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

4 Vectorizing Integration Flows
order to prove the theorem, we need to prove the two single claims of W (P ′′ ) ≤ W (P )
and W (P ′′ ) ≤ W (P ′ ).
For the proof of W (P ′′ ) ≤ W (P ), assume the worst case, where ∀o i : R o (o i ) = 1. If we
vectorize this to P ′′ , we need to compute the costs by W (b ′′
i ) = (R o(b ′′
i ))/(R e(b ′′
i )) · W (o i)
with R e (b ′′
i ) = 1/|b|. Due to the vectorized execution, W (P ′′ ) = max m i=1 W (b′′ i ), while
W (P ) = ∑ m
i=1 W (o i). Hence, we can write W (P ′′ ) = W (P ) if the condition ∀o i : R o (o i ) =
1 holds. This is the worst case. For each R o (o i ) < 1, we get W (P ′′ ) < W (P ).
In order to prove W (P ′′ ) ≤ W (P ′ ), we fix λ = 0. If we merge two buckets b i and b i+1 , we
see that R e (b ′′
i ) is increased from 1/|b| to 1/(|b|−1). Thus, we re-compute the costs W (b′′ i )
as mentioned before. In the worst case, W (b ′′
i ) = W (b′ i ), which is true iff R e(b ′ i ) = R o(b ′ i )
because then we also have R e (b ′′
i ) = R e(b ′ i ). Due to W (P ′′ ) = max m i=1 W (b′′ i ), we can state
W (P ′′ ) ≤ W (P ). Hence, Theorem 4.4 holds.
In conclusion, we cannot guarantee that the result of the A-CPV is the global optimum
because we cannot efficiently evaluate the effective resource consumption. However, we can
guarantee that each merging of execution buckets when solving the P-CPV with λ = 0
(where the costs of each bucket are lower than or equal to the highest operator costs)
improves the performance of the plan P .
4.3.3 Cost-Based Vectorization with Restricted Number of Buckets
Due to dynamically changing workload characteristics, we recommend using the cost-based
vectorization approach. However, there might exist scenarios where an explicit restriction
of k and thus, of the number of threads, is advantageous. Hence, in this subsection, we
discuss the necessary changes of the exhaustive and heuristic computation approaches
when using this constraint.
Exhaustive Computation Approach
With regard to the exhaustive cost-based computation approach (see Subsection 4.3.2),
only minor changes are required when restricting k. Due to the restricted number of
execution buckets, k = |b|, the search space is smaller than for the previously described
P-CPV. As already stated, for an operator sequence (best case), there are
|P ′′ | k =
k−1
∏
i=1
m − i
i
different possibilities, while for sets of operators, there are
|P ′′ | k = 1 k!
j=0
(4.16)
k∑
( )
(−1) k−j k
j m (4.17)
j
possibilities to distribute the m operators of plan P across k buckets. Hence, the enumeration
of candidate distribution schemes can be reused by simply invoking the recursive
Algorithm 4.2 only once for the given k. In addition, we change the optimality condition
for evaluating those candidates to
⎛ ⎛ ⎞⎞
l bi
φ = min ⎝ max
|b|=k ∑
⎝ W (o j ) ⎠⎠ , (4.18)
i=1
j=1
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