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

6.3 Re-Optimization
Algorithm 6.3 PlanOptTree Trigger Re-Optimization (A-PTR)
Require: invalid optimality conditions OC
1: clear memo
2: for all oc ∈ OC do // for each OCNode
3: for all oc1 ∈ oc.op1.ocnodes do // for each OCNode of operand 1
4: if oc1.θ = oc.θ and oc1.op2 = oc.op1 then
5: if ¬oc1.isOptimal(oc1.op1.agg, oc.op2.agg) then
6: OC ← OC ∪ oc1
7: rCheckTransitivity(oc, oc1.op1, left)
8: for all oc2 ∈ oc.op2.ocnodes do // for each OCNode of operand 2
9: if oc2.θ = oc.θ and oc2.op1 = oc.op2 then
10: if ¬oc2.isOptimal(oc2.op2.agg, oc.op1.agg) then
11: OC ← OC ∪ oc2
12: rCheckTransitivity(oc, oc2.op2, right)
13: PPOT ← optimizePlan(ptid, OC) // apply directed re-optimization
14: A-PPR(PPOT, OC) // update PlanOptTree
incrementally updated (line 14). It is important to note that the directed re-optimization
of operators involved in violated optimality conditions is equivalent to full re-optimization.
The directed re-optimization relies on monotonic cost functions. This ensures that no
local suboptima exist and that we will find the global optimum. The Picasso project
[RH05] showed that this assumption holds for complete cost diagrams over multiple alternative
plans of most queries [HDH07, HDH08, DBDH08]. In contrast, it always holds
for our cost model of integration flows with regard to a single plan (see Subsection 3.2.2).
However, due to the possibility of arbitrarily complex optimality conditions, even without
this property, we could still guarantee to trigger full re-optimization if a better plan exists.
Theorem 6.2 (Directed Re-Optimization). The directed re-optimization for all operators
o ′ ∈ P that have been identified by violated optimality conditions oc ′ of a PlanOptTree is
equivalent to the full re-optimization of all operators o ∈ P .
Proof. Assume all dependencies between operators o of plan P to be a directed graph
G = (V, A) of vertexes (operators) and arcs (dependencies). Then, the re-optimization of
P is a graph homomorphism f : G → H. In order to prove Theorem 6.2, we show that
∀o i /∈ o ′ (
: vpre(oi ) ∈ G ≡ v pre(oi ) ∈ H ) ∧ ( v suc(oi ) ∈ G ≡ v suc(oi ) ∈ H ) , (6.2)
where v pre(oi ) denotes the set of predecessors of operator o i and v suc(oi ) denotes the set of
successors of o i . (1) If there exists a homomorphism f : G → H such that
v j ≺ o i ∈ G ∧ o i ≺ v j ∈ H, (6.3)
then, the order v j ≺ o i is represented by an optimality condition oc with o i , v j ∈ oc or by a
transitive optimality condition toc with o i , v j ∈ toc. The same is true for successors of o i .
(2) All used cost functions are known to be monotonically non-decreasing w.r.t. the input
statistics. Hence, during re-optimization, f : G → H, the globally optimal solution will
be found. (3) Further, all operators o ′ included in violated optimality conditions ∀o i ∈ oc ′
or transitive optimality conditions ∀o i ∈ toc ′ are used by f : G → H. As a result,
∄ ( o i /∈ o ′ ∧ (( v pre(oi ) ∈ G ≠ v pre(oi ) ∈ H ) ∨ ( v suc(oi ) ∈ G ≠ v suc(oi ) ∈ H ))) , (6.4)
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