Introduction

1. Algorithmic Approaches for Efficient Enumeration of Candidate p-Cycles and Capacitated p-Cycle Network Design John Doucette1,2, Donna He3, Wayne D. Grover1,2, and Oliver Yang3 1TRLabs, Edmonton, AB, Canada 2University of Alberta, Edmonton, AB, Canada 3CNNR 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. Introduction • 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. SS,p = 3 SC,p = 9 AE(p) = 1.67 SS,p = 4 SC,p = 10 AE(p) = 1.80 A Priori p-Cycle Efficiency: AE(p) • AE(p) measures a cycle’s potential to provide protection relationships for working channels SC,p = # on cycle SS,p = # straddlers ci = unit cost of i Xp,i = 1 if on cycle Xp,i = 2 if straddler

4. A priori efficiency is: 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 • 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. (a) (b) (c) (d) (e) (f) (g) (h) 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

6. 4 4 4 4 1 1 1 1 7 7 7 7 3 3 3 3 6 6 6 6 11 11 11 11 2 2 2 2 9 9 9 9 10 10 10 10 5 5 5 5 8 8 8 8 SP-Add Operation to improve candidate cycles • Replace a span on an existing cycle with the shortest cycle-disjoint route connecting its end-nodes. SP-Add (5-6) SP-Add (6-7)

7. 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. 4 4 4 4 4 1 1 1 1 1 7 7 7 7 7 3 3 3 3 3 6 6 6 6 6 11 11 11 11 11 2 2 2 2 2 9 9 9 9 9 10 10 10 10 10 5 5 5 5 5 8 8 8 8 8 Expand equivalent Expand AE(p) = 1.0 AE(p) = 1.57 Expand Operation using SP-Add • Visit each span in the original cycle and perform SP-Add operation until no longer possible. SP-Add (6-9) SP-Add (7-10)

9. 4 4 4 4 4 1 1 1 1 1 7 7 7 7 7 3 3 3 3 3 6 6 6 6 6 11 11 11 11 11 2 2 2 2 2 9 9 9 9 9 10 10 10 10 10 5 5 5 5 5 8 8 8 8 8 Grow equivalent Grow AE(p) = 1.33 AE(p) = 1.67 Grow Operation using SP-Add • Perform the SP-Add operation and iterate on the new cycles until no further iterations are possible. SP-Add (6-7) SP-Add (7-10)

10. 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. 3 3 2 2 1 2 1 2 AE(p) = 1.67 Ew(p) = 3.78 AE(p) = 1.67 Ew(p) = 3.67 4 2 4 2 3 2 2 1 1 2 3 2 3 4 2 1 Demand-weighted p-Cycle Efficiency: Ew(p) • Ew(p) measures a cycle’s actual efficiency in providing protection relationships for uncovered working channels Xp,i = 1 if on cycle Xp,i = 2 if straddler wi = working on i ci = unit cost of i

12. Capacitated Iterative Design Algorithm • CIDA provides an iterated placement of p-cycles on a capacitated network based on Ew(p). (1) • It can be implemented with any pre-selected eligible p-cycle set. • Start with initial eligible cycle set (say by using Expand or Grow with SLA primary cycles) • Calculate Ew(p) for each eligible cycle and rank • Place one copy of cycle with highest Ew(p) • Update wi on each on-cycle and straddling span • Iterate steps 2 to 4 until all wi = 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. CIDA Results with Grow

14. Concluding Remarks • The Grow algorithm is effective at enumerating a good set of eligible cycles. • It will enumerate no more than S2·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 Doucette1,2, Donna He3, Wayne D. Grover1,2, and Oliver Yang3 1TRLabs, Edmonton, AB, Canada 2University of Alberta, Edmonton, AB, Canada 3CNNR 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. CIDA Results with Expand