## Abstract

We study streaming algorithms for the fundamental geometric problem of computing the cost of the Euclidean Minimum Spanning Tree (MST) on an n-point set X g d. In the streaming model, the points in X can be added and removed arbitrarily, and the goal is to maintain an approximation in small space. In low dimensions, (1+") approximations are possible in sublinear space [Frahling, Indyk, Sohler, SoCG '05]. However, for high dimensional spaces the best known approximation for this problem was Õ(logn), due to [Chen, Jayaram, Levi, Waingarten, STOC '22], improving on the prior O(log2 n) bound due to [Indyk, STOC '04] and [Andoni, Indyk, Krauthgamer, SODA '08]. In this paper, we break the logarithmic barrier, and give the first constant factor sublinear space approximation to Euclidean MST. For any "≥ 1, our algorithm achieves an Õ("-2) approximation in nO(") space. We complement this by proving that any single pass algorithm which obtains a better than 1.10-approximation must use ω(n) space, demonstrating that (1+") approximations are not possible in high-dimensions, and that our algorithm is tight up to a constant. Nevertheless, we demonstrate that (1+") approximations are possible in sublinear space with O(1/") passes over the stream. More generally, for any α ≥ 2, we give a α-pass streaming algorithm which achieves a (1+O(logα + 1/ α ")) approximation in nO(") dO(1) space. All our streaming algorithms are linear sketches, and therefore extend to the massively-parallel computation model (MPC). Thus, our results imply the first (1+")-approximation to Euclidean MST in a constant number of rounds in the MPC model. Previously, such a result was only known for low-dimensional space [Andoni, Nikolov, Onak, Yaroslavtsev, STOC '15], or required either O(logn) rounds or a O(logn) approximation.

Original language | English |
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Title of host publication | STOC 2023 - Proceedings of the 55th Annual ACM Symposium on Theory of Computing |

Editors | Barna Saha, Rocco A. Servedio |

Publisher | Association for Computing Machinery |

Pages | 156-169 |

Number of pages | 14 |

ISBN (Electronic) | 9781450399135 |

DOIs | |

State | Published - 2 Jun 2023 |

Externally published | Yes |

Event | 55th Annual ACM Symposium on Theory of Computing, STOC 2023 - Orlando, United States Duration: 20 Jun 2023 → 23 Jun 2023 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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ISSN (Print) | 0737-8017 |

### Conference

Conference | 55th Annual ACM Symposium on Theory of Computing, STOC 2023 |
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Country/Territory | United States |

City | Orlando |

Period | 20/06/23 → 23/06/23 |

### Bibliographical note

Funding Information:∗Xi Chen is supported by NSF grants CCF-1703925, IIS-1838154, CCF-2106429 and CCF-2107187. Erik Waingarten is supported by the National Science Foundation under Award No. 2002201 and Moses Charikar’s Simons Investigator award. This work was carried out while the author Amit Levi was a PhD student at the University of Waterloo.

Publisher Copyright:

© 2023 ACM.

## Keywords

- sublinear and near linear time algorithms
- Theory of computation → Streaming

## ASJC Scopus subject areas

- Software