On dimension partitions in discrete metric spaces

Abstract

Let (W,d) be a metric space and S = {s 1.s k } an ordered list of subsets of W. The distance between p â̂̂ W and s i â̂̂ S is d(p, s i ) = min { d(p,q): q â̂̂ s i }. S is a resolving set for W if d(x, s i ) = d(y, s i ) for all s i implies x = y. A metric basis is a resolving set of minimal cardinality, named the metric dimension of (W,d). The metric dimension has been extensively studied in the literature when W is a graph and S is a subset of points (classical case) or when S is a partition of W; the latter is known as the partition dimension problem. We have recently studied the case where W is the discrete space ℤn for a subset of points; in this paper, we tackle the partition dimension problem for classical Minkowski distances as well as polyhedral gauges and chamfer norms in ℤn . © 2013 Springer-Verlag Berlin Heidelberg.