1. 1 Efficient QoS Partition and Routing of Unicast and Multicast Dean H.Lorenz,Ariel Orda,Danny Raz,Yuval Shavitt Proceeding of IWQoS 2000, Pittsburgh, PA, June 2000

2. 2 Outline Background,model and problems Pseudo-polynomial algorithms Approximation techniques Lower & upper bounds Conclusion

3. 3 Background Supporting QoS : –Routing : Finding a path or tree with minimum cost –Resource allocation : Mapping end-to-end QoS into local ones Problem : –How to provide the required QoS with minimum cost?

4. 4 Model Each link offers several QoS guarantees –Associated with different cost Integer cost functions –Delay and cost are integer Focus on additive QoS requirements –Harder than bottleneck ones

5. 5 QoS Routing Problem Given : –Network graph G(V,E) –End-to-end delay requirement D –link cost functions Objective : –Find minimum cost route Cost of optimal resource allocation

6. 6 QoS Partition Problem Given : –A route (path or tree) –End-to-end delay requirement D –link cost functions Objective : –Find delay requirement for each link With minimum cost Satisfy end-to-end delay requirement

7. 7 Cost –Ensuring a specific guarantee on a route Various considerations –Link perspective Resources reserved or consumption –Network perspective performance –User perspective pricing scheme Feasible cheapest route

8. 8 Cost Function General integer cost functions –The (delay,cost) pairs are integer –Always decreasing d C l (d) c d (d,c) D

9. 9 Outline Background,model and problems Pseudo-polynomial algorithms Approximation techniques Lower & upper bounds Conclusion

10. 10 Restricted Shortest Path (RSP) Problem Given : –Network graph : G(V,E) –End-to-end QoS requirement : D –Single delay and cost for each link –Upper bound of optimal cost value : U Find : –Minimum cost path that satisfies QoS requirement delay(p) D, p means a path

11. 11 Recursive Form of RSP Problem D(v,i) : –minimum delay from source to v with cost no more than i Recursive formula until : – s v (d l,c l ) l i-c l i N(v) : neighbors of v i : end-to-end cost

12. 12 Optimal QoS Partition & Routing (OPQR) Problem Given : –Network graph : G(V,E) –End-to-end requirement : D –Delay/cost function for each link –Upper bound of optimal value : U Find : –Minimum cost path p and partition that satisfies QoS requirement D

13. 13 OPQR Problem Idea –View each link l as set of links –Corresponding all possible cost 1, …,U –Delay Minimum value achieve specified cost l l1l1 l2l2 lUlU

14. 14 Recursive Form of OPQR Problem Recursive formula until : –for j=1,2, …,i l i (1,d l (1)) (i,d l (i)) i-j N(v) : neighbors of v i : end-to-end cost s v

15. 15 Multicast Optimal QoS Partition(M- OPQ) Problem Given : –A tree :T –Delay/cost function for each link : –End-to-end delay requirement : D Find : –Optimal partition satisfying end-to-end delay requirement

16. 16 M-OPQ Problem Idea : –Use same technique in OPQR problem Notation : –X,Y,Z are tables holding best delay for each cost –Size is U s t1 t2 x y z

17. 17 Merge Procedure Find best allocation between Two branching sub-trees A sub-tree and its root link Merge two branching sub-trees –For c=1, …,U Merge sub-tree and its root link – For c=1, …,U

18. 18 Outline Background,model and problems Pseudo-polynomial algorithms Approximation techniques Lower & upper bounds Conclusion

19. 19 Logarithmic Sampling and Linear Scaling Idea –Log Sampling Improve methods of OPQR Check delays only corresponding to specific costs Specific costs are 1,,, …,U –Scaling Applied to all costs Smaller costs for OPQR problem Scale factor :

20. 20 Outline Background,model and problems Pseudo-polynomial algorithms Approximation techniques Lower & upper bounds Conclusion

21. 21 Finding Upper & Lower Bounds Test procedure Test( ): –Check whether is a valid upper bound General idea –Call Test( ) for = {1,2,4,8, … } –For some Test( ) return fail and Test(2 ) succeed –f-Approximated test procedure : Bound C by  C  f(2 )

22. 22 Test Procedure TEST ( ) For each link e –Set d e ( )  min{ d | c e (d)   } –Put on each link Find Shortest-Path p w.r.t. {d e ( )} If Delay(p)  D C  n = f( ) Else < C

23. 23 Outline Background,model and problems Pseudo-polynomial algorithms Approximation techniques Lower & upper bounds Conclusion

24. 24 Conclusion Characteristics : –Establish fully polynomial approximation schemes for problem OPQR –First FPAS for problem M-OPQ Future Works : –Cost model –Multicast routing problem under this framework