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just a property of absolute values

This is just what's called a triangular inequality; this is just a property of absolute values.

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the property that guarantees that d(A, B) ≤ d(A, C) + d(B, C)

Then the triangular inequality is the property that guarantees that d(A, B) ≤ d(A, C) + d(B, C).

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part of the definition of any metric distance

Since the triangular inequality is part of the definition of any metric distance, the upper bounds can be applied to any metric space.

the origin of the term triangular inequality

This is the origin of the term triangular inequality.

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a defining property of a metric space

The triangular inequality is a defining property of a metric space, while the stronger ultrametric inequality is a defining property of an ultrametric space.

The fourth requirement from distance functions

The fourth requirement from distance functions is called triangular inequality.

by a weaker one

In particular, the triangular inequality is replaced by a weaker one, which is satisfied by only those points which are situated on some path included in the graphical structure associated with...

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The last criterium

The last criterium is the most important one and is called triangular inequality.

e.g.f(ab+ f(bc=> , , ) , ) f(a, c)

Triangular Inequality requires that the distance between any two points, measured from either point, must be equal or less than the distance between these measured through a third point; e.g., f(a, b) + f(b, c) => f(a, c).

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a procedure in which an edge between two vertices is eliminated whenever these vertices are connected indirectly through one or more additional vertex

Generally, the triangular inequality reduction is a procedure in which an edge between two vertices is eliminated whenever these vertices are connected indirectly through one or more additional vertex.

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