12. Implementation Methods

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12. Implementation Methods

• Evaluation with gap-graphs
• Gap-graphs visually represents a conjunctions
  of difference constraints

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• Shortcut u ab v if u a x and x
  b v

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• Merge union vertices and edges from both
• input gap-order graphs. If some edge occurs
• but different labels, then keep larger label.
• Relation algebra on set of gap-graphs can
• be defined based on shortcut and merge.

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Evaluation with matricesMatrix representation of
first gap-graph
0 x1 x2 x3 x4 x5 x6
0 - ? - ? - ? - ? - ? - ? 1
x1 5 - ? -6 - ? - ? - ? - ?
x2 2 - ? - ? 3 - ? - ? - ?
x3 - ? 2 - ? - ? -5 - ? - ?
x4 - ? - ? - ? -3 - ? 3 - ?
x5 - ? -9 - ? - ? - ? - ? - ?
x6 - ? - ? - ? - ? - ? 3 - ?

• Question how can we test satisfiability of a
  gap-
• graph ?

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Question how can we shortcut a vertex ?
0 x2 x3 x4 x5 x6
0 - ? - ? - ? - ? - ? 1
x2 2 - ? 3 - ? - ? - ?
x3 3 8 - ? -5 - ? - ?
x4 - ? - ? -3 - ? 3 - ?
x5 - 4 - 15 - ? - ? - ? - ?
x6 - ? - ? - ? - ? 3 - ?

• Question how can we merge two gap-graph ?

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• Question how do we evaluate datalog
• queries ?
• Question how do we find complement and
• difference ?

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• Boolean constraints
• Set-graphs represent conjunction of subset
  constraints

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Pairs of matrices for each set-graph

• Question how do we test satisfiability ?

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• Optimization of relational algebra
• Perform selection and projection as early as
  possible by applying algebraic rewrite rules.