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

4.3 Cost-Based Vectorization
the related complexity analysis, (2) we introduced the parameter λ in order to adjust the
computation approaches, and (3) we integrated this heuristic algorithm into our overall
cost-based optimization framework.
Optimality Analysis
As already mentioned, the optimality of the vectorized plan depends on (1) the costs
of the single operators, (2) the CPU utilization of each operator and (3) the available
resources (possible parallelism). However, the A-CPV only takes the costs from (1) into
consideration. Nevertheless, we can give optimality guarantees for this heuristic approach.
The algorithm can be parameterized with respect to the hardware resources (3). If
we want to force the single-threaded execution, we simply set λ to λ ≥ ∑ m
i=1 W (o i) −
max m i=1 W (o i). If we want to force the highest meaningful degree of parallelism (this is
not necessarily a full vectorization), we simply set λ = 0.
Now, assuming the given λ configuration, the question is, which optimality guarantee we
can give for the solution of the cost-based plan vectorization algorithm 10 . For this purpose,
R e (o i ) denotes the empirical CPU utilization (measured with a specific configuration) of an
operator o i with 0 ≤ R e (o i ) ≤ 1, and R o (o i ) denotes the maximal resource consumption of
an operator o i with 0 ≤ R o (o i ) ≤ 1. Here, R o (o i ) = 1 means that the operator o i exhibits
an average CPU utilization of 100 percent. In fact, the condition ∑ m
i=1 R e(o i ) ≤ 1 must
hold.
Obviously, for an instance-based plan P , we can write R e (o i ) = R o (o i ) because all
operators are executed in sequence and thus, do not influence each other. When we
vectorize P to a fully vectorized plan P ′ , with a maximum of R e (o ′ i ) = 1/m, we have to
compute the costs with W (o ′ i ) = R o(o i )/R e (o ′ i ) · W (o i). When we merge two execution
buckets b ′ i and b′ i+1 during cost-based plan vectorization, we compute the effective CPU
utilization R e (b ′′
i ) = 1/|b|, the maximal CPU utilization R o(b ′′
i ) = (W (b′ i )·R o(b ′ i )+W (b′ i+1 )·
R o (b ′ i+1 ))/(W (b′ i ) + W (b′ i+1 )), and the cost
⎧
R e (b
⎪⎨
′ i )
W (b ′′ R
i ) = e (b ′′
i ) · W (b′ i) + R e(b ′ i+1 )
R e (b ′′
i+1 ) · W (b′ i+1) R e (b ′′
i ) ≤ R o(b ′′
i )
R o (b ⎪⎩
′ i )
R o (b ′′
i ) · W (b′ i) + R o(b ′ i+1 )
(4.15)
R o (b ′′
i+1 ) · W (b′ i+1) otherwise.
We made the assumption that each execution bucket gets the same maximal empirical
CPU utilization R e (b ′′
i ), that resources are not exchanged between those buckets, and we
do not take the temporal overlap into consideration. However, we can give the following
guarantee, while optimality cannot be ensured due to the heuristic character of the A-CPV
(see Theorem 4.3).
Theorem 4.4. The A-CPV solves (1) the P-CPV, and (2) the constrained P-CPV with
guarantees of W (P ′′ ) ≤ W (P ) and W (P ′′ ) ≤ W (P ′ ) under the restriction of λ = 0.
Proof. As a precondition, it is important to note that, for the case of λ = 0, the A-CPV
cannot result in a plan with k = 1 (although this is a special case of the P-CPV) due to
the maximum rule of ∑ |b k |
c=1 W (o c) + W (o j ) ≤ max (Algorithm 4.3, line 10). Hence, in
10 For this optimality analysis, we use a slightly different notation of operators o i and execution buckets
b i in order to clearly distinguish the three different execution models. Let o i denote instance-based
execution, o ′ i denote full vectorized execution, and o ′′
i denote cost-based vectorized execution.
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