Optimal Channel Choice for Collaborative Ad-Hoc Dissemination

Liang HuTechnical University

of Denmark

Jean-Yves Le BoudecEPFL

Milan VojnovićMicrosoft Research

IEEE Infocom 2010, San Diego, CA, March 2010

2

Delivery of Information Streamsthrough the infrastructure and device-to-device transfers

channelsusers infrastructure

3

Outlook

• System welfare objective

• Optimal GREEDY algorithm for solving the system welfare problem

• Distributed Metropolis-Hastings algorithm

• Simulation results

• Conclusion

4

Assignment of channels to users for dissemination

• User u subscribed to a set of channels S(u)• xuj = 1 if user forwards channel j, xuj = 0 otherwise

• Constraint: each user u forwards at most Cu channels

users channels

uj

• Find: an assignment of users to channels that maximizes a system welfare objective

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System Welfare Problem

= dissemination time for channel j under assignment x

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System Welfare Problem (cont’d)

• In this paper we consider the problem under assumption

for every channel j

i.e. utility of channel j is a function of the fraction of users that forward channel j

• For example, the assumption holds under random mixing mobility where each pair of nodes is in contact at some common positive rate

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System Welfare Problem (cont’d)

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Dissemination Time for Random Mixing Mobility

Fraction of subscribers of channel j that received the message by time t

Fraction of forwarders of channel j that received the message by time t

Access rate at which channel j content is downloaded from the infrastructure

Fraction of subscribers of channel jFraction of forwarders of channel j

Time for the message to reach a fraction of subscribers:

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Dissemination Time ... (cont’d)

Also observed in real-world mobility traces (Cambridge dataset):

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System Welfare Problem (cont’d)

• Polyhedron:

where

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System Welfare Problem (cont’d)

• Proof sketch: max-flow min-cut arguments

jus t

Cu - |S(u)|

1

0

user u subscribed to this channel

users channels

• For every subset of channels A:

= flow

v(A) = min-cut

• max-flow achieved by an integral assignment

Aj

jH

12

Outlook

• System welfare objective

• Optimal GREEDY algorithm for solving the system welfare problem

• Distributed Metropolis-Hastings algorithm

• Simulation results

• Conclusion

13

GREEDY

Init: Hj = 0 for every channel j

while 1 doFind a channel J for which incrementing HJ by one (if feasible) increases the systemwelfare the most

if no such J exists then break

HJ ← HJ + 1

end while

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GREEDY is Optimal

• Proof sketch: - objective function is concave- polyhedron is submodular

validating the conditions for optimality of the greedy procedure (Federgruen & Groenevelt, 1986)

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When Vj(f) is concave?

dj

-

Uj(t)

t dj

Uj(t)

t

16

Outlook

• System welfare objective

• Optimal GREEDY algorithm for solving the system welfare problem

• Distributed Metropolis-Hastings algorithm

• Simulation results

• Conclusion

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Distributed Algorithm• Metropolis-Hastings sampling – Choose a candidate assignment x’ with prob. Q(x, x’)

where x is the current assignment– Switch to x’ with prob.

where

normalization constant

temperature

u v

An example local rewiringwhen users u and v in contact:

User u samples a candidate assignment where user u switched to forwarding a randomly picked channel forwarded by user v

- Requires knowing fractions fj (can be estimated locally)

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User’s Battery Level• The system welfare objective extended to

• Additional factor for the acceptance probabilityfor our example rewiring:

battery level for user u

b

Wu,j(b)

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Simulation Results

• Cambridge mobility trace• Vj(f) = - tj(f) for every

channel j• J = 40 channels,

20 channels fwd per user, 10 subs. per user

• Subscriptions per channel ~ Zipf(2/3)

UNI = pick a channel to help uniformly at random

TOP = pick a channel to help in decreasing order of channel popularity

Dissemination time per channel in minutes

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Conclusion• Formulated a system welfare objective for optimizing

dissemination of multiple information streams– For cases where the dissemination time of a channel is a function of the

fraction of forwarders

• Showed that the problem is a concave optimization problem that can be solved by a greedy algorithm

• Distributed algorithm via Metropolis-Hastings sampling

• Simulations confirm benefits over heuristic approaches

• Future work – optimizing a system welfare objective under general user mobility?