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Spectral sparsification via random spanners

Published: 08 January 2012 Publication History

Abstract

In this paper we introduce a new notion of distance between nodes in a graph that we refer to as robust connectivity. Robust connectivity between a pair of nodes u and v is parameterized by a threshold k and intuitively captures the number of paths between u and v of length at most k. Using this new notion of distances, we show that any black box algorithm for constructing a spanner can be used to construct a spectral sparsifier. We show that given an undirected weighted graph G, simply taking the union of spanners of a few (polylogarithmically many) random subgraphs of G obtained by sampling edges at different probabilities, after appropriate weighting, yields a spectral sparsifier of G. We show how this be done in Õ(m) time, producing a sparsifier with Õ(n2) edges. While the cut sparsifiers of Benczur and Karger are based on weighting edges according to (inverse) strong connectivity, and the spectral sparsifiers are based on resistance, our method weights edges using the robust connectivity measure. The main property that we use is that this new measure is always greater than the resistance when scaled by a factor of O(k) (k is chosen to be O(log n)), but, just like resistance and connectivity, has a bounded sum, i.e. Õ(n), over all the edges of the graph.

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  • (2024)K-SpecPart: Supervised Embedding Algorithms and Cut Overlay for Improved Hypergraph PartitioningIEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems10.1109/TCAD.2023.333226843:4(1232-1245)Online publication date: Apr-2024
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cover image ACM Conferences
ITCS '12: Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
January 2012
516 pages
ISBN:9781450311151
DOI:10.1145/2090236
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 08 January 2012

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ITCS '12
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ITCS '12: Innovations in Theoretical Computer Science
January 8 - 10, 2012
Massachusetts, Cambridge

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ITCS '12 Paper Acceptance Rate 39 of 93 submissions, 42%;
Overall Acceptance Rate 172 of 513 submissions, 34%

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Cited By

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  • (2024)Tight Lower Bounds for Directed Cut Sparsification and Distributed Min-CutProceedings of the ACM on Management of Data10.1145/36511482:2(1-18)Online publication date: 14-May-2024
  • (2024)Decentralized Low-Stretch Trees via Low Diameter Graph DecompositionsSIAM Journal on Computing10.1137/22M148903453:2(247-286)Online publication date: 13-Mar-2024
  • (2024)K-SpecPart: Supervised Embedding Algorithms and Cut Overlay for Improved Hypergraph PartitioningIEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems10.1109/TCAD.2023.333226843:4(1232-1245)Online publication date: Apr-2024
  • (2023)A Combinatorial Cut-Toggling Algorithm for Solving Laplacian Linear SystemsAlgorithmica10.1007/s00453-023-01154-885:12(3680-3716)Online publication date: 1-Aug-2023
  • (2022)Nearly optimal vertex fault-tolerant spanners in optimal time: sequential, distributed, and parallelProceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3519935.3520047(1080-1092)Online publication date: 9-Jun-2022
  • (2022)SpecPartProceedings of the 41st IEEE/ACM International Conference on Computer-Aided Design10.1145/3508352.3549390(1-9)Online publication date: 30-Oct-2022
  • (2022)Quantum Speedup for Graph Sparsification, Cut Approximation, and Laplacian SolvingSIAM Journal on Computing10.1137/21M139101851:6(1703-1742)Online publication date: 16-Dec-2022
  • (2021)Optimal vertex fault-tolerant spanners in polynomial timeProceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3458064.3458238(2924-2938)Online publication date: 10-Jan-2021
  • (2021)Ultra-Sparse Near-Additive EmulatorsProceedings of the 2021 ACM Symposium on Principles of Distributed Computing10.1145/3465084.3467926(235-246)Online publication date: 21-Jul-2021
  • (2020)Graph Sparsification, Spectral Sketches, and Faster Resistance Computation via Short Cycle DecompositionsSIAM Journal on Computing10.1137/19M124763252:6(FOCS18-85-FOCS18-157)Online publication date: 9-Jun-2020
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