Stochastic Multicast with Network Coding

Ajay Gopinathan, Zongpeng Li

Department of Computer ScienceUniversity of Calgary

ICDCS 2009, June 24 2009, Montreal

Outline

• Capacity planning at multicast service provider

• Solution 1 – Heuristic– Usually but not always good solutions

• Solution 2 – Sampling– Provable performance bound

• Simulations• Conclusion

Problem Statement

NetworkNetwork

SLAContent

Provider

Content

Provider

Network

Service Provide

r

Network

Service Provide

r

Potential CustomersPotential Customers

Usage beyond SLA incurs penalties!

negotiate

negotiate

P(t)

The Content Provider’s Dilemma

• Content provider’s goal:– Minimize expected cost

• 2-stage stochastic optimization

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

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

Why multicast?

• Exploits replicable property of information– Reduce redundant transmissions– Efficient bandwidth usage => cost savings!

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

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”

Network Model

– Directed graph– Edge has cost and capacity– Receiver has set of paths to the source

Multicast Routing LP

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 -

Two-stage optimization

Two-stage optimization

• Optimal

• But intractable!– Exponentially sized– #P-Hard in general

• Can we approximate the optimal solution?

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

Solution 1 - Heuristic

• Simulations show excellent performance in most cases

• No provable performance bound– In fact, it is unbounded

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

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)

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

Multicast LP dual formulation

A 2-strict cost sharing scheme

• TheoremThe 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!

Simulations

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

Steiner Trees