1. Optimal Query Processing Meets Information Theory
Dan Suciu – University of Washington
Joint w/ Mahmoud Abo-Khamis and Hung Ngo
RelationalAI Inc

2. Basic Question
What is the optimal runtime to compute a query Q on a database D?
Data complexity: Q is fixed, runtime = f(D)
Applications: CSP where the nodes of Q are the variables of the CSP problem, its hyperedges are the constraints, and D represents the valid assignments
Databases: …

3. Example select * -- natural join from R, S, T
Q(X,Y,Z) = R(X,Y) ∧ S(Y,Z) ∧ T(Z,X)
Will prove later: |Q(D)| ≤ N3/2
select * natural join
from R, S, T
where R.Y = S.Y and S.Z = T.Z and T.X = R.X

4. Example select * -- natural join from R, S, T
Q(X,Y,Z) = R(X,Y) ∧ S(Y,Z) ∧ T(Z,X)
Will prove later: |Q(D)| ≤ N3/2
select * natural join
from R, S, T
where R.Y = S.Y and S.Z = T.Z and T.X = R.X
Query plan ⨝ ⨝
T(Z,X)
R(X,Y)
S(Y,Z)

5. Example select * -- natural join from R, S, T
Q(X,Y,Z) = R(X,Y) ∧ S(Y,Z) ∧ T(Z,X)
Will prove later: |Q(D)| ≤ N3/2
select * natural join
from R, S, T
where R.Y = S.Y and S.Z = T.Z and T.X = R.X R: X Y 1 2 3
...
N/2
Query plan ⨝ ⨝
T(Z,X) N
R(X,Y)
S(Y,Z)

6. Example select * -- natural join from R, S, T
Q(X,Y,Z) = R(X,Y) ∧ S(Y,Z) ∧ T(Z,X)
Will prove later: |Q(D)| ≤ N3/2
select * natural join
from R, S, T
where R.Y = S.Y and S.Z = T.Z and T.X = R.X R: X Y 1 2 3
...
N/2
S: (same as R) Y Z 1 2 3
...
N/2
T: (same as R) Z X 1 2 3
...
N/2
Query plan ⨝ ⨝
T(Z,X) N
R(X,Y)
S(Y,Z)

7. Example select * -- natural join from R, S, T
Q(X,Y,Z) = R(X,Y) ∧ S(Y,Z) ∧ T(Z,X)
Will prove later: |Q(D)| ≤ N3/2
select * natural join
from R, S, T
where R.Y = S.Y and S.Z = T.Z and T.X = R.X R: X Y 1 2 3
...
N/2
S: (same as R) Y Z 1 2 3
...
N/2
T: (same as R) Z X 1 2 3
...
N/2
Query plan ⨝
Θ(N2) tuples ⨝
T(Z,X) N
R(X,Y)
S(Y,Z)

8. Example select * -- natural join from R, S, T
Q(X,Y,Z) = R(X,Y) ∧ S(Y,Z) ∧ T(Z,X)
Will prove later: |Q(D)| ≤ N3/2
select * natural join
from R, S, T
where R.Y = S.Y and S.Z = T.Z and T.X = R.X
0 tuples R: X Y 1 2 3
...
N/2
S: (same as R) Y Z 1 2 3
...
N/2
T: (same as R) Z X 1 2 3
...
N/2
Query plan ⨝
Θ(N2) tuples ⨝
T(Z,X) N
R(X,Y)
S(Y,Z)

9. Example select * -- natural join from R, S, T
Q(X,Y,Z) = R(X,Y) ∧ S(Y,Z) ∧ T(Z,X)
Will prove later: |Q(D)| ≤ N3/2
select * natural join
from R, S, T
where R.Y = S.Y and S.Z = T.Z and T.X = R.X
0 tuples R: X Y 1 2 3
...
N/2
S: (same as R) Y Z 1 2 3
...
N/2
T: (same as R) Z X 1 2 3
...
N/2
Query plan ⨝
Θ(N2) tuples ⨝
T(Z,X) N
R(X,Y)
S(Y,Z)
Every plan costs Θ(N2)!

10. Outline
Motivation
Full CQ
Boolean CQ
Conclusions

11. Maximum Output Size maxD satisfies stats (|Q(D)|)
E.g. R(X,Y) ∧ S(Y,Z), |R|, |S| ≤ N
No other info: |Q(D)| ≤ N2
S.Y is a key: |Q(D)| ≤ N
S.Y has degree ≤ d: |Q(D)| ≤ d×N
E.g. R(X,Y) ∧ S(Y,Z) ∧ T(Z,X) No other info: |Q(D)| ≤ N3/2

12. Maximum Output Size maxD satisfies stats (|Q(D)|)
E.g. R(X,Y) ∧ S(Y,Z), |R|, |S| ≤ N
No other info: |Q(D)| ≤ N2
S.Y is a key: |Q(D)| ≤ N
S.Y has degree ≤ d: |Q(D)| ≤ d×N
E.g. R(X,Y) ∧ S(Y,Z) ∧ T(Z,X) No other info: |Q(D)| ≤ N3/2

13. Maximum Output Size maxD satisfies stats (|Q(D)|)
E.g. R(X,Y) ∧ S(Y,Z), |R|, |S| ≤ N
No other info: |Q(D)| ≤ N2
S.Y is a key: |Q(D)| ≤ N
S.Y has degree ≤ d: |Q(D)| ≤ d×N
E.g. R(X,Y) ∧ S(Y,Z) ∧ T(Z,X) No other info: |Q(D)| ≤ N3/2

14. Maximum Output Size maxD satisfies stats (|Q(D)|)
E.g. R(X,Y) ∧ S(Y,Z), |R|, |S| ≤ N
No other info: |Q(D)| ≤ N2
S.Y is a key: |Q(D)| ≤ N
S.Y has degree ≤ d: |Q(D)| ≤ d×N
E.g. R(X,Y) ∧ S(Y,Z) ∧ T(Z,X) No other info: |Q(D)| ≤ N3/2

15. Maximum Output Size maxD satisfies stats (|Q(D)|)
E.g. R(X,Y) ∧ S(Y,Z), |R|, |S| ≤ N
No other info: |Q(D)| ≤ N2
S.Y is a key: |Q(D)| ≤ N
S.Y has degree ≤ d: |Q(D)| ≤ d×N
E.g. R(X,Y) ∧ S(Y,Z) ∧ T(Z,X) No other info: |Q(D)| ≤ N3/2

16. Maximum Output Size maxD satisfies stats (|Q(D)|)
E.g. R(X,Y) ∧ S(Y,Z), |R|, |S| ≤ N
No other info: |Q(D)| ≤ N2
S.Y is a key: |Q(D)| ≤ N
S.Y has degree ≤ d: |Q(D)| ≤ d×N
E.g. R(X,Y) ∧ S(Y,Z) ∧ T(Z,X) No other info: |Q(D)| ≤ N3/2

17. Maximum Output Size maxD satisfies stats (|Q(D)|)
E.g. R(X,Y) ∧ S(Y,Z), |R|, |S| ≤ N
No other info: |Q(D)| ≤ N2
S.Y is a key: |Q(D)| ≤ N
S.Y has degree ≤ d: |Q(D)| ≤ d×N
E.g. R(X,Y) ∧ S(Y,Z) ∧ T(Z,X) No other info: |Q(D)| ≤ N3/2

18. Maximum Output Size maxD satisfies stats (|Q(D)|)
E.g. R(X,Y) ∧ S(Y,Z), |R|, |S| ≤ N
No other info: |Q(D)| ≤ N2
S.Y is a key: |Q(D)| ≤ N
S.Y has degree ≤ d: |Q(D)| ≤ d×N
E.g. R(X,Y) ∧ S(Y,Z) ∧ T(Z,X) No other info: |Q(D)| ≤ N3/2

19. Main Principle
Find information-theoretic proof of the upper bound, or the submodular width
Convert proof to algorithm

20. Proof of Upper Bound
Q(X,Y,Z) = R(X,Y) ∧ S(Y,Z) ∧ T(Z,X) Database D  entropic function H

21. Proof of Upper Bound
Q(X,Y,Z) = R(X,Y) ∧ S(Y,Z) ∧ T(Z,X) Database D  entropic function H
Database D
R(X,Y)
S(Y,Z)
T(Z,X) X Y a 3 2 b d Y Z 3 m 2 q Z X m a q b d

22. Proof of Upper Bound
Q(X,Y,Z) = R(X,Y) ∧ S(Y,Z) ∧ T(Z,X) Database D  entropic function H
Output Q(D)
Database D
R(X,Y)
S(Y,Z)
T(Z,X) X Y Z a 3 m 2 q b d X Y a 3 2 b d Y Z 3 m 2 q Z X m a q b d

23. Proof of Upper Bound
Q(X,Y,Z) = R(X,Y) ∧ S(Y,Z) ∧ T(Z,X) Database D  entropic function H
Output Q(D)
Database D
R(X,Y)
S(Y,Z)
T(Z,X) X Y Z a 3 m
1/5 2 q b d X Y a 3 2 b d Y Z 3 m 2 q Z X m a q b d
H(XYZ) = log |Q(D)|

24. Proof of Upper Bound
Q(X,Y,Z) = R(X,Y) ∧ S(Y,Z) ∧ T(Z,X) Database D  entropic function H
Output Q(D)
Database D
R(X,Y)
S(Y,Z)
T(Z,X) X Y Z a 3 m
1/5 2 q b d X Y a 3
2/5 2
1/5 b d Y Z 3 m
2/5 2 q
1/5 Z X m a
1/5 q
2/5 b d
H(XYZ) = log |Q(D)|

25. Proof of Upper Bound
Q(X,Y,Z) = R(X,Y) ∧ S(Y,Z) ∧ T(Z,X) Database D  entropic function H
Output Q(D)
Database D
R(X,Y)
S(Y,Z)
T(Z,X) X Y Z a 3 m
1/5 2 q b d X Y a 3
2/5 2
1/5 b d Y Z 3 m
2/5 2 q
1/5 Z X m a
1/5 q
2/5 b d
H(XYZ) = log |Q(D)|
H(XY) ≤ log NR
H(YZ) ≤ log NS
H(Z|Y) ≤ log degS(z|y)
H(XZ) ≤ log NT
Cardinalitites, functional dependences, max degrees

26. Proof of Upper Bound
Q(X,Y,Z) = R(X,Y) ∧ S(Y,Z) ∧ T(Z,X) |R|,|S|,|T| ≤ N  |Q(D)|≤ N3/2

27. Proof of Upper Bound
Q(X,Y,Z) = R(X,Y) ∧ S(Y,Z) ∧ T(Z,X) |R|,|S|,|T| ≤ N  |Q(D)|≤ N3/2
3 log N ≥ h(XY) + h(YZ) + h(XZ)
≥ h(XYZ) + h(Y) + h(XZ) ≥ h(XYZ) + h(XYZ) + h(∅) = 2 h(XYZ) = 2 log |Output|

28. Proof of Upper Bound
Q(X,Y,Z) = R(X,Y) ∧ S(Y,Z) ∧ T(Z,X) |R|,|S|,|T| ≤ N  |Q(D)|≤ N3/2
submodularity
3 log N ≥ h(XY) + h(YZ) + h(XZ)
≥ h(XYZ) + h(Y) + h(XZ) ≥ h(XYZ) + h(XYZ) + h(∅) = 2 h(XYZ) = 2 log |Output|

29. Proof of Upper Bound
Q(X,Y,Z) = R(X,Y) ∧ S(Y,Z) ∧ T(Z,X) |R|,|S|,|T| ≤ N  |Q(D)|≤ N3/2
submodularity
3 log N ≥ h(XY) + h(YZ) + h(XZ)
≥ h(XYZ) + h(Y) + h(XZ) ≥ h(XYZ) + h(XYZ) + h(∅) = 2 h(XYZ) = 2 log |Output|

30. Proof of Upper Bound
Q(X,Y,Z) = R(X,Y) ∧ S(Y,Z) ∧ T(Z,X) |R|,|S|,|T| ≤ N  |Q(D)|≤ N3/2
submodularity
3 log N ≥ h(XY) + h(YZ) + h(XZ)
≥ h(XYZ) + h(Y) + h(XZ) ≥ h(XYZ) + h(XYZ) + h(∅) = 2 h(XYZ) = 2 log |Output|
submodularity

31. Proof of Upper Bound
Q(X,Y,Z) = R(X,Y) ∧ S(Y,Z) ∧ T(Z,X) |R|,|S|,|T| ≤ N  |Q(D)|≤ N3/2
submodularity
3 log N ≥ h(XY) + h(YZ) + h(XZ)
≥ h(XYZ) + h(Y) + h(XZ) ≥ h(XYZ) + h(XYZ) + h(∅) = 2 h(XYZ) = 2 log |Output|
submodularity

32. Proof of Upper Bound
Q(X,Y,Z) = R(X,Y) ∧ S(Y,Z) ∧ T(Z,X) |R|,|S|,|T| ≤ N  |Q(D)|≤ N3/2
submodularity
3 log N ≥ h(XY) + h(YZ) + h(XZ)
≥ h(XYZ) + h(Y) + h(XZ) ≥ h(XYZ) + h(XYZ) + h(∅) = 2 h(XYZ) = 2 log |Q(D)|
submodularity

33. Shearer’s inequality h(XY) + h(YZ) + h(XZ) ≥ 2 h(XYZ)
Proof of Upper Bound
Q(X,Y,Z) = R(X,Y) ∧ S(Y,Z) ∧ T(Z,X) |R|,|S|,|T| ≤ N  |Q(D)|≤ N3/2
submodularity
3 log N ≥ h(XY) + h(YZ) + h(XZ)
≥ h(XYZ) + h(Y) + h(XZ) ≥ h(XYZ) + h(XYZ) + h(∅) = 2 h(XYZ) = 2 log |Q(D)|
submodularity
Shearer’s inequality h(XY) + h(YZ) + h(XZ) ≥ 2 h(XYZ)

34. Proof to Algorithm + Q(X,Y,Z) = R(X,Y) ∧ S(Y,Z) ∧ T(Z,X)
h(XY) + h(YZ) + h(XZ) ≥ 2 h(XYZ)
h(XY)+h(YZ)+h(XZ)
h(XYZ)
h(Y) + h(XZ) +
Proof
h(XYZ)

35. Proof to Algorithm + Runtime Õ(N3/2)
Q(X,Y,Z) = R(X,Y) ∧ S(Y,Z) ∧ T(Z,X)
h(XY) + h(YZ) + h(XZ) ≥ 2 h(XYZ)
Algorithm
h(XY)+h(YZ)+h(XZ)
R(X,Y)∧S(Y,Z)∧T(Z,X)
h(XYZ)
h(Y) + h(XZ) +
Proof
h(XYZ)

36. Proof to Algorithm + Runtime Õ(N3/2)
Q(X,Y,Z) = R(X,Y) ∧ S(Y,Z) ∧ T(Z,X)
h(XY) + h(YZ) + h(XZ) ≥ 2 h(XYZ)
Algorithm
h(XY)+h(YZ)+h(XZ)
R(X,Y)∧S(Y,Z)∧T(Z,X)
N3/2
h(XYZ)
h(Y) + h(XZ) +
Rlight(X,Y)∧S(Y,Z)
Proof
h(XYZ)
Rlight or Rheavy: degree(Y) ≤ or > N1/2

37. Proof to Algorithm + ∪ Runtime Õ(N3/2)
Q(X,Y,Z) = R(X,Y) ∧ S(Y,Z) ∧ T(Z,X)
h(XY) + h(YZ) + h(XZ) ≥ 2 h(XYZ)
Algorithm
h(XY)+h(YZ)+h(XZ)
R(X,Y)∧S(Y,Z)∧T(Z,X)
N3/2
h(XYZ)
h(Y) + h(XZ) +
N3/2
Rlight(X,Y)∧S(Y,Z) ∪
Proof
h(XYZ)
Rheavy(Y)∧T(X,Z)
Rlight or Rheavy: degree(Y) ≤ or > N1/2

38. Proof to Algorithm + ∪ Runtime Õ(N3/2)
Q(X,Y,Z) = R(X,Y) ∧ S(Y,Z) ∧ T(Z,X)
h(XY) + h(YZ) + h(XZ) ≥ 2 h(XYZ)
Algorithm
h(XY)+h(YZ)+h(XZ)
R(X,Y)∧S(Y,Z)∧T(Z,X)
N3/2
h(XYZ)
h(Y) + h(XZ) +
N3/2
Rlight(X,Y)∧S(Y,Z) ∪
Proof
h(XYZ)
Rheavy(Y)∧T(X,Z)
Rlight or Rheavy: degree(Y) ≤ or > N1/2

39. Another Quick Example Runtime Õ(N3/2)
Q(X,Y,Z) = R(X,Y) ∧ S(Y,Z) ∧ T(Z,U) ∧ A(X,Z,U) ∧ B(X,Y,U)
3 log N ≥ h(XY)+h(YZ)+h(XZ)+h(U|XZ)+h(X|YU)
≥ h(XY)+h(YZ)+h(XZ)+h(U|XYZ)+h(X|YZU)
h(Y)+h(XZ)+h(X|YZU) ≥
h(XYZ)+h(U|XYZ)
h(XYZU)
h(XYZ)+h(X|YZU)
h(XYZU)

40. Enumeration Problem: Discussion
Cardinalities: [Atserias,Grohe,Marx’08, Ngo,Re,Rudra’13]
Entropic bound = polymatroid bound
Algorithm for Q(D) has single log factor
Cardinalities + FDs + max degrees:
Entropic bound ≨ polymatroid bound
Algorithm for Q(D) has polylog factor

41. Outline
Motivation
Full CQ
Boolean CQ
Conclusions

42. Background: Tree Decomposition
Informally: TD = a tree where each node t represents an enumeration problem
Fractional hypetree width [Grohe,Marx’14] mintree maxnode t maxD
Submodular width [Marx’2013] maxD mintree maxnode t

43. Q() = ∃x∃y∃z∃u R(x,y)∧S(y,z)∧T(z,u)∧K(u,x)
|R|,|S|,|T|,|K| ≤ N O(N3/2) algorithm [Alon,Yuster,Zwick’97]

44. mintree maxnode t maxD Q() = ∃x∃y∃z∃u R(x,y)∧S(y,z)∧T(z,u)∧K(u,x)
|R|,|S|,|T|,|K| ≤ N O(N3/2) algorithm [Alon,Yuster,Zwick’97]
mintree maxnode t maxD

45. mintree maxnode t maxD Q() = ∃x∃y∃z∃u R(x,y)∧S(y,z)∧T(z,u)∧K(u,x)
|R|,|S|,|T|,|K| ≤ N O(N3/2) algorithm [Alon,Yuster,Zwick’97]
mintree maxnode t maxD
R(x,y),S(y,z)
T(z,u),K(u,x)
Tree decompositions
S(y,z),T(z,u)
K(u,x),R(x,y)

46. mintree maxnode t maxD Runtime Õ(N2) (suboptimal)Z X Y
Q() = ∃x∃y∃z∃u R(x,y)∧S(y,z)∧T(z,u)∧K(u,x)
|R|,|S|,|T|,|K| ≤ N O(N3/2) algorithm [Alon,Yuster,Zwick’97]
mintree maxnode t maxD
R(x,y),S(y,z)
T(z,u),K(u,x)
Tree decompositions
S(y,z),T(z,u)
K(u,x),R(x,y)
Runtime Õ(N2)
(suboptimal)

47. mintree maxnode t maxD Runtime Õ(N2) (suboptimal)Z X Y
Q() = ∃x∃y∃z∃u R(x,y)∧S(y,z)∧T(z,u)∧K(u,x)
|R|,|S|,|T|,|K| ≤ N O(N3/2) algorithm [Alon,Yuster,Zwick’97]
mintree maxnode t maxD
R(x,y),S(y,z)
T(z,u),K(u,x)
Tree decompositions
S(y,z),T(z,u)
K(u,x),R(x,y)
Runtime Õ(N2)
(suboptimal)

48. maxD mintree maxnode t Q() = ∃x∃y∃z∃u R(x,y)∧S(y,z)∧T(z,u)∧K(u,x)
|R|,|S|,|T|,|K| ≤ N O(N3/2) algorithm [Alon,Yuster,Zwick’97]
maxD mintree maxnode t
R(x,y),S(y,z)
T(z,u),K(u,x)
Tree decompositions
S(y,z),T(z,u)
K(u,x),R(x,y)

49. maxD mintree maxnode t Q() = ∃x∃y∃z∃u R(x,y)∧S(y,z)∧T(z,u)∧K(u,x)
|R|,|S|,|T|,|K| ≤ N O(N3/2) algorithm [Alon,Yuster,Zwick’97]
maxD mintree maxnode t
R(x,y),S(y,z)
T(z,u),K(u,x)
Tree decompositions
S(y,z),T(z,u)
K(u,x),R(x,y) T1
min( max(h(xyz),h(zux)), max(h(yzu),h(uxy))) =

50. maxD mintree maxnode t Q() = ∃x∃y∃z∃u R(x,y)∧S(y,z)∧T(z,u)∧K(u,x)
|R|,|S|,|T|,|K| ≤ N O(N3/2) algorithm [Alon,Yuster,Zwick’97]
maxD mintree maxnode t
R(x,y),S(y,z)
T(z,u),K(u,x)
Tree decompositions
S(y,z),T(z,u)
K(u,x),R(x,y) T1
min( max(h(xyz),h(zux)), max(h(yzu),h(uxy))) = T2

51. maxD mintree maxnode t Q() = ∃x∃y∃z∃u R(x,y)∧S(y,z)∧T(z,u)∧K(u,x)
|R|,|S|,|T|,|K| ≤ N O(N3/2) algorithm [Alon,Yuster,Zwick’97]
maxD mintree maxnode t
R(x,y),S(y,z)
T(z,u),K(u,x)
Tree decompositions
S(y,z),T(z,u)
K(u,x),R(x,y) T1
min( max(h(xyz),h(zux)), max(h(yzu),h(uxy))) = T2

52. maxD mintree maxnode t Q() = ∃x∃y∃z∃u R(x,y)∧S(y,z)∧T(z,u)∧K(u,x)
|R|,|S|,|T|,|K| ≤ N O(N3/2) algorithm [Alon,Yuster,Zwick’97]
maxD mintree maxnode t
R(x,y),S(y,z)
T(z,u),K(u,x)
Tree decompositions
S(y,z),T(z,u)
K(u,x),R(x,y) T1
min( max(h(xyz),h(zux)), max(h(yzu),h(uxy))) = T2
= max(min(h(xyz),h(yzu)), min(h(xyz),h(uxy)), min(h(zux),h(yzu)), min(h(zux),h(uxy)))

53. 3 log N ≥ h(xy) + h(yz) + h(zu)
Q() = ∃x∃y∃z∃u R(x,y)∧S(y,z)∧T(z,u)∧K(u,x)
|R|,|S|,|T|,|K| ≤ N O(N3/2) algorithm [Alon,Yuster,Zwick’97]
maxD mintree maxnode t
R(x,y),S(y,z)
T(z,u),K(u,x)
Tree decompositions
S(y,z),T(z,u)
K(u,x),R(x,y) T1
min( max(h(xyz),h(zux)), max(h(yzu),h(uxy))) =
3 log N ≥ h(xy) + h(yz) + h(zu) T2
= max(min(h(xyz),h(yzu)), min(h(xyz),h(uxy)), min(h(zux),h(yzu)), min(h(zux),h(uxy)))

54. 3 log N ≥ h(xy) + h(yz) + h(zu) ≥ h(xyz) + h(y) + h(zu)
Q() = ∃x∃y∃z∃u R(x,y)∧S(y,z)∧T(z,u)∧K(u,x)
|R|,|S|,|T|,|K| ≤ N O(N3/2) algorithm [Alon,Yuster,Zwick’97]
maxD mintree maxnode t
R(x,y),S(y,z)
T(z,u),K(u,x)
Tree decompositions
S(y,z),T(z,u)
K(u,x),R(x,y) T1
min( max(h(xyz),h(zux)), max(h(yzu),h(uxy))) =
3 log N ≥ h(xy) + h(yz) + h(zu)
≥ h(xyz) + h(y) + h(zu) T2
= max(min(h(xyz),h(yzu)), min(h(xyz),h(uxy)), min(h(zux),h(yzu)), min(h(zux),h(uxy)))

55. 3 log N ≥ h(xy) + h(yz) + h(zu) ≥ h(xyz) + h(y) + h(zu)
Q() = ∃x∃y∃z∃u R(x,y)∧S(y,z)∧T(z,u)∧K(u,x)
|R|,|S|,|T|,|K| ≤ N O(N3/2) algorithm [Alon,Yuster,Zwick’97]
maxD mintree maxnode t
R(x,y),S(y,z)
T(z,u),K(u,x)
Tree decompositions
S(y,z),T(z,u)
K(u,x),R(x,y) T1
min( max(h(xyz),h(zux)), max(h(yzu),h(uxy))) =
3 log N ≥ h(xy) + h(yz) + h(zu)
≥ h(xyz) + h(y) + h(zu)
≥ h(xyz) + h(yzu) + h(∅) T2
= max(min(h(xyz),h(yzu)), min(h(xyz),h(uxy)), min(h(zux),h(yzu)), min(h(zux),h(uxy)))

56. 3 log N ≥ h(xy) + h(yz) + h(zu) ≥ h(xyz) + h(y) + h(zu)
Q() = ∃x∃y∃z∃u R(x,y)∧S(y,z)∧T(z,u)∧K(u,x)
|R|,|S|,|T|,|K| ≤ N O(N3/2) algorithm [Alon,Yuster,Zwick’97]
maxD mintree maxnode t
R(x,y),S(y,z)
T(z,u),K(u,x)
Tree decompositions
S(y,z),T(z,u)
K(u,x),R(x,y) T1
min( max(h(xyz),h(zux)), max(h(yzu),h(uxy))) =
3 log N ≥ h(xy) + h(yz) + h(zu)
≥ h(xyz) + h(y) + h(zu)
≥ h(xyz) + h(yzu) + h(∅) ≥ 2 min(h(xyz),h(yzu)) T2
= max(min(h(xyz),h(yzu)), min(h(xyz),h(uxy)), min(h(zux),h(yzu)), min(h(zux),h(uxy)))

57. 3 log N ≥ h(xy) + h(yz) + h(zu) ≥ h(xyz) + h(y) + h(zu)
Q() = ∃x∃y∃z∃u R(x,y)∧S(y,z)∧T(z,u)∧K(u,x)
|R|,|S|,|T|,|K| ≤ N O(N3/2) algorithm [Alon,Yuster,Zwick’97]
maxD mintree maxnode t
R(x,y),S(y,z)
T(z,u),K(u,x)
Tree decompositions
subw(Q) = 3/2 log N
S(y,z),T(z,u)
K(u,x),R(x,y) T1
min( max(h(xyz),h(zux)), max(h(yzu),h(uxy))) =
3 log N ≥ h(xy) + h(yz) + h(zu)
≥ h(xyz) + h(y) + h(zu)
≥ h(xyz) + h(yzu) + h(∅) ≥ 2 min(h(xyz),h(yzu)) T2
= max(min(h(xyz),h(yzu)), min(h(xyz),h(uxy)), min(h(zux),h(yzu)), min(h(zux),h(uxy))) ≤ 3/2 log N

58. Proof to Algorithm
Use the proof of: to compute the disjunctive datalog rule: (details omitted)
h(xyz)+h(yzu) ≤ h(xy) + h(yz) + h(zu)
A(x,y,z) ∨ B(y,z,u)  R(x,y) ∧ S(y,z) ∧ T(z,u)
Runtime Õ(N3/2)

59. Outline
Motivation
Full CQ
Boolean CQ
Conclusions

60. Conclusions Query evaluation summary: Information theory  Proof
Proof  Algorithm
Open problems:
Better “Proof  Algorithm”
Fine-grained lower bounds

61. Thank You! Questions?