Exchanging Third-Party Information in a Network

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Karin Copeland
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1 David J. Love 1 Bertrand M. Hochwald Kamesh Krishnamurthy 1 1 School of Electrical and Computer Engineering Purdue University Beceem Communications ITA Workshop, 007

2 Information Exchange Some applications require global knowledge at each node Each node has a local part of that knowledge Each node must learn third-party information, not local to itself A common channel is provided (carries 1-bit/channel use) Information is broadcast over this control channel, i.e., only one node transmits at a time and all other nodes listen

3 3 Node Example 0 a,b,c are either 0 or 1 they represent link information a b Common channel 1 c Through training, each node has knowledge of the link information between itself and other nodes x 0 = [ a b ], x 1 = [ a c ], x = [ b c ] Simple exchange strategy: All nodes transmit their entire vector ( bits each on the common channel meaning 6 total common channel uses) Bad strategy!

4 3 Node Example A Better Transmission Strategy: Node 0 1 (a b) b a c (c a) * Each row represents one common channel use Terms in parentheses are bits transmitted by corresponding node Other terms are bits learned by remaining nodes Making use of correlation in the source vectors saves 4 common channel uses Is this the best strategy? Arbitrary number of nodes?

5 Third-Party Slepian-Wolf Number of common channel uses N N (This follows from a more general result which we now present)

6 Achievable Rates A set of rates (R 0,R 1,...,R N 1 ) is achievable, if any receiver i can reconstruct the other source vectors with an arbitrarily small probability of error using the source vector X i as side information X ^ ^ X 1, X 3 Encoder Decoder R R R 3 ^ ^ X, X 3 Decoder 1 R 3 Encoder 3 X 3 X 1 Encoder 1 R 1 R 1 Decoder 3 ^ ^ X 1, X

7 Rate Region Rate region R 3rd p can be described as the closure of the set R 3rd p = N 1 i=0 R i,sw R i,sw is the rate region for the sub-network in which node i is fixed as the receiver and has X i as side information. R i,sw = S A i R i (S) ; A i = {0,1,,i 1,i+1,,N 1} R i (S) = { (R 0,R 1,,R N 1 ) : R(S) H(X S X S c,x i ) } R(S) = j S R j X S = { X j j S }

8 Four-Node Example 0 a b c 3 e 1 d f

9 Four-Node Rate Region Rate region for the network is the closure of the intersection of the rate regions of the Slepian-Wolf type sub-networks, with the decoder i having X i as side information. X 0 R 1 + R + R 3 H(X 1,X,X 3 X 0) = R 0 + R + R X 1 0 R 0 + R 1 + R 3 3 R 0 + R 1 + R X R 0 + R 1 + R + R 3 4 X

10 Four-Node Information Exchange Table: Transmission strategy for a four-node network Node (b c) b c b c e f (e f ) e f d d (b d) b, c e, f a a (a e) Each row represents one common channel use Terms in parentheses are bits transmitted by corresponding node Other terms are bits learned by remaining nodes Scheme reaches bound. Best strategy!

11 More Generally R 1 + R R N 1 R 0 + R R N 1. R 0 + R R N ( ) N N + 1 ( ) N N + 1 (. ) N N + 1 R 0 + R 1 + R...R N 1 N N Number of channel uses = N 1 j=0 R j

12 Special Case: Linear code reaches bound Special case of Link information exchange Link between nodes i and j denoted by a bit G i,j Transmission strategy for N-nodes [ ] C i = G i,mod(i+1,n) G i,mod(i+,n) G i,mod(i+1,n) G i,mod(i+ N,N) (N-1) (N/+1) (N/) (N/-1) (N/) Bits in G 0 (N/-1)

13 Conclusions Summary Correlation in 3 rd party information can be leveraged Derived rate region assuming common control channel In one special case: linear code reaches bound Future work Information exchange in the absence of a common channel Codes for other correlation structures Adaptive codes (bound still applies!)

Network coding for multicast relation to compression and generalization of Slepian-Wolf

Network coding for multicast relation to compression and generalization of Slepian-Wolf 1 Overview Review of Slepian-Wolf Distributed network compression Error exponents Source-channel separation issues