On approximating Euclidean metrics by weighted t-cost distances in arbitrary dimension

Abstract

In this work, we propose a new class of distance functions called weighted t-cost distances. This function maximizes the weighted contribution of different t-cost norms in n-dimensional space. With proper weight assignment, this class of function also generalizes m-neighbor and octagonal distances. A non-strict upper bound (denoted as R u in this work) of its relative error with respect to Euclidean norm is derived and an optimal weight assignment by minimizing R u is obtained. However, it is observed that the strict upper bound of weighted t-cost norm may be significantly lower than R u . For example, an inverse square root weight assignment leads to a good approximation of Euclidean norm in arbitrary dimension.