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

4.3 Cost-Based Vectorization
Proof. The problem of finding all possible plans of m operators that are not connected by
any data dependencies is reducible to the known problem of finding all possible partitions
of a set with m members, where the Bell’s numbers B m [Bel34a, Bel34b] represent the
total number of partitions. Note that similar to this known problem, we exclude the plan
with zero operators (B 0 = B 1 = 1). Thus, the number of possible plans can be recursively
computed by
|P ′′ | = B m =
m−1
∑
j=0
( m − 1
j
)
· B j . (4.13)
Furthermore, each Bell number is the sum of Stirling numbers of the second kind [Jr.68].
As a result, we are able to determine the number of plans |P ′′ | k for a given k by
m∑
B m = S(m, k) with S(m, k) = 1 k∑
( )
(−1) k−j k
j m
k!
j
k=0
j=0
|P ′′ | k = 1 k∑
( ) (4.14)
(−1) k−j k
j m
k!
j
j=0
In addition, many asymptotic limits for Bell numbers are known [Lov93]. However, in
general, we can state that the Bell numbers grow in O(2 cm ), where c is a constant factor.
Due to the linear complexity of O(m) for determining the costs of a plan, the cost-based
plan vectorization problem exhibits an exponential worst-case overall time complexity of
O (2 m ) = O (m · 2 cm ). Hence, Lemma 4.2 holds.
Now, we can combine the results for the best and the worst case to the general result.
Theorem 4.2. The cost-based plan vectorization problem exhibits an exponential time
complexity of O(2 m ).
Proof. The cost-based plan vectorization problem exhibits an exponential time complexity
of O(2 m ) for both the best-case plan (Lemma 4.1) and the worst-case plan (Lemma 4.2).
Hence, Theorem 4.2 holds.
4.3.2 Computation Approach
So far, we have analyzed the search space of the P-CPV. We now explain how the optimal
plan is computed with regard to the current execution statistics. In detail, we present
an exhaustive computation approach (thus, with exponential time complexity) as well
as a heuristic with linear time complexity that is used within our general cost-based
optimization framework. For simplicity of presentation, we use the sequence of operators.
However, the general version of the exhaustive and heuristic computation approaches
use recursive algorithms that contain many specific cases for arbitrary combinations of
subplans (sequences and sets) as well as cases for complex operators.
Exhaustive Computation Approach
The exhaustive computation approach has the following three steps:
1. Scheme Enumeration: Enumerate all 2 m−1 possible plan distribution schemes for
the sequence of operators.
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