Dynamic Index Coding User set N Packet set P Broadcast Station N N p p p Michael J. Neely, Arash Saber Tehrani, Zhen Zhang University of Southern California Paper available at: (i) (ii) Page 1

Initial Setup Want to expand wireless throughput. Wireless users download popular files. N users. P packets. Packets are all in the Broadcast Station (BS). All users hear the transmissions of the BS. Each user wants a different subset of packets. Each user already has a different subset of packets in its cache. Represent this by a directed bipartite demand graph (see page 1). Page 2

Information Theory Result Finding min clearance time T min is the unsolved index coding problem. However, we have the following result, which extends [Bar-Yossef, Birk, Jayram, Kol] to the case of general demand graphs: Theorem 1: If the bipartite demand graph is acyclic, then T min = P. Corollary 1: We get a simple max acyclic subgraph bound. Corollary 2: Coding can only help if the demand graph has cycles! Page 3

Cyclic Coding Page 4 “Cyclic” coding over K users clears K packets in K-1 slots. Message 1: A + B Message 2: B + C Message 3: C + D Want: D Have: A Want: A Have: B Want: B Have: C Want: C Have: D User 1 User 2 User 3 User 4 Decoding for user 2: B + (A + B) = A Decoding for user 1: A + (A + B) + (B + C) + (C+D) = D Theorem 2: “Unicast” graphs with disjoint cycles can be optimally solved by cyclic coding!

Packets arrive randomly, rates ( 1, …, M ). A = Abstract space of coding options. Example: A = {cyclic coding actions} Each action α in A has: T(α) = frame size of action α. (μ 1 (α), …, μ M (α)) = clearance vector of action α. Queue dynamic for frame k, traffic class m: Q m [k+1] = max[Q m [k] – μ m (α[k]), 0] + arrivals over frame k Dynamic Index Coding Page 5 Frame size T(α[k]) affects number of arrivals

Max-Weight Code Selection Alg 1 (Known λ): Every new frame k, choose α[k] in A to maximize: ∑ m Q m [k] [μ m (α[k]) – λ m T(α[k])] Max-Weight Code Selection Alg 2 (Unknown λ): Every new frame k, choose α[k] in A to maximize: ∑ m Q m [k] [μ m (α[k])/T(α[k])] Algorithms Page 6 Theorem 3: Both algorithms support any rate vector (λ 1, …, λ M ) in the Code-Constrained Capacity region (i.e., subject to using codes in set A ).

Simulation of Max-Weight Code Selection Details: 3 user system. Each user has packets arriving rate λ. Each packet is independently in cache of another user with prob ½. Total number of traffic types = M = 12. Page 7

Special case of Broadcast Relay Networks: Users want to send to other users via Broadcast Relay. Each packet contained as side information in exactly one user. Each packet has exactly one user as destination. Admits a simplified graphical structure, can often compute T min. (nodes = users, links labeled by # packets) 7 users 48 packets Tmin = 39 Graph with 3 disjoint cycles Page 8 Max Acyclic Subgraph

Results for Broadcast Relay Networks: N=2 (  is 2-dimensional) N=3 (  is 6-dimensional) Any N, provided that the following additional assumption holds: Each user desires to transmit to at most one other user. Coding : Max-Weight combinations of direct transmission, 2-cycle coding, 3-cycle coding, …, N-cycle coding. This is information-theoretically optimal in these cases: N-user capacity region  = {( ij )} has N(N-1) dimensions! Page 9