1. Survivable Paths in Multilayer Networks Marzieh Parandehgheibi Hyang-won Lee Eytan Modiano 46 th Annual Conference on Information Sciences and Systems 21-23 March 2012

2. Multilayer Network Logical Layer : IP layer Physical Layer: Optical Fibers 23 6 1 4 5 23 1 4 Physical Topology Logical Topology

3. Reliability in Multilayer Network 3 24 15 3 24 15 Physical LayerLogical Layer

4. Reliability in Multilayer Network 3 24 15 3 24 15 Physical LayerLogical Layer 3 24 15 3 24 15

5. Reliability in Multilayer Network 3 24 15 3 24 15 Physical LayerLogical Layer 3 24 15 3 24 15

6. Reliability in Multilayer Network 3 24 15 3 24 15 Physical LayerLogical Layer 3 24 15 Not Survivable 3 24 15

7. Reliability in Multilayer Network 3 24 15 3 24 15 Physical LayerLogical Layer 3 24 15 Not Survivable 3 24 15 Survivable

8. SRLG Disjoint Path Shared Risk Link Group: a group of logical links sharing a common risk are said to belong to the same SRLG Find two SRLG disjoint paths between source and destination 3 24 15 3 24 15

9. Challenges 1- SRLG-disjoint paths may not always exist. 1234 1234 a) Physical Topology c) Lightpaths routed on the fibers 14 b) Logical Topology

10. Challenges 2- A pair of SRLG-disjoint paths could be very long and thus vulnerable 1234 5 6 N a) Physical Topology 14 b) Logical Topology 1 2 3 4 1234 5 6 N c) Lightpath Routing

11. Our Solution Find a set of paths that together will survive any single physical link failure.

12. Our Solution Find a set of paths that together will survive any single physical link failure. 1234 1234 a) Physical Topology c) Lightpaths routed on the fibers 14 b) Logical Topology All of three paths together will survive any single failure

13. Our Solution Find a set of paths that together will survive any single physical link failure. 1234 1234 a) Physical Topology c) Lightpaths routed on the fibers 14 b) Logical Topology All of three paths together will survive any single failure Find the minimum number of paths that together survive any single failure in the physical topology.

14. Problem Formulation 1234 f1 f2 f3 f4 f5 f6 P1 P2 P3 f1 f2 f3 f4 f5 f6 P1 P2 P3

15. Complexity Theorem: Computing the minimum number of survivable paths in multilayer networks is NP-hard. In addition, this minimum value cannot be approximated within any constant factor, unless P = NP. Proof: Reduction from minimum set cover problem.

16. Restricted Versions In general setting, the total number of paths is exponential: – Path Length Restriction: Each logical path is restricted to use at most K fibers – WDM Restriction: Each fiber is used by at most W logical paths. In both cases, the total number of paths is polynomial in the number of fibers.

17. Path Length Restriction Lemma: Under the path length restriction, the optimal solution of MSP problem is at most K+1.

18. Path Length Restriction Lemma: Under the path length restriction, the optimal solution of MSP problem is at most K+1. Proof: Randomly Select a Path Pi At most K fibers are used At least m-K fibers are NOT used Path Pi survives the failure of any of these fibers For each of these fibers, select a path not using it Total of K+1 Paths K additional Paths

19. Path Length Restriction Theorem: The path length restricted version of the MSP problem can be solved in polynomial time.

20. Path Length Restriction Theorem: The path length restricted version of the MSP problem can be solved in polynomial time. Proof: Total number of Paths P is Polynomial in m At most K+1 Paths Exhaustive Search: all sets of size 2: O(P 2 ) all sets of size 3: O(P 3 ). all sets of size K+1: O(P K+1 ) Exhaustive Search: all sets of size 2: O(P 2 ) all sets of size 3: O(P 3 ). all sets of size K+1: O(P K+1 ) O(P K+1 ): Polynomial in m O(P K+1 ): Polynomial in m

21. Approximations Exhaustive search returns an optimal solution, But its running time can be prohibitive for large values of m and K. Approximations: – Greedy – Epsilon-net – Random Sweep Greedy – Randomized Rounding

22. Approximations Exhaustive search returns an optimal solution, But its running time can be prohibitive for large values of m and K. Approximations: Greedy Epsilon-net – Random Sweep Greedy – Randomized Rounding

23. Greedy Algorithm Iteratively add paths until any fiber failure can be survived Input: P i ={f j | f j survived by P i } Select Path P k with largest size Update paths: Remove fibers that are already survived All paths empty? Done! YES NO

24. Greedy Algorithm Theorem: The greedy algorithm provides an O(logK) approximation in polynomial time for the path length restricted version of MSP. Running Time: O(Km K ) K+1 iterations Search through all paths to find largest O(m K )

25. Epsilon-net Algorithm  Randomly select paths – Fibers not used by many paths: higher probability of survival – Increase the weight of fibers that are hard to survive Initialize: – weight of each path P j is – Weight of each fiber f i is Definition: A fiber is -survivable, if for some

26. Epsilon-net Algorithm Update probability distribution (New set of є-survivable fibers) Update probability distribution (New set of є-survivable fibers) Randomly select S paths w.r.t the probability distribution (We can select S such that є-Survivable fibers are survived w.h.p.) Randomly select S paths w.r.t the probability distribution (We can select S such that є-Survivable fibers are survived w.h.p.) Find fibers that are not survived yet: S’ Double weight of Paths that survive fibers in S’ All fibers survived? Done YES NO

27. Epsilon-net Algorithm Theorem: Let, where s is the size of random subset and c is a constant. Let ξ be the optimal value of MSP problem. The є-net algorithm finds a set of survivable paths of size O(log K log ξ)ξ, with high probability. Running Time: O(K log(K) log(m) log(log(K))) = O(K log 2 K logm)

28. Simulations AlgorithmApproximationRunning TimeType Exhaustive SearchExact SolutionO(m K(K+1) )Deterministic GreedyO(log K)O(Km K )Deterministic Epsilon-netO(log K log λ)O(K log(K) log(m) log(log(K)))Probabilistic

29. WDM Restriction Lemma: Under the WDM restriction, the optimal number of survivable paths is at most W+1.

30. WDM Restriction Lemma: Under the WDM restriction, the optimal number of survivable paths is at most W+1. Proof: Arbitrary Set of W+1 Paths Arbitrary fiber i Fiber i used by at most W of these paths At least One Path Survives the failure of fiber i

31. WDM Restriction Theorem: The Wavelength restricted version of the MSP problem can be solved in polynomial time. Proof: By Exhaustive Search

32. Simulations AlgorithmApproximationRunning TimeType Exhaustive SearchExact SolutionO(W W+1 m W+1 )Deterministic GreedyO(log m)O(W 2 m)Deterministic Epsilon-netO(log W log λ)O(W log(W) log(m) log(log(W)))Probabilistic

33. Summary  Minimum Survivable Paths (MSP) Formulate NP-Hard Approximations of Minimum Set Cover Problem Restricted Versions: o Length Restricted Polynomially Solvable Approximation Algorithms o WDM Polynomially Solvable Approximation Algorithms