1 12. Implementation Methods Evaluation with gap-graphs Gap-graphs – visually represents a conjunctions of difference constraints

2 Shortcut – u a+b v if u a x and x b v

3 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.

4 Evaluation with matrices Matrix representation of first gap-graph: Question: how can we test satisfiability of a gap- graph ? 0x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 0 -  1 x1x  x2x2 2 3 x3x  x4x  3 x5x  x6x6 3

5 Question: how can we shortcut a vertex ? Question: how can we merge two gap-graph ? 0x2x2 x3x3 x4x4 x5x5 x6x6 0 -  1 x2x2 2 3 x3x  x4x  3 x5x  x6x6 3

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

7 Boolean constraints –Set-graphs – represent conjunction of subset constraints

8 Pairs of matrices for each set-graph Question: how do we test satisfiability ?

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