Fast and Space Efficient Spectral Sparsification in Dynamic Streams

Kapralov, Michael; Mousavifar, Aida; Musco, Cameron; Musco, Christopher; Nouri, Navid; Sidford, Aaron; Tardos, Jakab

doi:10.1137/1.9781611975994.111

Cited by 13 publications

(7 citation statements)

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“…Graph Sparsification. Graph sparsification was introduced by Benczúr and Karger [8] ("for-all" cut sparsfiers), and has led to research in a number of directions: Fung et al [17] and Kapralov and Panigrahy [27] gave new algorithms for preserving cuts in a sparsifier; Spielman and Teng [46] generalized to spectral sparsfiers that preserved all quadratic forms, which led to further research both in improving the bounds on the size of the sparsifier [45,7] and also in the running time of spectral sparsification algorithms (e.g., [35,5,36,11,34,31,33,32]); faster algorithms for fundamental graph problems such as maximum flow utilized sparsification results (e.g., [8,43]); Ahn and Guha [1] introduced sparsification in the streaming model, which has led to a large body of work for both cut sparifiers (e.g., [2,3,18]) and spectral sparsifiers (e.g., [26,25,24,4]) in graph streams; both cut [30,40] and spectral [44] sparsification have been studied in hypergraphs; etc. For lower bounds, Andoni et al [6] showed that any data structure that (1± )-approximately stores the sizes of all cuts in an undirected graph must use Ω(n/ 2 ) bits.…”

Section: Related Workmentioning

confidence: 99%

Sparsification of Directed Graphs via Cut Balance

Cen,

Cheng,

Panigrahi

et al. 2020

Preprint

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Sparsification, where the cut values of an input graph are approximately preserved by a sparse graph (called a cut sparsifier) or a succinct data structure (called a cut sketch), has been an influential tool in graph algorithms. But, this tool is restricted to undirected graphs, because some directed graphs are known to not admit sparsification. Such examples, however, are structurally very dissimilar to undirected graphs in that they exhibit highly unbalanced cuts. This motivates us to ask: can we sparsify a balanced digraph?To make this question concrete, we define balance β of a digraph as the maximum ratio of the cut value in the two directions (Ene et al., STOC 2016). We show the following results: * Part of this work was done when the author was a postdoctoral researcher at Duke University and when he was visiting the Institute of Advanced Study.

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Section: Related Workmentioning

confidence: 99%

Sparsification of Directed Graphs via Cut Balance

Cen,

Cheng,

Panigrahi

et al. 2020

Preprint

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“…The goal is to develop algorithms that minimize different parameter values, with a special focus on minimizing the storage for the graph data structure. While space complexity is the main focus, significant effort is devoted to optimizing the runtime of streaming algorithms, specifically the time to process an edge update, as well as the time to recover the final solution (see, e.g., [138] and [122] for some recent developments). Typically the space requirement of graph streaming algorithms is O(n polylog n) (this is known as the semi-streaming model [75]), i.e., about the space needed to store a few spanning trees of the graph.…”

Section: Streaming Graph Algorithmsmentioning

confidence: 99%

“…The idea is to apply classical sketching techniques such as COUNTSKETCH [159] or distinct elements sketch (e.g., HYPERLOGLOG [80]) to the edge incidence matrix of the input graph. Existing results show how to approximate the connectivity and cut structure [12], [16], spectral structure [123], [122], shortest path metric [12], [124], or subgraph counts [119], [117] using small sketches. Extensions to some of these techniques to hypergraphs were also proposed [96].…”

Section: Graph Sketching and Dynamic Graph Streamsmentioning

confidence: 99%

Practice of Streaming Processing of Dynamic Graphs: Concepts, Models, and Systems

Besta¹,

Fischer²,

Kalavri³

et al. 2019

Preprint

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Graph processing has become an important part of various areas of computing, including machine learning, medical applications, social network analysis, computational sciences, and others. A growing amount of the associated graph processing workloads are dynamic, with millions of edges added or removed per second. Graph streaming frameworks are specifically crafted to enable the processing of such highly dynamic workloads. Recent years have seen the development of many such frameworks. However, they differ in their general architectures (with key details such as the support for the parallel execution of graph updates, or the incorporated graph data organization), the types of updates and workloads allowed, and many others. To facilitate the understanding of this growing field, we provide the first analysis and taxonomy of dynamic and streaming graph processing. We focus on identifying the fundamental system designs and on understanding their support for concurrency and parallelism, and for different graph updates as well as analytics workloads. We also crystallize the meaning of different concepts associated with streaming graph processing, such as dynamic, temporal, online, and time-evolving graphs, edge-centric processing, models for the maintenance of updates, and graph databases. Moreover, we provide a bridge with the very rich landscape of graph streaming theory by giving a broad overview of recent theoretical related advances, and by analyzing which graph streaming models and settings could be helpful in developing more powerful streaming frameworks and designs. We also outline graph streaming workloads and research challenges.

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“…They showed that many problems, such as Connectivity and Bipartiteness, can be solved using the same amount of space as in insertion-only streams up to poly-logarithmic factors. Various other works subsequently gave results of a similar flavor and presented insertion-deletion streaming algorithms with similar space complexity as their insertion-only counterparts for problems including Spectral Sparsification [22] and ∆ + 1-coloring [3]. Konrad [23] and Assadi et al [5] were the first to give a separation result between the insertion-only graph stream model and the insertion-deletion graph stream model: While it is known that a 2-approximation to Maximum Matching can be computed using space O(n log n) in insertion-only streams, Konrad showed that space Ω(n establishes that their lower bound is optimal (a different algorithm that matches this lower bound is given by Chitnis et al [8]).…”

Section: Introductionmentioning

confidence: 97%

Optimal Lower Bounds for Matching and Vertex Cover in Dynamic Graph Streams

Dark,

Konrad

2020

Preprint

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In this paper, we give simple optimal lower bounds on the one-way two-party communication complexity of approximate Maximum Matching and Minimum Vertex Cover with deletions. In our model, Alice holds a set of edges and sends a single message to Bob. Bob holds a set of edge deletions, which form a subset of Alice's edges, and needs to report a large matching or a small vertex cover in the graph spanned by the edges that are not deleted. Our results imply optimal space lower bounds for insertion-deletion streaming algorithms for Maximum Matching and Minimum Vertex Cover. Previously, Assadi et al. [SODA 2016] gave an optimal space lower bound for insertion-deletion streaming algorithms for Maximum Matching via the simultaneous model of communication. Our lower bound is simpler and stronger in several aspects: The lower bound of Assadi et al. only holds for algorithms that (1) are able to process streams that contain a triple exponential number of deletions in n, the number of vertices of the input graph; (2) are able to process multi-graphs; and(3) never output edges that do not exist in the input graph when the randomized algorithm errs. In contrast, our lower bound even holds for algorithms that (1) rely on short (O(n 2 )-length) input streams; (2) are only able to process simple graphs; and (3) may output non-existing edges when the algorithm errs.

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Fast and Space Efficient Spectral Sparsification in Dynamic Streams

Cited by 13 publications

References 0 publications

Sparsification of Directed Graphs via Cut Balance

Sparsification of Directed Graphs via Cut Balance

Practice of Streaming Processing of Dynamic Graphs: Concepts, Models, and Systems

Optimal Lower Bounds for Matching and Vertex Cover in Dynamic Graph Streams

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