Efficient Primal-Dual Graph Algorithms for MapReduce

Bahmani, Bahman; Goel, Ashish; Munagala, Kamesh

doi:10.1007/978-3-319-13123-8_6

Cited by 22 publications

(40 citation statements)

References 8 publications

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“…Their algorithm needs O(log 1+ǫ n) passes; i.e., it has to read through the sequence of edge insertions O(log 1+ǫ n) times. (Their algorithm was also extended to a MapReduce algorithm, which was later improved by [5].) In Section 3, we improve this result of Bahmani et al in two respects: (a) We can process a dynamic stream of updates, and (b) we need only a single pass.…”

Section: Previous Workmentioning

confidence: 96%

Space- and Time-Efficient Algorithm for Maintaining Dense Subgraphs on One-Pass Dynamic Streams

Bhattacharya

Henzinger

Nanongkai

et al. 2015

Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing

107

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While in many graph mining applications it is crucial to handle a stream of updates efficiently in terms of both time and space, not much was known about achieving such type of algorithm. In this paper we study this issue for a problem which lies at the core of many graph mining applications called densest subgraph problem. We develop an algorithm that achieves time- and space-efficiency for this problem simultaneously. It is one of the first of its kind for graph problems to the best of our knowledge. Given an input graph, the densest subgraph is the subgraph that maximizes the ratio between the number of edges and the number of nodes. For any ε>0, our algorithm can, with high probability, maintain a (4+ε)-approximate solution under edge insertions and deletions using ~O(n) space and ~O(1) amortized time per update; here, $n$ is the number of nodes in the graph and ~O hides the O(polylog_{1+ε} n) term. The approximation ratio can be improved to (2+ε) with more time. It can be extended to a (2+ε)-approximation sublinear-time algorithm and a distributed-streaming algorithm. Our algorithm is the first streaming algorithm that can maintain the densest subgraph in one pass. Prior to this, no algorithm could do so even in the special case of an incremental stream and even when there is no time restriction. The previously best algorithm in this setting required O(log n) passes [BahmaniKV12]. The space required by our algorithm is tight up to a polylogarithmic factor.

QC 20150811

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Section: Previous Workmentioning

confidence: 96%

Space- and Time-Efficient Algorithm for Maintaining Dense Subgraphs on One-Pass Dynamic Streams

Bhattacharya

Henzinger

Nanongkai

et al. 2015

Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing

107

View full text Add to dashboard Cite

QC 20150811

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“…For example, the maximum density subgraph problem has been studied in many computational models, e.g. [5,6,13,16,21,25,30,32,38,44,47,51,52]. The low outdegree orientation problem has been studied in the centralized context in [11,14,24,34,39,40].…”

Section: Pseudo-forest Decomposition and Low Outdegree Orientationmentioning

confidence: 99%

On the Locality of Nash-Williams Forest Decomposition and Star-Forest Decomposition

Harris

2021

Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing

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Given a graph = ( , ) with arboricity , we study the problem of decomposing the edges of into (1 + ) disjoint forests in the distributed LOCAL model. While there is a polynomial time centralized algorithm for -forest decomposition (e.g. [Imai, J. Operation Research Soc. of Japan '83]), it remains an open question how close we can get to this exact decomposition in the LOCAL model. Barenboim and Elkin [PODC '08] developed a LOCAL algorithm to compute a (2 + ) -forest decomposition in ( log ) rounds. Ghaffari and Su [SODA '17] made further progress by computing a (1 + ) -forest decomposition in ( log 3 4 ) rounds when = Ω( √︁ log ), i.e., the limit of their algorithm is an ( + Ω( √︁ log ))forest decomposition. This algorithm, based on a combinatorial construction of Alon, McDiarmid & Reed [Combinatorica '92], in fact provides a decomposition of the graph into star-forests, i.e., each forest is a collection of stars.Our main goal is to reduce the threshold of in (1 + ) -forest decomposition. We obtain the following main results:)-round algorithm when = Ω (1) in simple graphs and multigraphs, where > 0 is any arbitrary constant.• An ( log 4 log Δ )-round algorithm when = Ω( log Δ log log Δ ) in simple graphs and multigraphs.• An ( log 4)-round algorithm when = Ω(log ) in simple graphs and multigraphs. This also covers an extension of the forest-decomposition problem to list-edge-coloring.• An ()-round algorithm for star-forest decomposition for = Ω( √︁ log Δ + log ) in simple graphs. When ≥ Ω(log Δ), this also covers a list-edge-coloring variant.Our techniques also give an algorithm for (1 + ) -outdegreeorientation in (log 3 / ) rounds, which is the first algorithm with linear dependency on −1 .At a high level, the first three results come from a combination of network decomposition, load balancing, and a new structural result on local augmenting sequences. The fourth result uses a

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“…For densest-subgraph detection in unweighted graphs, max-ow-based exact algorithms [15,20] and greedy approximation algorithms [11,20] have been proposed. Extensions include adding size bounds [3], using alternative metrics [35], nding subgraphs with limited overlap [7,14], and extending to large-scale graphs [5,6] and dynamic graphs [9,13,25]. Other approaches include spectral methods [27] and frequent itemset mining [29].…”

Section: Related Workmentioning

confidence: 99%

DenseAlert

Shin

Hooi

Kim

et al. 2017

Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining

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Consider a stream of retweet events -how can we spot fraudulent lock-step behavior in such multi-aspect data (i.e., tensors) evolving over time? Can we detect it in real time, with an accuracy guarantee? Past studies have shown that dense subtensors tend to indicate anomalous or even fraudulent behavior in many tensor data, including social media, Wikipedia, and TCP dumps. us, several algorithms have been proposed for detecting dense subtensors rapidly and accurately. However, existing algorithms assume that tensors are static, while many real-world tensors, including those mentioned above, evolve over time.We propose D S , an incremental algorithm that maintains and updates a dense subtensor in a tensor stream (i.e., a sequence of changes in a tensor), and D A , an incremental algorithm spo ing the sudden appearances of dense subtensors. Our algorithms are: (1) Fast and 'any time': updates by our algorithms are up to a million times faster than the fastest batch algorithms, (2) Provably accurate: our algorithms guarantee a lower bound on the density of the subtensor they maintain, and (3) E ective: our D A successfully spots anomalies in realworld tensors, especially those overlooked by existing algorithms.

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Efficient Primal-Dual Graph Algorithms for MapReduce

Cited by 22 publications

References 8 publications

Space- and Time-Efficient Algorithm for Maintaining Dense Subgraphs on One-Pass Dynamic Streams

Space- and Time-Efficient Algorithm for Maintaining Dense Subgraphs on One-Pass Dynamic Streams

On the Locality of Nash-Williams Forest Decomposition and Star-Forest Decomposition

DenseAlert

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