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

Jörg Kliewer NJIT

On the Connection Between Multiple-Unicast Network Coding and Single-Source Single-Sink Network Error Correction

Joint work with Wentao Huang and Michael Langberg

Network Error Correction

2

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

Network Error Correction

2

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

s1 s2 s3 s4 t4 t3 t2 t1

Known Cases

3

Adversary controls z links in the network Single source multicast, equal capacity links

• Capacity:
• Code design, e.g., in [Cai & Yeung 06], [Koetter &

Kschischang 08], [Jaggi et al. 08], [Silva et al. 08], [Brito, Kliewer 13] mincut − 2z

s t4 t3 t2 t1

Less Known and Studied Cases

4

Single source multicast: different 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]

s1 s2 s3 s4 t4 t3 t2 t1

This Work

5

Single-source single-sink Acyclic network Edges may not have unit capacity

s t

This Work

5

Single-source single-sink Acyclic network Edges may not have unit capacity Adversary controls single link

s t

This Work

5

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

This Work

5

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

Results

6

Results

6

s t

Network error correction problem:

Results

6

Computing capacity is as hard as computing the capacity of an error-free multiple unicast network coding problem.

s t

Network error correction problem:

Results

6

Computing capacity is as hard as computing the capacity of an error-free multiple unicast network coding problem.

s t

Network error correction problem:

. . . . . . . . . . . .

a1 ak A1 Ak zk z1 z

1

z

k

t x1 y1 yk xk bk b1 B1 Bk

N

sk s1 tk t1 s

Results

6

Computing capacity is as hard as computing the capacity of an error-free multiple unicast network coding problem.

s t

Network error correction problem:

. . . . . . . . . . . .

a1 ak A1 Ak zk z1 z

1

z

k

t x1 y1 yk xk bk b1 B1 Bk

N

sk s1 tk t1 s

. . . . . .

N

sk s1 tk t1

Results

6

Computing capacity is as hard as computing the capacity of an error-free multiple unicast network coding problem.

s t

Network error correction problem:

. . . . . . . . . . . .

a1 ak A1 Ak zk z1 z

1

z

k

t x1 y1 yk xk bk b1 B1 Bk

N

sk s1 tk t1 s

. . . . . .

N

sk s1 tk t1

Results

6

Computing capacity is as hard as computing the capacity of an error-free multiple unicast network coding problem.

s t

Network error correction problem:

. . . . . . . . . . . .

a1 ak A1 Ak zk z1 z

1

z

k

t x1 y1 yk xk bk b1 B1 Bk

N

sk s1 tk t1 s

. . . . . .

N

sk s1 tk t1

Reductions

7

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, …

Reductions

7

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

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

Reductions

7

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, …

Identify canonical problems (central problems that are related to several other problems) Generate a clustering of network communication problems via reductions

Network communication problems

Reductions

Reductions

7

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, …

Identify canonical problems (central problems that are related to several other problems) Generate a clustering of network communication problems via reductions

Network communication problems

Reductions

Proof

8

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

. . . . . .

N

sk s1 tk t1

Proof

8

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 Is rate tuple achievable with zero error? N (1, 1, . . . , 1)

. . . . . .

N

sk s1 tk t1

. . . . . . . . . . . .

a1 ak A1 Ak zk z1 z

1

z

k

t x1 y1 yk xk bk b1 B1 Bk

N

sk s1 tk t1 s

Proof

8

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 Is rate tuple achievable with zero error? N (1, 1, . . . , 1) Reduction: Construct new network N

N

. . . . . .

N

sk s1 tk t1

. . . . . . . . . . . .

a1 ak A1 Ak zk z1 z

1

z

k

t x1 y1 yk xk bk b1 B1 Bk

N

sk s1 tk t1 s

Proof

8

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 Is rate tuple achievable with zero error? N (1, 1, . . . , 1) Reduction: Construct new network N Adversary can access any single link except links leaving s and t

X X X X N

Zero Error Case

9

Theorem Rates achievable on iff rate k is achievable on . (1, 1, . . . , 1) N N

Zero Error Case

9

Theorem Rates achievable on iff rate k is achievable on . (1, 1, . . . , 1) N N

. . . . . . . . . . . .

a1 ak A1 Ak zk z1 z

1

z

k

t x1 y1 yk xk bk b1 B1 Bk

N

sk s1 tk t1 s

Proof sketch:

Zero Error Case

9

Theorem Rates achievable on iff rate k is achievable on . (1, 1, . . . , 1) N N

. . . . . . . . . . . .

a1 ak A1 Ak zk z1 z

1

z

k

t x1 y1 yk xk bk b1 B1 Bk

N

sk s1 tk t1 s

Assume on Source sends information on ai

Proof sketch:

(1, 1, . . . , 1) N

Zero Error Case

9

Theorem Rates achievable on iff rate k is achievable on . (1, 1, . . . , 1) N N

. . . . . . . . . . . .

a1 ak A1 Ak zk z1 z

1

z

k

t x1 y1 yk xk bk b1 B1 Bk

N

sk s1 tk t1 s

Assume on Source sends information on ai

Proof sketch:

(1, 1, . . . , 1) One error may occur on xi, yi, zi, z

i

N

Zero Error Case

9

Theorem Rates achievable on iff rate k is achievable on . (1, 1, . . . , 1) N N

. . . . . . . . . . . .

a1 ak A1 Ak zk z1 z

1

z

k

t x1 y1 yk xk bk b1 B1 Bk

N

sk s1 tk t1 s

Assume on Source sends information on ai

Proof sketch:

(1, 1, . . . , 1) One error may occur on xi, yi, zi, z

i

Bi performs majority decoding Rate k is possible on N N

Zero Error Case

10

Zero Error Case

10

Proof sketch:

Zero Error Case

10

. . . . . . . . . . . .

a1 ak A1 Ak zk z1 z

1

z

k

t x1 y1 yk xk bk b1 B1 Bk

N

sk s1 tk t1 s

Proof sketch: M

Assume rate k achievable on : show rate on N (1, 1, . . . , 1) N

Zero Error Case

10

. . . . . . . . . . . .

a1 ak A1 Ak zk z1 z

1

z

k

t x1 y1 yk xk bk b1 B1 Bk

N

sk s1 tk t1 s

Proof sketch:

Full rate (cut): 1-1 corresp. between message M, a1, . . . , ak, b1, . . . , bk

M

Assume rate k achievable on : show rate on N (1, 1, . . . , 1) N

Zero Error Case

10

. . . . . . . . . . . .

a1 ak A1 Ak zk z1 z

1

z

k

t x1 y1 yk xk bk b1 B1 Bk

N

sk s1 tk t1 s

Proof sketch:

Full rate (cut): 1-1 corresp. between message M, a1, . . . , ak, b1, . . . , bk

M

Assume rate k achievable on : show rate on N (1, 1, . . . , 1) N Error correction: For if a then For terminal cannot distinguish between M1, M2 M1 ̸= M2 bi(M1) ̸= bi(M2) z′

i(M1) ̸= z′ i(M2)

z′

i(M1) = z′ i(M2)

Zero Error Case

10

. . . . . . . . . . . .

a1 ak A1 Ak zk z1 z

1

z

k

t x1 y1 yk xk bk b1 B1 Bk

N

sk s1 tk t1 s

Proof sketch:

Full rate (cut): 1-1 corresp. between message M, a1, . . . , ak, b1, . . . , bk

M

Assume rate k achievable on : show rate on N (1, 1, . . . , 1) N Error correction: For if a then For terminal cannot distinguish between M1, M2 M1 ̸= M2 bi(M1) ̸= bi(M2) z′

i(M1) ̸= z′ i(M2)

z′

i(M1) = z′ i(M2)

Zero Error Case

10

. . . . . . . . . . . .

a1 ak A1 Ak zk z1 z

1

z

k

t x1 y1 yk xk bk b1 B1 Bk

N

sk s1 tk t1 s

Proof sketch:

Full rate (cut): 1-1 corresp. between message M, a1, . . . , ak, b1, . . . , bk

M

Assume rate k achievable on : show rate on N (1, 1, . . . , 1) N Error correction: For if a then For terminal cannot distinguish between M1, M2 M1 ̸= M2 bi(M1) ̸= bi(M2) z′

i(M1) ̸= z′ i(M2)

z′

i(M1) = z′ i(M2)

Zero Error Case

10

. . . . . . . . . . . .

a1 ak A1 Ak zk z1 z

1

z

k

t x1 y1 yk xk bk b1 B1 Bk

N

sk s1 tk t1 s

Proof sketch:

Full rate (cut): 1-1 corresp. between message M, a1, . . . , ak, b1, . . . , bk

M

Assume rate k achievable on : show rate on N (1, 1, . . . , 1) N Error correction: For if a then For terminal cannot distinguish between M1, M2 M1 ̸= M2 bi(M1) ̸= bi(M2) z′

i(M1) ̸= z′ i(M2)

z′

i(M1) = z′ i(M2)

Zero Error Case

10

. . . . . . . . . . . .

a1 ak A1 Ak zk z1 z

1

z

k

t x1 y1 yk xk bk b1 B1 Bk

N

sk s1 tk t1 s

Proof sketch:

Full rate (cut): 1-1 corresp. between message M, a1, . . . , ak, b1, . . . , bk

M

Assume rate k achievable on : show rate on N (1, 1, . . . , 1) N Error correction: For if a then For terminal cannot distinguish between M1, M2 M1 ̸= M2 bi(M1) ̸= bi(M2) z′

i(M1) ̸= z′ i(M2)

z′

i(M1) = z′ i(M2)

1-1 correspondence between bi ↔ z′

i

Zero Error Case

11

. . . . . . . . . . . .

a1 ak A1 Ak zk z1 z

1

z

k

t x1 y1 yk xk bk b1 B1 Bk

N

sk s1 tk t1 s

Proof sketch:

Full rate (cut): 1-1 corresp. between message M, a1, . . . , ak, b1, . . . , bk

M

Assume rate k achievable on : show rate on N (1, 1, . . . , 1) N 1-1 correspondence between bi ↔ z′

i

Zero Error Case

11

. . . . . . . . . . . .

a1 ak A1 Ak zk z1 z

1

z

k

t x1 y1 yk xk bk b1 B1 Bk

N

sk s1 tk t1 s

Proof sketch:

Full rate (cut): 1-1 corresp. between message M, a1, . . . , ak, b1, . . . , bk

M

Assume rate k achievable on : show rate on N (1, 1, . . . , 1) N 1-1 correspondence between bi ↔ z′

i

Same argument: 1-1 correspondence between ai ↔ xi ↔ yi ↔ zi

Zero Error Case

11

. . . . . . . . . . . .

a1 ak A1 Ak zk z1 z

1

z

k

t x1 y1 yk xk bk b1 B1 Bk

N

sk s1 tk t1 s

Proof sketch:

Full rate (cut): 1-1 corresp. between message M, a1, . . . , ak, b1, . . . , bk

M

Assume rate k achievable on : show rate on N (1, 1, . . . , 1) N 1-1 correspondence between bi ↔ z′

i

Same argument: 1-1 correspondence between ai ↔ xi ↔ yi ↔ zi

Zero Error Case

11

. . . . . . . . . . . .

a1 ak A1 Ak zk z1 z

1

z

k

t x1 y1 yk xk bk b1 B1 Bk

N

sk s1 tk t1 s

Proof sketch:

Full rate (cut): 1-1 corresp. between message M, a1, . . . , ak, b1, . . . , bk

M

Assume rate k achievable on : show rate on N (1, 1, . . . , 1) N 1-1 correspondence between bi ↔ z′

i

Same argument: 1-1 correspondence between ai ↔ xi ↔ yi ↔ zi In summary: Implies : multiple unicast zi ↔ xi ↔ bi ↔ z′

i

zi ↔ z′

i

Zero Error Case

11

. . . . . . . . . . . .

a1 ak A1 Ak zk z1 z

1

z

k

t x1 y1 yk xk bk b1 B1 Bk

N

sk s1 tk t1 s

Proof sketch:

Full rate (cut): 1-1 corresp. between message M, a1, . . . , ak, b1, . . . , bk

M

Assume rate k achievable on : show rate on N (1, 1, . . . , 1) N 1-1 correspondence between bi ↔ z′

i

Same argument: 1-1 correspondence between ai ↔ xi ↔ yi ↔ zi In summary: Implies : multiple unicast zi ↔ xi ↔ bi ↔ z′

i

zi ↔ z′

i

Asymptotic Error Case

12

. . . . . . . . . . . .

a1 ak A1 Ak zk z1 z

1

z

k

t x1 y1 yk xk bk b1 B1 Bk

N

sk s1 tk t1 s

Proof sketch: M

Asymptotic Error Case

12

. . . . . . . . . . . .

a1 ak A1 Ak zk z1 z

1

z

k

t x1 y1 yk xk bk b1 B1 Bk

N

sk s1 tk t1 s

Proof sketch: M

Arguments are more complicated Essentially the same

Asymptotic Error Case

12

. . . . . . . . . . . .

a1 ak A1 Ak zk z1 z

1

z

k

t x1 y1 yk xk bk b1 B1 Bk

N

sk s1 tk t1 s

Proof sketch: M

Arguments are more complicated Essentially the same a We show:

• a
• a
• a lim

n,0 I(bi; z i)/n = 1

lim

n,0 I(ai; zi)/n = 1

lim

n,0 I(ai; bi)/n = 1

Asymptotic Error Case

12

. . . . . . . . . . . .

a1 ak A1 Ak zk z1 z

1

z

k

t x1 y1 yk xk bk b1 B1 Bk

N

sk s1 tk t1 s

Proof sketch: M

Arguments are more complicated Essentially the same a We show:

• a
• a
• a lim

n,0 I(bi; z i)/n = 1

lim

n,0 I(ai; zi)/n = 1

lim

n,0 I(ai; bi)/n = 1

Asymptotic Error Case

12

. . . . . . . . . . . .

a1 ak A1 Ak zk z1 z

1

z

k

t x1 y1 yk xk bk b1 B1 Bk

N

sk s1 tk t1 s

Proof sketch: M

Arguments are more complicated Essentially the same a We show:

• a
• a
• a lim

n,0 I(bi; z i)/n = 1

lim

n,0 I(ai; zi)/n = 1

lim

n,0 I(ai; bi)/n = 1

> 0 In summary: Block codes: MU with possible lim

n,0 I(zi; z i)/n = 1

Current Understanding

13

. . . . . . . . . . . .

a1 ak A1 Ak zk z1 z

1

z

k

t x1 y1 yk xk bk b1 B1 Bk

N

sk s1 tk t1 s

. . . . . .

N

sk s1 tk t1

Current Understanding

13

. . . . . . . . . . . .

a1 ak A1 Ak zk z1 z

1

z

k

t x1 y1 yk xk bk b1 B1 Bk

N

sk s1 tk t1 s

. . . . . .

N

sk s1 tk t1

Current Understanding

13

. . . . . . . . . . . .

a1 ak A1 Ak zk z1 z

1

z

k

t x1 y1 yk xk bk b1 B1 Bk

N

sk s1 tk t1 s

. . . . . .

N

sk s1 tk t1

feasible with zero error (1, 1, . . . , 1) Rate k feasible with zero error

Current Understanding

13

. . . . . . . . . . . .

a1 ak A1 Ak zk z1 z

1

z

k

t x1 y1 yk xk bk b1 B1 Bk

N

sk s1 tk t1 s

. . . . . .

N

sk s1 tk t1

feasible with zero error (1, 1, . . . , 1) Rate k feasible with zero error feasible with error (1, 1, . . . , 1)

• Rate k feasible

with error

Current Understanding

13

. . . . . . . . . . . .

a1 ak A1 Ak zk z1 z

1

z

k

t x1 y1 yk xk bk b1 B1 Bk

N

sk s1 tk t1 s

. . . . . .

N

sk s1 tk t1

feasible with zero error (1, 1, . . . , 1) Rate k feasible with zero error feasible with error (1, 1, . . . , 1)

• Rate k feasible

with error

• asymp.

feasible, not exactly (1, 1, . . . , 1) Rate k asymp. feasible, not exactly not

• asymp. feasible

(1, 1, . . . , 1) Rate k not asymp. feasible

Current Understanding

13

. . . . . . . . . . . .

a1 ak A1 Ak zk z1 z

1

z

k

t x1 y1 yk xk bk b1 B1 Bk

N

sk s1 tk t1 s

. . . . . .

N

sk s1 tk t1

feasible with zero error (1, 1, . . . , 1) Rate k feasible with zero error feasible with error (1, 1, . . . , 1)

• Rate k feasible

with error

• asymp.

feasible, not exactly (1, 1, . . . , 1) Rate k asymp. feasible, not exactly not

• asymp. feasible

(1, 1, . . . , 1) Rate k not asymp. feasible

Infeasibility of Multiple Unicast

14

Theorem There exist networks and such that rate k is asymptotically achievable on but is not asymp. achievable on . (1, 1, . . . , 1) N N N N

a1 a2 A1 A2 z2 z1 z

1

z

2

t x1 y1 y2 x2 b2 b1 B1 B2 s2 s1 t2 t1 s C D

Infeasibility of Multiple Unicast

14

Proof (constructive):

Theorem There exist networks and such that rate k is asymptotically achievable on but is not asymp. achievable on . (1, 1, . . . , 1) N N N N

a1 a2 A1 A2 z2 z1 z

1

z

2

t x1 y1 y2 x2 b2 b1 B1 B2 s2 s1 t2 t1 s C D

Infeasibility of Multiple Unicast

14

Proof (constructive):

Theorem There exist networks and such that rate k is asymptotically achievable on but is not asymp. achievable on . (1, 1, . . . , 1) N N N N Unit capacity edges, messages M = (M1, M2), H(M1) = H(M2) = n-1 bits, adversary can access one link Network code:

a1 a2 A1 A2 z2 z1 z

1

z

2

t x1 y1 y2 x2 b2 b1 B1 B2 s2 s1 t2 t1 s C D

Infeasibility of Multiple Unicast

14

Proof (constructive):

Theorem There exist networks and such that rate k is asymptotically achievable on but is not asymp. achievable on . (1, 1, . . . , 1) N N N N Unit capacity edges, messages M = (M1, M2), H(M1) = H(M2) = n-1 bits, adversary can access one link Network code: a1(M) = x1(M) = y1(M) = z1(M) = M1

a1 a2 A1 A2 z2 z1 z

1

z

2

t x1 y1 y2 x2 b2 b1 B1 B2 s2 s1 t2 t1 s C D

Infeasibility of Multiple Unicast

14

Proof (constructive):

Theorem There exist networks and such that rate k is asymptotically achievable on but is not asymp. achievable on . (1, 1, . . . , 1) N N N N Unit capacity edges, messages M = (M1, M2), H(M1) = H(M2) = n-1 bits, adversary can access one link Network code: a1(M) = x1(M) = y1(M) = z1(M) = M1 a2(M) = x2(M) = y2(M) = z2(M) = M2

a1 a2 A1 A2 z2 z1 z

1

z

2

t x1 y1 y2 x2 b2 b1 B1 B2 s2 s1 t2 t1 s C D

Infeasibility of Multiple Unicast

14

Proof (constructive):

Theorem There exist networks and such that rate k is asymptotically achievable on but is not asymp. achievable on . (1, 1, . . . , 1) N N N N Unit capacity edges, messages M = (M1, M2), H(M1) = H(M2) = n-1 bits, adversary can access one link Network code: a1(M) = x1(M) = y1(M) = z1(M) = M1 a2(M) = x2(M) = y2(M) = z2(M) = M2 z

1(M) = z 2(M) = M1 + M2

a1 a2 A1 A2 z2 z1 z

1

z

2

t x1 y1 y2 x2 b2 b1 B1 B2 s2 s1 t2 t1 s C D

Infeasibility of Multiple Unicast

15

Proof (cont.):

Theorem There exist networks and such that rate k is asymptotically achievable on but is not asymp. achievable on . (1, 1, . . . , 1) N N N N

a bi reserves one bit to indicate if case 1

• r 2 happens

a1 a2 A1 A2 z2 z1 z

1

z

2

t x1 y1 y2 x2 b2 b1 B1 B2 s2 s1 t2 t1 s C D

Infeasibility of Multiple Unicast

15

Proof (cont.):

Theorem There exist networks and such that rate k is asymptotically achievable on but is not asymp. achievable on . (1, 1, . . . , 1) N N N N bi(xi, xi, z

i) =

• xi

if xi = yi (case 1) z

i

if xi = yi (case 2)

a bi reserves one bit to indicate if case 1

• r 2 happens

a1 a2 A1 A2 z2 z1 z

1

z

2

t x1 y1 y2 x2 b2 b1 B1 B2 s2 s1 t2 t1 s C D

Infeasibility of Multiple Unicast

15

Proof (cont.):

Theorem There exist networks and such that rate k is asymptotically achievable on but is not asymp. achievable on . (1, 1, . . . , 1) N N N N bi(xi, xi, z

i) =

• xi

if xi = yi (case 1) z

i

if xi = yi (case 2)

t is able to decode (M1, M2) correctly at asymptotic rate 2 Multiple unicast with rate (1, 1) is not feasible

Take Aways

16

Error correction in a simple network setting Single-source network error correction is at least as hard as multiple-unicast (error-free) network coding Falls into broader framework that connects different network communication problems via reductions

Take Aways

16

Error correction in a simple network setting Single-source network error correction is at least as hard as multiple-unicast (error-free) network coding Falls into broader framework that connects different network communication problems via reductions [Huang, Langberg, Kliewer, accepted for ISIT 2015]