On Fully Dynamic Graph Sparsifiers

Abraham, Ittai; Durfee, David; Koutis, Ioannis; Krinninger, Sebastian; Peng, Richard

doi:10.1109/focs.2016.44

Cited by 30 publications

(81 citation statements)

References 56 publications

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“…This result is similar to the ones from [36], and improves the dynamic sparsification algorithms in [35] by a factor of log 2 n. Furthermore, its running time is within O(log log n) factor of numerically oriented routines based on random projections and solving linear systems [11]. As a result, we're optimistic about the practical potential of this sparsification approach.…”

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confidence: 69%

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A Framework for Analyzing Resparsification Algorithms

Kyng

Pachocki

Peng³

et al. 2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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A spectral sparsifier of a graph G is a sparser graph H that approximately preserves the quadratic form of G, i.e., for all vectors x, xL G and L H denote the respective graph Laplacians. Spectral sparsifiers generalize cut sparsifiers, and have found many applications in designing graph algorithms. In recent years, there has been interest in computing spectral sparsifiers in semi-streaming and dynamic settings. Natural algorithms in these settings often involve repeated sparsification of a graph, and in turn accumulation of errors across these steps. We present a framework for analyzing algorithms that perform repeated sparsifications that only incur error corresponding to a single sparsification step, leading to better results for many of these reseparsification based algorithms.As an application, we show how to maintain a spectral sparsifier in the semi-streaming setting: We present a simple algorithm that, for a graph G on n vertices and m edges, computes a spectral sparsifier of G with O(n log n) edges in a single pass over G, using only O(n log n) space, and O(m log 2 n) total time. This improves on previous best semi-streaming algorithms for both spectral and cut sparsifiers by a factor of log n in both space and runtime. The algorithm also extends to semi-streaming row sampling for general PSD matrices. As another application, we use this framework to combine a spectral sparsification algorithm by Koutis with improved spanner constructions to give a parallel algorithm for constructing O(n log 2 n log log n) sized spectral sparsifiers in O(m log 2 n log log n) time. This is the best combinatorial graph sparsification algorithm to date, and the size of the sparsifiers produced is only a factor log n log log n more than ones produced by numerical routines.

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confidence: 69%

“…This property is crucial to some uses of sparsifiers in combinatorial algorithms, such as their recent incorporation in data structures [35]. By accounting for the structure of the output of ProbabilisticSpanner given in Section 4.1, we have a similar property.…”

mentioning

confidence: 79%

A Framework for Analyzing Resparsification Algorithms

Kyng

Pachocki

Peng³

et al. 2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…In step 4, instead of computing the exact partial Cholesky factorization of L, we use the algorithm ApxPartialCholesky in Lemma 3.2 to obtain an approximate partial Cholesky factorization of L. However, if we pass the whole L to ApxPartialCholesky, it may change the edges in E (1) , to which we need to perform θ-deletions when deactivating them. Thus, instead, we first delete all edges in E ( 1) and pass the resulting L to ApxPartialCholesky, and then add those edges back to the approximate Schur complement S returned by it.…”

Section: The Reason That Inmentioning

confidence: 99%

“…Step 5 we can compute y C (S \ θ e) † y C for each e ∈ E (1) by recursion is that S is a Laplacian matrix whose edges are supported on C, and hence to compute y C (S \ θ e) † y C for all e ∈ E (1) is just a smaller-sized version of Problem 4.1 in which L = S and E Q = E (1) . In the rest of this subsection we give an algorithm that solves Problem 4.1 approximately.…”

Section: The Reason That Inmentioning

confidence: 99%

“…The former maintains the inverse of a matrix under updates, and is a critical routine for many graph algorithms [45,33,22]. However, this often leads to dense matrices, and we combine it with techniques from graph sparsification [49,1,30,35] to obtain our nearlylinear running times. The additional need to maintain dot-products against arbitrary vectors also leads us to incorporate iterative methods in our routine for approximating the vertex Kirchhoff centrality.…”

Section: Introductionmentioning

confidence: 99%

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Kirchhoff Index as a Measure of Edge Centrality in Weighted Networks: Nearly Linear Time Algorithms

Li¹,

Zhang²

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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Estimating the relative importance of vertices and edges is a fundamental issue in the analysis of complex networks, and has found vast applications in various aspects, such as social networks, power grids, and biological networks. Most previous work focuses on metrics of vertex importance and methods for identifying powerful vertices, while related work for edges is much lesser, especially for weighted networks, due to the computational challenge. In this paper, we propose to use the well-known Kirchhoff index as the measure of edge centrality in weighted networks, called θ-Kirchhoff edge centrality. The Kirchhoff index of a network is defined as the sum of effective resistances over all vertex pairs. The centrality of an edge e is reflected in the increase of Kirchhoff index of the network when the edge e is partially deactivated, characterized by a parameter θ. We define two equivalent measures for θ-Kirchhoff edge centrality. Both are global metrics and have a better discriminating power than commonly used measures, based on local or partial structural information of networks, e.g. edge betweenness and spanning edge centrality. Despite the strong advantages of Kirchhoff index as a centrality measure and its wide applications, computing the exact value of Kirchhoff edge centrality for each edge in a graph is computationally demanding. To solve this problem, for each of the θ-Kirchhoff edge centrality metrics, we present an efficient algorithm to compute its -approximation for all the m edges in nearly linear time in m. The proposed θ-Kirchhoff edge centrality is the first global metric of edge importance that can be provably approximated in nearly-linear time. Moreover, according to the θ-Kirchhoff edge centrality, we present a θ-Kirchhoff vertex centrality measure, as well as a fast algorithm that can compute -approximate Kirchhoff vertex centrality for all the n vertices in nearly linear time in m.

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Optimal Offline Dynamic 2, 3-Edge/Vertex Connectivity

Peng

Sandlund

Sleator

2019

Lecture Notes in Computer Science

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We give offline algorithms for processing a sequence of 2 and 3 edge and vertex connectivity queries in a fully-dynamic undirected graph. While the current best fully-dynamic online data structures for 3-edge and 3-vertex connectivity require O(n 2/3 ) and O(n) time per update, respectively, our per-operation cost is only O(log n), optimal due to the dynamic connectivity lower bound of Patrascu and Demaine. Our approach utilizes a divide and conquer scheme that transforms a graph into smaller equivalents that preserve connectivity information. This construction of equivalents is closely-related to the development of vertex sparsifiers, and shares important connections to several upcoming results in dynamic graph data structures, outside of just the offline model.

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On Fully Dynamic Graph Sparsifiers

Cited by 30 publications

References 56 publications

A Framework for Analyzing Resparsification Algorithms

A Framework for Analyzing Resparsification Algorithms

Kirchhoff Index as a Measure of Edge Centrality in Weighted Networks: Nearly Linear Time Algorithms

Optimal Offline Dynamic 2, 3-Edge/Vertex Connectivity

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