1. Optimal Channel Choice for Collaborative Ad-Hoc Dissemination Liang Hu Technical University of Denmark Jean-Yves Le Boudec EPFL Milan Vojnović Microsoft Research IEEE Infocom 2010, San Diego, CA, March 2010

2. Delivery of Information Streams through the infrastructure and device-to-device transfers channels usersinfrastructure 2

3. Outlook System welfare objective Optimal GREEDY algorithm for solving the system welfare problem Distributed Metropolis-Hastings algorithm Simulation results Conclusion 3

4. Assignment of channels to users for dissemination User u subscribed to a set of channels S(u) x uj = 1 if user forwards channel j, x uj = 0 otherwise Constraint: each user u forwards at most C u channels users channels u j Find: an assignment of users to channels that maximizes a system welfare objective 4

5. System Welfare Problem = dissemination time for channel j under assignment x 5

6. 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 6

7. System Welfare Problem (cont’d) 7

8. 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 j Fraction of forwarders of channel j Time for the message to reach  fraction of subscribers: 8

9. Dissemination Time... (cont’d) Also observed in real-world mobility traces (Cambridge dataset): 9

10. System Welfare Problem (cont’d) Polyhedron: where 10

11. System Welfare Problem (cont’d) Proof sketch: max-flow min-cut arguments j u s t C u - |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 11

12. Outlook System welfare objective Optimal GREEDY algorithm for solving the system welfare problem Distributed Metropolis-Hastings algorithm Simulation results Conclusion 12

13. GREEDY Init: H j = 0 for every channel j while 1 do Find a channel J for which incrementing H J by one (if feasible) increases the system welfare the most if no such J exists then break H J ← H J + 1 end while 13

14. GREEDY is Optimal Proof sketch: - objective function is concave - polyhedron is submodular validating the conditions for optimality of the greedy procedure (Federgruen & Groenevelt, 1986) 14

15. When V j (f) is concave? djdj -  Uj(t)Uj(t) tdjdj Uj(t)Uj(t) t 15

16. Outlook System welfare objective Optimal GREEDY algorithm for solving the system welfare problem Distributed Metropolis-Hastings algorithm Simulation results Conclusion 16

17. 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 17 normalization constant temperature uv An example local rewiring when 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 f j (can be estimated locally)

18. User’s Battery Level The system welfare objective extended to Additional factor for the acceptance probability for our example rewiring: 18 battery level for user u b W u,j (b)

19. Simulation Results Cambridge mobility trace V j (f) = - t j (f) for every channel j J = 40 channels, 20 channels fwd per user, 10 subs. per user Subscriptions per channel ~ Zipf(2/3) 19 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

20. 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? 20