On the Connection Between Multiple-Unicast Network Coding and - PowerPoint PPT Presentation

On the Connection Between Multiple-Unicast Network Coding and Single-Source Single-Sink Network Error Correction Jörg Kliewer NJIT Joint work with Wentao Huang and Michael Langberg

Network Error Correction Problem: Adversary has control of some edges in a network Objective: Design coding scheme that is resilient against adversary Which rates are achievable? 2

Network Error Correction Problem: Adversary has control of some edges in a network Objective: Design coding scheme that is resilient against adversary Which rates are achievable? s 1 t 1 s 2 t 2 t 3 s 3 s 4 t 4 2

Known Cases Adversary controls z links in the network Single source multicast , equal capacity links ‣ mincut − 2 z Capacity: ‣ Code design, e.g., in [Cai & Yeung 06], [Koetter & Kschischang 08], [Jaggi et al. 08], [Silva et al. 08], [Brito, Kliewer 13] t 1 t 2 s t 3 t 4 3

Less Known and Studied Cases Single source multicast: di ff erent edge capacities, node adversaries, restricted adversaries (e.g., [Kosut, Tong, Tse 09], [Kim et al. 11], [Wang, Silva, Kschischang 08]) Multiple sources and terminals: Upper and lower capacity bounds [Vyetrenko, Ho, Dikaliotis 10], [Liang, Agrawal, Vaidya 10] s 1 t 1 s 2 t 2 t 3 s 3 s 4 t 4 4

This Work Single-source single-sink Acyclic network Edges may not have unit capacity s t 5

This Work Single-source single-sink Acyclic network Edges may not have unit capacity Adversary controls single link s t 5

This Work Single-source single-sink Acyclic network Edges may not have unit capacity Adversary controls single link Some edges cannot be accessed by the adversary s t 5

This Work Single-source single-sink Acyclic network Edges may not have unit capacity Adversary controls single link Some edges cannot be accessed by the adversary Reliable communication rate? s t 5

Results 6

Results Network error correction problem: s t 6

Results Network error correction problem: s t Computing capacity is as hard as computing the capacity of an error-free multiple unicast network coding problem. 6

Results Network error correction problem: s t Computing capacity is as hard as computing the capacity of an error-free multiple unicast network coding problem. s a 1 a k A 1 A k z 1 z k . . . s 1 s k . . . x 1 y 1 y k x k N t 1 t k . . . z � z � 1 k B 1 B k . . . b 1 b k t 6

Results Network error correction problem: s t Computing capacity is as hard as computing the capacity of an error-free multiple unicast network coding problem. s a 1 a k A 1 A k z 1 z k . . . s 1 s k s 1 s k . . . . . . x 1 y 1 y k x k N N t 1 t k t 1 t k . . . . . . z � z � 1 k B 1 B k . . . b 1 b k t 6

Results Network error correction problem: s t Computing capacity is as hard as computing the capacity of an error-free multiple unicast network coding problem. s a 1 a k A 1 A k z 1 z k . . . s 1 s k s 1 s k . . . . . . x 1 y 1 y k x k N N t 1 t k t 1 t k . . . . . . z � z � 1 k B 1 B k . . . b 1 b k t 6

Results Network error correction problem: s t Computing capacity is as hard as computing the capacity of an error-free multiple unicast network coding problem. s a 1 a k A 1 A k z 1 z k . . . s 1 s k s 1 s k . . . . . . x 1 y 1 y k x k N N t 1 t k t 1 t k . . . . . . z � z � 1 k B 1 B k . . . b 1 b k t 6

Reductions Result proved by reduction Significant interest recently ‣ Index coding/network coding, index coding/interference alignment, network equivalence, multiple unicast vs. multiple multicast NC, edge removal problem, … 7

Reductions Result proved by reduction Significant interest recently ‣ Index coding/network coding, index coding/interference alignment, network equivalence, multiple unicast vs. multiple multicast NC, edge removal problem, … Generate a clustering of network communication problems via reductions Network communication problems 7

Reductions Result proved by reduction Significant interest recently ‣ Index coding/network coding, index coding/interference alignment, network equivalence, multiple unicast vs. multiple multicast NC, edge removal problem, … Generate a clustering of network communication problems via reductions Network communication problems Reductions 7

Reductions Result proved by reduction Significant interest recently ‣ Index coding/network coding, index coding/interference alignment, network equivalence, multiple unicast vs. multiple multicast NC, edge removal problem, … Generate a clustering of network communication problems via reductions Identify canonical problems (central problems that are related to several other problems) Network communication problems Reductions 7

Reductions Result proved by reduction Significant interest recently ‣ Index coding/network coding, index coding/interference alignment, network equivalence, multiple unicast vs. multiple multicast NC, edge removal problem, … Generate a clustering of network communication problems via reductions Identify canonical problems (central problems that are related to several other problems) Network communication problems Reductions 7

Proof Computing error correcting capacity is as hard as computing the capacity of an error-free multiple unicast network coding problem. Proof for zero error communication Result also holds for both the case of asymptotic rate and asymptotic error 8

Proof Computing error correcting capacity is as hard as computing the capacity of an error-free multiple unicast network coding problem. Proof for zero error communication Result also holds for both the case of asymptotic rate and asymptotic error Starting point: MU NC problem N Is rate tuple achievable ( 1 , 1 , . . . , 1 ) with zero error? s 1 s k . . . N t 1 t k . . . 8

Proof Computing error correcting capacity is as hard as computing the capacity of an error-free multiple unicast network coding problem. Proof for zero error communication Result also holds for both the case of asymptotic rate and asymptotic error s Starting point: MU NC problem N a 1 a k A 1 A k Is rate tuple achievable ( 1 , 1 , . . . , 1 ) N � z 1 z k with zero error? . . . Reduction: Construct new network N � s 1 s k s 1 s k . . . . . . x 1 y 1 y k x k N N t 1 t k t 1 t k . . . . . . z � z � 1 k B 1 B k . . . b 1 b k 8 t

Proof Computing error correcting capacity is as hard as computing the capacity of an error-free multiple unicast network coding problem. Proof for zero error communication Result also holds for both the case of asymptotic rate and asymptotic error s Starting point: MU NC problem N X X a 1 a k A 1 A k Is rate tuple achievable ( 1 , 1 , . . . , 1 ) N � z 1 z k with zero error? . . . Reduction: Construct new network N � s 1 s k s 1 s k . . . . . . x 1 y 1 y k x k N N t 1 t k t 1 t k Adversary can access any single . . . . . . link except links leaving s and t z � z � 1 k B 1 B k . . . X X b 1 b k 8 t

Zero Error Case Theorem Rates achievable on i ff rate k is achievable on . ( 1 , 1 , . . . , 1 ) N � N 9

Zero Error Case Theorem Rates achievable on i ff rate k is achievable on . ( 1 , 1 , . . . , 1 ) N � N s Proof sketch: a 1 a k A 1 A k z 1 z k . . . s 1 s k . . . x 1 y 1 y k x k N t 1 t k . . . z � z � k 1 B 1 B k . . . b 1 b k t 9

Zero Error Case Theorem Rates achievable on i ff rate k is achievable on . ( 1 , 1 , . . . , 1 ) N � N s Proof sketch: a 1 a k A 1 A k Assume on ( 1 , 1 , . . . , 1 ) N z 1 z k . . . Source sends information on a i s 1 s k . . . x 1 y 1 y k x k N t 1 t k . . . z � z � k 1 B 1 B k . . . b 1 b k t 9

Zero Error Case Theorem Rates achievable on i ff rate k is achievable on . ( 1 , 1 , . . . , 1 ) N � N s Proof sketch: a 1 a k A 1 A k Assume on ( 1 , 1 , . . . , 1 ) N z 1 z k . . . Source sends information on a i s 1 s k One error may occur on x i , y i , z i , z � . . . i x 1 y 1 y k x k N t 1 t k . . . z � z � k 1 B 1 B k . . . b 1 b k t 9

Zero Error Case Theorem Rates achievable on i ff rate k is achievable on . ( 1 , 1 , . . . , 1 ) N � N s Proof sketch: a 1 a k A 1 A k Assume on ( 1 , 1 , . . . , 1 ) N z 1 z k . . . Source sends information on a i s 1 s k One error may occur on x i , y i , z i , z � . . . i x 1 y 1 y k x k N t 1 t k . . . B i performs majority decoding Rate k is possible on N � z � z � k 1 B 1 B k . . . b 1 b k t 9

Zero Error Case 10

Zero Error Case Proof sketch: 10

Zero Error Case Proof sketch: Assume rate k achievable on : show rate on ( 1 , 1 , . . . , 1 ) N � N M s a 1 a k A 1 A k z 1 z k . . . s 1 s k . . . x 1 y 1 y k x k N t 1 t k . . . z � z � k 1 B 1 B k . . . b 1 b k t 10