Supermetric Search with the Four-Point Property
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 supermetricspace 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.
- 1.Brin, S.: Near neighbor search in large metric spaces. In 21th International Conference on Very Large Data Bases (VLDB 1995) (1995)Google Scholar
- 3.Chávez, E., Navarro, G.: Metric databases. In: Rivero, L.C., Doorn, J.H., Ferraggine, V.E. (eds.) Encyclopedia of Database Technologies and Applications, pp. 366–371. Idea Group, Hershey (2005)Google Scholar
- 4.Connor, R., Cardillo, F.A., Vadicamo, L., Rabitti, F.: Hilbert exclusion: improved metric search through finite isometric embeddings. ArXiv e-prints (accepted for publication ACM TOIS, July 2016), April 2016Google Scholar
- 5.Figueroa, K., Navarro, G., Chávez, E.: Metric spaces library. www.sisap.org/library/manual.pdf
- 6.Noltemeier, H., Verbarg, K., Zirkelbach, C.: Monotonous Bisector* Trees — a tool for efficient partitioning of complex scenes of geometric objects. In: Monien, B., Ottmann, Th (eds.) Data Structures and Efficient Algorithms. LNCS, vol. 594, pp. 186–203. Springer, Heidelberg (1992). doi:10.1007/3-540-55488-2_27 CrossRefGoogle Scholar