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Sketching cuts in graphs and hypergraphs
, 2014
"... Sketching and streaming algorithms are in the forefront of current research directions for cut problems in graphs. In the streaming model, we show that (1 − ε)approximation for MaxCut must use n1−O(ε) space; moreover, beating 4/5approximation requires polynomial space. For the sketching model, we ..."
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Sketching and streaming algorithms are in the forefront of current research directions for cut problems in graphs. In the streaming model, we show that (1 − ε)approximation for MaxCut must use n1−O(ε) space; moreover, beating 4/5approximation requires polynomial space. For the sketching model, we show that runiform hypergraphs admit a (1 + ε)cutsparsifier (i.e., a weighted subhypergraph that approximately preserves all the cuts) with O(ε−2n(r + log n)) edges. We also make first steps towards sketching general CSPs (Constraint Satisfaction Problems). 1
Semistreaming algorithms for annotated graph streams
 Electronic Colloquium on Computational Complexity (ECCC
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Vertex and Hyperedge Connectivity in Dynamic Graph Streams
"... A growing body of work addresses the challenge of processing dynamic graph streams: a graph is defined by a sequence of edge insertions and deletions and the goal is to construct synopses and compute properties of the graph while using only limited memory. Linear sketches have proved to be a powerfu ..."
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A growing body of work addresses the challenge of processing dynamic graph streams: a graph is defined by a sequence of edge insertions and deletions and the goal is to construct synopses and compute properties of the graph while using only limited memory. Linear sketches have proved to be a powerful technique in this model and can also be used to minimize communication in distributed graph processing. We present the first linear sketches for estimating vertex connectivity and constructing hypergraph sparsifiers. Vertex connectivity exhibits markedly different combinatorial structure than edge connectivity and appears to be harder to estimate in the dynamic graph stream model. Our hypergraph result generalizes the work of Ahn et al. (PODS 2012) on graph sparsification and has the added benefit of significantly simplifying the previous results. One of the main ideas is related to the problem of reconstructing subgraphs that satisfy a specific sparsity property. We introduce a more general notion of graph degeneracy and extend the graph reconstruction result of Becker et al. (IPDPS 2011). 1
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, 2014
"... The emergence of massive datasets has led to the rise of new computational paradigms where computation is limited. In the streaming model the input graph is presented to the algorithm as a stream of edges which is prohibitively large to be stored in its entirety (i.e., the algorithm’s space complexi ..."
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The emergence of massive datasets has led to the rise of new computational paradigms where computation is limited. In the streaming model the input graph is presented to the algorithm as a stream of edges which is prohibitively large to be stored in its entirety (i.e., the algorithm’s space complexity must be small relative to the stream size). After reading the stream, the algorithm should report a solution to a predetermined problem on the graph. In the sketching model, the input graph is summarized into a socalled sketch, which is short yet suffices for further processing without access to the original input. Cuts in graphs is a classical topic of both theoretical and practical interest, studied extensively for more than half a century. A graph cut is a partition of the vertices to two disjoint sets, and the value of the cut is the number of edges (or their total weight in case the graph is weighted) with one endpoint in each part of the partition. This definition can be extended to runiform hypergraphs, in which case hyperedges are sets of r vertices, and a hyperedge belongs to the cut if it intersects both parts of the vertex bipartition. We first address a natural question, whether the the value of the maximum cut in a graph admits
On Sketching Quadratic Forms
, 2015
"... We undertake a systematic study of sketching a quadratic form: given an n × n matrix A, create a succinct sketch sk(A) which can produce (without further access to A) a multiplicative (1 + ε)approximation to xTAx for any desired query x ∈ Rn. While a general matrix does not admit nontrivial sketch ..."
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We undertake a systematic study of sketching a quadratic form: given an n × n matrix A, create a succinct sketch sk(A) which can produce (without further access to A) a multiplicative (1 + ε)approximation to xTAx for any desired query x ∈ Rn. While a general matrix does not admit nontrivial sketches, positive semidefinite (PSD) matrices admit sketches of size Θ(ε−2n), via the JohnsonLindenstrauss lemma, achieving the “for each ” guarantee, namely, for each query x, with a constant probability the sketch succeeds. (For the stronger “for all” guarantee, where the sketch succeeds for all x’s simultaneously, again there are no nontrivial sketches.) We design significantly better sketches for the important subclass of graph Laplacian matrices, which we also extend to symmetric diagonally dominant matrices. A sequence of work culminating in that of Batson, Spielman, and Srivastava (SIAM Review, 2014), shows that by choosing and reweighting O(ε−2n) edges in a graph, one achieves the “for all ” guarantee. Our main results advance this front. 1. For the “for all ” guarantee, we prove that Batson et al.’s bound is optimal even when we
Declaration
, 2014
"... I Ilias Giechaskiel of Magdalene College, being a candidate for the M.Phil in Advanced Computer Science, hereby declare that this report and the work described in it are my own work, unaided except as may be specified below, and that the report does not contain material that has already been used to ..."
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I Ilias Giechaskiel of Magdalene College, being a candidate for the M.Phil in Advanced Computer Science, hereby declare that this report and the work described in it are my own work, unaided except as may be specified below, and that the report does not contain material that has already been used to any substantial extent for a comparable purpose. Total word count: 14,311 (excluding Appendices A and B)