1. Stochastic Multicast with Network Coding Ajay Gopinathan, Zongpeng Li Department of Computer Science University of Calgary ICDCS 2009, June 24 2009, Montreal

2. Outline Capacity planning at multicast service provider Solution 1 – Heuristic – Usually but not always good solutions Solution 2 – Sampling – Provable performance bound Simulations Conclusion

3. Problem Statement Network SLA Content Provider Network Service Provider Potential Customers Usage beyond SLA incurs penalties! negotiat e P(t)

4. The Content Provider’s Dilemma Content provider’s goal: – Minimize expected cost 2-stage stochastic optimization

5. Two-stage stochastic optimization Stage 1: – Estimate capacity needed – Purchase capacity at fixed initial pricing scheme Stage 2: – Set of multicast receivers revealed – Bandwidth price increases by factor – Augment stage 1 capacity, for sufficient capacity to serve everyone Stage 1 purchasing decision should minimize cost of both stages in expectation

6. The Content Provider’s Dilemma Content provider’s goal: – Minimize expected cost Obstacles – Set of customers is non-deterministic Assume probability of subscription Based on market analysis/historical usage patterns – Employ the cheapest method for data delivery Multicast

7. Why multicast? Exploits replicable property of information – Reduce redundant transmissions – Efficient bandwidth usage => cost savings!

8. Content Provider’s Routing Solution Traditional multicast Finding and packing Steiner trees – NP-Hard! Network coding Exploit encodable property of information Polynomial time solvable linear programming formulation

9. Multicast with network coding Take home message – Compute multicast as union of unicast flows – Union of flows do not compete for bandwidth Conceptual flows “A multicast rate of d is achievable if and only if d is a feasible unicast rate to each multicast receiver separately”

10. Network Model – Directed graph – Edge has cost and capacity – Receiver has set of paths to the source

11. Multicast Routing LP

12. How to minimize expected cost? First stage, buy capacity at unit cost Second stage, cost increases by – Unit capacity cost For every let be probability that set is the customer set in second stage Capacity bought in first stage – Capacity bought in second stage -

13. Two-stage optimization

14. Optimal But intractable! – Exponentially sized – #P-Hard in general Can we approximate the optimal solution?

15. Solution 1 - Heuristic Idea – Future is more expensive by – Buy units of capacity in stage one if probability of requiring is Algorithm overview – Compute optimal flow to all receivers – Compute probability of requiring amounts of capacity on each edge – Buy on if above condition is met

16. Solution 1 - Heuristic Simulations show excellent performance in most cases No provable performance bound – In fact, it is unbounded

17. Solution 2 - Sampling Basic idea – sample from probability distribution to get estimate of customer set Is sampling once enough? – Need to factor in inflation parameter Theorem [Gupta et al., ACM STOC 2004] – Optimal – sample times – Possible to prove bound on solution

18. Cost sharing schemes Method for allocating cost of solution to the service set (multicast receivers) Denote as the cost share of in A A -strict cost sharing scheme for any two disjoint sets A and B: 1) 2) 3)

19. Cost sharing schemes Theorem [Gupta et al., ACM STOC 2004] If there exists a -strict cost sharing scheme, then sampling provides a (1 + )-approximate solution Does network coded multicast have such a scheme? – Yes! Use dual variables of primal multicast linear program

20. Multicast LP dual formulation

21. A 2-strict cost sharing scheme Theorem The variables in the dual linear program for multicast constitute a 2-strict cost sharing scheme Proof using LP duality and sub-additivity Sampling guarantees a 3-approximate solution!

22. Simulations

23. Conclusion Problem – minimize expected cost for content provider when set of customers are stochastic Two solutions – Heuristic Performs well in most cases No performance bound – Sampling Performs less well than heuristic in simulations Guaranteed performance bound

24. Steiner Trees