The document defines key concepts for computing full disjunctions from a set of relations, including: - Universal and maximal tuples that integrate connected and join-consistent tuples from the relations. - The full disjunction, which is the set of all maximal integrated tuples that can be generated from tuples in the relations. - Matching graphs used to represent matches between a query and database, with strata used to extend cyclic matching graphs.

1. Computing Full Disjunctions Yaron Kanza Yehoshua Sagiv The Selim and Rachel Benin School of Engineering and Computer Science The Hebrew University of Jerusalem

2. A Formal Definitions of Full Disjunction

12. Computing Full Disjunctions

14. 1 2 4 5 6 title language 7 3 year 8 director 9 name 10 movie date of birth 11 1983 movie actor Zelig Antz 1998 English 1/12/1935 Woody Allen title year filmography item filmography item v 1 v 2 w 1 v 3 title actor movie director filmography item w 2 w 3 w 4 date of birth name language The product of the query and the database is the next graph

15. title language director name movie date of birth movie actor title filmography item filmography item V 1 , 1 V 2 , 2 V 2 , 3 V 3 , 4 w 1 , 5 w 2 , 6 w 1 , 8 w 3 , 10 w 4 , 11 There are additional nodes that are not reachable from the root

17. title language director name movie date of birth movie actor title filmography item filmography item V 1 , 1 V 2 , 2 V 2 , 3 V 3 , 4 w 1 , 5 w 2 , 6 w 1 , 8 w 3 , 10 w 4 , 11 An OR-matching graph It is also a weak-matching graph V 1 , 1 V 2 , 2 w 1 , 5 w 2 , 6 V 3 , 4 w 3 , 10 w 4 , 11

18. title language director name movie date of birth movie actor title filmography item filmography item V 1 , 1 V 2 , 2 V 2 , 3 V 3 , 4 w 1 , 5 w 2 , 6 w 1 , 8 w 3 , 10 w 4 , 11 V 1 , 1 V 3 , 4 w 3 , 10 w 4 , 11 Another OR-matching graph V 2 , 3 w 1 , 8 It is not a weak-matching graph since the “ director” edge of the query is not preserved

19. Matching Graphs Each OR-matching graph represents an OR-matching (and each weak-matching graph represent a weak matching) An OR-matching can be represented by many OR-matching graphs, but all these graphs have the same set of nodes and only differ by their edges (and the same it true for weak-matchings and weak-matching graphs) Matching

21. title language director name movie date of birth movie actor title filmography item filmography item V 1 , 1 V 2 , 2 V 2 , 3 V 3 , 4 w 1 , 5 w 2 , 6 w 1 , 8 w 3 , 10 w 4 , 11 Dividing the edges to strata … Stratum 1 Stratum 2 Stratum 3

25. No change is being done movie V 1 , O 1 V 2 , O 2 actor V 3 , O 4 title V 2 , O 2 V 1 , O 3 The target of the added edge has a node with a pair that includes the root of Q without the root of D 1 No change is being done movie V 1 , O 1 V 2 , O 2 actor V 3 , O 4 movie V 1 , O 1 V 2 , O 2 The graph already includes the added edge 2

26. No change is being done movie V 1 , O 1 V 2 , O 2 actor V 3 , O 4 title V 2 , O 3 W 1 , O 8 The graph does not include the source of the added edge 3 movie V 1 , O 1 V 2 , O 2 actor V 3 , O 4 title V 2 , O 2 W 1 , O 5 The graph includes the source of the added edge and no node with the variable of the target 4 movie V 1 , O 1 V 2 , O 2 actor V 3 , O 4 title W 1 , O 5 The edge is added to the graph and the new graph replaces the existing graph

27. movie V 1 , O 1 V 2 , O 2 actor V 3 , O 4 The graph already includes the source and the target of the added edge but does not include the added edge itself 5 title W 1 , O 3 a.k.a V 2 , O 2 W 1 , O 3 The edge is added to the graph and the new graph replaces the existing graph a.k.a

28. movie V 1 , O 1 V 2 , O 2 actor V 3 , O 4 film V 3 , O 4 V 2 , O 4 The graph includes the source of the added edge but also includes a node with the same variable as the variable in the target of the added edge 6 title W 1 , O 3 Different nodes with the same variable V 2 A new graph is created and being added to the existing graph, without replacing it movie V 1 , O 1 V 2 , O 2 actor V 3 , O 4 title W 1 , O 3 movie V 1 , O 1 V 2 , O 4 actor V 3 , O 4 film (V 2 ,O 2 ) is replaced by (V 2 ,O 4 ), and nodes that are not reachable from the root are being erased

29. Applying the algorithm to the movies example V 1 , 1 1 V 1 , 1 2 movie V 2 , 2 V 1 , 1 movie V 2 , 2 V 1 , 1 3 movie V 2 , 2 V 1 , 1 V 1 , 1 V 2 , 3 movie movie V 2 , 2 V 1 , 1 V 1 , 1 V 2 , 3 movie

30. 4 movie V 2 , 2 V 1 , 1 V 1 , 1 V 2 , 3 movie actor V 1 , 1 V 3 , 4 V 3 , 4 movie V 2 , 2 V 1 , 1 V 1 , 1 V 2 , 3 movie actor V 3 , 4 actor 5 title V 2 , 2 w 1 , 5 V 3 , 4 movie V 2 , 2 V 1 , 1 V 1 , 1 V 2 , 3 movie actor V 3 , 4 actor title w 1 , 5 V 3 , 4 movie V 2 , 2 V 1 , 1 V 1 , 1 V 2 , 3 movie actor V 3 , 4 actor

31. 6 language V 2 , 2 w 2 , 6 title w 1 , 5 V 3 , 4 movie V 2 , 2 V 1 , 1 V 1 , 1 V 2 , 3 movie actor V 3 , 4 actor language w 2 , 6 title w 1 , 5 V 3 , 4 movie V 2 , 2 V 1 , 1 V 1 , 1 V 2 , 3 movie actor V 3 , 4 actor 7 language w 2 , 6 title w 1 , 5 V 3 , 4 movie V 2 , 2 V 1 , 1 V 1 , 1 V 2 , 3 movie actor V 3 , 4 actor title w 1 , 5 V 2 , 3 language w 2 , 6 title w 1 , 5 V 3 , 4 movie V 2 , 2 V 1 , 1 V 1 , 1 V 2 , 3 movie actor V 3 , 4 actor title w 1 , 5

32. language w 2 , 6 title w 1 , 5 V 3 , 4 movie V 1 , 1 V 2 , 3 movie actor V 3 , 4 actor title w 1 , 5 8 V 2 , 2 V 1 , 1 name V 3 , 4 w 3 , 10 name w 3 , 10 name w 3 , 10 V 3 , 4 w 4 , 11 date of birth 9 date of birth w 4 , 11 date of birth w 4 , 11

33. language w 2 , 6 title w 1 , 5 V 3 , 4 movie V 1 , 1 V 2 , 3 movie actor V 3 , 4 actor title w 1 , 5 10 director V 2 , 2 V 3 , 4 V 2 , 2 V 1 , 1 name w 3 , 10 name w 3 , 10 date of birth w 4 , 11 date of birth w 4 , 11 language w 2 , 6 title w 1 , 5 V 3 , 4 movie V 2 , 2 V 1 , 1 V 1 , 1 V 2 , 3 movie actor V 3 , 4 actor title w 1 , 5 name w 3 , 10 name w 3 , 10 date of birth w 4 , 11 date of birth w 4 , 11 director

34. 11 filmography item V 3 , 4 V 2 , 2 language w 2 , 6 title w 1 , 5 V 3 , 4 movie V 2 , 2 V 1 , 1 V 1 , 1 V 2 , 3 movie actor V 3 , 4 actor title w 1 , 5 name w 3 , 10 name w 3 , 10 date of birth w 4 , 11 date of birth w 4 , 11 title w 1 , 5 movie V 2 , 2 language w 2 , 6 V 3 , 4 V 1 , 1 V 1 , 1 V 2 , 3 movie actor V 3 , 4 actor title w 1 , 5 name w 3 , 10 name w 3 , 10 date of birth w 4 , 11 date of birth w 4 , 11 filmography item director V 1 , 1 V 2 , 2 V 3 , 4 actor name w 3 , 10 date of birth w 4 , 11 filmography item Subsumed by the left matching graph

35. 12 V 1 , 1 V 2 , 3 movie V 3 , 4 actor title w 1 , 5 name w 3 , 10 date of birth w 4 , 11 title w 1 , 5 movie V 2 , 2 language w 2 , 6 V 3 , 4 V 1 , 1 actor name w 3 , 10 date of birth w 4 , 11 filmography item director filmography item V 3 , 4 V 2 , 3 title w 1 , 5 movie V 2 , 2 language w 2 , 6 V 3 , 4 V 1 , 1 V 1 , 1 V 2 , 3 movie actor V 3 , 4 actor title w 1 , 5 name w 3 , 10 name w 3 , 10 date of birth w 4 , 11 date of birth w 4 , 11 filmography item director filmography item V 2 , 3 V 3 , 4 V 1 , 1 actor name w 3 , 10 date of birth w 4 , 11 filmography item Subsumed by the right matching graph

36. title language name movie date of birth movie actor title filmography item filmography item V 1 , 1 V 2 , 2 V 2 , 3 V 3 , 4 w 1 , 5 w 2 , 6 w 1 , 8 w 3 , 10 w 4 , 11 The Product Graph director title w 1 , 5 movie V 2 , 2 language w 2 , 6 V 3 , 4 V 1 , 1 actor name w 3 , 10 date of birth w 4 , 11 filmography item director V 1 , 1 V 2 , 3 movie V 3 , 4 actor title w 1 , 5 name w 3 , 10 date of birth w 4 , 11 filmography item The OR-Matchings