Dynamic Graphs in the Sliding-Window Model

Crouch, Michael S.; McGregor, Andrew; Stubbs, Daniel M.

doi:10.1007/978-3-642-40450-4_29

Cited by 40 publications

(41 citation statements)

References 32 publications

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“…Identifying the connected components of a graph is a fundamental problem that has been studied in a variety of settings (see e.g. [2,33,28,61,65,57] and the references therein). This problem is also of great practical importance [60] with a wide range of applications, e.g.…”

Section: Introductionmentioning

confidence: 99%

Near-Optimal Massively Parallel Graph Connectivity

Behnezhad

Dhulipala

Esfandiari

et al. 2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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Identifying the connected components of a graph, apart from being a fundamental problem with countless applications, is a key primitive for many other algorithms. In this paper, we consider this problem in parallel settings. Particularly, we focus on the Massively Parallel Computations (MPC) model, which is the standard theoretical model for modern parallel frameworks such as MapReduce, Hadoop, or Spark. We consider the truly sublinear regime of MPC for graph problems where the space per machine is n δ for some desirably small constant δ ∈ (0, 1).We present an algorithm that for graphs with diameter D in the wide range [log n, n], takes O(log D) rounds to identify the connected components and takes O(log log n) rounds for all other graphs. The algorithm is randomized, succeeds with high probability 1 , does not require prior knowledge of D, and uses an optimal total space of O(m). We complement this by showing a conditional lower-bound based on the widely believed 2-Cycle conjecture that Ω(log D) rounds are indeed necessary in this setting.Studying parallel connectivity algorithms received a resurgence of interest after the pioneering work of Andoni et al. [FOCS 2018] who presented an algorithm with O(log D · log log n) round-complexity. Our algorithm improves this result for the whole range of values of D and almost settles the problem due to the conditional lower-bound.Additionally, we show that with minimal adjustments, our algorithm can also be implemented in a variant of the (CRCW) PRAM in asymptotically the same number of rounds. * A preliminary version of this paper is O(1) round algorithm if e.g. D = O( √ n). We refute this possibility and show that indeed for any choice of D ∈ [log 1+Ω(1) , n], there are family of graphs with diameter D on which Ω(log D) rounds are necessary in this regime of MPC, if the 2-Cycle conjecture holds.Theorem 2. Fix some D ≥ log 1+ρ n for a desirably small constant ρ ∈ (0, 1). Any MPC algorithm with n 1−Ω(1) space per machine that w.h.p. identifies each connected component of any given n-vertex graph with diameter D requires Ω(log D ) rounds, unless the 2-Cycle conjecture is wrong.We note that proving any unconditional super constant lower bound for any problem in P, in this regime of MPC, would imply NC 1 P which seems out of the reach of current techniques [59].Extention to PRAM. As a side result, we provide an implementation of our connectivity algorithm in O(log D + log log m/n n) depth in the multiprefix CRCW PRAM model, a parallel computation model that permits concurrent reads and concurrent writes. This implementation of our algorithm performs O((m+n)(log D +log log m/n n)) work and is therefore nearly work-efficient. The following theorem states our result. We defer further elaborations on this result to Appendix B.3.

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Section: Introductionmentioning

confidence: 99%

Near-Optimal Massively Parallel Graph Connectivity

Behnezhad

Dhulipala

Esfandiari

et al. 2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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“…In terms of estimating the size of the maximum matching, Chitnis et al [8] extended the estimation algorithms for sparse graphs from [15] to the settings of dynamic streams usingÕ(n 4/5 ) space. A bridge between dynamic graphs and the insertion-only streaming model is the sliding window model studied by Crouch et al [10]. The authors give a (3 + ε)-approximation algorithm for maximum matching.…”

Section: Introductionmentioning

confidence: 99%

Sublinear Estimation of Weighted Matchings in Dynamic Data Streams

Bury

Schwiegelshohn

2015

Lecture Notes in Computer Science

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This paper presents an algorithm for estimating the weight of a maximum weighted matching by augmenting any estimation routine for the size of an unweighted matching. The algorithm is implementable in any streaming model including dynamic graph streams. We also give the first constant estimation for the maximum matching size in a dynamic graph stream for planar graphs (or any graph with bounded arboricity) usingÕ(n 4/5 ) space which also extends to weighted matching. Using previous results by Kapralov, Khanna, and Sudan (2014) we obtain a polylog(n) approximation for general graphs using polylog(n) space in random order streams, respectively. In addition, we give a space lower bound of Ω(n 1−ε ) for any randomized algorithm estimating the size of a maximum matching up to a 1 + O(ε) factor for adversarial streams.

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“…We call these the active edges and we will consider the case where w ≥ n. The results in this section were proved by Crouch et al [22]. Note that some of samplingbased algorithms for counting small subgraphs are also applicable in this model.…”

Section: Sliding Windowmentioning

confidence: 89%

Graph stream algorithms

McGregor¹

2014

SIGMOD Rec.

Self Cite

333

226

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Over the last decade, there has been considerable interest in designing algorithms for processing massive graphs in the data stream model. The original motivation was two-fold: a) in many applications, the dynamic graphs that arise are too large to be stored in the main memory of a single machine and b) considering graph problems yields new insights into the complexity of stream computation. However, the techniques developed in this area are now finding applications in other areas including data structures for dynamic graphs, approximation algorithms, and distributed and parallel computation. We survey the state-of-the-art results; identify general techniques; and highlight some simple algorithms that illustrate basic ideas.

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Dynamic Graphs in the Sliding-Window Model

Cited by 40 publications

References 32 publications

Near-Optimal Massively Parallel Graph Connectivity

Near-Optimal Massively Parallel Graph Connectivity

Sublinear Estimation of Weighted Matchings in Dynamic Data Streams

Graph stream algorithms

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