particular, in the CONGEST model, the problem of finding K in a graph can be trivially solved in O(n) rounds, or in fact, in O(max u∈V deg G (u)) rounds, where deg G (u) denotes the degree of node u in G. Indeed, if each node sends its entire neighborhood to all its neighbors, then afterwards, each node will be aware of all its neighbors and of their neighbors. Therefore, in particular, each node will be able to detect all cliques it belongs to. Since for each node u, the task of sending its entire neighborhood to all its neighbors can be performed in O(deg G (u)) rounds in the CONGEST model, the total number of rounds for the entire network is O(max u∈V deg G (u)) = O(n) rounds. In view of this simple observation, the main challenge in the clique K detection problem is whether this task can be performed in a sublinear number of rounds. n i=1 X i ∧ Y i . In a seminal work, Kalyanasundaram and Schnitger [13] showed that in any randomized communication protocol, the players must exchange Ω(n) bits to solve the set disjointness problem with constant success probability.Theorem 1 ([13]). The randomized two-party communication comp...

show abstract

Section: Property Testing Of H-freenessmentioning

confidence: 99%

“…since K 2,2 contains 4 edges. To bound the variance V|K|, we use the identity V|K| = E|K| 2 − (E|K|) 2 :…”

Section: Analysis Of Algorithmmentioning

confidence: 99%

Detecting cliques in CONGEST networks

Czumaj

Konrad

2019

Distrib. Comput.

View full text Add to dashboard Cite

show abstract

“…Furthermore, given a violating edge, it is possible to detect violation in poly(1/ǫ) rounds. 6 Hence, by sampling Θ(log(n)/ǫ) non-tree edges in each G j and running the detection procedure on each, a violation is detected with probability 1 − 1/poly(1/n).…”

Section: A High-level Description Of Our Algorithm For Testing Planaritymentioning

confidence: 99%

“…The testing results can be compared with the Ω(log n) lower bound of Censor-Hillel et al [6] for distributed testing of these properties on general (bounded-degree) graphs (with constant success probability). The spanner result can be compared to the recent result of Elkin and Neiman [12].…”

Section: Implications and Applications For Minor-free Graphsmentioning

confidence: 99%

See 1 more Smart Citation

Property Testing of Planarity in the CONGEST model

Levi

Medina

Ron

2018

Proceedings of the 2018 ACM Symposium on Principles of Distributed Computing

View full text Add to dashboard Cite

We give a distributed algorithm in the CONGEST model for property testing of planarity with one-sided error in general (unbounded-degree) graphs. Following Censor-Hillel et al. (DISC 2016), who recently initiated the study of property testing in the distributed setting, our algorithm gives the following guarantee: For a graph G = (V, E) and a distance parameter ǫ, if G is planar, then every node outputs accept, and if G is ǫ-far from being planar (i.e., more than ǫ · |E| edges need to be removed in order to make G planar), then with probability 1 − 1/poly(n) at least one node outputs reject. The algorithm runs in O(log |V | · poly(1/ǫ)) rounds, and we show that this result is tight in terms of the dependence on |V |.Our algorithm combines several techniques of graph partitioning and local verification of planar embeddings. Furthermore, we show how a main subroutine in our algorithm can be applied to derive additional results for property testing of cycle-freeness and bipartiteness, as well as the construction of spanners, in minor-free (unweighted) graphs. * This article extends work presented at PODC 2018 [39]. We note that Ghaffari and Haeupler [23] consider a different, but related question of finding a planar embedding of a planar graph. They give a distributed algorithm for this problem using O(D · min{log n, D}) rounds (where D is the diameter and n is the number of nodes).

show abstract