Synonyms
Distance space
Definition
In mathematics, a metric space is a pair M = (D, d), where D is a domain of objects (or objects’; keys or indexed descriptors) and d is a total (distance) function. The properties of the function d: D × D ↦ R, sometimes called the metric space postulates, are typically characterized as:
(p1) |
\( \forall x,y\in D,d\left(x,y\right)\ge 0 \) |
Non-negativity, |
(p2) |
\( \forall x,y\in D,d\left(x,y\right)=d\left(y,x\right) \) |
Symmetry, |
(p3) |
\( \forall x\in D,d\left(x,x\right)=0 \) |
Reflexivity, |
(p4) |
\( \forall x,y\,{\in}\,D,x\,{\ne}\,y\,{\Rightarrow}\,d\left(x,y\right)\,{>}\,0 \) |
Positiveness, |
(p5) |
\( \begin{aligned}\forall x,y,z&{\in}\,D,d\left(x,z\right){\le}d\left(x,y\right)\\[-12pt]\end{aligned} \) |
Triangle inequality |
\( \begin{aligned}\quad\quad\quad\quad+d\left(y,z\right)\end{aligned} \) |
Key Points
Modifying or even abandoning some of the metric function properties leads to interesting concepts that can better suit the reality in many situations. A pseudo-metric...
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Recommended Reading
Burago D, Burago YD, Ivanov S. A course in metric geometry. Providence: American Mathematical Society; 2001.
Bryant V. Metric spaces: iteration and application. New York: Cambridge University Press; 1985.
Zezula P, Amato G, Dohnal V, Batko M. Similarity search: the metric space approach. Berlin: Springer; 2006.
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Zezula, P., Batko, M., Dohnal, V. (2018). Metric Space. In: Liu, L., Özsu, M.T. (eds) Encyclopedia of Database Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8265-9_218
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DOI: https://doi.org/10.1007/978-1-4614-8265-9_218
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