On the Connection Between Multiple-Unicast Network Coding and Single-Source Single-Sink Network Error Correction
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1 On he Connecion Beween Muliple-Unica ework Coding and Single-Source Single-Sink ework Error Correcion Jörg Kliewer JIT Join work wih Wenao Huang and Michael Langberg
2 ework Error Correcion Problem: Adverary ha conrol of ome edge in a nework Objecive: Deign coding cheme ha i reilien again adverary Which rae are achievable? 2
3 ework Error Correcion Problem: Adverary ha conrol of ome edge in a nework Objecive: Deign coding cheme ha i reilien again adverary Which rae are achievable?
4 Known Cae Adverary conrol z link in he nework Single ource mulica, equal capaciy link Capaciy: mincu 2z Code deign, e.g., in [Cai & Yeung 06], [Koeer & Kchichang 08], [Jaggi e al. 08], [Silva e al. 08], [Brio, Kliewer 13]
5 Le Known and Sudied Cae Single ource mulica: differen edge capaciie, node adverarie, rericed adverarie (e.g., [Kou, Tong, Te 09], [Kim e al. 11], [Wang, Silva, Kchichang 08]) Muliple ource and erminal: Upper and lower capaciy bound [Vyerenko, Ho, Dikalioi0], [Liang, Agrawal, Vaidya 10]
6 Thi Work Single-ource ingle-ink Acyclic nework Edge may no have uni capaciy 5
7 Thi Work Single-ource ingle-ink Acyclic nework Edge may no have uni capaciy Adverary conrol ingle link 5
8 Thi Work Single-ource ingle-ink Acyclic nework Edge may no have uni capaciy Adverary conrol ingle link Some edge canno be acceed by he adverary 5
9 Thi Work Single-ource ingle-ink Acyclic nework Edge may no have uni capaciy Adverary conrol ingle link Some edge canno be acceed by he adverary Reliable communicaion rae? 5
10 Reul 6
11 Reul ework error correcion problem: 6
12 Reul ework error correcion problem: Compuing capaciy i a hard a compuing he capaciy of an error-free muliple unica nework coding problem. 6
13 Reul ework error correcion problem: Compuing capaciy i a hard a compuing he capaciy of an error-free muliple unica nework coding problem. A 1 a 1 a k A k B x 1 y 1 y k x k k k B k b k 6
14 Reul ework error correcion problem: Compuing capaciy i a hard a compuing he capaciy of an error-free muliple unica nework coding problem. A 1 a 1 a k A k B x 1 y 1 y k x k k k B k.... k k b k 6
15 Reul ework error correcion problem: Compuing capaciy i a hard a compuing he capaciy of an error-free muliple unica nework coding problem. A 1 a 1 a k A k B x 1 y 1 y k x k k k B k.... k k b k 6
16 Reul ework error correcion problem: Compuing capaciy i a hard a compuing he capaciy of an error-free muliple unica nework coding problem. A 1 a 1 a k A k B x 1 y 1 y k x k k k B k.... k k b k 6
17 Reducion Reul proved by reducion Significan inere recenly Index coding/nework coding, index coding/inerference alignmen, nework equivalence, muliple unica v. muliple mulica C, edge removal problem, 7
18 Reducion Reul proved by reducion Significan inere recenly Index coding/nework coding, index coding/inerference alignmen, nework equivalence, muliple unica v. muliple mulica C, edge removal problem, Generae a cluering of nework communicaion problem via reducion ework communicaion problem 7
19 Reducion Reul proved by reducion Significan inere recenly Index coding/nework coding, index coding/inerference alignmen, nework equivalence, muliple unica v. muliple mulica C, edge removal problem, Generae a cluering of nework communicaion problem via reducion Reducion ework communicaion problem 7
20 Reducion Reul proved by reducion Significan inere recenly Index coding/nework coding, index coding/inerference alignmen, nework equivalence, muliple unica v. muliple mulica C, edge removal problem, Generae a cluering of nework communicaion problem via reducion Idenify canonical problem (cenral problem ha are relaed o everal oher problem) Reducion ework communicaion problem 7
21 Reducion Reul proved by reducion Significan inere recenly Index coding/nework coding, index coding/inerference alignmen, nework equivalence, muliple unica v. muliple mulica C, edge removal problem, Generae a cluering of nework communicaion problem via reducion Idenify canonical problem (cenral problem ha are relaed o everal oher problem) Reducion ework communicaion problem 7
22 Proof Compuing error correcing capaciy i a hard a compuing he capaciy of an error-free muliple unica nework coding problem. Proof for zero error communicaion Reul alo hold for boh he cae of aympoic rae and aympoic error 8
23 Proof Compuing error correcing capaciy i a hard a compuing he capaciy of an error-free muliple unica nework coding problem. Proof for zero error communicaion Reul alo hold for boh he cae of aympoic rae and aympoic error Saring poin: MU C problem I rae uple (1, 1,..., 1) wih zero error? achievable.... k k 8
24 Proof Compuing error correcing capaciy i a hard a compuing he capaciy of an error-free muliple unica nework coding problem. Proof for zero error communicaion Reul alo hold for boh he cae of aympoic rae and aympoic error Saring poin: MU C problem I rae uple (1, 1,..., 1) achievable wih zero error? Reducion: Conruc new nework a 1 a k A 1 A k B k k x 1 y 1 y k x k B k 8 b k
25 Proof Compuing error correcing capaciy i a hard a compuing he capaciy of an error-free muliple unica nework coding problem. Proof for zero error communicaion Reul alo hold for boh he cae of aympoic rae and aympoic error Saring poin: MU C problem I rae uple (1, 1,..., 1) wih zero error? achievable Reducion: Conruc new nework Adverary can acce any ingle link excep link leaving and 8 A 1 a k k a k A k x 1 y 1 y k x k B 1 X X X X b k B k
26 Zero Error Cae Theorem Rae (1, 1,..., 1) achievable on iff rae k i achievable on. 9
27 Zero Error Cae Theorem Rae (1, 1,..., 1) achievable on iff rae k i achievable on. Proof kech: A 1 a 1 a k A k B x 1 y 1 y k x k k k B k b k 9
28 Zero Error Cae Theorem Rae (1, 1,..., 1) achievable on iff rae k i achievable on. Proof kech: Aume (1, 1,..., 1) on Source end informaion on ai a 1 a k A 1 A k B x 1 y 1 y k x k k k B k b k 9
29 Zero Error Cae Theorem Rae (1, 1,..., 1) achievable on iff rae k i achievable on. Proof kech: Aume (1, 1,..., 1) on Source end informaion on ai One error may occur on x i, y i, z i, z i a 1 a k A 1 A k B x 1 y 1 y k x k k k B k b k 9
30 Zero Error Cae Theorem Rae (1, 1,..., 1) achievable on iff rae k i achievable on. Proof kech: Aume (1, 1,..., 1) on Source end informaion on ai One error may occur on x i, y i, z i, z i Bi perform majoriy decoding Rae k i poible on a 1 a k A 1 A k B x 1 y 1 y k x k k k B k b k 9
31 Zero Error Cae 10
32 Zero Error Cae Proof kech: 10
33 Zero Error Cae Proof kech: Aume rae k achievable on : how rae (1, 1,..., 1) on M A 1 a 1 a k A k B x 1 y 1 y k x k k k B k b k 10
34 Zero Error Cae Proof kech: Aume rae k achievable on : how rae (1, 1,..., 1) on Full rae (cu): 1-1 correp. beween meage M, a 1,...,a k,,...,b k a 1 M a k A 1 A k B x 1 y 1 y k x k k k B k b k 10
35 Zero Error Cae Proof kech: Aume rae k achievable on : how rae (1, 1,..., 1) on Full rae (cu): 1-1 correp. beween meage M, a 1,...,a k,,...,b k Error correcion: For M 1 M 2 if a b i (M 1 ) b i (M 2 ) hen z i (M 1 ) z i (M 2 ) For z i (M 1 )=z i (M 2 ) erminal canno diinguih beween M1, M2 a 1 a k A 1 A k B x 1 y 1 y k x k M k k B k b k 10
36 Zero Error Cae Proof kech: Aume rae k achievable on : how rae (1, 1,..., 1) on Full rae (cu): 1-1 correp. beween meage M, a 1,...,a k,,...,b k Error correcion: For M 1 M 2 if a b i (M 1 ) b i (M 2 ) hen z i (M 1 ) z i (M 2 ) For z i (M 1 )=z i (M 2 ) erminal canno diinguih beween M1, M2 a 1 a k A 1 A k B x 1 y 1 y k x k M k k B k b k 10
37 Zero Error Cae Proof kech: Aume rae k achievable on : how rae (1, 1,..., 1) on Full rae (cu): 1-1 correp. beween meage M, a 1,...,a k,,...,b k Error correcion: For M 1 M 2 if a b i (M 1 ) b i (M 2 ) hen z i (M 1 ) z i (M 2 ) For z i (M 1 )=z i (M 2 ) erminal canno diinguih beween M1, M2 a 1 a k A 1 A k B x 1 y 1 y k x k M k k B k b k 10
38 Zero Error Cae Proof kech: Aume rae k achievable on : how rae (1, 1,..., 1) on Full rae (cu): 1-1 correp. beween meage M, a 1,...,a k,,...,b k Error correcion: For M 1 M 2 if a b i (M 1 ) b i (M 2 ) hen z i (M 1 ) z i (M 2 ) For z i (M 1 )=z i (M 2 ) erminal canno diinguih beween M1, M2 a 1 a k A 1 A k B x 1 y 1 y k x k M k k B k b k 10
39 Zero Error Cae Proof kech: Aume rae k achievable on : how rae (1, 1,..., 1) on Full rae (cu): 1-1 correp. beween meage M, a 1,...,a k,,...,b k Error correcion: For M 1 M 2 if a b i (M 1 ) b i (M 2 ) hen z i (M 1 ) z i (M 2 ) For z i (M 1 )=z i (M 2 ) erminal canno diinguih beween M1, M2 1-1 correpondence beween b i z i a 1 a k A 1 A k B x 1 y 1 y k x k M k k B k b k 10
40 Zero Error Cae Proof kech: Aume rae k achievable on : how rae (1, 1,..., 1) on Full rae (cu): 1-1 correp. beween meage M, a 1,...,a k,,...,b k 1-1 correpondence beween b i z i a 1 a k A 1 A k B x 1 y 1 y k x k M k k B k b k 11
41 Zero Error Cae Proof kech: Aume rae k achievable on : how rae (1, 1,..., 1) on Full rae (cu): 1-1 correp. beween meage M, a 1,...,a k,,...,b k 1-1 correpondence beween b i z i Same argumen: 1-1 correpondence beween a i x i y i z i a 1 a k A 1 A k B x 1 y 1 y k x k M k k B k b k 11
42 Zero Error Cae Proof kech: Aume rae k achievable on : how rae (1, 1,..., 1) on Full rae (cu): 1-1 correp. beween meage M, a 1,...,a k,,...,b k 1-1 correpondence beween b i z i Same argumen: 1-1 correpondence beween a i x i y i z i a 1 a k A 1 A k B x 1 y 1 y k x k M k k B k b k 11
43 Zero Error Cae Proof kech: Aume rae k achievable on : how rae (1, 1,..., 1) on Full rae (cu): 1-1 correp. beween meage M, a 1,...,a k,,...,b k 1-1 correpondence beween b i z i Same argumen: 1-1 correpondence beween a i x i y i z i In ummary: z i x i b i z i Implie z i z i : muliple unica a 1 a k A 1 A k B x 1 y 1 y k x k M k k B k b k 11
44 Zero Error Cae Proof kech: Aume rae k achievable on : how rae (1, 1,..., 1) on Full rae (cu): 1-1 correp. beween meage M, a 1,...,a k,,...,b k 1-1 correpondence beween b i z i Same argumen: 1-1 correpondence beween a i x i y i z i In ummary: z i x i b i z i Implie z i z i : muliple unica a 1 a k A 1 A k B x 1 y 1 y k x k M k k B k b k 11
45 Aympoic Error Cae Proof kech: M A 1 a 1 a k A k B x 1 y 1 y k x k k k B k b k 12
46 Aympoic Error Cae Proof kech: Argumen are more complicaed Eenially he ame M A 1 a 1 a k A k B x 1 y 1 y k x k k k B k b k 12
47 Aympoic Error Cae Proof kech: Argumen are more complicaed Eenially he ame M a We how: A 1 a 1 a k A k a lim I(a i; b i )/n = 1 n, 0 a lim I(a i; z i )/n = 1 n, 0 a lim n, 0 I(b i; z i )/n = 1 B x 1 y 1 y k x k k k B k b k 12
48 Aympoic Error Cae Proof kech: Argumen are more complicaed Eenially he ame M a We how: A 1 a 1 a k A k a lim I(a i; b i )/n = 1 n, 0 a lim I(a i; z i )/n = 1 n, 0 a lim n, 0 I(b i; z i )/n = 1 B x 1 y 1 y k x k k k B k b k 12
49 Aympoic Error Cae Proof kech: Argumen are more complicaed Eenially he ame M a We how: A 1 a 1 a k A k a lim I(a i; b i )/n = 1 n, 0 a lim I(a i; z i )/n = 1 n, 0 a lim n, 0 I(b i; z i )/n = 1 In ummary: lim I(z i; z i )/n = 1 n, 0 Block code: MU wih poible > 0 B x 1 y 1 y k x k k k B k b k 12
50 Curren Underanding A 1 a 1 a k A k.... k k B k x 1 y 1 y k x k k B k b k 13
51 Curren Underanding A 1 a 1 a k A k.... k k B k x 1 y 1 y k x k k B k b k 13
52 Curren Underanding A 1 a 1 a k A k.... k k B k x 1 y 1 y k x k k B k (1, 1,...,1) feaible Rae k feaible wih zero error wih zero error b k 13
53 Curren Underanding A 1 a 1 a k A k.... k k B k x 1 y 1 y k x k k B k (1, 1,...,1) feaible Rae k feaible wih zero error wih zero error b k (1, 1,...,1) feaible Rae k feaible wih error wih error 13
54 Curren Underanding A 1 a 1 a k A k.... k k B k x 1 y 1 y k x k k B k (1, 1,...,1) feaible Rae k feaible wih zero error wih zero error b k (1, 1,...,1) feaible Rae k feaible wih error wih error (1, 1,...,1) aymp. Rae k aymp. feaible, no exacly feaible, no exacly (1, 1,...,1) no Rae k no aymp. aymp. feaible feaible 13
55 Curren Underanding A 1 a 1 a k A k.... k k B k x 1 y 1 y k x k k B k (1, 1,...,1) feaible Rae k feaible wih zero error wih zero error b k (1, 1,...,1) feaible Rae k feaible wih error wih error (1, 1,...,1) aymp. Rae k aymp. feaible, no exacly feaible, no exacly (1, 1,...,1) no Rae k no aymp. aymp. feaible feaible 13
56 Infeaibiliy of Muliple Unica Theorem There exi nework and uch ha rae k i aympoically achievable on bu (1, 1,..., 1) i no aymp. achievable on. 14
57 Infeaibiliy of Muliple Unica Theorem There exi nework and uch ha rae k i aympoically achievable on bu (1, 1,..., 1) i no aymp. achievable on. Proof (conrucive): a 1 a 2 A 1 A 2 z 2 C 2 x 1 y 1 y 2 x 2 D 2 z 2 B 1 B 2 b 2 14
58 Infeaibiliy of Muliple Unica Theorem There exi nework and uch ha rae k i aympoically achievable on bu (1, 1,..., 1) i no aymp. achievable on. Proof (conrucive): Uni capaciy edge, meage M = (M1, M2), H(M1) = H(M2) = n-1 bi, adverary can acce one link ework code: a 1 a 2 A 1 A 2 2 z 2 x 1 y 1 y 2 x 2 C D 2 z 2 B 1 B 2 b 2 14
59 Infeaibiliy of Muliple Unica Theorem There exi nework and uch ha rae k i aympoically achievable on bu (1, 1,..., 1) i no aymp. achievable on. Proof (conrucive): Uni capaciy edge, meage M = (M1, M2), H(M1) = H(M2) = n-1 bi, adverary can acce one link ework code: a 1 (M) =x 1 (M) =y 1 (M) = (M) =M 1 a 1 a 2 A 1 A 2 2 z 2 x 1 y 1 y 2 x 2 C D 2 z 2 B 1 B 2 b 2 14
60 Infeaibiliy of Muliple Unica Theorem There exi nework and uch ha rae k i aympoically achievable on bu (1, 1,..., 1) i no aymp. achievable on. Proof (conrucive): Uni capaciy edge, meage M = (M1, M2), H(M1) = H(M2) = n-1 bi, adverary can acce one link ework code: a 1 (M) =x 1 (M) =y 1 (M) = (M) =M 1 a 2 (M) =x 2 (M) =y 2 (M) =z 2 (M) =M 2 a 1 a 2 A 1 A 2 2 z 2 x 1 y 1 y 2 x 2 C D 2 z 2 B 1 B 2 b 2 14
61 Infeaibiliy of Muliple Unica Theorem There exi nework and uch ha rae k i aympoically achievable on bu (1, 1,..., 1) i no aymp. achievable on. Proof (conrucive): Uni capaciy edge, meage M = (M1, M2), H(M1) = H(M2) = n-1 bi, adverary can acce one link ework code: a 1 (M) =x 1 (M) =y 1 (M) = (M) =M 1 a 2 (M) =x 2 (M) =y 2 (M) =z 2 (M) =M 2 (M) =z 2 (M) =M 1 + M 2 14 a 1 a 2 A 1 A 2 z 2 x 1 y 1 y 2 x 2 C D z 2 B 1 B b 2
62 Infeaibiliy of Muliple Unica Theorem There exi nework and uch ha rae k i aympoically achievable on bu (1, 1,..., 1) i no aymp. achievable on. Proof (con.): a 1 a 2 A 1 A 2 z 2 C 2 x 1 y 1 y 2 x 2 D 2 z 2 B 1 B 2 b 2 15
63 Infeaibiliy of Muliple Unica Theorem There exi nework and uch ha rae k i aympoically achievable on bu (1, 1,..., 1) i no aymp. achievable on. Proof (con.): a b i (x i, x i, z i )= x i if x i = y i (cae 1) z i if x i = y i (cae 2) bi reerve one bi o indicae if cae 1 or 2 happen a 1 a 2 A 1 A 2 C 2 z 2 x 1 y 1 y 2 x 2 D 2 z 2 B 1 B 2 b 2 15
64 Infeaibiliy of Muliple Unica Theorem There exi nework and uch ha rae k i aympoically achievable on bu (1, 1,..., 1) i no aymp. achievable on. Proof (con.): a b i (x i, x i, z i )= x i if x i = y i (cae 1) z i if x i = y i (cae 2) bi reerve one bi o indicae if cae 1 or 2 happen i able o decode (M1, M2) correcly a aympoic rae 2 Muliple unica wih rae (1, 1) i no feaible 15 a 1 a 2 A 1 A 2 z 2 x 1 y 1 y 2 x 2 C D z 2 B 1 B b 2
65 Take Away Error correcion in a imple nework eing Single-ource nework error correcion i a lea a hard a muliple-unica (error-free) nework coding Fall ino broader framework ha connec differen nework communicaion problem via reducion 16
66 Take Away Error correcion in a imple nework eing Single-ource nework error correcion i a lea a hard a muliple-unica (error-free) nework coding Fall ino broader framework ha connec differen nework communicaion problem via reducion [Huang, Langberg, Kliewer, acceped for ISIT 2015] 16
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