## Abstract

A central question in the geometry of finite metric spaces is how well can an arbitrary metric space be "faithfully preserved" by a mapping into Euclidean space. In this paper we present an algorithmic embedding which obtains a new strong measure of faithful preservation: not only does it (approximately) preserve distances between pairs of points, but also the volume of any set of k points. Such embeddings are known as volume preserving embeddings. We provide the first volume preserving embedding that obtains constant average volume distortion for sets of any fixed size. Moreover, our embedding provides constant bounds on all bounded moments of the volume distortion while maintaining the best possible worst-case volume distortion. Feige, in his seminal work on volume preserving embeddings defined the volume of a set S = {v_{1}, ..., v_{k} of points in a general metric space: the product of the distances from v_{i} to {v_{1}, ..., v_{i-1}}, normalized by 1/(k-1)!, where the ordering of the points is that given by Prim's minimum spanning tree algorithm. Feige also related this notion to the maximal Euclidean volume that a Lipschitz embedding of S into Euclidean space can achieve. Syntactically this definition is similar to the computation of volume in Euclidean spaces, which however is invariant to the order in which the points are taken. We show that a similar robustness property holds for Feige's definition: the use of any other order in the product affects volume^{1/(k-1)} by only a constant factor. Our robustness result is of independent interest as it presents a new competitive analysis for the greedy algorithm on a variant of the online Steiner tree problem where the cost of buying an edge is logarithmic in its length. This robustness property allows us to obtain our results on volume preserving embedding.

Original language | American English |
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Pages (from-to) | 366-389 |

Number of pages | 24 |

Journal | Discrete and Computational Geometry |

Volume | 52 |

Issue number | 2 |

DOIs | |

State | Published - Sep 2014 |

### Bibliographical note

Funding Information:Yair Bartal was supported in part by a Grant from the Israeli Science Foundation (1609/11) and in part by a Grant from the National Science Foundation (NSF CCF-0652536). Ofer Neiman was supported in part by ISF Grant No. (523/12) and by the European Union’s Seventh Framework Programme (FP7/2007-2013) under Grant Agreement No. 303809. Leonard J. Schulman was supported in part by NSF CCF-0515342 and NSA H98230-06-1-0074. Part of the research was done while Yair Bartal was at the Center of the Mathematics of Information, Caltech, CA, USA.

## Keywords

- Metric embedding
- Volume preserving embedding