On dimension partitions in discrete metric spaces

  • Authors:

  • Fabien Rebatel;Édouard Thiel

  • Affiliations:

  • Laboratoire d'Informatique Fondamentale de Marseille (LIF, UMR 7279), Aix-Marseille Université, Marseille cedex 9, France;Laboratoire d'Informatique Fondamentale de Marseille (LIF, UMR 7279), Aix-Marseille Université, Marseille cedex 9, France

  • Venue:
  • DGCI'13 Proceedings of the 17th IAPR international conference on Discrete Geometry for Computer Imagery
  • Year:
  • 2013

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Abstract

Let (W,d) be a metric space and S={s1 …sk} an ordered list of subsets of W. The distance between p∈W and si∈S is d(p, si)= min { d(p,q) : q∈si }. S is a resolving set for W if d(x, si)=d(y, si) for all si 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.