Shannon-type Inequalities, Submodular Width, and Disjunctive Datalog

Transcription

1 Shannon-type Inequalities, Submodular Width, and Disjunctive Datalog Hung Q. Ngo (Stealth Mode) With Mahmoud Abo Khamis and Dan PODS 2017

2 Table of Contents Connecting the Dots Output Size Bounds and Information Theory Shannon-flow Inequalities and the PANDA Algorithm Wrapping it up Appendix

3 Table of Contents Connecting the Dots Output Size Bounds and Information Theory Shannon-flow Inequalities and the PANDA Algorithm Wrapping it up Appendix

4 Exciting Recent Results on Query Evaluation

5 A Query Evaluation Problem Hypergraph H = ([n], E), E 2 [n]

6 A Query Evaluation Problem Hypergraph H = ([n], E), E 2 [n] Attribute A i, i [n], A J := {A j j J}, J [n]

7 A Query Evaluation Problem Hypergraph H = ([n], E), E 2 [n] Attribute A i, i [n], A J := {A j j J}, J [n] Relation R F (A F ) for each F E, R F for short e.g. R 123 for R 123 (A 1, A 2, A 3 ) e.g. T 25 for T 25 (A 2, A 5 )

8 A Query Evaluation Problem Hypergraph H = ([n], E), E 2 [n] Attribute A i, i [n], A J := {A j j J}, J [n] Relation R F (A F ) for each F E, R F for short e.g. R 123 for R 123 (A 1, A 2, A 3 ) e.g. T 25 for T 25 (A 2, A 5 )

9 A Query Evaluation Problem Hypergraph H = ([n], E), E 2 [n] Attribute A i, i [n], A J := {A j j J}, J [n] Relation R F (A F ) for each F E, R F for short e.g. R 123 for R 123 (A 1, A 2, A 3 ) e.g. T 25 for T 25 (A 2, A 5 ) Problem (Boolean Conjunctive Query (BCQ)) Q : S() F E R F, // can all R F be satisfied at once?

10 A Query Evaluation Problem Hypergraph H = ([n], E), E 2 [n] Attribute A i, i [n], A J := {A j j J}, J [n] Relation R F (A F ) for each F E, R F for short e.g. R 123 for R 123 (A 1, A 2, A 3 ) e.g. T 25 for T 25 (A 2, A 5 ) Problem (Boolean Conjunctive Query (BCQ)) Q : S() F E R F, // can all R F be satisfied at once? Question How do we evaluate Q efficiently?

11 A Query Evaluation Problem Hypergraph H = ([n], E), E 2 [n] Attribute A i, i [n], A J := {A j j J}, J [n] Relation R F (A F ) for each F E, R F for short e.g. R 123 for R 123 (A 1, A 2, A 3 ) e.g. T 25 for T 25 (A 2, A 5 ) Problem (Boolean Conjunctive Query (BCQ)) Q : S() F E R F, // can all R F be satisfied at once? Question How do we evaluate Q efficiently? Conjunctive, count, aggregate queries are fine too.

12 What do we mean by efficiency? Database centric complexity framework Assumption 1: query size data size Data complexity Fixed parameter tractability (e.g. parameter = some function of query size) Õ(something) = O (f( query ) polylog( data ) something)

13 What do we mean by efficiency? Database centric complexity framework Assumption 1: query size data size Data complexity Fixed parameter tractability (e.g. parameter = some function of query size) Õ(something) = O (f( query ) polylog( data ) something) Assumption 2: known constraints on input relations Cardinalities of materialized relations, or upper bounds cardinality constraints (CC) Functional dependencies, the more the merrier FD constraints (FDC) Degree bounds, the more the merrier Degree constraints (DC)

14 Example Q : S() R 12 R 23 R 34 R 41 A 1 + A 2 = A 3. H = ([4], {12, 23, 34, 14, 123}) A 2 R 12 R 23 A 1 A 3 R 41 A 4 R 34 A 1 A A 2 A 3 1 c 1 d 2 c A 3 A 4 c 3 d 4 d 5 e 6 e 7 A 4 A Cardinalities: R 12 = 4, R 23 = 3,...

15 Example Q : S() R 12 R 23 R 34 R 41 A 1 + A 2 = A 3. H = ([4], {12, 23, 34, 14, 123}) A 2 R 12 R 23 A 1 A 3 R 41 A 4 R 34 A 1 A A 2 A 3 1 c 1 d 2 c A 3 A 4 c 3 d 4 d 5 e 6 e 7 A 4 A Cardinalities: R 12 = 4, R 23 = 3,... FD: {A 1, A 2 } A 3, {A 3, A 2 } A 1,...

16 Example Q : S() R 12 R 23 R 34 R 41 A 1 + A 2 = A 3. H = ([4], {12, 23, 34, 14, 123}) A 2 R 12 R 23 A 1 A 3 R 41 A 4 R 34 A 1 A A 2 A 3 1 c 1 d 2 c A 3 A 4 c 3 d 4 d 5 e 6 e 7 A 4 A Cardinalities: R 12 = 4, R 23 = 3,... FD: {A 1, A 2 } A 3, {A 3, A 2 } A 1,... Degree Bounds: deg 34 (A 4 A 3 = x) def = σ A3=x(R 34 ) 2, x,...

17 Degree- generalize cardinality- and FD-constraints DC CC FDC Degree Constraints (DC): deg F (A Y A X ) N Y X, X Y F E

18 Degree- generalize cardinality- and FD-constraints DC CC FDC Degree Constraints (DC): deg F (A Y A X ) N Y X, X Y F E Cardinality Constraints (CC): R F N deg F (A F A ) N F def = N.

19 Degree- generalize cardinality- and FD-constraints DC CC FDC Degree Constraints (DC): deg F (A Y A X ) N Y X, X Y F E Cardinality Constraints (CC): R F N deg F (A F A ) N F def = N. Functional Dependencies (FDC): A X A Y deg F (A Y A X ) N Y X def = 1.

20 Islands of tractability (redrawn from D. Marx s slides) Prior results with cardinality constraints not FPT FPT PTIME Bounded submodular width Bounded fractional hypertree width Bounded (generalized) Hypertree Width Bounded fractional edge cover number Bounded Treewidth

21 Islands of tractability (redrawn from D. Marx s slides) Prior results with cardinality constraints not FPT FPT PTIME Bounded submodular width Bounded fractional hypertree width Bounded (generalized) Hypertree Width Bounded fractional edge cover number Bounded Treewidth We want the same map with degree constraints.

22 A Meta Algorithm (i.e. Meta Query Plan) Database D F E R F itime Datalog rule Q i Q i (D) A propositional formula on relations T j, j J Q o otime overall time = Õ( input + itime + otime ) Output Answer

23 A Meta Algorithm (i.e. Meta Query Plan) Database D F E R F itime Datalog rule Q i Q i (D) A propositional formula on relations T j, j J Q o otime overall time = Õ( input + itime + otime ) Output Answer (a) otime = Q i (D) + answer

24 A Meta Algorithm (i.e. Meta Query Plan) Database D F E R F itime Datalog rule Q i Q i (D) A propositional formula on relations T j, j J Q o otime overall time = Õ( input + itime + otime ) Output Answer (a) otime = Q i (D) + answer i.e. Q o evaluatable in linear time

25 A Meta Algorithm (i.e. Meta Query Plan) Database D F E R F itime Datalog rule Q i Q i (D) A propositional formula on relations T j, j J Q o otime overall time = Õ( input + itime + otime ) Output Answer (a) otime = Q i (D) + answer i.e. Q o evaluatable in linear time (b) itime = max D =DC Q i(d)

26 A Meta Algorithm (i.e. Meta Query Plan) Database D F E R F itime Datalog rule Q i Q i (D) A propositional formula on relations T j, j J Q o otime overall time = Õ( input + itime + otime ) Output Answer (a) otime = Q i (D) + answer i.e. Q o evaluatable in linear time (b) itime = max D =DC Q i(d) i.e. Qi evaluatable within its worst-case output size

27 A Meta Algorithm (i.e. Meta Query Plan) Database D F E R F itime Datalog rule Q i Q i (D) A propositional formula on relations T j, j J Q o otime overall time = Õ( input + itime + otime ) Output Answer (a) otime = Q i (D) + answer i.e. Q o evaluatable in linear time (b) itime = max D =DC Q i(d) i.e. Qi evaluatable within its worst-case output size Design Q i s.t. (a) holds and (b) as small as possible

28 Option 1: Full Conjunctive Query Full conjunctive query Database D Q i Q i (D) Q o Output F E R F itime T [n] otime Answer

29 Option 1: Full Conjunctive Query Full conjunctive query Database D Q i Q i (D) Q o Output F E R F itime T [n] otime Answer otime = Q i (D) + answer trivially true

30 Option 1: Full Conjunctive Query Database D F E R F Full conjunctive query itime Q i Q i (D) T [n] Q o otime Output Answer otime = Q i (D) + answer itime = max Q i(d) D =DC trivially true worst-case optimal algorithm Known if all DC are cardinality constraints NPRR [Ngo, Porat, Ré, Rudra PODS 12] Leapfrog-Triejoin [Veldhuizen ICDT 14] Generic Join [Ngo, Ré, Rudra SIGMOD Records 2013] Unknown for general DC until our work

31 Detour: Tree Decompositions, Informally

32 Detour: Tree Decompositions, Informally S() R(a, b, d) c < d T (c, b, d) U(b, e) V (c, e) b + e = f W (b, e, g) X(i, j, h) e b = k.

33 Detour: Tree Decompositions, Informally S() R(a, b, d) c < d T (c, b, d) U(b, e) V (c, e) b + e = f W (b, e, g) X(i, j, h) e b = k. b, c, e bag a, b, c, d bag b, e, f, g, h bag h, i, j bag e, b, k bag

34 Detour: Tree Decompositions, Informally S() R(a, b, d) c < d T (c, b, d) U(b, e) V (c, e) b + e = f W (b, e, g) X(i, j, h) e b = k. b, c, e bag a, b, c, d bag b, e, f, g, h bag h, i, j bag e, b, k bag Every relation is covered by some bag

35 Detour: Tree Decompositions, Informally S() R(a, b, d) c < d T (c, b, d) U(b, e) V (c, e) b + e = f W (b, e, g) g/f = h X(i, j, h) e b = k. b, c, e bag a, b, c, d bag b, e, f, g, h bag h, i, j bag e, b, k bag Every relation is covered by some bag

36 Detour: Tree Decompositions, Informally S() R(a, b, d) c < d T (c, b, d) U(b, e) V (c, e) b + e = f W (b, e, g) X(i, j, h) e b = k. b, c, e bag a, b, c, d bag b, e, f, g, h bag h, i, j bag e, b, k bag Every relation is covered by some bag Bags conntaining a given variable are connected

37 Detour: Tree Decompositions, Formally Hypergraph H = ([n], E) A Tree Decomposition of H is a pair (T, χ) where T = (V (T ), E(T )) is a tree See [Gottlob et al 2016], Gems of PODS.

38 Detour: Tree Decompositions, Formally Hypergraph H = ([n], E) A Tree Decomposition of H is a pair (T, χ) where T = (V (T ), E(T )) is a tree χ : V (T ) 2 [n] assigns a bag χ(v) to each tree-node v Every hyperedge F E is covered by some bag (F χ(v)) Bags containing i [n] forms a subtree See [Gottlob et al 2016], Gems of PODS.

39 Option 2: A Single Tree Decomposition Fix (T, χ) Database D Multiple Conjunctive Rules itime R F T χ(v) F E v V (T ) Q i Q i (D) Q o otime Output Answer

40 Option 2: A Single Tree Decomposition Fix (T, χ) Database D Multiple Conjunctive Rules itime R F T χ(v) F E v V (T ) Q i Q i (D) Q o otime Output Answer Q i (D) is Olteanu s factorized database (FDB)

41 Option 2: A Single Tree Decomposition Fix (T, χ) Database D Multiple Conjunctive Rules itime R F T χ(v) F E v V (T ) Q i Q i (D) Q o otime Output Answer Q i (D) is Olteanu s factorized database (FDB) otime = Q i (D) + answer Yannakakis for join, FDB/InsideOut for aggregates

42 Option 2: A Single Tree Decomposition Fix (T, χ) Database D Multiple Conjunctive Rules itime R F T χ(v) F E v V (T ) Q i Q i (D) Q o otime Output Answer Q i (D) is Olteanu s factorized database (FDB) otime = Q i (D) + answer Yannakakis for join, FDB/InsideOut for aggregates itime = max Q i(d) max max P v(d) D =DC D =DC v V (T ) Pv : T χ(v) R F v V (T ) F E

43 Option 2: A Single Tree Decomposition Fix (T, χ) Database D Multiple Conjunctive Rules itime R F T χ(v) F E v V (T ) Q i Q i (D) Q o otime Output Answer Q i (D) is Olteanu s factorized database (FDB) otime = Q i (D) + answer Yannakakis for join, FDB/InsideOut for aggregates itime = max Q i(d) max max P v(d) D =DC D =DC v V (T ) Pv : T χ(v) R F v V (T ) F E

44 Option 2: A Single Tree Decomposition Fix (T, χ) Database D Multiple Conjunctive Rules itime R F T χ(v) F E v V (T ) Q i Q i (D) Q o otime Output Answer Q i (D) is Olteanu s factorized database (FDB) otime = Q i (D) + answer Yannakakis for join, FDB/InsideOut for aggregates itime = max Q i(d) max max P v(d) D =DC D =DC v V (T ) Pv : T χ(v) R F v V (T ) F E min max max P v(d) N fhtw(h) N ghtw(h) N tw(h)+1 (T,χ) D =CC v V (T )

45 Option 3: Multiple Tree Decompositions Database D Q i Q i (D) itime R F T χ(v) F E (T,χ) v V (T ) Q o otime Output Answer ranges over non-redundant TDs (T, χ)

46 Option 3: Multiple Tree Decompositions Database D Q i Q i (D) itime R F T χ(v) F E (T,χ) v V (T ) Q o otime Output Answer ranges over non-redundant TDs (T, χ) otime = Q i (D) + answer Union of Yannakakis on all TDs

47 Option 3: Multiple Tree Decompositions Database D How to evaluate this? Q i Q i (D) itime R F T χ(v) F E (T,χ) v V (T ) Q o otime Output Answer ranges over non-redundant TDs (T, χ) otime = Q i (D) + answer Union of Yannakakis on all TDs itime = max D =DC Q i(d)?

48 Example: Q : S() R 12 R 23 R 34 R 41 A 2 R 12 R 23 A 1 A 3 R 41 A 4 R 34

49 Example: Q : S() R 12 R 23 R 34 R 41 A 2 R 12 R 23 A 1 A 3 R 41 A 4 R 34 A 1 A 2 A 3 A 1 A 4 A 3 A 1 A 2 A 4 A 2 A 4 A 3

50 Example: Q : S() R 12 R 23 R 34 R 41 A 2 R 12 R 23 A 1 A 3 R 41 A 4 R 34 A 1 A 2 A 3 A 1 A 4 A 3 A 1 A 2 A 4 A 2 A 4 A 3 (T 123 T 134 ) (T 124 T 234 ) R 12 R 23 R 34 R 41

51 Example: Q : S() R 12 R 23 R 34 R 41 A 2 R 12 R 23 A 1 A 3 R 41 A 4 R 34 A 1 A 2 A 3 A 1 A 4 A 3 A 1 A 2 A 4 A 2 A 4 A 3 (T 123 T 134 ) (T 124 T 234 ) R 12 R 23 R 34 R 41 By distributivity, rewrite an equivalent the head: (T 123 T 124 ) (T 123 T 234 ) (T 134 T 124 ) (T 134 T 234 ) R 12 R 23 R 34 R 41 (Each clause has one bag per TD)

52 Option 3: Multiple Tree Decompositions Database D Multiple Disjunctive Datalog Rules! itime R F T B F E B B B Q i Q i (D) Q o otime Output Answer otime = Q i (D) + answer Union of Yannakakis on all TDs itime = max Q i(d) max D =DC B PB : T B B B F E R F D =DC max P B(D) disjunctive datalog rule

53 Option 3: Multiple Tree Decompositions Database D Multiple Disjunctive Datalog Rules! itime R F T B F E B B B Q i Q i (D) Q o otime Output Answer otime = Q i (D) + answer Union of Yannakakis on all TDs itime = max Q i(d) max D =DC B PB : T B B B F E R F D =DC max P B(D) disjunctive datalog rule max max P B(D) N subw(h) N fhtw(h) D =CC B subw = submodular width (Daniel Marx, JACM 2013)

54 Example: Q : S() R 12 R 23 R 34 R 41 Option 1: Q i : T 1234 R 12 R 23 R 34 R 41

55 Example: Q : S() R 12 R 23 R 34 R 41 Option 1: Option 2: Q i : T 1234 R 12 R 23 R 34 R 41 either Q i : T 123 T 134 R 12 R 23 R 34 R 41 or Q i : T 124 T 234 R 12 R 23 R 34 R 41

56 Example: Q : S() R 12 R 23 R 34 R 41 Option 1: Option 2: Q i : T 1234 R 12 R 23 R 34 R 41 Option 3: either Q i : T 123 T 134 R 12 R 23 R 34 R 41 or Q i : T 124 T 234 R 12 R 23 R 34 R 41 Q i : (T 123 T 134 ) (T 124 T 234 ) R 12 R 23 R 34 R 41 Equivalent to: P 123,124 : T 123 T 124 R 12 R 23 R 34 R 41 P 123,234 : T 123 T 234 R 12 R 23 R 34 R 41 P 134,124 : T 134 T 124 R 12 R 23 R 34 R 41 P 134,234 : T 134 T 234 R 12 R 23 R 34 R 41

57 Example: Q : S() R 12 R 23 R 34 R 41 Option 1: itime = max D =DC Q i(d) = N 2 Q i : T 1234 R 12 R 23 R 34 R 41 Option 2: Option 3: either Q i : T 123 T 134 R 12 R 23 R 34 R 41 or Q i : T 124 T 234 R 12 R 23 R 34 R 41 Q i : (T 123 T 134 ) (T 124 T 234 ) R 12 R 23 R 34 R 41 Equivalent to: P 123,124 : T 123 T 124 R 12 R 23 R 34 R 41 P 123,234 : T 123 T 234 R 12 R 23 R 34 R 41 P 134,124 : T 134 T 124 R 12 R 23 R 34 R 41 P 134,234 : T 134 T 234 R 12 R 23 R 34 R 41

58 Example: Q : S() R 12 R 23 R 34 R 41 Option 1: itime = max D =DC Q i(d) = N 2 Q i : T 1234 R 12 R 23 R 34 R 41 Option 2: itime = max D =DC Q i(d) = N 2 Option 3: either Q i : T 123 T 134 R 12 R 23 R 34 R 41 or Q i : T 124 T 234 R 12 R 23 R 34 R 41 Q i : (T 123 T 134 ) (T 124 T 234 ) R 12 R 23 R 34 R 41 Equivalent to: P 123,124 : T 123 T 124 R 12 R 23 R 34 R 41 P 123,234 : T 123 T 234 R 12 R 23 R 34 R 41 P 134,124 : T 134 T 124 R 12 R 23 R 34 R 41 P 134,234 : T 134 T 234 R 12 R 23 R 34 R 41

59 Example: Q : S() R 12 R 23 R 34 R 41 Option 1: itime = max D =DC Q i(d) = N 2 Q i : T 1234 R 12 R 23 R 34 R 41 Option 2: itime = max D =DC Q i(d) = N 2 either Q i : T 123 T 134 R 12 R 23 R 34 R 41 or Q i : T 124 T 234 R 12 R 23 R 34 R 41 Option 3: itime = max D =DC Q i(d) = N 3/2 Q i : (T 123 T 134 ) (T 124 T 234 ) R 12 R 23 R 34 R 41 Equivalent to: P 123,124 : T 123 T 124 R 12 R 23 R 34 R 41 P 123,234 : T 123 T 234 R 12 R 23 R 34 R 41 P 134,124 : T 134 T 124 R 12 R 23 R 34 R 41 P 134,234 : T 134 T 234 R 12 R 23 R 34 R 41

60 Roadmap Given degree constraints DC, and a disjunctive datalog rule P : B B T B F E R F

61 Roadmap Given degree constraints DC, and a disjunctive datalog rule P : B B T B F E R F Question (Worst-case Output Size Bound) Find a good upper-bound for max P (D) D =DC

62 Roadmap Given degree constraints DC, and a disjunctive datalog rule P : B B T B F E R F Question (Worst-case Output Size Bound) Find a good upper-bound for max P (D) D =DC Question (Algorithm) Design an algorithm evaluating P within the bound.

63 Roadmap Given degree constraints DC, and a disjunctive datalog rule P : B B T B F E R F Question (Worst-case Output Size Bound) Find a good upper-bound for max P (D) D =DC Question (Algorithm) Design an algorithm evaluating P within the bound. Question (Gathering fruits) Plug bound/algorithm into Meta Algorithm, what do we get?

64 Table of Contents Connecting the Dots Output Size Bounds and Information Theory Shannon-flow Inequalities and the PANDA Algorithm Wrapping it up Appendix

65 High-level View of the Bound Given degree constraints DC, a disjunctive datalog rule P : B B T B F E We shall prove bounds of the form R F max log P (D) some function of h D =DC s.t. h is (approximately) entropic and h satisfies degree constraints

66 An Idea From Gottlob-Lee-Valiant-Valiant, JACM 12 Q(A 1, A 2, A 3, A 4 ) R 12 R 23 R 34 R 41. A 2 R 12 R 23 A 1 A 3 R 41 A 4 R 34

67 An Idea From Gottlob-Lee-Valiant-Valiant, JACM 12 Q(A 1, A 2, A 3, A 4 ) R 12 R 23 R 34 R 41. A 2 R 12 R 23 A 1 A 3 R 41 A 4 R 34 A 1 A 2 a 1 b 1 b 2 A 2 A 3 1 c 1 d 2 c A 3 A 4 c 3 d 4 d 5 A 4 A 1 3 b 4 a 4 b

68 An Idea From Gottlob-Lee-Valiant-Valiant, JACM 12 Q(A 1, A 2, A 3, A 4 ) R 12 R 23 R 34 R 41. A 1 A 2 A 3 A 4 a 1 d 4 b 1 c 3 b 1 d 4 b 2 c 3 A 2 R 12 R 23 A 1 A 3 R 41 A 4 R 34 A 1 A 2 a 1 b 1 b 2 A 2 A 3 1 c 1 d 2 c A 3 A 4 c 3 d 4 d 5 A 4 A 1 3 b 4 a 4 b

69 An Idea From Gottlob-Lee-Valiant-Valiant, JACM 12 Q(A 1, A 2, A 3, A 4 ) R 12 R 23 R 34 R 41. A 1 A 2 A 3 A 4 a 1 d 4 1/4 b 1 c 3 1/4 b 1 d 4 1/4 b 2 c 3 1/4 A 2 R 12 R 23 A 1 A 3 R 41 A 4 R 34 A 1 A 2 a 1 b 1 b 2 A 2 A 3 1 c 1 d 2 c A 3 A 4 c 3 d 4 d 5 A 4 A 1 3 b 4 a 4 b

70 An Idea From Gottlob-Lee-Valiant-Valiant, JACM 12 Q(A 1, A 2, A 3, A 4 ) R 12 R 23 R 34 R 41. A 1 A 2 A 3 A 4 a 1 d 4 1/4 b 1 c 3 1/4 b 1 d 4 1/4 b 2 c 3 1/4 A 2 R 12 R 23 A 1 A 3 R 41 A 4 R 34 A 1 A 2 a 1 1/4 b 1 2/4 b 2 1/4 A 2 A 3 1 c 1/4 1 d 2/4 2 c 1/4 A 3 A 4 c 3 2/4 d 4 2/4 d 5 0 A 4 A 1 3 b 2/4 4 a 1/4 4 b 1/4

71 An Idea From Gottlob-Lee-Valiant-Valiant, JACM 12 Q(A 1, A 2, A 3, A 4 ) R 12 R 23 R 34 R 41. A 1 A 2 A 3 A 4 a 1 d 4 1/4 b 1 c 3 1/4 b 1 d 4 1/4 b 2 c 3 1/4 H unif (A 1 A 2 A 3 A 4 ) = log Q A 2 R 12 R 23 A 1 A 3 R 41 A 4 R 34 A 1 A 2 a 1 1/4 b 1 2/4 b 2 1/4 A 2 A 3 1 c 1/4 1 d 2/4 2 c 1/4 A 3 A 4 c 3 2/4 d 4 2/4 d 5 0 A 4 A 1 3 b 2/4 4 a 1/4 4 b 1/4

72 An Idea From Gottlob-Lee-Valiant-Valiant, JACM 12 Q(A 1, A 2, A 3, A 4 ) R 12 R 23 R 34 R 41. A 1 A 2 A 3 A 4 a 1 d 4 1/4 b 1 c 3 1/4 b 1 d 4 1/4 b 2 c 3 1/4 H unif (A 1 A 2 A 3 A 4 ) = log Q A 2 R 12 R 23 A 1 A 3 R 41 A 4 R 34 A 1 A 2 a 1 1/4 b 1 2/4 b 2 1/4 A 2 A 3 1 c 1/4 1 d 2/4 2 c 1/4 A 3 A 4 c 3 2/4 d 4 2/4 d 5 0 A 4 A 1 3 b 2/4 4 a 1/4 4 b 1/4 H unif (A 1 A 2 ) log R 12, H unif (A 2 A 3 ) log R 23, H unif (A 3 A 4 ) log R 34,...

73 An Idea From Gottlob-Lee-Valiant-Valiant, JACM 12 Q(A 1, A 2, A 3, A 4 ) R 12 R 23 R 34 R 41. A 1 A 2 A 3 A 4 a 1 d 4 1/4 b 1 c 3 1/4 b 1 d 4 1/4 b 2 c 3 1/4 H unif (A 1 A 2 A 3 A 4 ) = log Q A 2 R 12 R 23 A 1 A 3 R 41 A 4 R 34 A 1 A 2 a 1 1/4 b 1 2/4 b 2 1/4 A 2 A 3 1 c 1/4 1 d 2/4 2 c 1/4 A 3 A 4 c 3 2/4 d 4 2/4 d 5 0 A 4 A 1 3 b 2/4 4 a 1/4 4 b 1/4 H unif (A 1 A 2 ) log R 12, H unif (A 2 A 3 ) log R 23, H unif (A 3 A 4 ) log R 34,... H unif (A 2 A 1 = a ) log σa1 = a R 12, Hunif (A 2 A 1 = b ) log σa1 = b R 12,...

74 An Idea From Gottlob-Lee-Valiant-Valiant, JACM 12 Q(A 1, A 2, A 3, A 4 ) R 12 R 23 R 34 R 41. A 1 A 2 A 3 A 4 a 1 d 4 1/4 b 1 c 3 1/4 b 1 d 4 1/4 b 2 c 3 1/4 H unif (A 1 A 2 A 3 A 4 ) = log Q A 2 R 12 R 23 A 1 A 3 R 41 A 4 R 34 A 1 A 2 a 1 1/4 b 1 2/4 b 2 1/4 A 2 A 3 1 c 1/4 1 d 2/4 2 c 1/4 A 3 A 4 c 3 2/4 d 4 2/4 d 5 0 A 4 A 1 3 b 2/4 4 a 1/4 4 b 1/4 H unif (A 1 A 2 ) log R 12, H unif (A 2 A 3 ) log R 23, H unif (A 3 A 4 ) log R 34,... H unif (A 2 A 1 = a ) log σa1 = a R 12, Hunif (A 2 A 1 = b ) log σa1 = b R 12,... H unif (A 2 A 1 ) log max σa1 =xr 12 x }{{}

75 An Idea From Gottlob-Lee-Valiant-Valiant, JACM 12 Q(A 1, A 2, A 3, A 4 ) R 12 R 23 R 34 R 41. A 1 A 2 A 3 A 4 a 1 d 4 1/4 b 1 c 3 1/4 b 1 d 4 1/4 b 2 c 3 1/4 H unif (A 1 A 2 A 3 A 4 ) = log Q A 2 R 12 R 23 A 1 A 3 R 41 A 4 R 34 A 1 A 2 a 1 1/4 b 1 2/4 b 2 1/4 A 2 A 3 1 c 1/4 1 d 2/4 2 c 1/4 A 3 A 4 c 3 2/4 d 4 2/4 d 5 0 A 4 A 1 3 b 2/4 4 a 1/4 4 b 1/4 H unif (A 1 A 2 ) log R 12, H unif (A 2 A 3 ) log R 23, H unif (A 3 A 4 ) log R 34,... H unif (A 2 A 1 = a ) log σa1 = a R 12, Hunif (A 2 A 1 = b ) log σa1 = b R 12,... H unif (A 2 A 1 ) log max σa1 =xr 12 x }{{} deg R12 (A 2 A 1 )

76 Entropic Bound for Full Conjunctive Queries Q : T [n] F E R F, and degree constraints DC

77 Entropic Bound for Full Conjunctive Queries Q : T [n] F E R F, and degree constraints DC max log Q(D) sup h([n]) D =DC

78 Entropic Bound for Full Conjunctive Queries Q : T [n] F E R F, and degree constraints DC max log Q(D) sup h([n]) D =DC subject to (whatever H unif satisfies):

79 Entropic Bound for Full Conjunctive Queries Q : T [n] F E R F, and degree constraints DC max log Q(D) sup h([n]) D =DC subject to (whatever H unif satisfies): h is Entropic

80 Entropic Bound for Full Conjunctive Queries Q : T [n] F E R F, and degree constraints DC max log Q(D) sup h([n]) D =DC subject to (whatever H unif satisfies): h is Entropic There is some distribution on A [n] such that h(x) is the marginal entropy on A X, for all X

81 Entropic Bound for Full Conjunctive Queries Q : T [n] F E R F, and degree constraints DC max log Q(D) sup h([n]) D =DC subject to (whatever H unif satisfies): h is Entropic There is some distribution on A [n] such that h(x) is the marginal entropy on A X, for all X h satisfies DC

82 Entropic Bound for Full Conjunctive Queries Q : T [n] F E R F, and degree constraints DC max log Q(D) sup h([n]) D =DC subject to (whatever H unif satisfies): h is Entropic There is some distribution on A [n] such that h(x) is the marginal entropy on A X, for all X h satisfies DC h(y X) def = h(y ) h(x) log N Y X, X Y F E

83 Entropic Bound for Full Conjunctive Queries Q : T [n] F E R F, and degree constraints DC max log Q(D) sup h([n]) D =DC subject to (whatever H unif satisfies): h is Entropic There is some distribution on A [n] such that h(x) is the marginal entropy on A X, for all X h satisfies DC h(y X) def = h(y ) h(x) log N Y X, X Y F E Good Bound, but not computable!

84 Hierarchy of Set Functions h : 2 [n] R +, non-negative, monotone, h( ) = 0 h(x) h(y ) if X Y SA n := {h h is sub-additive} h(x Y ) h(x) + h(y ) Γ n := {h h is submodular} h(x Y ) + h(x Y ) h(x) + h(y ) Γ n: topological closure of Γ n Γ n = {h : h is entropic} M n : Modular h(x) = x x h(x)

85 Bounds for Full Conjunctive Query HDC def = { h h(y X) log N Y X, (X, Y, N Y X ) }

86 Bounds for Full Conjunctive Query HDC def = { h h(y X) log N Y X, (X, Y, N Y X ) } Then, max log Q(D) max h([n]) D =DC h Γ n HDC max h([n]) h Γ n HDC max h([n]) h SA n HDC entropic bound polymatroid bound sub-additive bound.

87 Size Bounds for Full Conjunctive Queries Bound Entropic Bound Polymatroid Bound Definition log Q max h Γ n HDC h([n]) log Q max h Γ n HDC h([n])

88 Size Bounds for Full Conjunctive Queries Bound Entropic Bound Polymatroid Bound Definition log Q max h Γ n HDC h([n]) log Q max h Γ n HDC h([n]) CC only AGM bound (Tight) [Atserias et al. FOCS 08] AGM bound (Tight) [Atserias et al. FOCS 08]

89 Size Bounds for Full Conjunctive Queries Bound Entropic Bound Polymatroid Bound Definition log Q max h Γ n HDC h([n]) log Q max h Γ n HDC h([n]) CC only CC + FD only AGM bound (Tight) AGM bound (Tight) [Atserias et al. FOCS 08] [Atserias et al. FOCS 08] Entropic Bound for FD Polymatroid Bound for FD [Gottlob et al. JACM 12] [Gottlob et al. JACM 12] (Tight [Gogacz et al. ICDT 17]) (Not tight [our work] )

90 Size Bounds for Full Conjunctive Queries Bound Entropic Bound Polymatroid Bound Definition log Q max h Γ n HDC h([n]) log Q max h Γ n HDC h([n]) CC only CC + FD only DC AGM bound (Tight) [Atserias et al. FOCS 08] Entropic Bound for FD [Gottlob et al. JACM 12] AGM bound (Tight) [Atserias et al. FOCS 08] Polymatroid Bound for FD [Gottlob et al. JACM 12] (Tight [Gogacz et al. ICDT 17]) (Not tight [our work] ) Entropic Bound for DC Polymatroid Bound for DC (Tight [our work] ) (Not tight [our work] )

91 Disjunctive Datalog: Size Bounds P : B B T B (A B ) F E R F (A F ) P (D) def = min T:T =P max B B T B Theorem ( our work ) max log P (D) max D =DC min h(b) h Γ B B n HDC }{{} Entropic bound max min h(b) h Γ n HDC B B }{{} Polymatroid bound Tight Not Tight Imply all known bounds for (Full) Conjunctive Queries!

92 Earlier Example P : B B T B (A B ) F E R F (A F ) P (D) def = min T:T =P max B B T B A 2 R 12 R 23 A 1 A 3 R 41 CC : R 12 N, R 23 N, R 34 N, R 41 N. P 123,234 : T 123 T 234 R 12 R 23 R 34 R 41. A 4 R 34

93 Earlier Example P : B B T B (A B ) F E R F (A F ) P (D) def = min T:T =P max B B T B A 2 R 12 R 23 A 1 A 3 R 41 CC : R 12 N, R 23 N, R 34 N, R 41 N. P 123,234 : T 123 T 234 R 12 R 23 R 34 R 41. A 4 R 34 max log P 123,234(D) max min{h(a 1A 2 A 3 ), h(a 2 A 3 A 4 )} D =CC h Γ n CC 1 max h Γ n CC max h Γ n CC 3 log N. 2 2 [h(a 1A 2 A 3 ), h(a 2 A 3 A 4 )] 1 2 [h(a 1A 2 ) + h(a 2 A 3 ) + h(a 3 A 4 )]

94 Table of Contents Connecting the Dots Output Size Bounds and Information Theory Shannon-flow Inequalities and the PANDA Algorithm Wrapping it up Appendix

95 Roadmap Given degree constraints DC and a disjunctive datalog rule P : B B T B F E R F Answer (Worst-case Output Size Bound) max log P (D) max D =DC min h Γ n HDC B B h(b) = polymatroid bound. Question (Algorithm) Compute a model for P within Õ(2polymatroid bound ) Question (Gathering fruits) Plug bound/algorithm into Meta Algorithm, what do we get?

96 Connection to Shannon-flow Inequalities Lemma (Linearize it) There exists non-negative λ = (λ B ) B B, with λ 1 = 1, s.t. max min h(b) = h Γ n HDC B B max h Γ n HDC B B λ B h(b) (1)

97 Connection to Shannon-flow Inequalities Lemma (Linearize it) There exists non-negative λ = (λ B ) B B, with λ 1 = 1, s.t. max min h(b) = h Γ n HDC B B max h Γ n HDC B B Lemma (Shannon-flow inequality) There exists δ 0 s.t. 2 polymatroid bound = λ B h(b) B B (X,Y,N Y X ) λ B h(b) (1) (X,Y,N Y X ) N δ Y X Y X, and δ Y X h(y X), h Γ n (2)

98 Connection to Shannon-flow Inequalities Lemma (Linearize it) There exists non-negative λ = (λ B ) B B, with λ 1 = 1, s.t. max min h(b) = h Γ n HDC B B max h Γ n HDC B B Lemma (Shannon-flow inequality) There exists δ 0 s.t. 2 polymatroid bound = λ B h(b) B B (X,Y,N Y X ) λ B h(b) (1) (X,Y,N Y X ) N δ Y X Y X, and δ Y X h(y X), h Γ n (2) (2) is a (vast) generalization of Shearer s lemma

99 PANDA (Proof-Assisted entropic Degree-Aware) What? Compute a model for our disjunctive datalog rule ( Run within Õ 2 polymatroid bound) = Õ (X,Y,N Y X ) N δ Y X Y X :

100 PANDA (Proof-Assisted entropic Degree-Aware) What? Compute a model for our disjunctive datalog rule ( Run within Õ 2 polymatroid bound) = Õ (X,Y,N Y X ) How? Proof as symbolic instructions Construct a Proof Sequence for the corresponding Shannon-flow inequality Proof steps relational operators. N δ Y X Y X :

101 Proof sequence Shannon-flow inequality: λ B h(b) B B h(y X) def = h(y ) h(x), X Y (X,Y,N Y X ) δ Y X h(y X) Proof sequence, convert RHS to LHS using following steps (In)equality Steps (X Y ) h(x) + h(y X) = h(y ) h(x) + h(y X) h(y ) h(y ) = h(x) + h(y X) h(y ) h(x) h(y ) h(x) + h(y X) h(y ) h(x) h(y X) h(y Z X Z) h(y X) h(y Z X Z)

102 Proof sequence Shannon-flow inequality: λ B h(b) B B h(y X) def = h(y ) h(x), X Y (X,Y,N Y X ) δ Y X h(y X) Proof sequence, convert RHS to LHS using following steps (In)equality Steps (X Y ) h(x) + h(y X) = h(y ) h(x) + h(y X) h(y ) h(y ) = h(x) + h(y X) h(y ) h(x) h(y ) h(x) + h(y X) h(y ) h(x) h(y X) h(y Z X Z) h(y X) h(y Z X Z) Theorem There is a proof sequence for every Shannon-flow inequality. The length is data-independent.

103 Proof Steps As Relational Operators Shannon-flow inequality: λ B h(b) δ Y X h(y X) B B (X,Y,N Y X ) Proof sequence, Steps (X Y ) h(x) + h(y X) h(y ) h(y ) h(x) + h(y X) h(y ) h(x) h(y X) h(y Z X Z) Relational Operator

104 Proof Steps As Relational Operators Shannon-flow inequality: λ B h(b) δ Y X h(y X) B B (X,Y,N Y X ) Proof sequence, Steps (X Y ) h(x) + h(y X) h(y ) h(y ) h(x) + h(y X) h(y ) h(x) h(y X) h(y Z X Z) Relational Operator (join)

105 Proof Steps As Relational Operators Shannon-flow inequality: λ B h(b) δ Y X h(y X) B B (X,Y,N Y X ) Proof sequence, Steps (X Y ) h(x) + h(y X) h(y ) h(y ) h(x) + h(y X) h(y ) h(x) h(y X) h(y Z X Z) Relational Operator (join) (data partition)

106 Proof Steps As Relational Operators Shannon-flow inequality: λ B h(b) δ Y X h(y X) B B (X,Y,N Y X ) Proof sequence, Steps (X Y ) h(x) + h(y X) h(y ) h(y ) h(x) + h(y X) h(y ) h(x) h(y X) h(y Z X Z) Relational Operator (join) (data partition) (projection)

107 Proof Steps As Relational Operators Shannon-flow inequality: λ B h(b) δ Y X h(y X) B B (X,Y,N Y X ) Proof sequence, Steps (X Y ) h(x) + h(y X) h(y ) h(y ) h(x) + h(y X) h(y ) h(x) h(y X) h(y Z X Z) Relational Operator (join) (data partition) (projection) (NOP)

108 Proof Steps As Relational Operators Shannon-flow inequality: λ B h(b) δ Y X h(y X) B B (X,Y,N Y X ) Proof sequence, Steps (X Y ) h(x) + h(y X) h(y ) h(y ) h(x) + h(y X) h(y ) h(x) h(y X) h(y Z X Z) Relational Operator (join) (data partition) (projection) (NOP) Theorem PANDA solves any disjunctive datalog rule P in time Õ(N + poly(log N) 2 polymatroid bound for P )

109 Example: P : T 123 T 234 R 12 R 23 R 34 R 41. R 12, R 23, R 34, R 41 N

110 Example: P : T 123 T 234 R 12 R 23 R 34 R 41. R 12, R 23, R 34, R 41 N P N 3/2

111 Example: P : T 123 T 234 R 12 R 23 R 34 R 41. R 12, R 23, R 34, R 41 N P N 3/2 log P min(h(a 1 A 2 A 3 ), h(a 2 A 3 A 4 )) (polymatroid bound) 1 2( h(a1 A 2 A 3 ) + h(a 2 A 3 A 4 ) ) (linearize) 1 2( h(a1 A 2 ) + h(a 2 A 3 ) + h(a 3 A 4 ) ) (Shannon-flow) 3 2 log N (Cardinality constraints)

112 Example: P : T 123 T 234 R 12 R 23 R 34 R 41. R 12, R 23, R 34, R 41 N P N 3/2 log P min(h(a 1 A 2 A 3 ), h(a 2 A 3 A 4 )) (polymatroid bound) 1 2( h(a1 A 2 A 3 ) + h(a 2 A 3 A 4 ) ) (linearize) 1 2( h(a1 A 2 ) + h(a 2 A 3 ) + h(a 3 A 4 ) ) (Shannon-flow) 3 2 log N (Cardinality constraints) Proof sequence h(a 1 A 2 ) + h(a 2 A 3 ) + h(a 3 A 4 ) Proof Step

113 Example: P : T 123 T 234 R 12 R 23 R 34 R 41. R 12, R 23, R 34, R 41 N P N 3/2 log P min(h(a 1 A 2 A 3 ), h(a 2 A 3 A 4 )) (polymatroid bound) 1 2( h(a1 A 2 A 3 ) + h(a 2 A 3 A 4 ) ) (linearize) 1 2( h(a1 A 2 ) + h(a 2 A 3 ) + h(a 3 A 4 ) ) (Shannon-flow) 3 2 log N (Cardinality constraints) Proof sequence h(a 1 A 2 ) + h(a 2 A 3 ) + h(a 3 A 4 ) Proof Step ( h(a3 A 4 ) h(a 4 A 3 ) + h(a 3 ) )

114 Example: P : T 123 T 234 R 12 R 23 R 34 R 41. R 12, R 23, R 34, R 41 N P N 3/2 log P min(h(a 1 A 2 A 3 ), h(a 2 A 3 A 4 )) (polymatroid bound) 1 2( h(a1 A 2 A 3 ) + h(a 2 A 3 A 4 ) ) (linearize) 1 2( h(a1 A 2 ) + h(a 2 A 3 ) + h(a 3 A 4 ) ) (Shannon-flow) 3 2 log N (Cardinality constraints) Proof sequence h(a 1 A 2 ) + h(a 2 A 3 ) + h(a 3 A 4 ) h(a 1 A 2 ) + h(a 2 A 3 ) + h(a 4 A 3 ) + h(a 3 ) Proof Step ( h(a3 A 4 ) h(a 4 A 3 ) + h(a 3 ) )

115 Example: P : T 123 T 234 R 12 R 23 R 34 R 41. R 12, R 23, R 34, R 41 N P N 3/2 log P min(h(a 1 A 2 A 3 ), h(a 2 A 3 A 4 )) (polymatroid bound) 1 2( h(a1 A 2 A 3 ) + h(a 2 A 3 A 4 ) ) (linearize) 1 2( h(a1 A 2 ) + h(a 2 A 3 ) + h(a 3 A 4 ) ) (Shannon-flow) 3 2 log N (Cardinality constraints) Proof sequence h(a 1 A 2 ) + h(a 2 A 3 ) + h(a 3 A 4 ) h(a 1 A 2 ) + h(a 2 A 3 ) + h(a 4 A 3 ) + h(a 3 ) Proof Step ( h(a3 A 4 ) h(a 4 A 3 ) + h(a 3 ) ) ( h(a4 A 3 ) h(a 4 A 2 A 3 ) )

116 Example: P : T 123 T 234 R 12 R 23 R 34 R 41. R 12, R 23, R 34, R 41 N P N 3/2 log P min(h(a 1 A 2 A 3 ), h(a 2 A 3 A 4 )) (polymatroid bound) 1 2( h(a1 A 2 A 3 ) + h(a 2 A 3 A 4 ) ) (linearize) 1 2( h(a1 A 2 ) + h(a 2 A 3 ) + h(a 3 A 4 ) ) (Shannon-flow) 3 2 log N (Cardinality constraints) Proof sequence h(a 1 A 2 ) + h(a 2 A 3 ) + h(a 3 A 4 ) h(a 1 A 2 ) + h(a 2 A 3 ) + h(a 4 A 3 ) + h(a 3 ) h(a 1 A 2 ) + h(a 2 A 3 ) + h(a 4 A 2 A 3 ) + h(a 3 ) Proof Step ( h(a3 A 4 ) h(a 4 A 3 ) + h(a 3 ) ) ( h(a4 A 3 ) h(a 4 A 2 A 3 ) )

117 Example: P : T 123 T 234 R 12 R 23 R 34 R 41. R 12, R 23, R 34, R 41 N P N 3/2 log P min(h(a 1 A 2 A 3 ), h(a 2 A 3 A 4 )) (polymatroid bound) 1 2( h(a1 A 2 A 3 ) + h(a 2 A 3 A 4 ) ) (linearize) 1 2( h(a1 A 2 ) + h(a 2 A 3 ) + h(a 3 A 4 ) ) (Shannon-flow) 3 2 log N (Cardinality constraints) Proof sequence h(a 1 A 2 ) + h(a 2 A 3 ) + h(a 3 A 4 ) h(a 1 A 2 ) + h(a 2 A 3 ) + h(a 4 A 3 ) + h(a 3 ) h(a 1 A 2 ) + h(a 2 A 3 ) + h(a 4 A 2 A 3 ) + h(a 3 ) Proof Step ( h(a3 A 4 ) h(a 4 A 3 ) + h(a 3 ) ) ( h(a4 A 3 ) h(a 4 A 2 A 3 ) ) ( h(a2 A 3 ) + h(a 4 A 2 A 3 ) h(a 2 A 3 A 4 ) )

118 Example: P : T 123 T 234 R 12 R 23 R 34 R 41. R 12, R 23, R 34, R 41 N P N 3/2 log P min(h(a 1 A 2 A 3 ), h(a 2 A 3 A 4 )) (polymatroid bound) 1 2( h(a1 A 2 A 3 ) + h(a 2 A 3 A 4 ) ) (linearize) 1 2( h(a1 A 2 ) + h(a 2 A 3 ) + h(a 3 A 4 ) ) (Shannon-flow) 3 2 log N (Cardinality constraints) Proof sequence h(a 1 A 2 ) + h(a 2 A 3 ) + h(a 3 A 4 ) h(a 1 A 2 ) + h(a 2 A 3 ) + h(a 4 A 3 ) + h(a 3 ) h(a 1 A 2 ) + h(a 2 A 3 ) + h(a 4 A 2 A 3 ) + h(a 3 ) h(a 1 A 2 ) + h(a 2 A 3 A 4 ) + h(a 3 ) Proof Step ( h(a3 A 4 ) h(a 4 A 3 ) + h(a 3 ) ) ( h(a4 A 3 ) h(a 4 A 2 A 3 ) ) ( h(a2 A 3 ) + h(a 4 A 2 A 3 ) h(a 2 A 3 A 4 ) )

119 Example: P : T 123 T 234 R 12 R 23 R 34 R 41. R 12, R 23, R 34, R 41 N P N 3/2 log P min(h(a 1 A 2 A 3 ), h(a 2 A 3 A 4 )) (polymatroid bound) 1 2( h(a1 A 2 A 3 ) + h(a 2 A 3 A 4 ) ) (linearize) 1 2( h(a1 A 2 ) + h(a 2 A 3 ) + h(a 3 A 4 ) ) (Shannon-flow) 3 2 log N (Cardinality constraints) Proof sequence h(a 1 A 2 ) + h(a 2 A 3 ) + h(a 3 A 4 ) h(a 1 A 2 ) + h(a 2 A 3 ) + h(a 4 A 3 ) + h(a 3 ) h(a 1 A 2 ) + h(a 2 A 3 ) + h(a 4 A 2 A 3 ) + h(a 3 ) h(a 1 A 2 ) + h(a 2 A 3 A 4 ) + h(a 3 ) Proof Step ( h(a3 A 4 ) h(a 4 A 3 ) + h(a 3 ) ) ( h(a4 A 3 ) h(a 4 A 2 A 3 ) ) ( h(a2 A 3 ) + h(a 4 A 2 A 3 ) h(a 2 A 3 A 4 ) ) ( h(a1 A 2 ) h(a 1 A 2 A 3 ) )

120 Example: P : T 123 T 234 R 12 R 23 R 34 R 41. R 12, R 23, R 34, R 41 N P N 3/2 log P min(h(a 1 A 2 A 3 ), h(a 2 A 3 A 4 )) (polymatroid bound) 1 2( h(a1 A 2 A 3 ) + h(a 2 A 3 A 4 ) ) (linearize) 1 2( h(a1 A 2 ) + h(a 2 A 3 ) + h(a 3 A 4 ) ) (Shannon-flow) 3 2 log N (Cardinality constraints) Proof sequence h(a 1 A 2 ) + h(a 2 A 3 ) + h(a 3 A 4 ) h(a 1 A 2 ) + h(a 2 A 3 ) + h(a 4 A 3 ) + h(a 3 ) h(a 1 A 2 ) + h(a 2 A 3 ) + h(a 4 A 2 A 3 ) + h(a 3 ) h(a 1 A 2 ) + h(a 2 A 3 A 4 ) + h(a 3 ) h(a 1 A 2 A 3 ) + h(a 2 A 3 A 4 ) + h(a 3 ) Proof Step ( h(a3 A 4 ) h(a 4 A 3 ) + h(a 3 ) ) ( h(a4 A 3 ) h(a 4 A 2 A 3 ) ) ( h(a2 A 3 ) + h(a 4 A 2 A 3 ) h(a 2 A 3 A 4 ) ) ( h(a1 A 2 ) h(a 1 A 2 A 3 ) )

121 Example: P : T 123 T 234 R 12 R 23 R 34 R 41. R 12, R 23, R 34, R 41 N P N 3/2 log P min(h(a 1 A 2 A 3 ), h(a 2 A 3 A 4 )) (polymatroid bound) 1 2( h(a1 A 2 A 3 ) + h(a 2 A 3 A 4 ) ) (linearize) 1 2( h(a1 A 2 ) + h(a 2 A 3 ) + h(a 3 A 4 ) ) (Shannon-flow) 3 2 log N (Cardinality constraints) Proof sequence h(a 1 A 2 ) + h(a 2 A 3 ) + h(a 3 A 4 ) h(a 1 A 2 ) + h(a 2 A 3 ) + h(a 4 A 3 ) + h(a 3 ) h(a 1 A 2 ) + h(a 2 A 3 ) + h(a 4 A 2 A 3 ) + h(a 3 ) h(a 1 A 2 ) + h(a 2 A 3 A 4 ) + h(a 3 ) h(a 1 A 2 A 3 ) + h(a 2 A 3 A 4 ) + h(a 3 ) Proof Step ( h(a3 A 4 ) h(a 4 A 3 ) + h(a 3 ) ) ( h(a4 A 3 ) h(a 4 A 2 A 3 ) ) ( h(a2 A 3 ) + h(a 4 A 2 A 3 ) h(a 2 A 3 A 4 ) ) ( h(a1 A 2 ) h(a 1 A 2 A 3 ) ) ( h(a1 A 2 A 3 ) + h(a 3 ) h(a 1 A 2 A 3 ) )

122 Example: P : T 123 T 234 R 12 R 23 R 34 R 41. R 12, R 23, R 34, R 41 N P N 3/2 log P min(h(a 1 A 2 A 3 ), h(a 2 A 3 A 4 )) (polymatroid bound) 1 2( h(a1 A 2 A 3 ) + h(a 2 A 3 A 4 ) ) (linearize) 1 2( h(a1 A 2 ) + h(a 2 A 3 ) + h(a 3 A 4 ) ) (Shannon-flow) 3 2 log N (Cardinality constraints) Proof sequence h(a 1 A 2 ) + h(a 2 A 3 ) + h(a 3 A 4 ) h(a 1 A 2 ) + h(a 2 A 3 ) + h(a 4 A 3 ) + h(a 3 ) h(a 1 A 2 ) + h(a 2 A 3 ) + h(a 4 A 2 A 3 ) + h(a 3 ) h(a 1 A 2 ) + h(a 2 A 3 A 4 ) + h(a 3 ) h(a 1 A 2 A 3 ) + h(a 2 A 3 A 4 ) + h(a 3 ) h(a 1 A 2 A 3 ) + h(a 2 A 3 A 4 ) Proof Step ( h(a3 A 4 ) h(a 4 A 3 ) + h(a 3 ) ) ( h(a4 A 3 ) h(a 4 A 2 A 3 ) ) ( h(a2 A 3 ) + h(a 4 A 2 A 3 ) h(a 2 A 3 A 4 ) ) ( h(a1 A 2 ) h(a 1 A 2 A 3 ) ) ( h(a1 A 2 A 3 ) + h(a 3 ) h(a 1 A 2 A 3 ) )

123 Example: P : T 123 T 234 R 12 R 23 R 34 R 41. R 12, R 23, R 34, R 41 N P N 3/2 h(a 3 A 4 ) h(a 4 A 3 ) + h(a 3 ) h(a 4 A 3 ) h(a 4 A 2 A 3 ) h(a 2 A 3 ) + h(a 4 A 2 A 3 ) h(a 2 A 3 A 4 ) h(a 1 A 2 ) h(a 1 A 2 A 3 ) h(a 1 A 2 A 3 ) + h(a 3 ) h(a 1 A 2 A 3 )

124 Example: P : T 123 T 234 R 12 R 23 R 34 R 41. R 12, R 23, R 34, R 41 N P N 3/2 h(a 3 A 4 ) h(a 4 A 3 ) + h(a 3 ) R 34 (A 3, A 4 ) R (l) 34 (A 3, A 4 ), R (h) 3 (A 3) h(a 4 A 3 ) h(a 4 A 2 A 3 ) h(a 2 A 3 ) + h(a 4 A 2 A 3 ) h(a 2 A 3 A 4 ) h(a 1 A 2 ) h(a 1 A 2 A 3 ) h(a 1 A 2 A 3 ) + h(a 3 ) h(a 1 A 2 A 3 )

125 Example: P : T 123 T 234 R 12 R 23 R 34 R 41. R 12, R 23, R 34, R 41 N P N 3/2 h(a 3 A 4 ) h(a 4 A 3 ) + h(a 3 ) h(a 4 A 3 ) h(a 4 A 2 A 3 ) R 34 (A 3, A 4 ) R (l) 34 (A 3, A 4 ), R (h) 3 (A 3) R (l) 34 (A 3, A 4 ) R (l) 34 (A 3, A 4 ) h(a 2 A 3 ) + h(a 4 A 2 A 3 ) h(a 2 A 3 A 4 ) h(a 1 A 2 ) h(a 1 A 2 A 3 ) h(a 1 A 2 A 3 ) + h(a 3 ) h(a 1 A 2 A 3 )

126 Example: P : T 123 T 234 R 12 R 23 R 34 R 41. R 12, R 23, R 34, R 41 N P N 3/2 h(a 3 A 4 ) h(a 4 A 3 ) + h(a 3 ) h(a 4 A 3 ) h(a 4 A 2 A 3 ) h(a 2 A 3 ) + h(a 4 A 2 A 3 ) h(a 2 A 3 A 4 ) R 34 (A 3, A 4 ) R (l) 34 (A 3, A 4 ), R (h) 3 (A 3) R (l) 34 (A 3, A 4 ) R (l) 34 (A 3, A 4 ) R 23 (A 2, A 3 ) R (l) 34 (A 3, A 4 ) T 234 (A 2, A 3, A 4 ) h(a 1 A 2 ) h(a 1 A 2 A 3 ) h(a 1 A 2 A 3 ) + h(a 3 ) h(a 1 A 2 A 3 )

127 Example: P : T 123 T 234 R 12 R 23 R 34 R 41. R 12, R 23, R 34, R 41 N P N 3/2 h(a 3 A 4 ) h(a 4 A 3 ) + h(a 3 ) h(a 4 A 3 ) h(a 4 A 2 A 3 ) h(a 2 A 3 ) + h(a 4 A 2 A 3 ) h(a 2 A 3 A 4 ) h(a 1 A 2 ) h(a 1 A 2 A 3 ) R 34 (A 3, A 4 ) R (l) 34 (A 3, A 4 ), R (h) 3 (A 3) R (l) 34 (A 3, A 4 ) R (l) 34 (A 3, A 4 ) R 23 (A 2, A 3 ) R (l) 34 (A 3, A 4 ) T 234 (A 2, A 3, A 4 ) R 12 (A 1, A 2 ) R 12 (A 1, A 2 ) h(a 1 A 2 A 3 ) + h(a 3 ) h(a 1 A 2 A 3 )

128 Example: P : T 123 T 234 R 12 R 23 R 34 R 41. R 12, R 23, R 34, R 41 N P N 3/2 h(a 3 A 4 ) h(a 4 A 3 ) + h(a 3 ) h(a 4 A 3 ) h(a 4 A 2 A 3 ) h(a 2 A 3 ) + h(a 4 A 2 A 3 ) h(a 2 A 3 A 4 ) h(a 1 A 2 ) h(a 1 A 2 A 3 ) h(a 1 A 2 A 3 ) + h(a 3 ) h(a 1 A 2 A 3 ) R 34 (A 3, A 4 ) R (l) 34 (A 3, A 4 ), R (h) 3 (A 3) R (l) 34 (A 3, A 4 ) R (l) 34 (A 3, A 4 ) R 23 (A 2, A 3 ) R (l) 34 (A 3, A 4 ) T 234 (A 2, A 3, A 4 ) R 12 (A 1, A 2 ) R 12 (A 1, A 2 ) R 12 (A 1, A 2 ) R (h) 3 (A 3) T 123 (A 1, A 2, A 3 )

129 Table of Contents Connecting the Dots Output Size Bounds and Information Theory Shannon-flow Inequalities and the PANDA Algorithm Wrapping it up Appendix

130 Roadmap Given degree constraints DC and a disjunctive datalog rule P : B B T B F E R F Answer (Worst-case Output Size Bound) Polymatroid bound max log P (D) max min h(b) D =DC h Γ n HDC B B Answer (Algorithm) PANDA computes a model for P within Õ(2polymatroid bound ) Question (Gathering fruits) Plug bound/algorithm into Meta Algorithm, what do we get?

131 Option 1: Full Conjunctive Query Full conjunctive query Database D Q i Q i (D) Q o Output F E R F itime T [n] otime Answer itime = max Q i(d) max D =DC h Γ 2h([n]) = PANDA s runtime n HDC PANDA is worst-case optimal whenever the polymatroid bound is tight!

132 Option 2: A Single Tree Decomposition Let (T, χ) be a tree decomposition of H to be chosen later Database D F E R F Multiple Conjunctive Rules itime Q i Q i (D) v V (T ) T χ(v) itime = max Q i(d) max max P v(d) D =DC D =DC v V (T ) Q o otime 2 max v V (T ) max h Γn DC h(χ(v)) = PANDA s runtime Pick the best (T, χ) before running PANDA: min max max (T,χ) v V (T ) h Γ n CC h(χ(v)) log N fhtw(q) PANDA evaluates Q within Õ(N fhtw(q) )-time. Output Answer

133 Option 3: Multiple Tree Decompositions Database D Multiple Disjunctive Datalog Rules! itime R F T B F E B B B Q i Q i (D) Q o otime Output Answer log itime = max D =DC log Q i(d) max max B = max max D =DC B max min h(b) = h Γ n HDC B B max min h Γ n HDC (T,χ) v V (T ) max max h Γ n HDC min (T,χ) v V (T ) log P B(D) max max min h(b) h Γ n HDC B B B h(χ(v)) = log(panda s runtime) h(χ(v)) log N subw(q) PANDA evaluates Q within Õ(N subw(q) )-time.

134 Quantities of Interests X {Γ n, Γ n, SA n } Y {HDC, HFD, HCC, log N ED, log N VD} Define LogSizeBound X Y (P ) def = max h X Y min B B h(b) Minimaxwidth X Y (Q) def = min max (T,χ) v V (T ) max h(χ(v)), h X Y Maximinwidth X Y (Q) def = max min max h(χ(v)). h X Y (T,χ) v V (T )

135 Summary of Bounds Z SAn Γn HDC HCC ED log N VD log N Γ n DAEB(Q) log ρ 2 AGM(Q) (Q) log 2 N log 2 VB(Q) DAPB(Q) log ρ 2 AGM(Q) (Q) log 2 N log 2 VB(Q) ρ(q, (NF )F E) ρ(q) log 2 N log 2 VB(Q) eda-fhtw(q) LogSizeBound X Y (Q) da-fhtw(q) eda-subw(q) fhtw(q) tw(q) + 1 fhtw(q) tw(q) + 1 ghtw(q) tw(q) + 1 Minimaxwidth X Y (Q) X da-subw(q) ghtw(q) subw(q) tw(q) + 1 tw(q) + 1 tw(q) + 1 Maximinwidth X Y (Q) Y

136 Many Open Questions Is the entropic bound computable under CC HDC or DC?

137 Many Open Questions Is the entropic bound computable under CC HDC or DC? Worst-case optimal algorithm for full conjunctive queries under CC HDC or DC

138 Many Open Questions Is the entropic bound computable under CC HDC or DC? Worst-case optimal algorithm for full conjunctive queries under CC HDC or DC Worst-case optimal algorithm for disjunctive datalog rules under CC HDC or DC

139 Many Open Questions Is the entropic bound computable under CC HDC or DC? Worst-case optimal algorithm for full conjunctive queries under CC HDC or DC Worst-case optimal algorithm for disjunctive datalog rules under CC HDC or DC Remove the annoying poly-log factor from PANDA

140 Many Open Questions Is the entropic bound computable under CC HDC or DC? Worst-case optimal algorithm for full conjunctive queries under CC HDC or DC Worst-case optimal algorithm for disjunctive datalog rules under CC HDC or DC Remove the annoying poly-log factor from PANDA Other choices for the propositional formula in the Meta Algorithm, perhaps tradding off itime and?

141 Many Open Questions Is the entropic bound computable under CC HDC or DC? Worst-case optimal algorithm for full conjunctive queries under CC HDC or DC Worst-case optimal algorithm for disjunctive datalog rules under CC HDC or DC Remove the annoying poly-log factor from PANDA Other choices for the propositional formula in the Meta Algorithm, perhaps tradding off itime and? What about negations?

142 Many Thanks! Questions?

143 Table of Contents Connecting the Dots Output Size Bounds and Information Theory Shannon-flow Inequalities and the PANDA Algorithm Wrapping it up Appendix

144 Semantics of Our Disjunctive Datalog Rule P : T B (A B ) R F (A F ) B B F E

145 Semantics of Our Disjunctive Datalog Rule P : T B (A B ) R F (A F ) B B F E A 2 R 12 R 23 A 1 A 3 R 41 P 123,234 : T 123 T 234 R 12 R 23 R 34 R 41. A 4 R 34

146 Semantics of Our Disjunctive Datalog Rule P : T B (A B ) R F (A F ) B B F E A 2 R 12 R 23 A 1 A 3 R 41 P 123,234 : T 123 T 234 R 12 R 23 R 34 R 41. A 4 R 34

147 Semantics of Our Disjunctive Datalog Rule P : T B (A B ) R F (A F ) B B F E A 2 R 12 R 23 A 1 A 3 R 41 P 123,234 : T 123 T 234 R 12 R 23 R 34 R 41. A 4 R 34

148 Semantics of Our Disjunctive Datalog Rule P : T B (A B ) R F (A F ) B B F E A 2 R 12 R 23 A 1 A 3 R 41 P 123,234 : T 123 T 234 R 12 R 23 R 34 R 41. A 4 R 34 A 1 A 2 a 1 b 1 b 2 A 2 A 3 1 c 1 d 2 c A 3 A 4 c 3 d 4 d 5 A 4 A 1 3 b 4 a 4 b

149 Semantics of Our Disjunctive Datalog Rule P : T B (A B ) R F (A F ) B B F E A 2 R 12 R 23 A 1 A 3 R 41 P 123,234 : T 123 T 234 R 12 R 23 R 34 R 41. A 4 R 34 A 1 A 2 a 1 b 1 b 2 A 2 A 3 1 c 1 d 2 c A 3 A 4 c 3 d 4 d 5 A 4 A 1 3 b 4 a 4 b A 1 A 2 A 3 A 4 a 1 d 4 b 1 c 3 b 1 d 4 b 2 c 3

150 Semantics of Our Disjunctive Datalog Rule P : T B (A B ) R F (A F ) B B F E A 2 R 12 R 23 A 1 A 3 R 41 P 123,234 : T 123 T 234 R 12 R 23 R 34 R 41. A 4 R 34 A 1 A 2 a 1 b 1 b 2 A 2 A 3 1 c 1 d 2 c A 3 A 4 c 3 d 4 d 5 A 4 A 1 3 b 4 a 4 b A 1 A 2 A 3 A 4 a 1 d 4 b 1 c 3 b 1 d 4 b 2 c 3 A 1 A 2 A 3 b 1 c b 2 c A 2 A 3 A 4 1 d 4 1 c 3 2 d 4

151 Semantics of Our Disjunctive Datalog Rule P : T B (A B ) R F (A F ) B B F E A 2 R 12 R 23 A 1 A 3 R 41 P 123,234 : T 123 T 234 R 12 R 23 R 34 R 41. A 4 R 34 A 1 A 2 a 1 b 1 b 2 A 2 A 3 1 c 1 d 2 c A 3 A 4 c 3 d 4 d 5 A 4 A 1 3 b 4 a 4 b A 1 A 2 A 3 A 4 a 1 d 4 b 1 c 3 b 1 d 4 b 2 c 3 A 1 A 2 A 3 b 1 c b 2 c A 2 A 3 A 4 1 d 4 1 c 3 2 d 4

152 Semantics of Our Disjunctive Datalog Rule P : T B (A B ) R F (A F ) B B F E A 2 R 12 R 23 A 1 A 3 R 41 P 123,234 : T 123 T 234 R 12 R 23 R 34 R 41. A 4 R 34 A 1 A 2 a 1 b 1 b 2 A 2 A 3 1 c 1 d 2 c A 3 A 4 c 3 d 4 d 5 A 4 A 1 3 b 4 a 4 b A 1 A 2 A 3 A 4 a 1 d 4 b 1 c 3 b 1 d 4 b 2 c 3 A 1 A 2 A 3 b 1 c b 2 c Inclusive Disjunction A 2 A 3 A 4 1 d 4 1 c 3 2 d 4

153 Semantics of Our Disjunctive Datalog Rule P : T B (A B ) R F (A F ) B B F E A 2 R 12 R 23 A 1 A 3 R 41 P 123,234 : T 123 T 234 R 12 R 23 R 34 R 41. A 4 R 34 A 1 A 2 a 1 b 1 b 2 A 2 A 3 1 c 1 d 2 c A 3 A 4 c 3 d 4 d 5 A 4 A 1 3 b 4 a 4 b A 1 A 2 A 3 A 4 a 1 d 4 b 1 c 3 b 1 d 4 b 2 c 3 A 1 A 2 A 3 b 1 c b 2 c A 2 A 3 A 4 1 d 4 1 c 3 2 d 4 No minimal model requirement

154 Semantics of Our Disjunctive Datalog Rule P : T B (A B ) R F (A F ) B B F E A 2 R 12 R 23 A 1 A 3 R 41 P 123,234 : T 123 T 234 R 12 R 23 R 34 R 41. A 4 R 34 A 1 A 2 a 1 b 1 b 2 A 2 A 3 1 c 1 d 2 c A 3 A 4 c 3 d 4 d 5 A 4 A 1 3 b 4 a 4 b A 1 A 2 A 3 A 4 a 1 d 4 b 1 c 3 b 1 d 4 b 2 c 3 A 1 A 2 A 3 b 1 c b 2 c A 2 A 3 A 4 1 d 4 1 c 3 2 d 4 Model size is max( T 123, T 234 ) = 3

155 Semantics of Our Disjunctive Datalog Rule P : T B (A B ) R F (A F ) B B F E A 2 R 12 R 23 A 1 A 3 R 41 P 123,234 : T 123 T 234 R 12 R 23 R 34 R 41. A 4 R 34 A 1 A 2 a 1 b 1 b 2 A 2 A 3 1 c 1 d 2 c A 3 A 4 c 3 d 4 d 5 A 4 A 1 3 b 4 a 4 b A 1 A 2 A 3 A 4 a 1 d 4 b 1 c 3 b 1 d 4 b 2 c 3 A 1 A 2 A 3 b 1 c b 2 c A 2 A 3 A 4 1 d 4 1 c 3 2 d 4 Output size is the minimum over all models

156 Semantics of Our Disjunctive Datalog Rule P : T B (A B ) R F (A F ) B B F E A 2 R 12 R 23 A 1 A 3 R 41 P 123,234 : T 123 T 234 R 12 R 23 R 34 R 41. A 4 R 34 A 1 A 2 a 1 b 1 b 2 A 2 A 3 1 c 1 d 2 c A 3 A 4 c 3 d 4 d 5 A 4 A 1 3 b 4 a 4 b A 1 A 2 A 3 A 4 a 1 d 4 b 1 c 3 b 1 d 4 b 2 c 3 A 1 A 2 A 3 b 1 c b 2 c A 2 A 3 A 4 1 d 4 A minimum-sized model of size 2

157 Semantics of Our Disjunctive Datalog Rule P : T B (A B ) R F (A F ) B B F E A 2 R 12 R 23 A 1 A 3 R 41 P 123,234 : T 123 T 234 R 12 R 23 R 34 R 41. A 4 R 34 A 1 A 2 a 1 b 1 b 2 A 2 A 3 1 c 1 d 2 c A 3 A 4 c 3 d 4 d 5 A 4 A 1 3 b 4 a 4 b A 1 A 2 A 3 A 4 a 1 d 4 b 1 c 3 b 1 d 4 b 2 c 3 A 1 A 2 A 3 b 1 c b 2 c A 2 A 3 A 4 1 d 4 A minimum-sized model of size 2 Hence, the output size is 2

158 Output Size of a Disjunctive Datalog Rule P : B B T B (A B ) F E R F (A F ) P (D) def = min T:T =P max B B T B

159 Output Size of a Disjunctive Datalog Rule P : B B T B (A B ) F E R F (A F ) P (D) def = min T:T =P max B B T B A 2 R 12 R 23 A 1 A 3 R 41 A 4 R 34

160 Output Size of a Disjunctive Datalog Rule P : B B T B (A B ) F E R F (A F ) P (D) def = min T:T =P max B B T B A 2 R 12 R 23 A 1 A 3 R 41 CC : R 12 N, R 23 N, R 34 N, R 41 N. A 4 R 34

161 Output Size of a Disjunctive Datalog Rule P : B B T B (A B ) F E R F (A F ) P (D) def = min T:T =P max B B T B A 2 R 12 R 23 A 1 A 3 R 41 CC : R 12 N, R 23 N, R 34 N, R 41 N. A 4 R 34 P 123,234 : T 123 T 234 R 12 R 23 R 34 R 41.

162 Output Size of a Disjunctive Datalog Rule P : B B T B (A B ) F E R F (A F ) P (D) def = min T:T =P max B B T B A 2 R 12 R 23 A 1 A 3 R 41 CC : R 12 N, R 23 N, R 34 N, R 41 N. A 4 R 34 P 123,234 : T 123 T 234 R 12 R 23 R 34 R 41. max P 123,234(D) N 3/2 D =CC

163 Output Size of a Disjunctive Datalog Rule P : B B T B (A B ) F E R F (A F ) P (D) def = min T:T =P max B B T B A 2 R 12 R 23 A 1 A 3 R 41 CC : R 12 N, R 23 N, R 34 N, R 41 N. A 4 R 34 P 123,234 : T 123 T 234 R 12 R 23 R 34 R 41. max P 123,234(D) N 3/2 D =CC P : T 123 R 12 R 23 R 34 R 41.

164 Output Size of a Disjunctive Datalog Rule P : B B T B (A B ) F E R F (A F ) P (D) def = min T:T =P max B B T B A 2 R 12 R 23 A 1 A 3 R 41 CC : R 12 N, R 23 N, R 34 N, R 41 N. A 4 R 34 P 123,234 : T 123 T 234 R 12 R 23 R 34 R 41. max P 123,234(D) N 3/2 D =CC P : T 123 R 12 R 23 R 34 R 41. P (D) = N 2, for some D

165 Output Size of a Disjunctive Datalog Rule P : B B T B (A B ) F E R F (A F ) P (D) def = min T:T =P max B B T B A 2 R 12 R 23 A 1 A 3 R 41 CC : R 12 N, R 23 N, R 34 N, R 41 N. A 4 R 34 P 123,234 : T 123 T 234 R 12 R 23 R 34 R 41. max P 123,234(D) N 3/2 D =CC P : T 123 R 12 R 23 R 34 R 41. P (D) = N 2, for some D Using Option 3, 4-cycle query answerable in Õ(N 3/2 )-time, matching [Alon, Yuster, Zwick 97]

166 Polymatroid Bound: Examples Q(A 1, A 2, A 3, A 4 ) R 12 (A 1, A 2 ), R 23 (A 2, A 3 ), R 34 (A 3, A 4 ), R 41 (A 4, A 1 ). A 2 R 12 R 23 A 1 A 3 R 41 A 4 R 34 R 12, R 23, R 34, R 41 N Q N 2

167 Polymatroid Bound: Examples Q(A 1, A 2, A 3, A 4 ) R 12 (A 1, A 2 ), R 23 (A 2, A 3 ), R 34 (A 3, A 4 ), R 41 (A 4, A 1 ). A 2 R 12 R 23 A 1 A 3 R 41 A 4 R 34 R 12, R 23, R 34, R 41 N Q N 2 log Q = h(a 1 A 2 A 3 A 4 ) h(a 1 A 2 ) + h(a 3 A 4 ) 2 log N

168 Polymatroid Bound: Examples Q(A 1, A 2, A 3, A 4 ) R 12 (A 1, A 2 ), R 23 (A 2, A 3 ), R 34 (A 3, A 4 ), R 41 (A 4, A 1 ). A 2 R 12 R 23 A 1 A 3 R 41 A 4 R 34 R 12, R 23, R 34, R 41 N Q N 2 log Q = h(a 1 A 2 A 3 A 4 ) h(a 1 A 2 ) + h(a 3 A 4 ) 2 log N deg 12 (A 1 A 2 A 1 ), deg 12 (A 1 A 2 A 2 ) D Q D N 3/2

169 Polymatroid Bound: Examples Q(A 1, A 2, A 3, A 4 ) R 12 (A 1, A 2 ), R 23 (A 2, A 3 ), R 34 (A 3, A 4 ), R 41 (A 4, A 1 ). A 2 R 12 R 23 A 1 A 3 R 41 A 4 R 34 R 12, R 23, R 34, R 41 N Q N 2 log Q = h(a 1 A 2 A 3 A 4 ) h(a 1 A 2 ) + h(a 3 A 4 ) 2 log N deg 12 (A 1 A 2 A 1 ), deg 12 (A 1 A 2 A 2 ) D Q D N 3/2 2 log Q = 2h(A 1 A 2 A 3 A 4 ) h(a 2 A 3 ) + h(a 3 A 4 ) + h(a 4 A 1 ) + h(a 2 A 1 ) + h(a 1 A 2 ) 3 log N + 2 log D

170 Polymatroid Bound: Examples Q(A 1, A 2, A 3, A 4 ) R 12 (A 1, A 2 ), R 23 (A 2, A 3 ), R 34 (A 3, A 4 ), R 41 (A 4, A 1 ). A 2 R 12 R 23 A 1 A 3 R 41 A 4 R 34 R 12, R 23, R 34, R 41 N Q N 2 log Q = h(a 1 A 2 A 3 A 4 ) h(a 1 A 2 ) + h(a 3 A 4 ) 2 log N deg 12 (A 1 A 2 A 1 ), deg 12 (A 1 A 2 A 2 ) D Q D N 3/2 2 log Q = 2h(A 1 A 2 A 3 A 4 ) h(a 2 A 3 ) + h(a 3 A 4 ) + h(a 4 A 1 ) + h(a 2 A 1 ) + h(a 1 A 2 ) 3 log N + 2 log D A 1 A 2, A 2 A 1 Q N 3/2

171 Polymatroid Bound is not tight!

172 Polymatroid Bound is not tight! Proof Strategy

173 Polymatroid Bound is not tight! Proof Strategy Take a non-shannon inequality (modified from [Zhang-Yeung 98]) 11h(ABXY C) 3h(XY ) + 3h(AX) + 3h(AY ) + h(bx) + h(by ) + 5h(C) + h(abxy C AB) + h(abxy C AC) + 4h(ABXY C AXY ) + h(abxy C BXY ) + 2h(ABXY C XC) + 2h(ABXY C Y C) (3)

174 Polymatroid Bound is not tight! Proof Strategy Take a non-shannon inequality (modified from [Zhang-Yeung 98]) 11h(ABXY C) 3h(XY ) + 3h(AX) + 3h(AY ) + h(bx) + h(by ) + 5h(C) + h(abxy C AB) + h(abxy C AC) + 4h(ABXY C AXY ) + h(abxy C BXY ) + 2h(ABXY C XC) + 2h(ABXY C Y C) (3) Construct a query Q where (3) gives the (entropic) bound 11 log Q = 11h(ABXY C) 11 log N log N 2 = 43 log N