Why Hausdorff Distance Is Natural in Interval Computations

Several different metrics have been proposed to describe distance between intervals and, more generally, between compact sets. In this paper, we show that from the viewpoint of interval computations, the most adequate distance is the Hausdorff distance dH(A,A ′) – the smallest value ε > 0 for which every element a ∈ A is ε-close to some element a′ ∈ A′, and every element a′ ∈ A′ is ε-close to some element a ∈ A. 1 Formulation of the Problem Uncertainty-motivates sets as extensions of points. One of the main objectives of interval computations is to deal with uncertainty. Because of uncertainty, instead of the exact value a of a physical quantity, we only know a set A (usually, an interval) of possible values. In the cases of several variables, instead of a tuple a = (a1, . . . , an) consisting of their exact values, we only know a set A of possible tuples. In general, instead of the exact state a of the corresponding system, we only know a set A of possible states. The case of complete knowledge can be viewed as a “degenerate” case, when the corresponding set A consists of a single element a: A = {a}. Need to define distance between sets. In many practical situations, on the set X of all possible states, we have a physically meaningful distance d(a, b). For example, on the set of real numbers, a natural distance is usually d(a, b) = |a−b|. It is desirable to extend this distance from elements (i.e., degenerate sets) to a more general case of distance between sets. There are many ways to define the distance between sets. In general, there are many ways to extend a function to its original domain to a larger domain. In particular, there are many ways to extend distance from a given set X (i.e., from the class of all 1-element subsets of the set X) to a larger class of subsets of X.