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Single Pass Spectral Sparsification in Dynamic Streams [article]

Michael Kapralov and Yin Tat Lee and Cameron Musco and Christopher Musco and Aaron Sidford
2015 arXiv   pre-print
We present the first single pass algorithm for computing spectral sparsifiers of graphs in the dynamic semi-streaming model.  ...  work, the best known single pass algorithm for maintaining spectral sparsifiers in dynamic streams required sketches of dimension Omega((1/epsilon^2) n^(5/3)) [AGM14].  ...  Acknowledgements We would like to thank Richard Peng for pointing us to the recursive row sampling algorithm contained in [MP12] , which became a critical component of our streaming algorithm.  ... 
arXiv:1407.1289v3 fatcat:swm5tbcppnglxmurbz7r5q7y6i

Single Pass Spectral Sparsification in Dynamic Streams

M. Kapralov, Y. T. Lee, C. N. Musco, C. P. Musco, A. Sidford
2017 SIAM journal on computing (Print)  
We present the first single pass algorithm for computing spectral sparsifiers of graphs in the dynamic semi-streaming model.  ...  best known single pass algorithm for maintaining spectral sparsifiers in dynamic streams required sketches of dimension Ω( 1 2 n 5/3 ) [AGM14].  ...  Acknowledgements We would like to thank Richard Peng for pointing us to the recursive row sampling algorithm contained in [MP12] , which became a critical component of our streaming algorithm.  ... 
doi:10.1137/141002281 fatcat:nehqritumnhtnoh4edudgvzjaq

Single Pass Spectral Sparsification in Dynamic Streams

Michael Kapralov, Yin Tat Lee, Cameron Musco, Christopher Musco, Aaron Sidford
2014 2014 IEEE 55th Annual Symposium on Foundations of Computer Science  
We present the first single pass algorithm for computing spectral sparsifiers of graphs in the dynamic semi-streaming model.  ...  best known single pass algorithm for maintaining spectral sparsifiers in dynamic streams required sketches of dimension Ω( 1 2 n 5/3 ) [AGM14].  ...  Acknowledgements We would like to thank Richard Peng for pointing us to the recursive row sampling algorithm contained in [MP12] , which became a critical component of our streaming algorithm.  ... 
doi:10.1109/focs.2014.66 dblp:conf/focs/KapralovLMMS14 fatcat:3hbqkrd4lzc2dmxfhh6jefbppi

Spectral Sparsification in Dynamic Graph Streams [chapter]

Kook Jin Ahn, Sudipto Guha, Andrew McGregor
2013 Lecture Notes in Computer Science  
(SODA 2012, PODS 2012 gives rise to the first data stream algorithm for the construction of spectral sparsifiers in the dynamic setting where edges can be added or removed from the stream.  ...  cut information (in the case of combinatorial sparsifiers) and additional spectral information (in the case of spectral sparsifiers).  ...  This will be performed in a single-pass over a dynamic graph stream using O( −2 n 5/3 polylog n) space. Let S be the set of samples returned. 2.  ... 
doi:10.1007/978-3-642-40328-6_1 fatcat:agzu7blp3zc45c3dc2qot7mcui

A Framework for Analyzing Resparsification Algorithms [article]

Rasmus Kyng, Jakub Pachocki, Richard Peng, Sushant Sachdeva
2016 arXiv   pre-print
In recent years, there has been interest in computing spectral sparsifiers in semi-streaming and dynamic settings.  ...  of G with O(n n) edges in a single pass over G, using only O(n n) space, and O(m ^2 n) total time.  ...  Kelner and Levin [KL13] gave a simple single-pass algorithm for constructing a spectral sparsifier.  ... 
arXiv:1611.06940v1 fatcat:nziw3pob55dvlmhdhpqu4pbcsi

Graph stream algorithms

Andrew McGregor
2014 SIGMOD record  
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  ...  Over the last decade, there has been considerable interest in designing algorithms for processing massive graphs in the data stream model.  ...  Min-Cut and Sparsification In this section we revisit graph sparsification in the context of dynamic graphs.  ... 
doi:10.1145/2627692.2627694 fatcat:r4jv5erzozel7epluhnduywjze

Single pass sparsification in the streaming model with edge deletions [article]

Ashish Goel and Michael Kapralov and Ian Post
2012 arXiv   pre-print
In this paper we give a construction of cut sparsifiers of Benczur and Karger in the dynamic streaming setting in a single pass over the data stream.  ...  Previous constructions either required multiple passes or were unable to handle edge deletions. We use Õ(1/^2) time for each stream update and Õ(n/^2) time to construct a sparsifier.  ...  The focus of this paper is to provide a single-pass implementation of cut sparsification on dynamic streams in the semi-streaming model.  ... 
arXiv:1203.4900v1 fatcat:qjjlfeaoonhsjgafpjnhw6aebq

Dynamic Streaming Spectral Sparsification in Nearly Linear Time and Space [article]

Michael Kapralov and Navid Nouri and Aaron Sidford and Jakab Tardos
2019 arXiv   pre-print
In this paper we consider the problem of computing spectral approximations to graphs in the single pass dynamic streaming model.  ...  recovers a spectral sparsifier in Õ(n) time.  ...  A sketch is called oblivious if its distribution is independent of the input -such sketches yield efficient single pass dynamic streaming algorithms for sparsification.  ... 
arXiv:1903.12150v1 fatcat:55r4ia2nuzdbvjiwhaudzz4xse

Communication-Optimal Distributed Dynamic Graph Clustering

Chun Jiang Zhu, Tan Zhu, Kam-Yiu Lam, Song Han, Jinbo Bi
2019 PROCEEDINGS OF THE THIRTIETH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE AND THE TWENTY-EIGHTH INNOVATIVE APPLICATIONS OF ARTIFICIAL INTELLIGENCE CONFERENCE  
the message passing and the blackboard models.  ...  We consider the problem of clustering graph nodes over large-scale dynamic graphs, such as citation networks, images and web networks, when graph updates such as node/edge insertions/deletions are observed  ...  Dynamic Graph Streams.  ... 
doi:10.1609/aaai.v33i01.33015957 fatcat:pvzm4e4mrjefjhcwlvnjg55s74

Communication-Optimal Distributed Dynamic Graph Clustering [article]

Chun Jiang Zhu, Tan Zhu, Kam-Yiu Lam, Song Han, Jinbo Bi
2018 arXiv   pre-print
the message passing and the blackboard models.  ...  We consider the problem of clustering graph nodes over large-scale dynamic graphs, such as citation networks, images and web networks, when graph updates such as node/edge insertions/deletions are observed  ...  Dynamic Graph Streams.  ... 
arXiv:1811.06072v1 fatcat:5w6oqvitjzas5jdlj3hklols5a

Spectral Sparsification in the Semi-streaming Setting

Jonathan A. Kelner, Alex Levin
2012 Theory of Computing Systems  
The strongest commonly-used notion of sparsification is spectral sparsification; H is a spectral sparsifier of G if the quadratic forms induced by the Laplacians of G and H approximate one another well  ...  only one pass over G.  ...  They then, using an ingenious construction and careful analysis, find a way to implement this in a single pass.  ... 
doi:10.1007/s00224-012-9396-1 fatcat:g2pmivg6gfhzjgiej7yf62k4da

Graph Sparsification via Refinement Sampling [article]

Ashish Goel and Michael Kapralov and Sanjeev Khanna
2010 arXiv   pre-print
In this paper, we introduce a new technique for graph sparsification, namely refinement sampling, that gives an Õ(m) time semi-streaming algorithm for graph sparsification.  ...  Specifically, we show that refinement sampling can be used to design a one-pass streaming algorithm for sparsification that takes O( n) time per edge, uses O(^2 n) space per node, and outputs an -sparsifier  ...  Also, REFINE can be implemented using a single pass over the set of edges. A scheme of refinement relations between S l,k is given in Fig. 1 .  ... 
arXiv:1004.4915v1 fatcat:wn7uc7e2svglbakiozlllcjtsi

Spanners and sparsifiers in dynamic streams

Michael Kapralov, David Woodruff
2014 Proceedings of the 2014 ACM symposium on Principles of distributed computing - PODC '14  
In this paper, we concentrate on the problem of constructing linear sketches of graphs that (approximately) preserve the spectral information of the graph in a few passes over the stream.  ...  Linear sketching is a popular technique for computing in dynamic streams, where one needs to handle both insertions and deletions of elements.  ...  O(N/D)-ADDITIVE SPANNERS INÕ(ND) SPACE IN A SINGLE PASS In this section we give an algorithm for constructing additive spanners in the dynamic stream model.  ... 
doi:10.1145/2611462.2611497 dblp:conf/podc/KapralovW14 fatcat:blore6f7rvczjo2a6vulb2yznm

On Fully Dynamic Graph Sparsifiers

Ittai Abraham, David Durfee, Ioannis Koutis, Sebastian Krinninger, Richard Peng
2016 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)  
First, we give a fully dynamic algorithm for maintaining a (1 ±ϵ) -spectral sparsifier with amortized update time poly(n, ϵ^-1).  ...  We initiate the study of dynamic algorithms for graph sparsification problems and obtain fully dynamic algorithms, allowing both edge insertions and edge deletions, that take polylogarithmic time after  ...  Thus, the full dynamic update process of all data structures takes O(poly(log n, −1 )) time per dynamic update ofG.  ... 
doi:10.1109/focs.2016.44 dblp:conf/focs/AbrahamDKKP16 fatcat:tu35ze66lncbdbg32fjv63az5u

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

Arnold Filtser, Michael Kapralov, Navid Nouri
2020 arXiv   pre-print
then, and has led to near optimal dynamic streaming algorithms for many fundamental problems such as connectivity, cut and spectral sparsifiers and matchings.  ...  In particular, no single pass algorithm is known, and the optimal tradeoff between the number of passes, stretch and space complexity is open.  ...  We prove the following theorem in Section 4, as a corollary we obtain a positive answer to the question above (as spectral sparsifier can be computed in a single dynamic stream pass [KLM + 14]).  ... 
arXiv:2007.14204v2 fatcat:afissnbsu5flxftswo7fbwujp4
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