## Abstract

The Euclidean Minimum Spanning Tree problem is to decide whether a given graph G = (P, E) on a set of points in the two-dimensional plane is a minimum spanning tree with respect to the Euclidean distance. Czumaj et al. [A. Czumaj, C. Sohler, M. Ziegler, Testing Euclidean Minimum Spanning Trees in the plane, Unpublished, Part II of ESA 2000 paper, downloaded from http://web.njit.edu/~czumaj/] gave a 1-sided-error non-adaptive property-tester for this task of query complexity over(O, ̃) (sqrt(n)). We show that every non-adaptive (not necessarily 1-sided-error) property-tester for this task has a query complexity of Ω (sqrt(n)), implying that the test in [A. Czumaj, C. Sohler, M. Ziegler, Testing Euclidean Minimum Spanning Trees in the plane, Unpublished, Part II of ESA 2000 paper, downloaded from http://web.njit.edu/~czumaj/] is of asymptotically optimal complexity. We further prove that every adaptive property-tester has query complexity of Ω (n^{1 / 3}). Those lower bounds hold even when the input graph is promised to be a bounded degree tree.

Original language | English |
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Pages (from-to) | 219-225 |

Number of pages | 7 |

Journal | Information Processing Letters |

Volume | 102 |

Issue number | 6 |

DOIs | |

State | Published - 15 Jun 2007 |

### Bibliographical note

Funding Information:✩ Research supported in part by grant 55/03 from the Israel Science Foundation. * Corresponding author. E-mail addresses: orenb@technion.ac.il (O. Ben-Zwi), loded@cs.haifa.ac.il (O. Lachish), ilan@cs.haifa.ac.il (I. Newman).

## Keywords

- Computational geometry
- Graph algorithms
- Randomized algorithms

## ASJC Scopus subject areas

- Theoretical Computer Science
- Signal Processing
- Information Systems
- Computer Science Applications