# On Dimension Partitions in Discrete Metric Spaces

• Fabien Rebatel
• Édouard Thiel
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7749)

## Abstract

Let (W,d) be a metric space and S = {s 1s k } an ordered list of subsets of W. The distance between p ∈ W and s i  ∈ S is d(p, s i ) =  min { d(p,q) : q ∈ 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 ℤ n 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 ℤ n .

## Keywords

dimension partition metric dimension distance geometry discrete distance norm

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