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Supermetric Search with the Four-Point Property

Part of the Lecture Notes in Computer Science book series (LNISA,volume 9939)


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.

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    All of the (Java) code for these experiments can be downloaded from

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    Which we therefore believe makes the best performance yet published for these metric/dataset combinations.


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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.

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Correspondence to Richard Connor .

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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.

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  • Print ISBN: 978-3-319-46758-0

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