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

4.2 Plan Vectorization
Receive (o1)
[service: s5, out: msg1]
Assign (o2)
[in: msg1, out: msg2]
Invoke (o3)
[service: s4, in: msg2, out: msg3]
Join (o4)
[in: msg1,msg3, out: msg4]
Assign (o5)
[in: msg4, out: msg5]
Invoke (o6)
[service s3, in: msg5]
D
δ msg1
D
δ msg2
D
δ msg3
D
δ msg4
D
δ msg5
δ msg1
D
D (set of dependencies)
δ D msg1 (o 2 o 1,q 1)
δ D msg1 (o 4 o 1,q 2)
δ D msg2 (o 3 o 2,q 3)
δ D msg3 (o 4 o 3,q 4)
δ D msg4 (o 5 o 4,q 5)
δ D msg5 (o 6 o 5,q 6)
Copy (oc)
[in: msg1, out: msg1]
Assign (o2)
[in: msg1, out: msg2]
Invoke (o3)
[service: s4, in: msg2, out: msg3]
Join (o4)
[in: msg1,msg3, out: msg4]
Assign (o5)
[in: msg4, out: msg5]
Invoke (o6)
[service s3, in: msg5]
(a) Dependency Graph DG(P 2) of Plan P 2
(b) Vectorized Plan P ′ 2
Figure 4.5: Example Plan Vectorization
we create a queue q i for each dependency and connect the operators with these queues.
Finally, we simply remove all temporal dependencies and get P ′ .
Although the A-PV is only executed once during the initial deployment of an integration
flow, it is important to note its time complexity with increasing number of operators m.
Theorem 4.1. The A-PV exhibits a cubic worst-case time complexity of O(m 3 ).
Proof. Basically, we prove the complexity for the two algorithm parts: the dependency
analysis and the graph creation. Assume an operator sequence o. We fix the number of
operators m with m = |o|. Then, an arbitrary operator o i with 1 ≤ i ≤ m can—in the
worst case—be the target of i − 1 data dependencies δi − , and it can be the source of m − i
data dependencies δ
i + . Based on the equivalence of δ− = δ + and thus, |δ − | = |δ + |, there
are at most
m∑
(i − 1) =
i=1
m−1
∑
i=1
i =
m · (m − 1)
2
(4.4)
dependencies between arbitrary operators of this sequence. Hence, the dependency analysis
is computable with quadratic complexity of O(m 2 ).
When evaluating operator o i , there are at most ∑ m−1
i=1
i = m · (m − 1)/2 (for cases
i = m − 1 and i = m) dependencies in D. During the graph creation, for each operator,
each dependency d ∈ D must be evaluated in order to connect queues and execution
buckets. In summary, for all operators, we must check at most
m∑
i=1 j=1
i∑
(m − j) = m ·
m · (m + 1)
2
= m2 · (m + 1)
2
+
−
m∑
i · (m − i − 1)
i=1
m∑
i 2 −
i=1
m∑
m · i +
i=1
m∑
i=1
i = m3 − m
3
(4.5)
dependencies for graph creation. Hence, the algorithm part of graph creation is computed
with cubic complexity of O(m 3 ). In summary, the plan vectorization algorithm exhibits a
cubic worst-case time complexity of O(m 3 ) = O(m 3 +m 2 ). Hence, Theorem 4.1 holds.
Note that this is the complexity analysis of our A-PV algorithm, while the P-PV problem
can be solved with quadratic time complexity of O(m 2 ). For example, one can use
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