Low Complexity Encoding for Network Codes

1. Low Complexity Encoding for Network Codes Yuval Cassuto Michelle Effros Sidharth Jaggi

2. Obligatory Example/History s [ACLY00] [ACLY00] Characterization Non-constructive b1 b2 C=2 [LYC03], [KM02] Constructive (linear) Exp-time design S I M P L E R b1 b2 [JCJ03], [SET03] Poly-time design Centralized design b1 b1 b2 b1+b2 [HKMKE03], [JCJ03]Decentralized design . . . b1 b1 b1+b2 b1+b2 [This work]All the above, plus optimal implementation complexity. t1 t2 (b1,b2) b1 (b1,b2)

3. Complexity F(2m)-linear network [KM02],[HKMKE03],… b1 b2 bm Source:- Group together `m’ bits, Every node:- Perform linear combinations over finite field F(2m) β1 β2 βk

4. Complexity …,[JEHM04],… [HKMKE03],… = “Thm”: For any algebraic code, at least half the β matrices in F(2m) have at least m2/2 non-zero elements Randomly chosen algebraic encoders require O(m2) bit operations [KKHRM05]

5. Simplicity – Permute-and-add Permutation matrix (sparse) = “Thm”: With “high” probability, permute-and-add codes have “almost” the same performance as algebraic codes Permute-and-add encoders require O(m) bit operations Tight! “Thm”: To achieve capacity, needO(m) bit operations

6. Simplicity – Permute-and-add Loss of information Loss of information = “Thm”: With “high” probability, permute-and-add codes have “almost” the same performance as algebraic codes Permute-and-add encoders require O(m) bit operations Tight! “Thm”: To achieve capacity, need O(m) bit operations

7. Permute-and-add Codes m “sufficiently” large b1 b2 b2 b1 bm bm  ’ b’1 b’1 b’m b’2 b’m b’2  ’’ b’’m b’’1 b’’2 b’’1 b’’m b’’2 Uniformly at random

8. Permute-and-add Codes Transfer matrix b2 b1 bm   ’ b’1 b’m b’2   b’’m b’’1 b’’2 ’’ Uniformly at random

9. Permute-and-add Codes Percolate transfer matrices across successive cutsets (in header) Not true, with probability c > 0 If each transfer matrix full rank, Final transfer matrices full rank Decode by inverting final transfer matrix, QED

10. Permute-and-add Codes R=C- |E|εm-εm m “sufficiently” large Thm: Permute-and-add codes achieve R=C-(|E|+1)εm , Pr > 1-(|T||E|+1)2-O(mεm) Each transfer matrix “almost” full rank, (1-εm) fraction Final transfer matrices “almost” full rank (1-|E|εm) fraction Decode by inverting final transfer matrix, QED Prπ [Row rank > (1-|E|εm) fraction] > 1-|T||E|2-O(mεm) Prπ [Final transform invertible] > 1-(|T||E|+1)2-O(mεm) Prπ [Row rank > (1-εm) fraction] > 1-2-O(mεm)

11. Proof of Lemma Transform Gaussian Elimination I “Almost” full rank, w.h.p. L1,L2

12. The End/Where now? • Low-complexity decoding? • Fewer packets encoded together at nodes? • Same permutation at each node? • Zero-error/Deterministic? • Permute-and-add Vs. Algebraic [HKMKE03] • Rate Almost Same (ε loss) • Probability of error Almost same (smaller exp) • Block-length Almost same (1/ε increase) • Simple Distributed Design Ditto • Implementation Complexity Quadratically better • Randomness required Quadratically better