Cost-Based Optimization of Integration Flows - Datenbanken ...

6 On-Demand Re-Optimization
Join Enumeration Example
Recall our join enumeration heuristic for the optimization technique WD10: Join Enumeration,
presented in Subsection 3.3.3, where we assumed a left-deep join tree (R ⋊⋉ S) ⋊⋉ T
of n = 3 data sets (without cross products and only one join implementation in the form
of a nested loop join) with the following n! = 6 possible plans:
P a (opt) : (R ⋊⋉ S) ⋊⋉ T P c : (R ⋊⋉ T ) ⋊⋉ S P e : (S ⋊⋉ T ) ⋊⋉ R
P b : (S ⋊⋉ R) ⋊⋉ T P d : (T ⋊⋉ R) ⋊⋉ S P f : (T ⋊⋉ S) ⋊⋉ R.
The costs of a (nested loop) join are computed by C(R ⋊⋉ S) = |R| + |R| · |S|. Further,
the join output cardinality can be derived by |R ⋊⋉ S| = f R,S · |R| · |S| with a join filter
selectivity of f R,S = |R ⋊⋉ S|/(|R| · |S|). Thus, the costs of the complete plan (R ⋊⋉ S) ⋊⋉ T
are given by C((R ⋊⋉ S) ⋊⋉ T ) = |R| + |R| · |S| + f R,S · |R| · |S| + f R,S · |R| · |S| · |T |.
Assuming variable selectivities and cardinalities, the optimality conditions for arbitrary
left-deep join trees (see Figure 6.9(a)) are specified as follows. First, fix two base relations
with the commutativity optimality condition of oc 1 : |R| ≤ |S|. Second, the optimality of
executing R ⋊⋉ S before ∗ ⋊⋉ T is given if the following optimality condition holds:
oc 2 : |R| + |R| · |S| + f R,S · |R| · |S| + f R,S · |R| · |S| · |T |
≤|R| + |R| · |T | + f R,T · |R| · |T | + f R,T · |R| · |T | · |S|.
oc ′ 2 : |S| + f R,S · |S| + f R,S · |S| · |T | ≤ |T | + f R,T · |T | + f R,T · |T | · |S|,
(6.5)
where oc 2 has been algebraically simplified to oc ′ 2 by subtracting |R| and subsequently
dividing by |R|. Note that it is possible to monitor all cardinalities |R|, |S|, and |T |
but only the selectivities f R,S and f (R⋊⋉S),T . To estimate f R,T , we need to derive it with
f R,T θ f (R⋊⋉S),T , where θ is a mapping function representing the correlation. If we assume
statistical independence of selectivities, we can set f R,T = f 20 (R⋊⋉S),T .
oc 2
oc 3
oc 4
V
W
R S * T
|R| |S| |R S| |T| |* T|
T
fR,S
fR,T=f(R ⋈ S),T
R S
oc 1
(a) Optimality Conditions
≤
(oc1)
C1
≤
(oc’2)
C2
(b) Example POT
(c) Complexity Analysis
Figure 6.9: Example Join Enumeration
The PlanOptTree for this example is illustrated in Figure 6.9(b). Here, the PlanOptTree
contains the two mentioned optimality conditions, which use a hierarchy of atomic and
complex statistics. All input cardinalities of base relations and the output cardinalities
of join operators are used in the form of atomic statistic nodes. The join selectivities
(complex statistic nodes) are computed from the atomic statistics. Both terms of the
inequality oc ′ 2 are computed using atomic and complex statistics.
20 Another approach would be to use serial histograms [Ioa93] or exact frequency matrices [Pol05].
182