## Abstract

In the Steiner point removal problem, we are given a weighted graph G = (V, E) and a set of terminals K \subset V of size k. The objective is to find a minor M of G with only the terminals as its vertex set, such that distances between the terminals will be preserved up to a small multiplicative distortion. Kamma, Krauthgamer, and Nguyen [SIAM J. Comput., 44 (2015), pp. 975-995] devised a ball-growing algorithm with exponential distributions to show that the distortion is at most O(log^{5} k). Cheung [Proceedings of the 29th Annual ACM/SIAM Symposium on Discrete Algorithms, 2018, pp. 1353-1360] improved the analysis of the same algorithm, bounding the distortion by O(log^{2} k). We devise a novel and simpler algorithm (called the Relaxed-Voronoi algorithm) which incurs distortion O(log k). This algorithm can be implemented in almost linear time (O(| E| log | V | )).

Original language | English |
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Pages (from-to) | 249-278 |

Number of pages | 30 |

Journal | SIAM Journal on Computing |

Volume | 48 |

Issue number | 2 |

DOIs | |

State | Published - 2019 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2019 Society for Industrial and Applied Mathematics

## Keywords

- Distortion
- Metric embedding
- Minor graph
- Randomized algorithm
- Steiner point removal (SPR)