CPSC-608 Database Systems Fall 2011 Instructor: Jianer Chen Office: HRBB 315C Phone: Notes #8

LQP Optimization with Size 2

Two techniques: 3

LQP Optimization with Size Two techniques: Estimating sizes of immediate relations For natural join: T(R(X, y) S(y, Z)) = T(R)T(S)/max{V(R, y), V(S, y)} 4

LQP Optimization with Size Two techniques: Estimating sizes of immediate relations For natural join: T(R(X, y) S(y, Z)) = T(R)T(S)/max{V(R, y), V(S, y)} Consider different order of an operation (((R S) T) U) = (R U) (S T) 5

Consider: A(a, b): T(A) = 1000, V(A, a) = 100, V(A, b) = 200 B(b, c): T(B) = 1000, V(B, b) = 100, V(B, c) = 500 C(c, d): T(C) = 1000, V(C, c) = 20, V(C, d) = 1000 D(d, a): T(D) = 1000, V(D, d) = 1000, V(D, a) = 50 We want to have a good LQP for A B C D 6

Left-deep join tree 7

8 ? ? ? ?

Left-deep join tree (all 4! = 24 permutations) 9 AB C D BA C D AB D C BA D C CA B D CA D B BC A D CB A D DA B C DA C B DB A C DB C A DC A B DC B A CB D A CD A B CD B A BC D A BD A C BD C A AC B D AC D B AD B C AD C B

10 AB C D BA C D AB D C BA D C CA B D CA D B BC A D CB A D DA B C DA C B DB A C DB C A DC A B DC B A CB D A CD A B CD B A BC D A BD A C BD C A AC B D AC D B AD B C AD C B

Left-deep join tree (all 4! = 24 permutations) 11 AB C D BA C D AB D C BA D C CA B D CA D B BC A D CB A D DA B C DA C B DB A C DB C A DC A B DC B A CB D A CD A B CD B A BC D A BD A C BD C A AC B D AC D B AD B C AD C B

Left-deep join tree (all 4!/2 = 12 permutations) 12 AB C D BA C D AB D C BA D C CA B D CA D B BC A D CB A D DA B C DA C B DB A C DB C A DC A B DC B A CB D A CD A B CD B A BC D A BD A C BD C A AC B D AC D B AD B C AD C B

Left-deep join tree (all 4!/2 = 12 permutations) 13 AB C D BA C D AB D C BA D C CA B D CA D B BC A D CB A D DA B C DA C B DB A C DB C A DC A B DC B A CB D A CD A B CD B A BC D A BD A C BD C A AC B D AC D B AD B C AD C B

Left-deep join tree 14 A B C D B D A C A(a, b): T(A) = 1000, V(A, a) = 100, V(A, b) = 200 B(b, c): T(B) = 1000, V(B, b) = 100, V(B, c) = 500 C(c, d): T(C) = 1000, V(C, c) = 20, V(C, d) = 1000 D(d, a): T(D) = 1000, V(D, d) = 1000, V(D, a) = 50 T(R(X, y) S(y, Z)) = T(R)T(S)/max{V(R, y), V(S, y)}

Left-deep join tree 15 A B C D B D A C A(a, b): T(A) = 1000, V(A, a) = 100, V(A, b) = 200 B(b, c): T(B) = 1000, V(B, b) = 100, V(B, c) = 500 C(c, d): T(C) = 1000, V(C, c) = 20, V(C, d) = 1000 D(d, a): T(D) = 1000, V(D, d) = 1000, V(D, a) = 50 T(R(X, y) S(y, Z)) = T(R)T(S)/max{V(R, y), V(S, y)} 5000 V(*, c) = 500

Left-deep join tree 16 A B C D B D A C A(a, b): T(A) = 1000, V(A, a) = 100, V(A, b) = 200 B(b, c): T(B) = 1000, V(B, b) = 100, V(B, c) = 500 C(c, d): T(C) = 1000, V(C, c) = 20, V(C, d) = 1000 D(d, a): T(D) = 1000, V(D, d) = 1000, V(D, a) = 50 T(R(X, y) S(y, Z)) = T(R)T(S)/max{V(R, y), V(S, y)} 5000 V(*, c) =

Left-deep join tree 17 A B C D B D A C A(a, b): T(A) = 1000, V(A, a) = 100, V(A, b) = 200 B(b, c): T(B) = 1000, V(B, b) = 100, V(B, c) = 500 C(c, d): T(C) = 1000, V(C, c) = 20, V(C, d) = 1000 D(d, a): T(D) = 1000, V(D, d) = 1000, V(D, a) = 50 T(R(X, y) S(y, Z)) = T(R)T(S)/max{V(R, y), V(S, y)} 5000 V(*, c) = cost = 15000

Left-deep join tree 18 A B C D B D A C A(a, b): T(A) = 1000, V(A, a) = 100, V(A, b) = 200 B(b, c): T(B) = 1000, V(B, b) = 100, V(B, c) = 500 C(c, d): T(C) = 1000, V(C, c) = 20, V(C, d) = 1000 D(d, a): T(D) = 1000, V(D, d) = 1000, V(D, a) = 50 T(R(X, y) S(y, Z)) = T(R)T(S)/max{V(R, y), V(S, y)} 5000 V(*, c) = V(*, a) = 50 V(*,b) = cost = cost =

Left-deep join tree (all 4!/2 = 12 permutations) 19 AB C D BA C D AB D C BA D C CA B D CA D B BC A D CB A D DA B C DA C B DB A C DB C A DC A B DC B A CB D A CD A B CD B A BC D A BD A C BD C A AC B D AC D B AD B C AD C B

Left-deep join tree (all 4!/2 = 12 permutations) 20 AB C D BA C D AB D C BA D C CA B D CA D B BC A D CB A D DA B C DA C B DB A C DB C A DC A B DC B A CB D A CD A B CD B A BC D A BD A C BD C A AC B D AC D B AD B C AD C B

Left-deep tree: general algorithm Input: n relations R 1, R 2, …, R n Output: the best left-deep join of R 1, R 2, …, R n 1.Construct a left-deep tree T of n leaves; 2.For each P of the permutations of the n relations R 1, R 2, …, R n Do assign the n relations to the leaves of T in order of P; evaluate the cost of the plan; 3. Pick the plan with the permutation that gives the minimum cost. 21

Dynamic Programming Consider all tree structures. 22

Dynamic Programming Consider all tree structures. Again consider A B C D Five tree structures: Each of (a)-(d) has 12 different assignments, and (e) has 3 different assignments. So totally there are 51 different ways to join the 4 relations. Too many when the number of relations is relatively large. 23 (a) (e) (d) (c)(b)

Dynamic Programming Consider 24 D D D D A A A A B B B B C C C C

Dynamic Programming Consider 25 D D D D A A A A B B B B C C C C

Dynamic Programming Consider We really only need to find the best way to join A B C, then join D with this best join. 26 D D D D A A A A B B B B C C C C

Dynamic Programming Consider We really only need to find the best way to join A B C, then join D with this best join. How do we find the best join of A B C? 27 D D D D A A A A B B B B C C C C

Dynamic Programming Consider We really only need to find the best way to join A B C, then join D with this best join. How do we find the best join of A B C? We consider all possible ways: (A B) C, (A C) B, (B C) A. 28 D D D D A A A A B B B B C C C C

Dynamic programming: general algorithm Input: n relations R 1, R 2, …, R n Output: the best join of R 1, R 2, …, R n 1.FOR each R i DO {cost(R i ) = 0; size(R i ) = 0}; 2.FOR each pair of R i and R j DO {cost(R i, R j ) = 0; compute size(R i R j )}; 3.FOR k = 3 TO n DO FOR any k relations S 1, S 2, …, S k of R 1, R 2, …, R n DO FOR each partition P = {(S i 1, …, S i j ), (S i j+1,…, S i k )} of S 1, S 2, …, S k DO cost(P) = cost(S i 1, …, S i j ) + size(S i 1 … S i j ) + cost(S i j+1, …, S i k ) + size(S i j+1 … S i k ); let cost(S 1, S 2, …, S k ) be the smallest cost(P) among the above partitions; computer size(S 1 S 2 … S k ) (and remember this partition P); 4. Return cost(R 1, R 2, …, R n ). 29

Dynamic Programming: Example A(a, b): T(A) = 1000, V(A, a) = 100, V(A, b) = 200 B(b, c): T(B) = 1000, V(B, b) = 100, V(B, c) = 500 C(c, d): T(C) = 1000, V(C, c) = 20, V(C, d) = 1000 D(d, a): T(D) = 1000, V(D, d) = 1000, V(D, a) = 50 T(R(X, y) S(y, Z)) = T(R)T(S)/max{V(R, y), V(S, y)} 30 A cost = 0 size = 0 D cost = 0 size = 0 C cost = 0 size = 0 B cost = 0 size = 0

Dynamic Programming: Example A(a, b): T(A) = 1000, V(A, a) = 100, V(A, b) = 200 B(b, c): T(B) = 1000, V(B, b) = 100, V(B, c) = 500 C(c, d): T(C) = 1000, V(C, c) = 20, V(C, d) = 1000 D(d, a): T(D) = 1000, V(D, d) = 1000, V(D, a) = 50 T(R(X, y) S(y, Z)) = T(R)T(S)/max{V(R, y), V(S, y)} 31 A cost = 0 size = 0 D cost = 0 size = 0 C cost = 0 size = 0 B cost = 0 size = 0 A, B cost = 0 size = 5000 C, D cost = 0 size = 1000 B, D cost = 0 size = B, C cost = 0 size = 2000 A, D cost = 0 size = A, C cost = 0 size =

Dynamic Programming: Example A(a, b): T(A) = 1000, V(A, a) = 100, V(A, b) = 200 B(b, c): T(B) = 1000, V(B, b) = 100, V(B, c) = 500 C(c, d): T(C) = 1000, V(C, c) = 20, V(C, d) = 1000 D(d, a): T(D) = 1000, V(D, d) = 1000, V(D, a) = 50 T(R(X, y) S(y, Z)) = T(R)T(S)/max{V(R, y), V(S, y)} 32 A cost = 0 size = 0 D cost = 0 size = 0 C cost = 0 size = 0 B cost = 0 size = 0 A, B cost = 0 size = 5000 C, D cost = 0 size = 1000 B, D cost = 0 size = B, C cost = 0 size = 2000 A, D cost = 0 size = A, C cost = 0 size = A, B, CB, C, DA, C, DA, B, D

Dynamic Programming: Example A(a, b): T(A) = 1000, V(A, a) = 100, V(A, b) = 200 B(b, c): T(B) = 1000, V(B, b) = 100, V(B, c) = 500 C(c, d): T(C) = 1000, V(C, c) = 20, V(C, d) = 1000 D(d, a): T(D) = 1000, V(D, d) = 1000, V(D, a) = 50 T(R(X, y) S(y, Z)) = T(R)T(S)/max{V(R, y), V(S, y)} 33 A cost = 0 size = 0 D cost = 0 size = 0 C cost = 0 size = 0 B cost = 0 size = 0 A, B cost = 0 size = 5000 C, D cost = 0 size = 1000 B, D cost = 0 size = B, C cost = 0 size = 2000 A, D cost = 0 size = A, C cost = 0 size = A, B, CB, C, DA, C, DA, B, D

Dynamic Programming: Example A(a, b): T(A) = 1000, V(A, a) = 100, V(A, b) = 200 B(b, c): T(B) = 1000, V(B, b) = 100, V(B, c) = 500 C(c, d): T(C) = 1000, V(C, c) = 20, V(C, d) = 1000 D(d, a): T(D) = 1000, V(D, d) = 1000, V(D, a) = 50 T(R(X, y) S(y, Z)) = T(R)T(S)/max{V(R, y), V(S, y)} 34 A cost = 0 size = 0 D cost = 0 size = 0 C cost = 0 size = 0 B cost = 0 size = 0 A, B cost = 0 size = 5000 C, D cost = 0 size = 1000 B, D cost = 0 size = B, C cost = 0 size = 2000 A, D cost = 0 size = A, C cost = 0 size = A, B, CB, C, DA, C, DA, B, D CB B D B C C D D

Dynamic Programming: Example A(a, b): T(A) = 1000, V(A, a) = 100, V(A, b) = 200 B(b, c): T(B) = 1000, V(B, b) = 100, V(B, c) = 500 C(c, d): T(C) = 1000, V(C, c) = 20, V(C, d) = 1000 D(d, a): T(D) = 1000, V(D, d) = 1000, V(D, a) = 50 T(R(X, y) S(y, Z)) = T(R)T(S)/max{V(R, y), V(S, y)} 35 A cost = 0 size = 0 D cost = 0 size = 0 C cost = 0 size = 0 B cost = 0 size = 0 A, B cost = 0 size = 5000 C, D cost = 0 size = 1000 B, D cost = 0 size = B, C cost = 0 size = 2000 A, D cost = 0 size = A, C cost = 0 size = A, B, CB, C, DA, C, DA, B, D CB B D B C C D D

Dynamic Programming: Example A(a, b): T(A) = 1000, V(A, a) = 100, V(A, b) = 200 B(b, c): T(B) = 1000, V(B, b) = 100, V(B, c) = 500 C(c, d): T(C) = 1000, V(C, c) = 20, V(C, d) = 1000 D(d, a): T(D) = 1000, V(D, d) = 1000, V(D, a) = 50 T(R(X, y) S(y, Z)) = T(R)T(S)/max{V(R, y), V(S, y)} 36 A cost = 0 size = 0 D cost = 0 size = 0 C cost = 0 size = 0 B cost = 0 size = 0 A, B cost = 0 size = 5000 C, D cost = 0 size = 1000 B, D cost = 0 size = B, C cost = 0 size = 2000 A, D cost = 0 size = A, C cost = 0 size = A, B, CB, C, DA, C, DA, B, D CB B D B C C D D

Dynamic Programming: Example A(a, b): T(A) = 1000, V(A, a) = 100, V(A, b) = 200 B(b, c): T(B) = 1000, V(B, b) = 100, V(B, c) = 500 C(c, d): T(C) = 1000, V(C, c) = 20, V(C, d) = 1000 D(d, a): T(D) = 1000, V(D, d) = 1000, V(D, a) = 50 T(R(X, y) S(y, Z)) = T(R)T(S)/max{V(R, y), V(S, y)} 37 A cost = 0 size = 0 D cost = 0 size = 0 C cost = 0 size = 0 B cost = 0 size = 0 A, B cost = 0 size = 5000 C, D cost = 0 size = 1000 B, D cost = 0 size = B, C cost = 0 size = 2000 A, D cost = 0 size = A, C cost = 0 size = A, B, CB, C, DA, C, DA, B, D CB B D B C C D D 2000

Dynamic Programming: Example A(a, b): T(A) = 1000, V(A, a) = 100, V(A, b) = 200 B(b, c): T(B) = 1000, V(B, b) = 100, V(B, c) = 500 C(c, d): T(C) = 1000, V(C, c) = 20, V(C, d) = 1000 D(d, a): T(D) = 1000, V(D, d) = 1000, V(D, a) = 50 T(R(X, y) S(y, Z)) = T(R)T(S)/max{V(R, y), V(S, y)} 38 A cost = 0 size = 0 D cost = 0 size = 0 C cost = 0 size = 0 B cost = 0 size = 0 A, B cost = 0 size = 5000 C, D cost = 0 size = 1000 B, D cost = 0 size = B, C cost = 0 size = 2000 A, D cost = 0 size = A, C cost = 0 size = A, B, CB, C, DA, C, DA, B, D CB B D B C C D D 2000

Dynamic Programming: Example A(a, b): T(A) = 1000, V(A, a) = 100, V(A, b) = 200 B(b, c): T(B) = 1000, V(B, b) = 100, V(B, c) = 500 C(c, d): T(C) = 1000, V(C, c) = 20, V(C, d) = 1000 D(d, a): T(D) = 1000, V(D, d) = 1000, V(D, a) = 50 T(R(X, y) S(y, Z)) = T(R)T(S)/max{V(R, y), V(S, y)} 39 A cost = 0 size = 0 D cost = 0 size = 0 C cost = 0 size = 0 B cost = 0 size = 0 A, B cost = 0 size = 5000 C, D cost = 0 size = 1000 B, D cost = 0 size = B, C cost = 0 size = 2000 A, D cost = 0 size = A, C cost = 0 size = A, B, CB, C, DA, C, DA, B, D CB B D B C C D D

Dynamic Programming: Example A(a, b): T(A) = 1000, V(A, a) = 100, V(A, b) = 200 B(b, c): T(B) = 1000, V(B, b) = 100, V(B, c) = 500 C(c, d): T(C) = 1000, V(C, c) = 20, V(C, d) = 1000 D(d, a): T(D) = 1000, V(D, d) = 1000, V(D, a) = 50 T(R(X, y) S(y, Z)) = T(R)T(S)/max{V(R, y), V(S, y)} 40 A cost = 0 size = 0 D cost = 0 size = 0 C cost = 0 size = 0 B cost = 0 size = 0 A, B cost = 0 size = 5000 C, D cost = 0 size = 1000 B, D cost = 0 size = B, C cost = 0 size = 2000 A, D cost = 0 size = A, C cost = 0 size = A, B, CB, C, DA, C, DA, B, D CB B D B C C D D

Dynamic Programming: Example A(a, b): T(A) = 1000, V(A, a) = 100, V(A, b) = 200 B(b, c): T(B) = 1000, V(B, b) = 100, V(B, c) = 500 C(c, d): T(C) = 1000, V(C, c) = 20, V(C, d) = 1000 D(d, a): T(D) = 1000, V(D, d) = 1000, V(D, a) = 50 T(R(X, y) S(y, Z)) = T(R)T(S)/max{V(R, y), V(S, y)} 41 A cost = 0 size = 0 D cost = 0 size = 0 C cost = 0 size = 0 B cost = 0 size = 0 A, B cost = 0 size = 5000 C, D cost = 0 size = 1000 B, D cost = 0 size = B, C cost = 0 size = 2000 A, D cost = 0 size = A, C cost = 0 size = A, B, CB, C, D cost = 1000 size = 2000 A, C, DA, B, D B C D

Dynamic Programming: Example A(a, b): T(A) = 1000, V(A, a) = 100, V(A, b) = 200 B(b, c): T(B) = 1000, V(B, b) = 100, V(B, c) = 500 C(c, d): T(C) = 1000, V(C, c) = 20, V(C, d) = 1000 D(d, a): T(D) = 1000, V(D, d) = 1000, V(D, a) = 50 T(R(X, y) S(y, Z)) = T(R)T(S)/max{V(R, y), V(S, y)} A cost = 0 size = 0 D cost = 0 size = 0 C cost = 0 size = 0 B cost = 0 size = 0 A, B cost = 0 size = 5000 C, D cost = 0 size = 1000 B, D cost = 0 size = B, C cost = 0 size = 2000 A, D cost = 0 size = A, C cost = 0 size = A, B, C cost = 2000 size = B, C, D cost = 1000 size = 2000 A, C, D cost = 1000 size = A, B, D cost = 5000 size = B C D

Dynamic Programming: Example A(a, b): T(A) = 1000, V(A, a) = 100, V(A, b) = 200 B(b, c): T(B) = 1000, V(B, b) = 100, V(B, c) = 500 C(c, d): T(C) = 1000, V(C, c) = 20, V(C, d) = 1000 D(d, a): T(D) = 1000, V(D, d) = 1000, V(D, a) = 50 T(R(X, y) S(y, Z)) = T(R)T(S)/max{V(R, y), V(S, y)} 43 A cost = 0 size = 0 D cost = 0 size = 0 C cost = 0 size = 0 B cost = 0 size = 0 A, B cost = 0 size = 5000 C, D cost = 0 size = 1000 B, D cost = 0 size = B, C cost = 0 size = 2000 A, D cost = 0 size = A, C cost = 0 size = A, B, C cost = 2000 size = B, C, D cost = 1000 size = 2000 A, C, D cost = 1000 size = A, B, D cost = 5000 size = A, B, C, D B C D

Dynamic Programming: Example A(a, b): T(A) = 1000, V(A, a) = 100, V(A, b) = 200 B(b, c): T(B) = 1000, V(B, b) = 100, V(B, c) = 500 C(c, d): T(C) = 1000, V(C, c) = 20, V(C, d) = 1000 D(d, a): T(D) = 1000, V(D, d) = 1000, V(D, a) = 50 T(R(X, y) S(y, Z)) = T(R)T(S)/max{V(R, y), V(S, y)} 44 A cost = 0 size = 0 D cost = 0 size = 0 C cost = 0 size = 0 B cost = 0 size = 0 A, B cost = 0 size = 5000 C, D cost = 0 size = 1000 B, D cost = 0 size = B, C cost = 0 size = 2000 A, D cost = 0 size = A, C cost = 0 size = A, B, C cost = 2000 size = B, C, D cost = 1000 size = 2000 A, C, D cost = 1000 size = A, B, D cost = 5000 size = A, B, C, D A {B,C,D}DC B {A,C,D} {A,B,D} {A,B,C} {A,B} {C,D} {A,C} {B,D}{A,D} {B,C} B C D

Dynamic Programming: Example A(a, b): T(A) = 1000, V(A, a) = 100, V(A, b) = 200 B(b, c): T(B) = 1000, V(B, b) = 100, V(B, c) = 500 C(c, d): T(C) = 1000, V(C, c) = 20, V(C, d) = 1000 D(d, a): T(D) = 1000, V(D, d) = 1000, V(D, a) = 50 T(R(X, y) S(y, Z)) = T(R)T(S)/max{V(R, y), V(S, y)} 45 A cost = 0 size = 0 D cost = 0 size = 0 C cost = 0 size = 0 B cost = 0 size = 0 A, B cost = 0 size = 5000 C, D cost = 0 size = 1000 B, D cost = 0 size = B, C cost = 0 size = 2000 A, D cost = 0 size = A, C cost = 0 size = A, B, C cost = 2000 size = B, C, D cost = 1000 size = 2000 A, C, D cost = 1000 size = A, B, D cost = 5000 size = A, B, C, D A {B,C,D}DC B {A,C,D} {A,B,D} {A,B,C} {A,B} {C,D} {A,C} {B,D}{A,D} {B,C} 3000 B C D

Dynamic Programming: Example A(a, b): T(A) = 1000, V(A, a) = 100, V(A, b) = 200 B(b, c): T(B) = 1000, V(B, b) = 100, V(B, c) = 500 C(c, d): T(C) = 1000, V(C, c) = 20, V(C, d) = 1000 D(d, a): T(D) = 1000, V(D, d) = 1000, V(D, a) = 50 T(R(X, y) S(y, Z)) = T(R)T(S)/max{V(R, y), V(S, y)} 46 A cost = 0 size = 0 D cost = 0 size = 0 C cost = 0 size = 0 B cost = 0 size = 0 A, B cost = 0 size = 5000 C, D cost = 0 size = 1000 B, D cost = 0 size = B, C cost = 0 size = 2000 A, D cost = 0 size = A, C cost = 0 size = A, B, C cost = 2000 size = B, C, D cost = 1000 size = 2000 A, C, D cost = 1000 size = A, B, D cost = 5000 size = A, B, C, D A {B,C,D}DC B {A,C,D} {A,B,D} {A,B,C} {A,B} {C,D} {A,C} {B,D}{A,D} {B,C} B C D

Dynamic Programming: Example A(a, b): T(A) = 1000, V(A, a) = 100, V(A, b) = 200 B(b, c): T(B) = 1000, V(B, b) = 100, V(B, c) = 500 C(c, d): T(C) = 1000, V(C, c) = 20, V(C, d) = 1000 D(d, a): T(D) = 1000, V(D, d) = 1000, V(D, a) = 50 T(R(X, y) S(y, Z)) = T(R)T(S)/max{V(R, y), V(S, y)} 47 A cost = 0 size = 0 D cost = 0 size = 0 C cost = 0 size = 0 B cost = 0 size = 0 A, B cost = 0 size = 5000 C, D cost = 0 size = 1000 B, D cost = 0 size = B, C cost = 0 size = 2000 A, D cost = 0 size = A, C cost = 0 size = A, B, C cost = 2000 size = B, C, D cost = 1000 size = 2000 A, C, D cost = 1000 size = A, B, D cost = 5000 size = A, B, C, D A {B,C,D}DC B {A,C,D} {A,B,D} {A,B,C} {A,B} {C,D} {A,C} {B,D}{A,D} {B,C} B C D

Dynamic Programming: Example A(a, b): T(A) = 1000, V(A, a) = 100, V(A, b) = 200 B(b, c): T(B) = 1000, V(B, b) = 100, V(B, c) = 500 C(c, d): T(C) = 1000, V(C, c) = 20, V(C, d) = 1000 D(d, a): T(D) = 1000, V(D, d) = 1000, V(D, a) = 50 T(R(X, y) S(y, Z)) = T(R)T(S)/max{V(R, y), V(S, y)} 48 A cost = 0 size = 0 D cost = 0 size = 0 C cost = 0 size = 0 B cost = 0 size = 0 A, B cost = 0 size = 5000 C, D cost = 0 size = 1000 B, D cost = 0 size = B, C cost = 0 size = 2000 A, D cost = 0 size = A, C cost = 0 size = A, B, C cost = 2000 size = B, C, D cost = 1000 size = 2000 A, C, D cost = 1000 size = A, B, D cost = 5000 size = A, B, C, D A {B,C,D}DC B {A,C,D} {A,B,D} {A,B,C} {A,B} {C,D} {A,C} {B,D}{A,D} {B,C} B C D

Dynamic Programming: Example A(a, b): T(A) = 1000, V(A, a) = 100, V(A, b) = 200 B(b, c): T(B) = 1000, V(B, b) = 100, V(B, c) = 500 C(c, d): T(C) = 1000, V(C, c) = 20, V(C, d) = 1000 D(d, a): T(D) = 1000, V(D, d) = 1000, V(D, a) = 50 T(R(X, y) S(y, Z)) = T(R)T(S)/max{V(R, y), V(S, y)} 49 A cost = 0 size = 0 D cost = 0 size = 0 C cost = 0 size = 0 B cost = 0 size = 0 A, B cost = 0 size = 5000 C, D cost = 0 size = 1000 B, D cost = 0 size = B, C cost = 0 size = 2000 A, D cost = 0 size = A, C cost = 0 size = A, B, C cost = 2000 size = B, C, D cost = 1000 size = 2000 A, C, D cost = 1000 size = A, B, D cost = 5000 size = A, B, C, D A {B,C,D}DC B {A,C,D} {A,B,D} {A,B,C} {A,B} {C,D} {A,C} {B,D}{A,D} {B,C} A B C D

Dynamic Programming: Example A(a, b): T(A) = 1000, V(A, a) = 100, V(A, b) = 200 B(b, c): T(B) = 1000, V(B, b) = 100, V(B, c) = 500 C(c, d): T(C) = 1000, V(C, c) = 20, V(C, d) = 1000 D(d, a): T(D) = 1000, V(D, d) = 1000, V(D, a) = 50 T(R(X, y) S(y, Z)) = T(R)T(S)/max{V(R, y), V(S, y)} 50 A cost = 0 size = 0 D cost = 0 size = 0 C cost = 0 size = 0 B cost = 0 size = 0 A, B cost = 0 size = 5000 C, D cost = 0 size = 1000 B, D cost = 0 size = B, C cost = 0 size = 2000 A, D cost = 0 size = A, C cost = 0 size = A, B, C cost = 2000 size = B, C, D cost = 1000 size = 2000 A, C, D cost = 1000 size = A, B, D cost = 5000 size = A, B, C, D A {B,C,D}DC B {A,C,D} {A,B,D} {A,B,C} {A,B} {C,D} {A,C} {B,D}{A,D} {B,C} A B C D

Dynamic Programming: Example A(a, b): T(A) = 1000, V(A, a) = 100, V(A, b) = 200 B(b, c): T(B) = 1000, V(B, b) = 100, V(B, c) = 500 C(c, d): T(C) = 1000, V(C, c) = 20, V(C, d) = 1000 D(d, a): T(D) = 1000, V(D, d) = 1000, V(D, a) = 50 T(R(X, y) S(y, Z)) = T(R)T(S)/max{V(R, y), V(S, y)} 51 A cost = 0 size = 0 D cost = 0 size = 0 C cost = 0 size = 0 B cost = 0 size = 0 A, B cost = 0 size = 5000 C, D cost = 0 size = 1000 B, D cost = 0 size = B, C cost = 0 size = 2000 A, D cost = 0 size = A, C cost = 0 size = A, B, C cost = 2000 size = B, C, D cost = 1000 size = 2000 A, C, D cost = 1000 size = A, B, D cost = 5000 size = A, B, C, D A {B,C,D}DC B {A,C,D} {A,B,D} {A,B,C} {A,B} {C,D} {A,C} {B,D}{A,D} {B,C} A B C D B C D

Dynamic Programming: Example A(a, b): T(A) = 1000, V(A, a) = 100, V(A, b) = 200 B(b, c): T(B) = 1000, V(B, b) = 100, V(B, c) = 500 C(c, d): T(C) = 1000, V(C, c) = 20, V(C, d) = 1000 D(d, a): T(D) = 1000, V(D, d) = 1000, V(D, a) = 50 T(R(X, y) S(y, Z)) = T(R)T(S)/max{V(R, y), V(S, y)} 52 A cost = 0 size = 0 D cost = 0 size = 0 C cost = 0 size = 0 B cost = 0 size = 0 A, B cost = 0 size = 5000 C, D cost = 0 size = 1000 B, D cost = 0 size = B, C cost = 0 size = 2000 A, D cost = 0 size = A, C cost = 0 size = A, B, C cost = 2000 size = B, C, D cost = 1000 size = 2000 A, C, D cost = 1000 size = A, B, D cost = 5000 size = A, B, C, D cost = 3000 A {B,C,D}DC B {A,C,D} {A,B,D} {A,B,C} {A,B} {C,D} {A,C} {B,D}{A,D} {B,C} A B C D

LQP Optimization with Size: Summary Estimating sizes of immediate relations Consider different order of an operation left-deep tree dynamic programming 53