1. Extraction Operators

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1. Extraction Operators

• Projection
• Selection
• Unorder
• Join
• Cartesian Product
• Union
• Difference
• Intersection

set-like operators (commutativity)
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• Projection
• ?type, name(e expression)
• follows the edges of a given type and name
  from the input collection of vertices. The result
  is the collection of target vertices of the
  followed edges. The order of the output
  collection depends on the order of the input
  collection.
• type E, A, R, D or disjunctions () of these
• name regular expression over strings
• Example. ?E, (Pp)ainters)(e) produces all
  the target vertices of element containment edges
  that have names starting with Painter, painter,
  Painters, or painters, and that originate from
  the vertices in e.

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• Selection
• ?condition(e expression)
• selects input collection vertices fulfilling
  the given condition. The order of the output
  collection depends on the order of the input
  collection.
• In condition one can use projection operator,
  comparison operators (, lt, gt etc.), and logical
  operators (and, or, not).
• Example. ??A, nameDali(e expression)
  selects all the vertices that have an attribute
  called name with the value Dali.

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• Unorder
• ?(e expression)
• unorders an input collection.
• Unordering collections enables the definition of
  set-like (commutative) operators.
• ?, ?, ?, ?, and ? are considering the order of
  the input collections in building the result.
  Unordering input collections enables the usage of
  their set-like variants (order is not important).
  
• Two vertices are equal if they have the same
  value.

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• Join
• (x expression) ?condition (y expression)
• ?condition(x ? y)
• the cartesian product is defined as two loops
  the external loop traverses the left input
  collection and the internal loop traverses the
  right input collection.
• The pairs that fulfill the join condition
  form virtual vertices that have as outgoing
  edges, first the outgoing edges of the vertex
  from the left input collection and then the
  outgoing edges of the vertex from the right input
  collection preserving the original edge order.

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• Join (ctnd)
• Example. (x ?E, person(people))
• ??A, id(x) ?A, name(y)
• (y ?E, painter(painters)) pairs
  person and painter vertices in virtual vertices
  based on the equality of the id attribute value
  of a person and the name attribute value of a
  painter.
• Cartesian Product
• (x expression) ? (y expression)
• join without condition.

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• Difference
• (x expression) ? (yexpression)
• the result is a collection containing vertices
  that exist in first collection but not in second
  one preserving the original vertex order of the
  left input collection.
• Intersection
• (x expression) ? (yexpression)
• the result is a collection containing vertices
  that exist in both input collections preserving
  the original vertex order of the left input
  collection.

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• Union
• (x expression) ? (yexpression)
• the result is a collection that contains first
  the left input collection, then the right input
  collection preserving the original vertex order.
• It can be easily generalized to an n-ary
  operator.
• Flexibility ensured by the fact that the input
  collections do not have to be union compatible
  as it was the case in relational algebra.

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2. Meta-operators

• Map
• mapf(e expression)
• applies a given function f to each element of
  the input collection. The function results are
  concatenated in the output collection.
• All unary extraction operators have an inherent
  map operator associated to them. In the
  construction phase, the map operator is used
  explicitly to iterate not just over collection of
  vertices but also collection of edges.

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• Kleene Star
• ?f(e expression)
• repeats a given function f possibly infinite
  times starting with the given input collection.
  At each iteration the results of the function are
  added to the next function input. The order of
  the result collection depends on the order of the
  input collection and the recursion order.
• Compared with the map operator it includes in the
  result the input collection.
• When a fix point is reached (output input) the
  repetition stops.

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3. Construction Operators

• Create vertex
• vertextype(value)
• adds a new vertex with the given type and
  value to the graph.
• Example. v1 vertexelement(null) creates a
  vertex of type element which has a new value (id)
  given by the system (by a skolem function) and
  v2 vertexstring(Dali) creates a vertex of
  type string and value Dali.

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• Create edge
• edgetype, name, parent(child)
• adds a new edge with a given type, name,
  parent, and child vertex to the graph.
• Before inserting a new edge of type E, the
  resulting graph is checked for presence of loops
  among element containment edges.
• If such a loop is formed the operation is
  unsuccessful.
• Example. edgeE, painter, v1(v2) creates an edge
  of type E with name painter between two vertices
  (the vertices from the previous example).

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Examples

• Example. Copying a vertex v means creating a new
  vertex with the type and value of the original
  vertex vertextype(v)(value(v)).
• Example. Copying an edge e, involves copying also
  the parent and children vertices
• edgetype(e), name(e), vp(vc) where
• vp vertextype(parent(e))(value(parent(e))
  , and
• vc vertextype(child(e))(value(child(e))).

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• Example. Copying a complete graph starting from
  the vertex v can be done by copying all the graph
  edges and their associated vertices
• mapedgetype(e), name(e),
• vertextype(parent(e))(value
  (parent(e)))
• (vertextype(child(e))(value
  (child(e))))
• (e) where
• e parentedge(?EAD, (child(x)))
• (x parentedge(?EAD, (v)))
• e represents all the edges in the original
  graph.
• For each original vertex a unique new vertex
  is created using a skolem function.