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 space 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.
- Four-point Property
- Triangle Inequality Property
- Finite Embedding Property
- Balanced Indexing Structure
- Threshold Queries
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The term supermetric space has previously been used in the domains of particle physics and evolutionary biology as a pseudonym for the mathematical term ultra-metric, a concept of no interest in metric search; we believe our concept is of sufficient importance to the domain to justify its reuse with a different meaning.
This is a preview of subscription content, access via your institution.
Tax calculation will be finalised at checkout
Purchases are for personal use onlyLearn about institutional subscriptions
All of the (Java) code for these experiments can be downloaded from https://bitbucket.org/richardconnor/metric-space-framework/.
Which we therefore believe makes the best performance yet published for these metric/dataset combinations.
Brin, S.: Near neighbor search in large metric spaces. In 21th International Conference on Very Large Data Bases (VLDB 1995) (1995)
Chávez, E., Ludueña, V., Reyes, N., Roggero, P.: Faster proximity searching with the distal SAT. Inf. Syst. 59, 15–47 (2016)
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)
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 2016
Figueroa, K., Navarro, G., Chávez, E.: Metric spaces library. www.sisap.org/library/manual.pdf
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
Novak, D., Batko, M., Zezula, P.: Metric index: an efficient and scalable solution for precise and approximate similarity search. Inf. Syst. 36(4), 721–733 (2011). Selected Papers from the 2nd International Workshop on Similarity Search and Applications SISAP (2009)
Zezula, P., Amato, G., Dohnal, V., Batko, M.: Similarity Search: The Metric Space Approach. Advances in Database Systems, vol. 32. Springer, New York (2006)
We would like to thank the anonymous referees for helpful comments on an earlier version of this paper. Richard Connor would like to acknowledge support by the National Research Council of Italy (CNR) for a Short-term Mobility Fellowship (STM) in June 2015, which funded a stay at ISTI-CNR in Pisa during which much of this work was conceived.
Editors and Affiliations
© 2016 Springer International Publishing AG
About this paper
Cite this paper
Connor, R., Vadicamo, L., Cardillo, F.A., Rabitti, F. (2016). Supermetric Search with the Four-Point Property. In: Amsaleg, L., Houle, M., Schubert, E. (eds) Similarity Search and Applications. SISAP 2016. Lecture Notes in Computer Science(), vol 9939. Springer, Cham. https://doi.org/10.1007/978-3-319-46759-7_4
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-46758-0
Online ISBN: 978-3-319-46759-7