Decomposing a graph into expanding subgraphs

Moshkovitz, Guy; Shapira, A.

doi:10.1002/rsa.20727

Cited by 10 publications

(11 citation statements)

References 36 publications

(141 reference statements)

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“…We mention that this bound is far from tight and indeed, without too much effort, it can be shown that the number of cycles of length s in Q d is at most 2 d sd 2 s 2 (see [39,Claim 3.2]).…”

Section: Property (Ii)mentioning

confidence: 99%

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Expansion, long cycles, and complete minors in supercritical random subgraphs of the hypercube

Erde¹,

Kang²,

Krivelevich³

2021

Preprint

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Analogous to the case of the binomial random graph G(d + 1, p), it is known that the behaviour of a random subgraph of a d-dimensional hypercube, where we include each edge independently with probability p, which we denote by Q d p , undergoes a phase transition when p is around 1 d . More precisely, standard arguments show that significantly below this value of p, with probability tending to one as d → ∞ (whp for short) all components of this graph have order O(d), whereas Ajtai, Komlós and Szemerédi [1] showed that significantly above this value, in the supercritical regime, whp there is a unique 'giant' component of order Θ 2 d . In G(d + 1, p) much more is known about the complex structure of the random graph which emerges in this supercritical regime. For example, it is known that in this regime whp G(d + 1, p) contains paths and cycles of length Ω(d), as well as complete minors of order Ω d . In this paper we obtain analogous results in Q d p . In particular, we show that for supercritical p, i.e., when p = 1+ǫ d for a positive constant ǫ, whp Q d p contains a cycle of length Ω 2 d d 3 (log d ) 3 and a complete minor of order Ω 2 d 2 d 3 (log d ) 3 .In order to prove these results, we show that whp the largest component of Q d p has good edge-expansion properties, a result of independent interest. We also consider the genus of Q d p and show that, in this regime of p, whp the genus is Ω 2 d .

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Section: Property (Ii)mentioning

confidence: 99%

“…In fact, it is perhaps surprising that the expansion properties of L 1 Q d p are useful at all, since random subgraphs of the hypercube have been used by Moshkowitz and Shapira [39] to construct graphs which have in some way the worst possible expansion properties.…”

Section: Introductionmentioning

confidence: 99%

Expansion, long cycles, and complete minors in supercritical random subgraphs of the hypercube

Erde¹,

Kang²,

Krivelevich³

2021

Preprint

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“…We note that the regularity condition is crucial for proving Theorem 3. The following theorem was proved in [29]. Theorem 4 ([29]).…”

Section: A Regular Non-expanding Graphmentioning

confidence: 99%

“…We note that each of the parameters in Theorem 3 is essentially optimal (see [29] for further discussion).…”

Section: For Every T-vertex Subgraph H Of G That Is Not a Forestmentioning

confidence: 99%

Constructing near spanning trees with few local inspections

Levi

Moshkovitz

Ron

et al. 2016

Random Struct Algorithms

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Constructing a spanning tree of a graph is one of the most basic tasks in graph theory.Motivated by several recent studies of local graph algorithms, we consider the following variant of this problem. Let G be a connected bounded-degree graph. Given an edge e in G we would like to decide whether e belongs to a connected subgraph G consisting of (1+ )n edges (for a prespecified constant > 0), where the decision for different edges should be consistent with the same subgraph G . Can this task be performed by inspecting only a constant number of edges in G? Our main results are:• We show that if every t-vertex subgraph of G has expansion 1/(log t) 1+o(1) then one can (deterministically) construct a sparse spanning subgraph G of G using few inspections. To this end we analyze a "local" version of a famous minimum-weight spanning tree algorithm.• We show that the above expansion requirement is sharp even when allowing randomization. To this end we construct a family of 3-regular graphs of high girth, in which every t-vertex subgraph has expansion 1/(log t) 1−o(1) . We prove that for this family of graphs, any local algorithm for the sparse spanning graph problem requires inspecting a number of edges which is proportional to the girth.

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“…It is well known that any graph can be decomposed into connected components of conductance Ω(ǫ/ log n) (and hence poly(ǫ −1 , log n) mixing time) after removing at most an ǫ-fraction of the edges [5,33,39,41]. Moshkovitz and Shapira [31] showed that this bound is essentially tight. In particular, removing any constant fraction ǫ of the edges, the remaining components have conductance at most O((log log n) 2 / log n).…”

Section: Introductionmentioning

confidence: 99%

Distributed Triangle Detection via Expander Decomposition

Chang

Pettie

Zhang

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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We present improved distributed algorithms for triangle detection and its variants in the CONGEST model. We show that Triangle Detection, Counting, and Enumeration can be solved inÕ(n 1/2 ) rounds. In contrast, the previous state-of-the-art bounds for Triangle Detection and Enumeration wereÕ(n 2/3 ) andÕ(n 3/4 ), respectively, due to Izumi and LeGall (PODC 2017).The main technical novelty in this work is a distributed graph partitioning algorithm. We show that inÕ(n 1−δ ) rounds we can partition the edge set of the network G• Each connected component induced by E m has minimum degree Ω(n δ ) and conductance Ω(1/polylog(n)). As a consequence the mixing time of a random walk within the component is O(polylog(n)).• The subgraph induced by E s has arboricity at most n δ .• |E r | ≤ |E|/6.All of our algorithms are based on the following generic framework, which we believe is of interest beyond this work. Roughly, we deal with the set E s by an algorithm that is efficient for low-arboricity graphs, and deal with the set E r using recursive calls. For each connected component induced by E m , we are able to simulate CONGESTED-CLIQUE algorithms with small overhead by applying a routing algorithm due to Ghaffari, Kuhn, and Su (PODC 2017) for high conductance graphs.We consider Triangle Detection problems in distributed networks. In the LOCAL model [34], which has no limit on bandwidth, all variants of Triangle Detection can be solved in exactly one round of communication: every vertex v simply announces its neighborhood N (v) to all neighbors. However, in models that take bandwidth into account, e.g., CONGEST, Triangle Detection becomes significantly more complicated. Whereas many graph optimization problems studied in the CONGEST model are intrinsically "global" (i.e., require at least diameter time) [2,11,12,14,15,20,26], Triangle Detection is somewhat unusual in that it can, in principle, be solved using only locally available information.The CONGEST Model. The underlying distributed network is represented as an undirected graph G = (V, E), where each vertex corresponds to a computational device, and each edge corresponds to a bi-directional communication link. We assume each v ∈ V initially knows some global parameters such as n = |V |, ∆ = max v∈V deg(v), and D = diameter(G). Each vertex v has a distinct Θ(log n)-bit identifier ID(v). The computation proceeds according to synchronized rounds.In each round, each vertex v can perform unlimited local computation, and may send a distinct O(log n)-bit message to each of its neighbors. Throughout the paper we only consider the randomized variant of CONGEST. Each vertex is allowed to generate unlimited local random bits, but there is no global randomness.The Congested Clique Model. The CONGESTED-CLIQUE model is a variant of CONGEST that allows all-to-all communication. Each vertex initially knows its adjacent edges and the set of vertex IDs, which we can assume w.l.o.g. is {1, . . . , |V |}. In each round, each vertex transmits n − 1 O(log n)-bit messages, one addressed to ...

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Decomposing a graph into expanding subgraphs

Cited by 10 publications

References 36 publications

Expansion, long cycles, and complete minors in supercritical random subgraphs of the hypercube

Expansion, long cycles, and complete minors in supercritical random subgraphs of the hypercube

Constructing near spanning trees with few local inspections

Distributed Triangle Detection via Expander Decomposition

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