26
Carlo Combi, Sara Migliorini, Barbara Oliboni, and Alberto Belussi
• S is a set of strings to be used as atomic values.
• G = G 2 ∪ G 3 where G 2 is the set of all possible pairs (x, y) of real numbers
that represents coordinates in the 2D space, and G 3 is the set of all possible
triples (x, y, z) of real numbers that represents coordinates in the 3D space. In
particular, the possible value for a spatial component can be a single point or
a list of coordinates that defines a simple polyline, or a set of coordinates that
defines a closed polyline that identifies a simple polygon.
• V is a set of temporal elements, and as for edges, they are used to keep track of
the time intervals during which a node exists in reality.
The function can be seen as composed of the following five single-valued functions:
T n , , L n , , S , , V , , G .
⊥ means ‘undefined,’ and in the following, when the context is clear, if an element of
a label is mapped in ⊥, it will be omitted.
Each GeoMTGM graph must satisfy the following integrity constraints in order
to correctly represent real data. In particular, some of them ensure the correctness of
the general structure of the graph (constraints 1–7), some of them ensure the consistency of the temporal elements specified on atomic and complex nodes (constraints
8 and 9), and finally, constraints 10–15 ensure the correctness of the spatial data in
geoAtomic and GeoField nodes, and of the geographical and composed-by edges.
Definition 2.2 (Integrity Constraints of a GeoMTGM graph).
The following conditions must be satisfied by each GeoMTGM graph:
1. Root. The root of the graph is a geographical complex node with the name
“Schema” whose spatial attribute is a polygon containing all the territory managed by the graph. It is connected to an atomic node named “SRS” that indicates
the considered spatial reference system.
2. Values. Values are associated to atomic and stream nodes only.
∀x ∈ N( T n (x) = complex ∨
T n (x) = geoComplex ∨
T n (x) = geoAtomic) → S (x) = ⊥)
3. General Relational Edges. Relational edges can be defined only between two
(geographical) complex nodes or between a complex node and its atomic or
stream nodes, or between a geographical complex node and its atomic, stream
nodes, or geoAtomic nodes.
∀∀m, relational, RelName, II, n ∈ E
(( T n (m) = T n (n) = complex) ∨
( T n (m) = T n (n) = geoComplex) ∨
( T n (m) = complex ∧
( T n (n) = atomic ∨ T n (n) = stream )) ∨
Carlo Combi, Sara Migliorini, Barbara Oliboni, and Alberto Belussi
• S is a set of strings to be used as atomic values.
• G = G 2 ∪ G 3 where G 2 is the set of all possible pairs (x, y) of real numbers
that represents coordinates in the 2D space, and G 3 is the set of all possible
triples (x, y, z) of real numbers that represents coordinates in the 3D space. In
particular, the possible value for a spatial component can be a single point or
a list of coordinates that defines a simple polyline, or a set of coordinates that
defines a closed polyline that identifies a simple polygon.
• V is a set of temporal elements, and as for edges, they are used to keep track of
the time intervals during which a node exists in reality.
The function can be seen as composed of the following five single-valued functions:
T n , , L n , , S , , V , , G .
⊥ means ‘undefined,’ and in the following, when the context is clear, if an element of
a label is mapped in ⊥, it will be omitted.
Each GeoMTGM graph must satisfy the following integrity constraints in order
to correctly represent real data. In particular, some of them ensure the correctness of
the general structure of the graph (constraints 1–7), some of them ensure the consistency of the temporal elements specified on atomic and complex nodes (constraints
8 and 9), and finally, constraints 10–15 ensure the correctness of the spatial data in
geoAtomic and GeoField nodes, and of the geographical and composed-by edges.
Definition 2.2 (Integrity Constraints of a GeoMTGM graph).
The following conditions must be satisfied by each GeoMTGM graph:
1. Root. The root of the graph is a geographical complex node with the name
“Schema” whose spatial attribute is a polygon containing all the territory managed by the graph. It is connected to an atomic node named “SRS” that indicates
the considered spatial reference system.
2. Values. Values are associated to atomic and stream nodes only.
∀x ∈ N( T n (x) = complex ∨
T n (x) = geoComplex ∨
T n (x) = geoAtomic) → S (x) = ⊥)
3. General Relational Edges. Relational edges can be defined only between two
(geographical) complex nodes or between a complex node and its atomic or
stream nodes, or between a geographical complex node and its atomic, stream
nodes, or geoAtomic nodes.
∀∀m, relational, RelName, II, n ∈ E
(( T n (m) = T n (n) = complex) ∨
( T n (m) = T n (n) = geoComplex) ∨
( T n (m) = complex ∧
( T n (n) = atomic ∨ T n (n) = stream )) ∨