Optimal Channel Choice for Collaborative Ad-Hoc Dissemination
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Transcript of Optimal Channel Choice for Collaborative Ad-Hoc Dissemination
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
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Delivery of Information Streamsthrough the infrastructure and device-to-device transfers
channelsusers infrastructure
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Outlook
• System welfare objective
• Optimal GREEDY algorithm for solving the system welfare problem
• Distributed Metropolis-Hastings algorithm
• Simulation results
• Conclusion
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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
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Outlook
• System welfare objective
• Optimal GREEDY algorithm for solving the system welfare problem
• Distributed Metropolis-Hastings algorithm
• Simulation results
• Conclusion
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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
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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?