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ICDCS 2009, June 24 2009, Montreal. Stochastic Multicast with Network Coding. Ajay Gopinathan, Zongpeng Li Department of Computer Science University of Calgary. Outline. Capacity planning at multicast service provider Solution 1 – Heuristic Usually but not always good solutions

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Stochastic Multicast with Network Coding

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Stochastic multicast with network coding

ICDCS 2009, June 24 2009, Montreal

Stochastic Multicast with Network Coding

Ajay Gopinathan, Zongpeng Li

Department of Computer Science

University of Calgary



  • 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

Problem Statement

Network Service Provider

Content Provider




Usage beyond SLA incurs penalties!



Potential Customers

The content provider s dilemma

The Content Provider’s Dilemma

  • Content provider’s goal:

    • Minimize expectedcost

      • 2-stage stochastic optimization

Two 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 dilemma1

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

Why multicast?

  • Exploits replicable property of information

    • Reduce redundant transmissions

    • Efficient bandwidth usage => cost savings!

Content provider s routing solution

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

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

Network Model

  • Directed graph

  • Edge has cost and capacity

  • Receiver has set of paths to the source

Multicast routing lp

Multicast Routing LP

How to minimize expected cost

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

Two stage optimization1

Two-stage optimization

  • Optimal

  • But intractable!

    • Exponentially sized

    • #P-Hard in general

  • Can we approximate the optimal solution?

Solution 1 heuristic

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 heuristic1

Solution 1 - Heuristic

  • Simulations show excellent performance in most cases

  • No provable performance bound

    • In fact, it is unbounded

Solution 2 sampling

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

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:




Cost sharing schemes1

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

Multicast LP dual formulation

A 2 strict cost sharing scheme

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!





  • 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

Steiner Trees

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