### Abstract

We present the first single pass algorithm for computing spectral sparsifiers of graphs in the dynamic semi-streaming model. Given a single pass over a stream containing insertions and deletions of edges to a graph, G, our algorithm maintains a randomized linear sketch of the incidence matrix into dimension O((1/ε2) n polylog(n)). Using this sketch, the algorithm can output a (1 +/-ε) spectral sparsifier for G with high probability. While O((1/ε2) n polylog(n)) space algorithms are known for computing cut sparsifiers in dynamic streams [AGM12b, GKP12] and spectral sparsifiers in insertion-only streams [KL11], prior to our work, the best known single pass algorithm for maintaining spectral sparsifiers in dynamic streams required sketches of dimension ω((1/ε2) n(5/3)) [AGM14]. To achieve our result, we show that, using a coarse sparsifier of G and a linear sketch of G's incidence matrix, it is possible to sample edges by effective resistance, obtaining a spectral sparsifier of arbitrary precision. Sampling from the sketch requires a novel application of ℓ_{2}/ℓ_{2} sparse recovery, a natural extension of the ℓ_{0} methods used for cut sparsifiers in [AGM12b]. Recent work of [MP12] on row sampling for matrix approximation gives a recursive approach for obtaining the required coarse sparsifiers. Under certain restrictions, our approach also extends to the problem of maintaining a spectral approximation for a general matrix AT A given a stream of updates to rows in A.

Original language | English (US) |
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Title of host publication | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |

Publisher | IEEE Computer Society |

Pages | 561-570 |

Number of pages | 10 |

ISBN (Electronic) | 9781479965175 |

DOIs | |

State | Published - Dec 7 2014 |

Event | 55th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2014 - Philadelphia, United States Duration: Oct 18 2014 → Oct 21 2014 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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ISSN (Print) | 0272-5428 |

### Other

Other | 55th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2014 |
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Country | United States |

City | Philadelphia |

Period | 10/18/14 → 10/21/14 |

### All Science Journal Classification (ASJC) codes

- Computer Science(all)

### Keywords

- dimensionality reduction
- sketching
- sparse recovery
- sparsification
- streaming

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## Cite this

*Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS*(pp. 561-570). [6979041] (Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS). IEEE Computer Society. https://doi.org/10.1109/FOCS.2014.66