## Abstract

We study streaming algorithms for two fundamental geometric problems: computing the cost of a Minimum Spanning Tree (MST) of an n-point set X g {1,2 "}d, and computing the Earth Mover Distance (EMD) between two multi-sets A,B g {1,2 "}d of size n. We consider the turnstile model, where points can be added and removed. We give a one-pass streaming algorithm for MST and a two-pass streaming algorithm for EMD, both achieving an approximation factor of Õ(logn) and using (n,d,")-space only. Furthermore, our algorithm for EMD can be compressed to a single pass with a small additive error. Previously, the best known sublinear-space streaming algorithms for either problem achieved an approximation of O(min{ logn , log(Δd)} logn). For MST, we also prove that any constant space streaming algorithm can only achieve an approximation of ω(logn), analogous to the ω(logn) lower bound for EMD. Our algorithms are based on an improved analysis of a recursive space partitioning method known generically as the Quadtree. Specifically, we show that the Quadtree achieves an Õ(logn) approximation for both EMD and MST, improving on the O(min{ logn , log(Δd)} logn) approximation.

Original language | English |
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Title of host publication | STOC 2022 - Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing |

Editors | Stefano Leonardi, Anupam Gupta |

Publisher | Association for Computing Machinery |

Pages | 222-233 |

Number of pages | 12 |

ISBN (Electronic) | 9781450392648 |

DOIs | |

State | Published - 6 Sep 2022 |

Externally published | Yes |

Event | 54th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2022 - Rome, Italy Duration: 20 Jun 2022 → 24 Jun 2022 |

### Publication series

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

### Conference

Conference | 54th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2022 |
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Country/Territory | Italy |

City | Rome |

Period | 20/06/22 → 24/06/22 |

### 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:

© 2022 Owner/Author.

## Keywords

- earth-mover's distance
- minimum spanning tree
- sketching
- streaming

## ASJC Scopus subject areas

- Software