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 followingweaker version ofthe triangle inequality: "d"("x", "z") ≤ $ho$ max{"d"("x", "y"), "d"("y", "z")} for some parameter $ho$ ≥ 1.A set with an inframetric is called an inframetric space. This notion subsumes bothstandard metric spaces (1 ≤ $ho$ ≤ 2) and
ultrametric spaces ($ho$ = 1). Inframetrics were notably introduced to model
internet round-trip delay times.

Definition

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

"d" : "X" × "X" → R

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

# "d"("x", "y") ≥ 0 ("non-negativity")
# "d"("x", "y") = 0 if and only if "x" = "y" ("identity of indiscernibles")
# "d"("x", "y") = "d"("y", "x") ("symmetry")
# "d"("x", "z") ≤ $ho$ max{"d"("x", "y"), "d"("y", "z")} ("$ho$-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"("x", "z") ≤ 2 max{"d"("x", "y"), "d"("y", "z")}. Any metric is thus a 2-inframetric. The definition of 1-inframetric is equivalent to that of ultrametric.

References

" [http://ieeexplore.ieee.org/Xplore/login.jsp?url=/iel5/4509594/4509595/04509758.pdf?isnumber=4509595&prod=CNF&arnumber=4509758&arSt=1085&ared=1093&arAuthor=Fraigniaud%2C+P.%3B+Lebhar%2C+E.%3B+Viennot%2C+L. The Inframetric Model for the Internet] ", Pierre Fraigniaud, Emmanuelle Lebhar and Laurent Viennot, IEEE INFOCOM, pp. 1085-1093, April 2008.