Low Complexity Algebraic Multicast Network Codes Sidharth “Sid” Jaggi Philip Chou Kamal Jain

The Multicast Problem Model Network with directed edges, nodes. Single source, rate R. |T| sinks, desired rate R, identical information. Examples – 1. Online news broadcasts. 2. Online gaming. s t1t1 t2t2 t |T| Network R R R R

History-I Assumptions Acyclic graph. Each link has unit capacity. Links have zero delay. Arithmetic operations allowed at all nodes. Upper bound for multicast capacity C, C ≤ min{C i } s t1t1 t2t2 t |T| C |T| C1C1 C2C2 Network Multicast capacity C achievable! (Random coding argument, [ACLY 2000])

Example s t1t1 t2t2 b1b1 b2b2 b2b2 b2b2 b1b1 b1b1 (b 1,b 2 ) b 1 +b 2 (b 1,b 2 ) Example due to [ACLY2000]

History-II b1b1 b2b2 bmbm β1β1 β2β2 βkβk F(2 m )-linear network can achieve multicast capacity C! F(2 m )-linear network [KM2001] Source:- Group together `m’ bits, Any node:- Perform linear combinations over finite field F(2 m ) Local Coding Vector: [β 1 β 2... β k ] 2 m >|T|C, Computational complexity high.

Results (Ours and Others’) Importance Linear encoding/decoding. [SET2003] Capacity gap without coding arbitrarily large. Lower bound on field size 2 m.. [LL(preprint)]  Lower bound on alphabet size ; finding smallest alphabet NP-hard.  Multicast the only “easy and interesting” case. Can be implemented as binary block-linear codes. Randomized construction (Also [SET2003],[HMKKE2003])  Faster design,  More robust (Single code, zero error, small rate loss, arbitrary failure pattern). Main Result (Also [SET2003]): For 2 m ≥ |T|, exists an F(2 m )- linear network which can be designed in polynomial time.

Minimum Block Length... Thus, for 2 n (2 n +1)/2 receivers, minimum block length = n. Therefore minimum block length≈0.5(log(|T|)) Each subspace at some U node has 2 n vectors. Total space has 2 2n vectors. s intermediate nodes receivers Subspaces can't interesect

Example: Local Coding Vectors s b1b1 b2b2 b1b1 b1b1 b 1 +b 2 Local Coding Vector: [1] Local Coding Vector: [1 1] length = number of incoming edges (variable)

Idea: Global Coding Vectors s b1b1 b 1 +b 2 Global Coding Vector: [1 0] Global Coding Vector: [0 1] Information carried by edge ↔ (g.c.v.) edge.[s 1 s 2...s C ] T Global Coding Vector: [1 1] length = capacity (fixed) b2b2

Idea: Linear Independence TASK: Find local coding vectors so that each receiver can decode. METHOD: Find local coding vectors sequentially so that global coding vectors on every cut-set to every receiver are linearly independent. s t1t1 t2t2 PREPROCESSING: Find a set of C paths from s to each t i. b2b2 b1b1 b 1 +b 2 T 2 = [1 1] [0 1] T 1 = [1 0] [1 1] OUTPUT: Local coding vectors, Final global coding vectors. Decoding: [x 1...x k ]=[T i ] -1 [y 1...y k ] T

Add edges {e -1,e -2,..,e -C } feeding into source node t Initialize these edges as {x 1,x 2,..,x C } e -1 e -2 b1b1 b2b2 s e0e0 e1e1 e2e2 e3e3 e4e4 e5e5 e6e6 e7e7 e8e8 t1t1 t2t2 Example: Design Algorithm Take the union of the edges to form a network flow N. Colour edges appropriately. Partial order on edges Use Ford-Fulkerson algorithm to find a set of paths from s to each t.

e -1 e -2 b1b1 b2b2 TASK: Find local coding vectors so that each receiver can decode. s e0e0 e1e1 e2e2 e3e3 e4e4 e5e5 e6e6 e7e7 e8e8 METHOD: Find local coding vectors sequentially so that global coding vectors on every cut-set to every receiver are linearly independent. t1t1 t2t2 g.c.v.'s on cutset to t 1 g.c.v.'s on cutset to t 2 Denote above matrices by [T i ] Example: Design Algorithm Edge 6

Encoding/Decoding Decoding: Receiver t i receives symbols [y 1...y k ], output [x 1...x k ]=[T i ] -1 [y 1...y k ] T β1β1 β2β2 βkβk Encoding: Local coding vectors used to encode. Final global coding vector matrices

One edge at a time… e1e1 e2e2 e4e4 e5e5 e3e3 M 1 ({1,4}) = [1 0] [0 1] b2b2 b1b1 [1 0] [0 1] [1 1] M 2 ({3,2}) = [1 0] [0 1] M 2 ({2,5}) = [0 1] [1 1] M 1 ({1,5}) = [1 0] [1 1] Design g.c.v. for edge 5 b 1 +b 2

Choosing local coding vectors appropriately Let v 1,…,v k be global coding vectors feeding into e. Let M i (n) be matrices of global coding vectors on n th cutset to t i. Inductive hypothesis – each M i (n) has rank C e v1v1 v2v2 vkvk Let L = span {v 1,…,v k }, L S j = span{rows(M j (n))- v j } Then we wish to find v in L such that v not in S j for all S j (and therefore rank of each M i is still C) v1v1 vkvk V SkSk S1S1 L

Cool Lemma Lemma: V S1S1 L Proof: Consider V L Hence Proved. (Quick deterministic algorithm, Faster randomized algorithms.) SkSk

Practical Optimal Network Codes. Multicast the only “easy and interesting” case for network coding problems. Capacity achieving codes. Linear encoding/decoding. Small field sizes.  At most quadratic gap.  Smallest field-size determination hard. Polynomial design complexity. (Random code design) Robustness. (Random code design) Joint paper being prepared for submission to IT Trans.: Jaggi, Sanders, Chou, Effros, Egner, Jain, Tolhuizen