1. Algorithmic Approaches for Efficient Enumeration of Candidate p-Cycles and Capacitated p-Cycle Network Design John Doucette 1,2, Donna He 3, Wayne D. Grover 1,2, and Oliver Yang 3 1 TRLabs, Edmonton, AB, Canada 2 University of Alberta, Edmonton, AB, Canada 3 CNNR Lab, SITE, University of Ottawa, Ottawa, ON, Canada john.doucette@trlabs.ca, dhE098@site.uottawa.ca, grover@trlabs.ca, yang@site.uottawa.ca DRCN 2003 Banff, AB, Canada 19-22 October 2003

2. John Doucette, Donne He, Wayne D. Grover, Oliver Yang DRCN 2003, Banff, AB, Canada, 19-22 October 2003 2 Algorithmic Approaches for Efficient Enumeration of Candidate p-Cycles and Capacitated p-Cycle Network DesignIntroduction ILP methods of p-cycle network design –Requires enumeration of a large sample set of cycles, even with pre-selection methods –The number of cycles in a network grows exponentially –A suitably sized cycle set is difficult to produce for large networks, and ILP computation time suffers Hence motivation to study heuristic methods –Need a fast and simple method of producing a good cycle set –Need an efficient near-optimal algorithm for p-cycle network design

3. John Doucette, Donne He, Wayne D. Grover, Oliver Yang DRCN 2003, Banff, AB, Canada, 19-22 October 2003 3 Algorithmic Approaches for Efficient Enumeration of Candidate p-Cycles and Capacitated p-Cycle Network Design A Priori p-Cycle Efficiency: AE(p) AE(p) measures a cycle’s potential to provide protection relationships for working channels X p,i = 1 if on cycle X p,i = 2 if straddler S S,p = 3 S C,p = 9 AE(p) = 1.67 S S,p = 4 S C,p = 10 AE(p) = 1.80 S C,p = # on cycle S S,p = # straddlers c i = unit cost of i

4. John Doucette, Donne He, Wayne D. Grover, Oliver Yang DRCN 2003, Banff, AB, Canada, 19-22 October 2003 4 Algorithmic Approaches for Efficient Enumeration of Candidate p-Cycles and Capacitated p-Cycle Network Design Straddling Link Algorithm (Zhang and Yang, ICC 2002) For each span in the network: –Find the shortest pair of node-disjoint routes connecting its end-nodes (excluding the span itself) –Construct a cycle by combining those two routes A priori efficiency is: Advantages: –Avoids enumeration of a large number of cycles Disadvantages: –Cycles are inefficient and collectively ill suited for capacitated designs (most are single-straddler cycles)

5. John Doucette, Donne He, Wayne D. Grover, Oliver Yang DRCN 2003, Banff, AB, Canada, 19-22 October 2003 5 Algorithmic Approaches for Efficient Enumeration of Candidate p-Cycles and Capacitated p-Cycle Network Design Straddling Link Algorithm (Illustration) Sample SLA results: –Eight spans have a primary cycle –Seven unique cycles with one straddler by intent, two sometimes by side-effect (a)(b)(c)(d) (e)(f)(g)(h)

6. John Doucette, Donne He, Wayne D. Grover, Oliver Yang DRCN 2003, Banff, AB, Canada, 19-22 October 2003 6 Algorithmic Approaches for Efficient Enumeration of Candidate p-Cycles and Capacitated p-Cycle Network Design SP-Add Operation to improve candidate cycles Replace a span on an existing cycle with the shortest cycle-disjoint route connecting its end- nodes. 1 2 345 6 7 8 9 10 11 SP-Add (5-6) 1 2 345 6 7 8 9 10 11 1 2 345 6 7 8 9 10 11 SP-Add (6-7) 1 2 345 6 7 8 9 10 11

7. John Doucette, Donne He, Wayne D. Grover, Oliver Yang DRCN 2003, Banff, AB, Canada, 19-22 October 2003 7 Algorithmic Approaches for Efficient Enumeration of Candidate p-Cycles and Capacitated p-Cycle Network Design SP-Add Operation (2) Advantages: –Cycle sets have higher average AE(p) than SLA Disadvantages: –Cycle sets produced are still too inefficient and collectively ill suited for capacitated designs. Only AE is being improved, not actual design efficiencies.

8. John Doucette, Donne He, Wayne D. Grover, Oliver Yang DRCN 2003, Banff, AB, Canada, 19-22 October 2003 8 Algorithmic Approaches for Efficient Enumeration of Candidate p-Cycles and Capacitated p-Cycle Network Design Expand Operation using SP-Add Visit each span in the original cycle and perform SP-Add operation until no longer possible. 1 2 345 6 7 8 9 10 11 SP-Add (6-9) 1 2 345 6 7 8 9 10 11 SP-Add (7-10) SP-Add (7-10) 1 2 345 6 7 8 9 10 11 Expand equivalent AE(p) = 1.57AE(p) = 1.0 Expand 1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11

9. John Doucette, Donne He, Wayne D. Grover, Oliver Yang DRCN 2003, Banff, AB, Canada, 19-22 October 2003 9 Algorithmic Approaches for Efficient Enumeration of Candidate p-Cycles and Capacitated p-Cycle Network Design Grow Operation using SP-Add Perform the SP-Add operation and iterate on the new cycles until no further iterations are possible. 1 2 345 6 7 8 9 10 11 SP-Add (6-7) 1 2 345 6 7 8 9 10 11 SP-Add (7-10) SP-Add (7-10) 1 2 345 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11 Grow equivalent AE(p) = 1.67AE(p) = 1.33 Grow 1 2 3 4 5 6 7 8 9 10 11

10. John Doucette, Donne He, Wayne D. Grover, Oliver Yang DRCN 2003, Banff, AB, Canada, 19-22 October 2003 10 Algorithmic Approaches for Efficient Enumeration of Candidate p-Cycles and Capacitated p-Cycle Network Design Expand and Grow Operations: Analysis The Expand operation results in 10.3% and 17.9% increases in AE(p) versus SLA. The Grow operation results in 24.8% and 41.4% increases in AE(p) versus SLA.

11. John Doucette, Donne He, Wayne D. Grover, Oliver Yang DRCN 2003, Banff, AB, Canada, 19-22 October 2003 11 Algorithmic Approaches for Efficient Enumeration of Candidate p-Cycles and Capacitated p-Cycle Network Design Demand-weighted p-Cycle Efficiency: E w (p) E w (p) measures a cycle’s actual efficiency in providing protection relationships for uncovered working channels 1 3 2 32 1 4 1 2 2 2 4 AE(p) = 1.67 E w (p) = 3.78 1 3 2 3 2 3 2 1 2 2 2 4 AE(p) = 1.67 E w (p) = 3.67 X p,i = 1 if on cycle X p,i = 2 if straddler w i = working on i c i = unit cost of i

12. John Doucette, Donne He, Wayne D. Grover, Oliver Yang DRCN 2003, Banff, AB, Canada, 19-22 October 2003 12 Algorithmic Approaches for Efficient Enumeration of Candidate p-Cycles and Capacitated p-Cycle Network Design Capacitated Iterative Design Algorithm CIDA provides an iterated placement of p-cycles on a capacitated network based on E w (p). (1) It can be implemented with any pre-selected eligible p-cycle set. 1.Start with initial eligible cycle set (say by using Expand or Grow with SLA primary cycles) 2.Calculate E w (p) for each eligible cycle and rank 3.Place one copy of cycle with highest E w (p) 4.Update w i on each on-cycle and straddling span 5.Iterate steps 2 to 4 until all w i = 0 (1) W. D. Grover, Mesh-based Survivable Networks: Options and Strategies for Optical, MPLS, SONET and ATM Networking, Prentice Hall PTR, 2004, pp.699-700.

13. John Doucette, Donne He, Wayne D. Grover, Oliver Yang DRCN 2003, Banff, AB, Canada, 19-22 October 2003 13 Algorithmic Approaches for Efficient Enumeration of Candidate p-Cycles and Capacitated p-Cycle Network Design CIDA Results with Grow

14. John Doucette, Donne He, Wayne D. Grover, Oliver Yang DRCN 2003, Banff, AB, Canada, 19-22 October 2003 14 Algorithmic Approaches for Efficient Enumeration of Candidate p-Cycles and Capacitated p-Cycle Network Design Concluding Remarks The Grow algorithm is effective at enumerating a good set of eligible cycles. –It will enumerate no more than S 2 ·N cycles. The CIDA algorithm quickly and efficiently places p-cycles in a capacitated network. –It is generally within 5% of ILP solutions when given the same cycle set. The CIDA-Grow combination is the best overall alternative (yet identified) to full-blown ILP design with enumeration of a large cycle set. Iterated design assembly is not the problem. –CIDA alone almost matches ILP on the same set of cycles. –The challenge remains to find a way to generate a small set of candidate cycles that always enables a near-optimal design assembly.

15. Algorithmic Approaches for Efficient Enumeration of Candidate p-Cycles and Capacitated p-Cycle Network Design John Doucette 1,2, Donna He 3, Wayne D. Grover 1,2, and Oliver Yang 3 1 TRLabs, Edmonton, AB, Canada 2 University of Alberta, Edmonton, AB, Canada 3 CNNR Lab, SITE, University of Ottawa, Ottawa, ON, Canada john.doucette@trlabs.ca, dhE098@site.uottawa.ca, grover@trlabs.ca, yang@site.uottawa.ca DRCN 2003 Banff, AB, Canada 19-22 October 2003

16. John Doucette, Donne He, Wayne D. Grover, Oliver Yang DRCN 2003, Banff, AB, Canada, 19-22 October 2003 16 Algorithmic Approaches for Efficient Enumeration of Candidate p-Cycles and Capacitated p-Cycle Network Design CIDA Results with Expand