1. RankSQL: Query Algebra and Optimization for Relational Top-k Queries
AUTHORS: Chengkai Li
Kevin Chen-Chuan Chang
Ihab F. Ilyas
Sumin Song
Presenter: Roman Yarovoy
October 3, 2007

2. Before RankSQL
Ranking (top-k) queries: Query result is sorted by rank and limited to top k results.
Support for ranking was lacking from RDBMS.
Previously, isolated cases of top-k query processing were studied.
No way to integrate top-k operations with other relational operations.

3. Previous (traditional) approach
Query processing without ranking support:
Evaluate select-project-join (SPJ) query and materialize the result.
Sort the result according to a given ranking function.
Take only top k tuples.
Associated problems:
No interest in total order of all the results.
Evaluating ranking function(s) can be expensive.

4. Key contribution
Li et al. proposed:
Extending relational algebra to support ranking as a first-class database construct.
Consequence: Rank-aware relational query engine  Rank-aware query optimization.

5. Top-k query: Example 1 R T TID a1 a2 p1 p2 p3 r1 27 50 0.5 0.3 0.25 r217 60
0.6
0.65 r3 47 90
0.7
0.95 r4 87 10
0.8
0.1
0.35 r5 70
0.9
0.75
TID b1 b2 p4 p5 t1 47 55
0.5
0.1 t2 66 65
0.7 t3 27 15
0.8 t4 99 95
0.2

6. WHERE r.a1=t.b1 AND r.a2>t.b2 ORDER-BY p1+p2+p3+p4+p5 LIMIT 2
Example 1 (cont’d)
SELECT *
FROM R r, T t
WHERE r.a1=t.b1 AND r.a2>t.b2
ORDER-BY p1+p2+p3+p4+p5
LIMIT 2
(where F = p1+p2+p3+p4+p5)
TID a1 a2 b1 b2 F
r3/t1 47 90 55
2.95
r1/t3 27 50 15
2.35
r5/t1 70

7. Rank-relational algebra
There was no way to express such query in relational algebra.
Extend relational algebra by adding rank as a first-class operation.
Based on the observations of first-class constructs (eg. selection), two requirements are needed to support ranking:
Splitting – Predicate-by-predicate rank evaluation.
Interleaving – Swapping rank operator with other operators (i.e. ranking is not only applied after filtering).

8. Ranking Principle
Def: Given a ranking function F and a set of evaluated predicates P={p1, p2, … , pn}, maximal-possible score of a tuple t is defined as:
Ranking Principle: If FP[t1] > FP[t2], then t1 must be ranked before t2.

9. Rank-Relation
Def: For monotonic scoring function F(p1, …, pn) and a subset P of {p1, …, pn}, a relation R augmented with ranking induced by P is called a rank-relation, denoted by RP.
Implicit attribute of RP is the score of tuple t, that is FP[t].
Order relationship of RP :
For all t1, t2 Є RP : t1 < RP t2 ↔ FP[t1] < FP[t2]

10. Operators of rank-relations
Rank (or μ) operator “adds” a predicate p to set P.
i.e. μp(RP) ≡ R P U{p}.
Example 2: μp1(R{p2}) ≡ R{p1, p2}, where F=∑(p1, p2, p3).
TID a1 a2 p1 p2 p3
F{p1, p2} r3 47 90
0.7
0.95
2.4 r2 17 60
0.6
0.5
0.65
2.1 r5 70
0.9
0.1
0.75
2.0 r4 87 10
0.8
0.35
1.9 r1 27 50
0.3
0.25
1.8

11. Extended operators

12. Example 3: Extended Join
πa1,a2,b2(σc (R{p1, p2 p3} JOIN T{p4, p5}))
SELECT r.a1, r.a2, t.b1
FROM R r, T t
WHERE c
ORDER-BY F
LIMIT 2
(F = ∑ P and c = r.a1+r.a2 < t.b1)
TID a1 a2 b2
F {p1, p2, p3, p4, p5}
r2/t4 17 60 99
2.45
r4/t4 87 10
1.95
r1/t4 27 50
1.75

13. Extended operators (cont’d)
Note:
Cartesian product is defined similarly to join, but not discussed in the paper.
Projection operator π has not changed.
Computation is based on both Boolean and ranking logical properties.
Perform Boolean operations and maintain the order induced by all given ranking predicates.

14. Equivalence relations
In the extended rank-relational model, ranking is a first-class construct.
Can derive algebraic equivalences from the definitions of operators (Proofs are omitted).
Example 4:
σc(RP) ≡ (σcR)P
RP1 ∩ TP2 ≡ (R ∩ T)P1 U P2
Thus, we can interleave the rank operator with other operators (i.e. push μ down across operators).

15. Equivalence relations (cont’d)

16. Equivalence relations (cont’d)
Note:
Proposition 1 states that ranking can be done in stages (i.e. one predicate at the time).
By Propositions 2, 3, and 4, the relations hold commutative and associative laws.
By Propositions 4 and 5, μ can be swapped with other operators.

17. Incremental execution
Blocking operators (eg. sort) lead to materialization of intermediate results.
Goal: To avoid materialization and implement a pipelining execution strategy.
We want to split rank computation into stages and to reduce the number of tuples considered in the upcoming stages.
We can output (i.e. advance to the next stage) a tuple t, whenever t has a score which is greater or equal to the score of any future tuple t′′ .

18. Incremental execution (cont’d)
Apply μp to RP and maintain priority queue ordered by P U{p}.
Let X = set of tuples from preceding stage.
Draw t′ from X.
If FP U{p}[t] ≥ FP[t′] and FP[t′] ≥ FP[t′′] for any future t′′ drawn from x,
then FP U{p}[t] ≥ FP U{p}[t′′] and t can be output (proceed to next stage).

19. Example 5: Top 2 of W idxScanp6(W) μp7 μp8 Given F = AVG(p6, p7, p8) 3
TID x p6 p7 p8 F w1 3
0.9
0.8
0.2
29/30 w2 7
0.7
0.1
14/15 w3 5
0.6
9/10 w4 1
0.5
0.4
5/6
TID x F w1 3
9/10 w2 7
5/6 w3 5
23/30 w4 1
19/30
TID x F w1 3
19/30 w4 1
3/5 w2 7
8/15 w3 5
7/15

20. Different evaluation plans
There exist algorithms to implement rank-aware operators as well as incremental evaluation.
Efficiency of query evaluation will now depend not only on the regular operators, but also on the rank-aware operators.
Due to algebraic equivalence laws, we can define additional evaluation plans.
Hence, we want a query optimizer to take additional execution plans into consideration.

21. Rank-aware optimizer Extended algebra  Extended search space.
Impact on enumeration algorithm:
Li et al. designed a 2-dimension enumeration algorithm: Dimension 1 = Join size, Dimension 2 = Ranking predicates.
The algorithm is exponential in both dimensions.
Heuristics applied to reduce search space.
Impact on cost model:
For ranking queries, it is more difficult to estimate the query cardinality of the intermediate results, whose accuracy is the core of the cost model.
Authors proposed to estimate cardinality by randomly sampling tuples.
Def: The term cardinality of a query refers to the number of rows in the query result whereas selectivity refers to the probability of a row being selected.

22. Critique
Erroneous examples.
No example of “tie-breaking” function.
Bad explanation of incremental evaluation.

23. Future research directions
Cardinality estimation: New/improved techniques for random sampling over joins.
Dynamically determined/chosen k.
Exploring physical properties of rank-aware execution plans.