2 GeoMTGM
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the corresponding representations in grid-based or vector-based structures usually
adopted in spatial database systems.
2.3.2 GeoMTGM Formal Definition
In this section, we provide a formal description of the GeoMTGM. This definition
is based on the formal definition of MTGM presented in [10], so we omit the multimedia aspect and constraints about the multimedia part of the model and explain in
detail only the geographical part.
Definition 2.1 (GeoMTGM Graph). A geographical multimedia temporal graph
(or GeoMTGM graph) is a directed labeled graph N, E, , with a single root where:
Nodes
N is a finite set of nodes.
Edges
E is a set of labeled edges of the form m, label, n, with m, n ∈ N, and label is the
string label associated to a specific edge, in particular, it has the form
label ∈ (T e × L e ) × (L MMProp ∪ {⊥}) × (F ∪ {⊥}) × (V ∪ {⊥}))
where:
• T e = {relational, geographical, composed-by} is the set of types for edges.
• L e is the set of strings to be used as name for edges.
• L MMProp = {[frame s, frame e]} ∪ {[time s,time e]} ∪ {[x, y, width, height]} is
the set of descriptions to define a subpart of a stream object. In particular,
[frame s, frame e] stands for a pair of real numbers that describe the start and
end frame of a video, [time s, time e] is a pair of real numbers that define the
limit of an audio stream, and [x, y, width, height] are four real numbers that
define a subpart of an image object.
• F is the set of strings to be used as value for the value of a geographical property
represented in the edges that start from a GeoField node.
• V is a set of temporal elements of the form: [t 1 , t 2 ) ∪ [t 3 , t 4 ) ∪ · · · ∪ [t n−1 , t n ),
with n > 1. We use a temporal element to keep track of the time intervals during
which an edge exists in the reality.
Node Labels
is a function that associates a unique label to each node.
: N −→ (T n ∪ {⊥}) × (L n ∪ {⊥}) × (S ∪ {⊥}) × (G ∪ {⊥}) × (V ∪ {⊥})
where:
• T n = {complex, geoComplex, atomic, stream, geoAtomic} is the set of types for
nodes.
• L n is the set of strings to be used as name for all nodes of the graph.
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the corresponding representations in grid-based or vector-based structures usually
adopted in spatial database systems.
2.3.2 GeoMTGM Formal Definition
In this section, we provide a formal description of the GeoMTGM. This definition
is based on the formal definition of MTGM presented in [10], so we omit the multimedia aspect and constraints about the multimedia part of the model and explain in
detail only the geographical part.
Definition 2.1 (GeoMTGM Graph). A geographical multimedia temporal graph
(or GeoMTGM graph) is a directed labeled graph N, E, , with a single root where:
Nodes
N is a finite set of nodes.
Edges
E is a set of labeled edges of the form m, label, n, with m, n ∈ N, and label is the
string label associated to a specific edge, in particular, it has the form
label ∈ (T e × L e ) × (L MMProp ∪ {⊥}) × (F ∪ {⊥}) × (V ∪ {⊥}))
where:
• T e = {relational, geographical, composed-by} is the set of types for edges.
• L e is the set of strings to be used as name for edges.
• L MMProp = {[frame s, frame e]} ∪ {[time s,time e]} ∪ {[x, y, width, height]} is
the set of descriptions to define a subpart of a stream object. In particular,
[frame s, frame e] stands for a pair of real numbers that describe the start and
end frame of a video, [time s, time e] is a pair of real numbers that define the
limit of an audio stream, and [x, y, width, height] are four real numbers that
define a subpart of an image object.
• F is the set of strings to be used as value for the value of a geographical property
represented in the edges that start from a GeoField node.
• V is a set of temporal elements of the form: [t 1 , t 2 ) ∪ [t 3 , t 4 ) ∪ · · · ∪ [t n−1 , t n ),
with n > 1. We use a temporal element to keep track of the time intervals during
which an edge exists in the reality.
Node Labels
is a function that associates a unique label to each node.
: N −→ (T n ∪ {⊥}) × (L n ∪ {⊥}) × (S ∪ {⊥}) × (G ∪ {⊥}) × (V ∪ {⊥})
where:
• T n = {complex, geoComplex, atomic, stream, geoAtomic} is the set of types for
nodes.
• L n is the set of strings to be used as name for all nodes of the graph.