## Abstract

We give a deterministic, nearly logarithmic-space algorithm for mild spectral sparsification of undirected graphs. Given a weighted, undirected graph G on n vertices described by a binary string of length N, an integer k ≤ logn, and an error parameter ϵ > 0, our algorithm runs in space Õ(k log(N · w_{max}/w_{min})), where w_{max} and w_{min} are the maximum and minimum edge weights in G, and produces a weighted graph H with Õ(n^{1+2/k}/ϵ^{2}) edges that spectrally approximates G, in the sense of Spielman and Teng, up to an error of ϵ. Our algorithm is based on a new bounded-independence analysis of Spielman and Srivastava's effective resistance-based edge sampling algorithm and uses results from recent work on space-bounded Laplacian solvers. In particular, we demonstrate an inherent trade-off (via upper and lower bounds) between the amount of (bounded) independence used in the edge sampling algorithm, denoted by k above, and the resulting sparsity that can be achieved.

Original language | English |
---|---|

Article number | 7 |

Journal | ACM Transactions on Computation Theory |

Volume | 16 |

Issue number | 2 |

DOIs | |

State | Published - 14 Mar 2024 |

## Keywords

- Derandomization
- graph sparsification
- space-bounded computation

## ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics