1. Abstract Temporal Graph
Minimization Algorithms

2. minimize_full (ATG)
1. ConvertFull (ATG)
2. PropagateFull(ATG)

3. ConvertFull(ATG)
/* Nodes n1, n2, …., nm */
for i=1 to m
for j=1 to m
if i=j add self-referencing arc at ni
with label {self_ref}
else if there is an arc from ni to nj
do nothing
else if there is an arc from nj to ni
add an arc from ni to nj
with label tcji-1
else add arc from ni to nj set to C

4. The critical step is executed m3 times
PropagateFull(ATG)
/* Nodes n1, n2, …., nm */
repeat
for k=1 to m
for i=1 to m
for j=1 to m
/* critical step */
tcij ← match(tcij, propagate(tcik,tckj))
if tcij = {} a conflict is raised
until no arc label is reduced /* no change */
The critical step is executed m3 times

5. if i=j remove self-referencing arc at ni
PropagateFull cont
/* remove self-referencing and inverse arcs */
for i=1 to m
for j=i to m
if i=j remove self-referencing arc at ni
else tcij ← match(tcij,tcji-1)
if tcij = {} a conflict is raised
else delete arc from nj to ni

6. ni nk nj

7. minimize_ordered (ATG)
1. ConvertOrdered (ATG)
2. PropagateOrdered(ATG)

8. ConvertOrdered(ATG)
/* Let node ordering be: n1, n2, …., nm */
for i=1 to (m-1)
for j=i+1 to m
if nodes ni to nj are unconnected
add an arc from ni to nj with label C
else if there is only an arc from ni to nj
do nothing
else if there is only an arc from nj to ni
add arc from ni to nj with label tcji-1
and delete the arc from nj to ni
else tcij ← match(tcij,tcji-1)
if tcij = {} a conflict is raised
else delete arc from nj to ni

9. The critical step is executed (m3 – 3m2 + 2m)/6 times
PropagateOrdered(ATG)
/* Let node ordering be: n1, n2, …., nm */
repeat
/* for every intermediate node nk */
for k=2 to (m-1)
/* for every incoming arc to nk */
for i=1 to (k-1)
/* for every outgoing arc from nk */
for j=(K+1) to m
/* critical step */
tcij ← match(tcij, propagate(tcik,tckj))
if tcij = {} a conflict is raised
until no arc label is reduced /* no change */
The critical step is executed (m3 – 3m2 + 2m)/6 times

10. Point of caution:
In the fully connected ATG there are at least two distinct paths between every pair of nodes and the propagation is bidirectional.
In the ordered ATG the propagation is unidirectional and thus the labels of the basic arcs stay invariant under the propagation n1 n2 n3

11. C = {<, =, >} self_ref is set to = Transitivity Table
Example of a simple ATG – Instantaneous events and relative constraints
C = {<, =, >}
self_ref is set to =
Transitivity Table
Cik
cij
<  =
> 
<
<, =, >
Ckj

12. A simple example temporal graph
<  N1 N2 = = N5 N3 N4
<, >
< 

13. Original Temporal Graph
n1 ---> n2 { < }
n1 ---> n3 { = }
n2 ---> n4 { = }
n3 ---> n4 { < }
n4 ---> n5 { < > }

14. Fully Connected Temporal Graph

15. Minimized Temporal Graph
n1 ---> n2 { < }
n1 ---> n3 { = }
n1 ---> n4 { < }
n1 ---> n5 { < = > }
n2 ---> n3 { > }
n2 ---> n4 { = }
n2 ---> n5 { < > }
n3 ---> n4 { < }
n3 ---> n5 { < = > }
n4 ---> n5 { < > }

16. <  N1 N2 = = N5 N3 N4
<, >
< 

17. Original Temporal Graph
n1 ---> n2 { < }
n1 ---> n3 { = }
n2 ---> n4 { = }
n3 ---> n4 { < }
n4 ---> n5 { < > }

18. Ordered Temporal Graph
n1 ---> n2 { < }
n1 ---> n3 { = }
n1 ---> n4 { < = > }
n1 ---> n5 { < = > }
n2 ---> n3 { < = > }
n2 ---> n4 { = }
n2 ---> n5 { < = > }
n3 ---> n4 { < }
n3 ---> n5 { < = > }
n4 ---> n5 { < > }

19. Minimized Temporal Graph
n1 ---> n2 { < }
n1 ---> n3 { = }
n1 ---> n4 { < }
n1 ---> n5 { < = > }
n2 ---> n3 { < = > }
n2 ---> n4 { = }
n2 ---> n5 { < > }
n3 ---> n4 { < }
n3 ---> n5 { < = > }
n4 ---> n5 { < > }

20. Another simple example temporal graph
<  N1 N2 =
<  N3

21. Original Temporal Graph
n1 ---> n2 { < }
n1 ---> n3 { = }
n2 ---> n3 { < }

22. Temporal Graph
n1 ---> n1 { = }
n1 ---> n2 { < }
n1 ---> n3 { = }
n2 ---> n1 { > }
n2 ---> n2 { = }
n2 ---> n3 { < }
n3 ---> n1 { = }
n3 ---> n2 { > }
n3 ---> n3 { = }
Conflict detected between nodes n2 and n3

23. Another Example Diagnosis reached (n1)
Discussion about therapy (n2 – n3)
Consensus on best therapy regime (n4)
Patient agreed on proposed therapy (n5)
Application of therapy (n6 – n7)
Adjustment of some therapy parameters (n8)
Patient monitoring (n9 – n10)

24. Original Temporal Graph
n1 ---> n3 { < = }
n2 ---> n1 { < = }
n4 ---> n3 { = > }
n5 ---> n4 { = > }
n6 ---> n5 { = > }
n8 ---> n6 { > }
n8 ---> n7 { < }
n9 ---> n2 { < = }
n10 ---> n7 { > }

25. Derived Temporal Relations
n1 ---> n2 { = > }
n1 ---> n3 { < = }
n1 ---> n4 { < = }
n1 ---> n5 { < = }
n1 ---> n6 { < = }
n1 ---> n7 { < }
n1 ---> n8 { < }
n1 ---> n9 { = > }
n1 ---> n10 { < }
n2 ---> n3 { < = }
n2 ---> n4 { < = }
n2 ---> n5 { < = }
n2 ---> n6 { < = }
n2 ---> n7 { < }
n2 ---> n8 { < }
n2 ---> n9 { = > }
n2 ---> n10 { < }

26. Derived Temporal Relations cont
n3 ---> n4 { < = }
n3 ---> n5 { < = }
n3 ---> n6 { < = }
n3 ---> n7 { < }
n3 ---> n8 { < }
n3 ---> n9 { = > }
n3 ---> n10 { < }
n4 ---> n5 { < = }
n4 ---> n6 { < = }
n4 ---> n7 { < }
n4 ---> n8 { < }
n4 ---> n9 { = > }
n4 ---> n10 { < }
n5 ---> n6 { < = }
n5 ---> n7 { < }
n5 ---> n8 { < }
n5 ---> n9 { = > }
n5 ---> n10 { < }

27. Derived Temporal Relations cont
n6 ---> n7 { < }
n6 ---> n8 { < }
n6 ---> n9 { = > }
n6 ---> n10 { < }
n7 ---> n8 { > }
n7 ---> n9 { > }
n7 ---> n10 { < }
n8 ---> n9 { > }
n8 ---> n10 { < }
n9 ---> n10 { < }