Graph Sketching and Streaming: New Approaches for Analyzing Massive Graphs

McGregor, Andrew

doi:10.1007/978-3-319-58747-9_4

Cited by 7 publications

(5 citation statements)

References 28 publications

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“…Streaming algorithms are well-studied with too many results to list and we refer the reader to [McG14,McG17] for a survey of streaming algorithms. The idea of linear graph sketching was introduced in a seminal paper of Ahn, Guha, and McGregror [AGM12b].…”

Section: Related Workmentioning

confidence: 99%

Graph Spanners by Sketching in Dynamic Streams and the Simultaneous Communication Model

Filtser¹,

Kapralov²,

Nouri³

2020

Preprint

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Graph sketching is a powerful technique introduced by the seminal work of Ahn, Guha and McGregor'12 on connectivity in dynamic graph streams that has enjoyed considerable attention in the literature since then, and has led to near optimal dynamic streaming algorithms for many fundamental problems such as connectivity, cut and spectral sparsifiers and matchings. Interestingly, however, the sketching and dynamic streaming complexity of approximating the shortest path metric of a graph is still far from well-understood. Besides a direct k-pass implementation of classical spanner constructions, the state of the art amounts to a O(log k)-pass algorithm of Ahn, Guha and McGregor'12, and a 2-pass algorithm of Kapralov and Woodruff'14. In particular, no single pass algorithm is known, and the optimal tradeoff between the number of passes, stretch and space complexity is open.In this paper we introduce several new graph sketching techniques for approximating the shortest path metric of the input graph. We give the first single pass sketching algorithm for constructing graph spanners: we show how to obtain a O(n 2 3 )-spanner using O(n) space, and in general a O(n 2 3 (1−α) )-spanner using O(n 1+α ) space for every α ∈ [0, 1], a tradeoff that we think may be close optimal. We also give new spanner construction algorithms for any number of passes, simultaneously improving upon all prior work on this problem. Finally, we note that unlike the original sketching approach of Ahn, Guha and McGregor'12, none of the existing spanner constructions yield simultaneous communication protocols with low per player information. We give the first such protocols for the spanner problem that use a small number of rounds.

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Section: Related Workmentioning

confidence: 99%

Graph Spanners by Sketching in Dynamic Streams and the Simultaneous Communication Model

Filtser¹,

Kapralov²,

Nouri³

2020

Preprint

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“…The past decade has seen a large amount of work on the space complexity of graph problems in the streaming model of computation (see, e.g. the recent survey by McGregor [McG17]). The semistreaming model of computation, which allows Õ(n) space to process a graph on n vertices, has been extensively studied, with space efficient algorithms known for many fundamental graph problems such as spanning trees [AGM12a], sparsifiers [AG09, KL11, AGM12b, KLM + 14], matchings [AG11, AG13, GKK12, Kap13, GO12, HRVZ15, Kon15, AKLY15], spanners [AGM12b,KW14].…”

Section: Related Workmentioning

confidence: 99%

The Sketching Complexity of Graph and Hypergraph Counting

Kallaugher¹,

Kapralov²,

Price³

2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

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Subgraph counting is a fundamental primitive in graph processing, with applications in social network analysis (e.g., estimating the clustering coefficient of a graph), database processing and other areas. The space complexity of subgraph counting has been studied extensively in the literature, but many natural settings are still not well understood. In this paper we revisit the subgraph (and hypergraph) counting problem in the sketching model, where the algorithm's state as it processes a stream of updates to the graph is a linear function of the stream. This model has recently received a lot of attention in the literature, and has become a standard model for solving dynamic graph streaming problems.In this paper we give a tight bound on the sketching complexity of counting the number of occurrences of a small subgraph H in a bounded degree graph G presented as a stream of edge updates. Specifically, we show that the space complexity of the problem is governed by the fractional vertex cover number of the graph H. Our subgraph counting algorithm implements a natural vertex sampling approach, with sampling probabilities governed by the vertex cover of H. Our main technical contribution lies in a new set of Fourier analytic tools that we develop to analyze multiplayer communication protocols in the simultaneous communication model, allowing us to prove a tight lower bound. We believe that our techniques are likely to find applications in other settings. Besides giving tight bounds for all graphs H, both our algorithm and lower bounds extend to the hypergraph setting, albeit with some loss in space complexity.

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“…In the vertex input semi-streaming model, Goel et al [26] give a deterministic 1−1/e approximation and Kapralov [32] proves that no semi-streaming algorithm can improve upon this ratio. (See also the recent survey by McGregor [41].) The difference between semi-streaming algorithms and online algorithms in the sense of competitive analysis is that streaming algorithms do not have to make online decisions but must maintain small space throughout the computation while online algorithms must make irrevocable decisions for each input item but have no space requirement.…”

Section: Introductionmentioning

confidence: 99%

On Conceptually Simple Algorithms for Variants of Online Bipartite Matching

2019

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We present a series of results regarding conceptually simple algorithms for bipartite matching in various online and related models.We first consider a deterministic adversarial model. The best approximation ratio possible for a one-pass deterministic online algorithm is 1/2, which is achieved by any greedy algorithm. Dürr et al. [20] recently presented a 2-pass algorithm called Category-Advice that achieves approximation ratio 3/5. We extend their algorithm to multiple passes. We prove the exact approximation ratio for the k-pass Category-Advice algorithm for all k ≥ 1, and show that the approximation ratio converges to the inverse of the golden ratio 2/(1 + √ 5) ≈ 0.618 as k goes to infinity. The convergence is extremely fast -the 5-pass Category-Advice algorithm is already within 0.01% of the inverse of the golden ratio.We then consider a natural greedy algorithm in the online stochastic IID model -Min-Degree. This algorithm is an online version of a well-known and extensively studied offline algorithm MinGreedy. MinGreedy is known to have excellent empirical performance, as well as excellent performance under various input models. Alas, recently it was shown to have approximation ratio 1/2 in the adversarial offline setting -the approximation ratio achieved by any greedy algorithm. We obtain a result in the online known IID model that, in spirit, is similar to the offline result, but the proof is different. Namely, we show that MinDegree cannot achieve an approximation ratio better than 1 − 1/e, which is guaranteed by any consistent greedy algorithm in the known IID model.Finally, following the work in Besser and Poloczek [7], we depart from an adversarial or stochastic ordering and investigate a natural randomized algorithm (MinRanking) in the priority model. Although the priority model allows the algorithm to choose the input ordering in a general but well defined way, this natural algorithm cannot obtain the approximation of the Ranking algorithm in the ROM model. * Research is supported by NSERC.

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Graph Sketching and Streaming: New Approaches for Analyzing Massive Graphs

Cited by 7 publications

References 28 publications

Graph Spanners by Sketching in Dynamic Streams and the Simultaneous Communication Model

Graph Spanners by Sketching in Dynamic Streams and the Simultaneous Communication Model

The Sketching Complexity of Graph and Hypergraph Counting

On Conceptually Simple Algorithms for Variants of Online Bipartite Matching

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