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Table 2: Workflow verification algorithms Comparison.
Completeness Transformation Complexity
Sadiq-Orlowska No No O((|N| + |F|)
2
)
Lin-Zhao-Li-Chen Yes Yes O((|N| + |F|)
2
.|N|
2
)
Mahanti-Sinnakkrishnan Yes No O(|F|
2
)
Tour
´
e-Ba
¨
ına Yes No O(|K|
2
+ |F
′
|
2
)
Taking into account properties above and what is
exposed higher in this paper, we can affirm that (1) for
termination, our algorithm finishes by convergence
of the reduction and the traversal; (2) for correctness,
our algorithm answers YES when it is correct and NO
otherwise; (3) for completeness, our algorithm ver-
ifies the correctness of any graph, same complexes;
and (4) for transformation, our algorithm does not
transform the graph because it does not change nodes
order and does not introduce new node. Our algo-
rithm begins with the application of the four first rules
R
1
...R
4
reduce graph size. This choice takes into ac-
count the fact that if the workflow graph G
i
is the
graph obtained after application of i iterations of the
rules R
1
to R
4
on G, then: reduction rules do not pro-
duce structural conflicts, that means, if G
i
is correct
then G
i+1
is correct; reduction rules do not delete
structural conflicts, that means, if G
i
is incorrect then
G
i+1
is incorrect; and, If the application of rules R
1
to R
4
cannot reduce G to an empty graph then G
i
has
to be a graph split-join otherwise G will always be re-
duced to an empty graph. In case where graph is not
completely reducible, the rule R
5
allows to verify the
correctness of first level join nodes by making a partial
and localized traversal of the graph. For others join
nodes, we apply rules R
6
, R
7
or a simple deduction by
using predecessors correct join nodes behavior.
Algorithm complexity is O((|N| + |F|)
2
) where |N| +
|F| represents the number of nodes and transition in
the workflow graph (Sadiq and Orlowska, 2000), in
case the graph is completely reducible by rules R
1
to
R
4
. The average case complexity is much lower in
O((|N| + |F|)
2
), since the first iterations reduce radi-
cally workflow graph size.
In case where the graph is not reducible (split-join
graph), the worse case is that R
5
is the only applica-
ble rule to verify join nodes correctness. In that case,
we would be obliged to traverse the entire new graph.
But, as this case does not exist in a split-join graph
then our algorithm complexity is theoretically lower
than that of the graph traversal in depth algorithm.
Thus more efficient with a complexity O(|F
′
|
2
) where
|F
′
| is the number of transition between join first level
nodes and their split S
0
. In the final, in the worst
of the cases, reduction O(|K|
2
) + traversal O(|F
′
|
2
)
gives an algorithm in O(|K|
2
+ |F
′
|
2
) where |K| is the
number of reducible node and transition and |F
′
| is
the number of transition between first level join nodes
and their split S
0
. Table 2 compares verification algo-
rithms according to defined properties.
6 CONCLUSIONS
Through this paper you discovered a new effective
and complete algorithm that verifies the structural
correctness of any acyclic workflow graph by hy-
bridizing the graph reduction and traversal. In our
perspective, we work on our algorithm application for
cyclic workflow graphs structural verification through
a workflow graph structural verification tool.
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