![](bg7.png)
structure, while, for each arc, references to two MC-
components and to a non-manifold vertex are stored.
Hence, the storage cost of the compact MC-graph
is equal to S
C
IA
∗
= (d + 2)s
d
+ 5s
0
− 4(d + 1). Ex-
perimental results in Table 2 (see columns S
C
IA
∗
and
S
IQM
) show that the compact MC-graph tends to be
more cost effective than the IQM data structure for
any dimension. For instance, S
IQM
≈ 1.14 × S
C
IA
∗
and
S
IQM
≈ 1.16 ×S
C
IA
∗
for, respectively, simplicial 4- and
5-complexes. Note that these representations coincide
when encoding manifolds, since they are equivalent to
the IA data structure, plus one additional reference to
a top simplex in the input shape.
In addition, our experimental results in Table 2
(see columns S
IQM
and S
IG
) show that also the IQM
data structure is more compact than the incidence
graph in any dimension, e.g., S
IG
≈ 8 ×S
IQM
for sim-
plicial 4-complexes.
7 CONCLUDING REMARKS
We have presented a structural model for non-ma-
nifold shapes, which are decomposed into a collection
of MC-components, a decidable superclass of mani-
folds of any dimension. We have designed and imple-
mented the Compact MC-graph, a graph-based repre-
sentation for the MC-decomposition (Hui and De Flo-
riani, 2007), which can be combined with any topo-
logical data structure representing non-manifolds.
We have combined the compact MC-graph with
all the topological data structures, which are currently
implemented in the Mangrove TDS Library (Canino
and De Floriani, 2012). Our tests show that the com-
pact MC-graph, if combined with the IS (De Flori-
ani et al., 2010) and the IA
∗
(Canino et al., 2011)
data structures, is more compact than the incidence
graph (Edelsbrunner, 1987), which is a widely used
data structure in several applications. However, this
latter, unlike our compact MC-graph, does not ex-
pose the structure of a shape explicitly, and does not
support the identification of non-manifold singulari-
ties efficiently (Canino, 2012). Our tests also show
that the compact MC-graph is more cost effective than
the IQM-decomposition (De Floriani et al., 2003) and
than the DLD data structure (Hui et al., 2006), even
for high dimensions.
There is an increasing interest in quad and un-
structured hexahedral meshes in geometry process-
ing, animation, and numerical simulations. Some data
structures, specific for simplicial complexes, like the
IS and IA* data structures, can be easily extended to
deal with such shapes, since all the simplifying as-
sumptions, that make the two data structures com-
pact in the case of simplicial complexes, hold also
for quad and hexahedral meshes. Thus, also the MC-
decomposition can be extended to such meshes and
also to more general cell complexes .
Finally, we are designing new graph-based rep-
resentations for the IQM-decomposition. The prop-
erties of the IQM components may allow for a very
compact encoding. In fact, an IQM-component is al-
most manifold, thus it may be representable through
very compact data structures, specific for manifolds,
like (Gurung and Rossignac, 2009; Gurung et al.,
2011a; Gurung et al., 2011b), just to mention few.
ACKNOWLEDGEMENTS
This work has been partially supported by the Italian
Ministry of Education and Research under the PRIN
2009 program, and by the National Science Founda-
tion under grant number IIS-1116747.
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