Inframetric

Inframetric

In mathematics, an inframetric is a distance function between elements of a set that generalizes the notion of metric. It is defined by the following weaker version of the triangle inequality: d(x, z) ≤ rho max{d(x, y), d(y, z)} for some parameter rho ≥ 1. A set with an inframetric is called an inframetric space. This notion subsumes both standard metric spaces (1 ≤ rho ≤ 2) and ultrametric spaces (rho = 1). Inframetrics were notably introduced to model internet round-trip delay times.

Definition

For a given parameter rho ≥ 1, a rho-inframetric on a set X is a function (called the distance function or simply distance)

d : X × XR

(where R is the set of real numbers). For all x, y, z in X, this function is required to satisfy the following conditions:

  1. d(x, y) ≥ 0     (non-negativity)
  2. d(x, y) = 0   if and only if   x = y     (identity of indiscernibles)
  3. d(x, y) = d(y, x)     (symmetry)
  4. d(x, z) ≤ rho max{d(x, y), d(y, z)}     (rho-inframetric inequality).

Note that only the last axiom differs from the metric definition. The classical triangle inequality d(x, z) ≤ d(x, y) + d(y, z) implies d(xz) ≤ 2 max{d(xy), d(yz)}. Any metric is thus a 2-inframetric. The definition of 1-inframetric is equivalent to that of ultrametric.

References

The Inframetric Model for the Internet, Pierre Fraigniaud, Emmanuelle Lebhar and Laurent Viennot, IEEE INFOCOM, pp. 1085-1093, April 2008.

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