stochastic multicast with network coding ajay gopinathan, zongpeng li department of computer science...

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