Error analysis of octagonal distances defined by periodic neighborhood sequences for approximating Euclidean metrics in arbitrary dimension

In this paper, we consider approximation of Euclidean metrics by octagonal distances defined by periodic neighborhood sequences in arbitrary dimension. We derive an expression for maximum relative error (MRE) of an octagonal distance approximated by a weighted t-cost distance (WtD) function, with respect to the Euclidean metric in n-dimensional space. For this, we have used a general expression of MRE reported previously for a class of distances, in the form of a linear combination of weighted t-cost (WtD) and weighted (or chamfering) distances (CWD) and derived the expressions for specific cases of WtDs and CWDs. Further, this has also been applied to obtain theoretical expressions of MRE for m-neighbor distances (mND) in arbitrary dimension, and it improves the previously reported results regarding optimum value of m in an n-dimensional space. We also considered the adjustment of MRE values choosing an optimum scale factor. Computing theoretical values of scale adjusted MRE, we have reported good octagonal distances for approximating Euclidean metrics in different dimensional spaces. Previously, only a few such distances were reported for 2-D and 3-D spaces.