### Abstract

Metric indexing research is concerned with the efficient evaluation of queries in metric spaces. In general, a large space of objects is arranged in such a way that, when a further object is presented as a query, those objects most similar to the query can be efficiently found. Most such mechanisms rely upon the triangle inequality property of the metric governing the space. The triangle inequality property is equivalent to a finite embedding property, which states that any three points of the space can be isometrically embedded in two-dimensional Euclidean space. In this paper, we examine a class of semimetric space which is finitely 4-embeddable in three-dimensional Euclidean space. In mathematics this property has been extensively studied and is generally known as the four-point property.

All spaces with the four-point property are metric spaces, but they also have some stronger geometric guarantees. We coin the term supermetric as, in terms of metric search, they are significantly more tractable. We show some stronger geometric guarantees deriving from the four-point property which can be used in indexing to great effect, and show results for two of the SISAP benchmark searches that are substantially better than any previously published.

All spaces with the four-point property are metric spaces, but they also have some stronger geometric guarantees. We coin the term supermetric as, in terms of metric search, they are significantly more tractable. We show some stronger geometric guarantees deriving from the four-point property which can be used in indexing to great effect, and show results for two of the SISAP benchmark searches that are substantially better than any previously published.

Original language | English |
---|---|

Title of host publication | 9th International Conference on Similiarty Search and Applications |

Publisher | Springer-Verlag |

Pages | 51-64 |

Number of pages | 14 |

Volume | 9939 |

ISBN (Print) | 9783319467580 |

DOIs | |

Publication status | Published - 24 Oct 2016 |

Event | SISAP 2016 - 9th International Conference on Similarity Searches and Applications - Tokyo, Japan Duration: 24 Oct 2016 → 26 Oct 2016 |

### Publication series

Name | Lecture Notes in Computing Science |
---|---|

Publisher | Springer-Verlag |

ISSN (Print) | 0302-9743 |

### Conference

Conference | SISAP 2016 - 9th International Conference on Similarity Searches and Applications |
---|---|

Country | Japan |

City | Tokyo |

Period | 24/10/16 → 26/10/16 |

### Keywords

- metric search
- Hilbert embedding
- four-point property

## Cite this

Connor, R., Vadicamo, L., Cardillo, F. A., & Rabitti, F. (2016). Supermetric search with the four-point property. In

*9th International Conference on Similiarty Search and Applications*(Vol. 9939, pp. 51-64). [2] (Lecture Notes in Computing Science). Springer-Verlag. https://doi.org/10.1007/978-3-319-46759-7