1. An Efficient Algorithm for the Computation of Optimum Paths in Weighted Graphs Radu Iosif, Marius Bozga, Vassiliki Sfyrla Univ. de Grenoble ’’Joseph Fourier’’ & VERIMAG MEMICS 2007 October 2007 Znojmo(CZ)

2. Verimag Leading research center in embedded systems Contributes in synchronous languages, verification, testing and modelling

3. Verimag Leading research center in embedded systems Contributes in synchronous languages, verification, testing and modelling Verification ’’Are we building the product right? ’’ or ’’Does the product conform to the specifications? ’’

4. Verification using counter automata  Counter automata represents an important class of infinite-state systems  used for verification of protocols, software, etc…  Flat counter automata are a subclass of counter automata  without nested loops  reachability and termination are decidable  In particular, reachability problem of flat counter automata is related to finding the optimal path of given length in weighted graphs [CJ’98, BIL’06] x=x+1 A BCD y=y+2 x=0 Y=0 [x>y]

5. Description of the problem (1) Weighted Directed Graph G(V,E,w) Vertices V := {A,B,C,D,E} Edges E := {AB, BC, CD, DE, …} integer weight w := {w(AB)=1, w(BC)=2,… }

6. Description of the problem (2) Which is the weight of the optimal path from the node A to the node E in 1 step ? in 2 steps ? in 3 steps ? … in n steps ?

7. Description of the problem (3) LengthPathsOptimal Path Weight 1-

8. Description of the problem LengthPathsOptimal Path Weight 1- 2- Description of the problem (3)

9. Description of the problem LengthPathsOptimal Path Weight 1- 2- 3ACDE 5 Description of the problem (3)

10. LengthPathsOptimal Path Weight 1- 2- 3ACDE 5 4ABCDE ACCDE ABCDE6 Description of the problemDescription of the problem (3)

11. LengthPathsOptimal Path Weight 1- 2- 3ACDE 5 4ABCDE ACCDE ABCDE6 5 ABACDE ACDCDE ACCCDE ACDCDE8 Description of the problemDescription of the problem (3)

12. LengthPathsOptimal Path Weight 1- 2- 3ACDE 5 4ABCDE ACCDE ABCDE6 5 ABACDE ACDCDE ACCCDE ACDCDE8 … nw Description of the problemDescription of the problem (3)

13. Known Solution  The set can be expressed using Presburger arithmetic  This method is general but costly the size of the formula is linear to the size of the graph there are no efficient methods to simplify it

14. Our approach (1)  finds simultaneously all sets as the least fixpoint solution of the following system of equations: Definitions:

15. Our approach (2) Solving the system of equations iteratively  termination is not guaranteed  presence of loops in the weighted graph => accelerate, consider an augmented system, where loops are taken into account at every step:

16. However, ’’optimal ratio loops’’ are enough to ensure termination  in principle, much fewer than « all » loops Our approach (2) Solving the system of equations iteratively  termination is not guaranteed  presence of loops in the weighted graph => accelerate, consider an augmented system, where loops are taken into account at every step:

17. Implementation  Represent sets as semilinear sets on : where  Finding optimal ratio loops using the Howard Algorithm[TCGG’98]  Implementing the min operator using rewriting rules on semilinear sets

18. Conclusions  We introduce an innovative solution for the computation of the weight of the optimal paths of given length in weighted graphs  The results provide a technique for the analysis of flat counter automata  The method has been implemented in Java and the results are very satisfying (to be released soon)