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Published in:
2013 | OriginalPaper | Chapter
On Dimension Partitions in Discrete Metric Spaces
Authors : Fabien Rebatel, Édouard Thiel
Published in: Discrete Geometry for Computer Imagery
Publisher: Springer Berlin Heidelberg
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Let (
W
,
d
) be a metric space and
S
= {
s
1
…
s
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
.