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

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

3. 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. 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 x1x1 5 -6 -  x2x2 2 3 x3x3 -2-2 -5 -  x4x4 -3 -  3 x5x5 -9 -  x6x6 3

5. 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 x3x3 3-8-8 -5 -  x4x4 -3 -  3 x5x5 - 4- 15 -  x6x6 3

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

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

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

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