By Michel Marie Deza

ISBN-10: 3642002331

ISBN-13: 9783642002335

ISBN-10: 364200234X

ISBN-13: 9783642002342

Distance metrics and distances became a necessary software in lots of parts of natural and utilized arithmetic, and this encyclopedia is the 1st one to regard the topic in complete. The e-book appears to be like simply as learn intensifies into metric areas and particularly, distance layout for functions. those distances are really an important, for instance, in computational biology, photo research, speech acceptance, and knowledge retrieval. the following, an evaluate of the sensible questions coming up in the course of collection of a ''good'' distance functionality has been left apart in want of a entire directory of the most to be had distances, a great tool for the gap layout neighborhood. This reader-friendly reference bargains either self sufficient introductions and definitions, whereas while making cross-referencing effortless via hyperlink-like boldfaced references to unique definitions. This top of the range ebook is a mixture of reference source and coffee-table booklet

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**Example text**

The size of the smallest -covering is at most the size of the largest 2 packing. , M is also an -net. • Congruence order of metric space A metric space (X, d) has congruence order n if every ﬁnite metric space which is not isometrically embeddable in (X, d) has a subspace with at most n points which is not isometrically embeddable in (X, d). For example, the congruence order of l2n is n + 3 (Menger 1928); it is 4 for the path metric of a tree. , the graph with the vertex-set X and the edge-set {xy : d(x, y) ∈ D}.

Editing metric Given a ﬁnite set X and a ﬁnite set O of (unary) editing operations on X, the editing metric on X is the path metric of the graph with the vertex-set X and xy being an edge if y can be obtained from x by one of the operations from O. • Gallery metric A chamber system is a set X (whose elements are referred to as chambers) equipped with n equivalence relations ∼i , 1 ≤ i ≤ n. A gallery is a sequence of chambers x1 , . . , xm such that xi ∼j xi+1 for every i and some j depending on i.

Dominating metric Given metrics d and d1 on a set X, d1 dominates d if d1 (x, y) ≥ d(x, y) for all x, y ∈ X. Cf. non-contractive mapping (or dominating mapping). • Metric transform A metric transform is a distance obtained as a function of a given metric (cf. Chap. 4). , for any > 0, there exists n0 ∈ N such that d(xn , x∗ ) < for any n > n0 . A sequence {xn }n , xn ∈ X, is called a Cauchy sequence if, for any > 0, there exists n0 ∈ N such that d(xn , xm ) < for any m, n > n0 . A metric space (X, d) is called a complete metric space if every Cauchy sequence in it converges.

### Encyclopedia of Distances by Michel Marie Deza

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