Stochastic Multicast with Network Coding

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

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 Service Provider Content Provider SLA negotiate negotiate Usage beyond SLA incurs penalties! Network P(t) Potential Customers

4. The Content Provider’s Dilemma • Content provider’s goal: • Minimize expectedcost • 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. Two-stage optimization • 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 Aand 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