1. Network Coding and Reliable Communications Group Algebraic Network Coding Approach to Deterministic Wireless Relay Networks MinJi Kim, Muriel Médard

2. Network Coding and Reliable Communications Group Wireless Network Open problem: capacity & code construction for wireless relay networks – Channel noise – Interference High SNR – Noise → 0 & large gain – Large transmit power High SNR rate region: – TDM, Noise free additive channel [Ray et al. ‘03] – Note: This holds for higher field size (not just binary) [Ray et al. ‘03] [Avestimehr et al. ‘07]“Deterministic model” (ADT model) – Interference – Model noise deterministically – Use binary field – In essence, high SNR regime Model as error free links R1R1 R2R2 Y(e 1 ) Y(e 2 ) e1e1 e2e2 e3e3 Y(e 3 ) Y(e 3 ) = Y(e 1 ) + Y(e 2 ) R1R1 R2R2 log(1+P 2 /N) log(1+P 1 /N) log(1+P 2 /(P 1 +N)) log(1+P 1 /(P 2 +N)) R1R1 R2R2 High SNR

3. Network Coding and Reliable Communications Group ADT Network Background Min-cut: minimal rank of an incidence matrix of a certain cut between the source and destination [Avestimehr et al. ‘07] – Requires optimization over a large set of matrices – Min-cut Max-flow Theorem holds for unicast/multicast sessions Matroidal [Goemans et al. ’09] – Algebraic Network Coding is also Matroidal [Dougherty et al. ’07] Unicast code construction algorithms [Goemans et al. ‘09][Yazdi & Savari ‘09][Amaudruz & Fragouli ‘09] Multicast code construction algorithms [Erez et al. ‘10][Ebrahimi & Fragouli ‘10]

4. Network Coding and Reliable Communications Group Our Contributions Connection to Algebraic Network Coding [Koetter & Médard ‘03]: – Use of higher field size [Ray et al. ’03] 1. 2.Can’t achieve capacity for multicast with just binary field [Feder et al. ’03][Rasala-Lehman & Lehman ’04][Fragouli et al. ‘04] 3.[Jaggi et al. ‘06] “permute-and-add”: Show that network codes in higher field size F q can be converted to binary-vector code in (F 2 ) n without loss in performance R1R1 R2R2 log(1+P 2 /N) log(1+P 1 /N) log(1+P 2 /(P 1 +N)) log(1+P 1 /(P 2 +N)) R1R1 R2R2 High SNR

5. Network Coding and Reliable Communications Group Our Contributions Connection to Algebraic Network Coding [Koetter & Médard ‘03]: – Use of higher field size [Ray et al. ’03] – Model broadcast constraint with hyper-edges – Capture ADT network problem with a single system matrix M Prove that min-cut of ADT networks = max rank( M ) Prove Min-cut Max-flow for unicast/multicast holds Extend optimality of linear operations to certain non-multicast sessions Incorporate failures and erasures [Lun et al. ‘04] Incorporate cycles – Show that random linear network coding achieves capacity [Ho et al. ‘03] – Do not address ADT network model’s ability to approximate the wireless networks; but show that ADT network problems can be captured by the algebraic network coding framework

6. Network Coding and Reliable Communications Group ADT Network Model Original ADT model (Binary field) – Broadcast: multiple edges (bit pipes) from the same node – Interference: additive MAC over binary field Higher SNR: S-V 1 Higher SNR: S-V 2 broadcast interference Algebraic model:

7. Network Coding and Reliable Communications Group Algebraic Framework Assume higher field size X(S, i) : source process i Y(e) : process at port e Z(T, i) : destination process i Assume linear operations – at the source S : α(i, e j ) – at the nodes V : β(e j, e j’ ) – at the destination T : ε(e j, (T, i))

8. Network Coding and Reliable Communications Group System Matrix M= A(I – F ) -1 B T Linear operations – Encoding at the source S : α(i, e j ) – Decoding at the destination T : ε(e j, (T, i)) e1e1 e2e2 e3e3 e4e4 e5e5 e6e6 e7e7 e8e8 e9e9 e 10 e 11 e 12 a b c d f

9. Network Coding and Reliable Communications Group System Matrix M= A(I – F ) -1 B T Linear operations – Coding at the nodes V : β(e j, e j’ ) – F represents physical structure of the ADT network – F k : non-zero entry = path of length k between nodes exists – (I-F) -1 = I + F + F 2 + F 3 + … : connectivity of the network (impulse response of the network) e1e1 e2e2 e3e3 e4e4 e5e5 e6e6 e7e7 e8e8 e9e9 e 10 e 11 e 12 a b c d f F = Broadcast constraint (hyperedge) MAC constraint (addition) Internal operations (network code) Linear code with some coefficients fixed by the network!

10. Network Coding and Reliable Communications Group System Matrix M = A(I – F ) -1 B T Z = X(S) M e1e1 e2e2 e3e3 e4e4 e5e5 e6e6 e7e7 e8e8 e9e9 e 10 e 11 e 12 a b c d f Input-output relationship of the network Captures rate Captures network code, topology (Field size as well)

11. Network Coding and Reliable Communications Group Theorem: Min-cut of ADT Networks From [Avestimehr et al. ‘07] – Requires optimizing over ALL cuts between S and T – Not constructive: assumes infinite block length, internal node operations not considered Show that the rank of M is equivalent – System matrix captures the structure of the network – Constructive: the assignment of variables gives a network code e1e1 e2e2 e3e3 e4e4 e5e5 e6e6 e7e7 e8e8 e9e9 e 10 e 11 e 12 a b c d f

12. Network Coding and Reliable Communications Group Min-cut Max-flow Theorem For a unicast/multicast connection from source S to destination T, the following are equivalent: 1.A unicast/multicast connection of rate R is feasible 2. mincut(S,T i ) ≥ R for all destinations T i 3.There exists an assignment of variables such that M is invertible Proof idea: 1. & 2. equivalent by previous work 3.→1. If M is invertible, then connection has been established 1.→3. If connection established, M = I. Therefore, M is invertible Alternate proof of sufficiency of linear operations for multicast in ADT networks [Avestimehr et al. ‘07]

13. Network Coding and Reliable Communications Group Corollaries Extend Min-cut Max-flow theorem to other connections: – [Multiple multicast]: Sources S 1 S 2 … S k wants to transmit to all destinations T 1 T 2 … T N – [Disjoint multicast]: – [Two-level multicast]: Two sets of destinations, a set T m for multicast connection, another set T d for disjoint multicast connection. S2S2 T1T1 TNTN Network S1S1 SkSk S T1T1 T3T3 a, b, c, d T2T2 a b, c d Destination wants S T1T1 T3T3 a, b, c, d Destination wants T4T4 T6T6 T5T5 a b, c d Network

14. Network Coding and Reliable Communications Group Corollaries Extend Min-cut Max-flow theorem to other connections: – [Multiple multicast]: Sources S 1 S 2 … S k wants to transmit to all destinations T 1 T 2 … T N – [Disjoint multicast]: – [Two-level multicast]: Two sets of destinations, a set T m for multicast connection, another set T d for disjoint multicast connection. Random linear network coding achieves capacity for unicast, multicast, and above connections. Extend results to ADT networks with… – Delay – Cycles – Erasures/Failures

15. Network Coding and Reliable Communications Group Conclusions ADT network can be expressed with Algebraic Network Coding Formulation – Use of higher field size – Model broadcast constraint with hyper-edge – Capture ADT network problem with a single system matrix M Prove an algebraic definition of min-cut = max rank( M ) Prove Min-cut Max-flow for unicast/multicast holds Show that random linear network coding achieves capacity Extend optimality of linear operations to non-multicast sessions – Disjoint multicast, Two-level multicast, multiple source multicast, generalized min-cut max-flow theorem – Random linear network coding achieves capacity Incorporate delay and failures (allows cycles within the network)