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

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Jörg Kliewer NJIT

On the Connection Between Multiple-Unicast

Network Coding and Single-Source Single-Sink

Network Error Correction

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Network Error Correction

Problem:

Adversary has control of some edges in a network

Objective: Design coding scheme that is resilient against

adversary

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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

s

₂

s

₃

s

t

₃

t

₂

t

₁

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Known Cases

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

t

₄

t

₃

t

₂

t

₁

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Less Known and Studied Cases

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]

s

₁

s

₂

s

₃

s

t

₃

t

₂

t

₁

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This Work

Single-source single-sink Acyclic network

Edges may not have unit capacity

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This Work

Edges may not have unit capacity Adversary controls single link

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This Work

Some edges cannot be accessed by the adversary

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This Work

Some edges cannot be accessed by the adversary

Reliable communication rate?

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Results

s t

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Results

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

s t

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Results

s t

Network error correction problem:

.. . .. . .. . a₁ a_k A₁ A_k z_k z₁ x₁ y₁ s1_N sk y_k x_k t_k t₁ s

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Results

s t

.. . .. . .. . .. . a₁ a_k A₁ A_k z_k z₁ z₁ z_k x₁ y₁ y_k x_k b_k b₁ B₁ B_k N s_k s₁ t_k t₁ s .. . .. . N s_k s₁ t_k t₁

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Results

s t

.. . .. . .. . a₁ a_k A₁ A_k z_k z₁ x₁ y₁ s1_N sk y_k x_k t_k t₁ s .. . .. . N s_k s₁ t_k t₁

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Results

s t

.. . .. . .. . .. . a₁ a_k A₁ A_k z_k z₁ z₁ z_k x₁ y₁ y_k x_k b_k b₁ B₁ B_k N s_k s₁ t_k t₁ s .. . .. . N s_k s₁ t_k t₁

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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, …

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Reductions

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Generate a clustering of network communication problems via reductions

Network communication problems

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Reductions

‣

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Reductions

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Identify canonical problems (central problems that are related to several other problems)

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Reductions

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Identify canonical problems (central problems that are related to several other problems)

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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

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.. . .. . N s_k s₁ t_k t₁

Proof

Starting point: MU NC problem

Is rate tuple achievable with zero error?

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

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.. . .. . N s_k s₁ t_k t₁ .. . .. . .. . .. . a₁ a_k A₁ A_k z_k z₁ z₁ z_k x₁ y₁ y_k x_k B₁ B_k N s_k s₁ t_k t₁ s

Proof

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

Reduction: Construct new network _N

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.. . .. . N s_k s₁ t_k t₁ .. . .. . .. . a₁ a_k A₁ A_k z_k z₁ x₁ y₁ s1_N sk y_k x_k t_k t₁ s

Proof

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

Reduction: Construct new network _N

Adversary can access any single link except links leaving s and t

X

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Zero Error Case

Theorem

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Zero Error Case

Theorem

Rates achievable on i(1, 1, . . . , 1) N ﬀ rate k is achievable on .N

.. . .. . .. . .. . a₁ a_k A₁ A_k z_k z₁ z₁ z_k x₁ y₁ y_k x_k B₁ B_k N s_k s₁ t_k t₁ s Proof sketch:

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Zero Error Case

Theorem

.. . .. . .. . .. . a₁ a_k A₁ A_k z_k z₁ z₁ z_k t x₁ y₁ y_k x_k b_k b₁ B₁ B_k N s_k s₁ t_k t₁ s Assume on

Source sends information on ai Proof sketch:

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Zero Error Case

Theorem

.. . .. . .. . .. . a₁ a_k A₁ A_k z_k z₁ z₁ z_k x₁ y₁ y_k x_k B₁ B_k N s_k s₁ t_k t₁ s Assume on

(1, 1, . . . , 1)

One error may occur on x_i, y_i, z_i, z_i

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Zero Error Case

Theorem

.. . .. . .. . .. . a₁ a_k A₁ A_k z_k z₁ z₁ z_k t x₁ y₁ y_k x_k b_k b₁ B₁ B_k N s_k s₁ t_k t₁ s Assume on

(1, 1, . . . , 1)

One error may occur on x_i, y_i, z_i, z_i

Bi performs majority decoding

Rate k is possible on _N

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Zero Error Case

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Zero Error Case

.. . .. . .. . .. . a₁ a_k A₁ A_k z_k z₁ z₁ z_k x₁ y₁ y_k x_k B B N s_k s₁ t_k t₁ s Proof sketch: M

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Zero Error Case

.. . .. . .. . .. . a₁ a_k A₁ A_k z_k z₁ z₁ z_k t x₁ y₁ y_k x_k b_k b₁ B₁ B_k N s_k s₁ t_k t₁ s Proof sketch:

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

M

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Zero Error Case

.. . .. . .. . .. . a₁ a_k A₁ A_k z_k z₁ z₁ z_k x₁ y₁ y_k x_k B B N s_k s₁ t_k t₁ s Proof sketch:

M

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

Error correction: For if

a then

For terminal

cannot distinguish between M1, M2

M₁ _̸= M2

b_i(M1) ̸= bi(M2) z′_i(M1) ̸= z′_i (M2)

z′

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Zero Error Case

M

a then

For terminal

M₁ _̸= M2

b_i(M1) ̸= bi(M2) z′_i(M1) ̸= z′_i (M2)

z′

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Zero Error Case

M

a then

For terminal

M₁ _̸= M2

b_i(M1) ̸= bi(M2) z′_i(M1) ̸= z′_i (M2)

z′

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Zero Error Case

M

a then

For terminal

M₁ _̸= M2

b_i(M1) ̸= bi(M2) z′_i(M1) ̸= z′_i (M2)

z′

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Zero Error Case

M

a then

For terminal

M₁ _̸= M2 b_i(M1) ̸= bi(M2) z′_i(M1) ̸= z′_i (M2) z′ i (M1) = z′i(M2) 1-1 correspondence between b_i _↔ z′ i

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Zero Error Case

M

1-1 correspondence between

b_i _↔ z′

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Zero Error Case

M

b_i _↔ z′

i

Same argument: 1-1 correspondence between a_i _↔ x_i _↔ y_i _↔ z_i

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Zero Error Case

M

b_i _↔ z′

i

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Zero Error Case

M

b_i _↔ z′

i

In summary:

Implies : multiple unicast

z_i _↔ x_i _↔ b_i _↔ z′

i

z_i _↔ z′

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Zero Error Case

M

b_i _↔ z′

i

In summary:

Implies : multiple unicast

z_i _↔ x_i _↔ b_i _↔ z′

i

z_i _↔ z′

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Asymptotic Error Case

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Asymptotic Error Case

.. . .. . .. . .. . a₁ a_k A₁ A_k z_k z₁ z₁ z_k t x₁ y₁ y_k x_k b_k b₁ B₁ B_k N s_k s₁ t_k t₁ s Proof sketch: M

Arguments are more complicated Essentially the same

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a We show:

‣

a

‣

a

‣

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

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.. . .. . .. . .. . a₁ a_k A₁ A_k z_k z₁ z₁ z_k t x₁ y₁ y_k x_k b_k b₁ B₁ B_k N s_k s₁ t_k t₁ s Proof sketch: M

a We show:

‣

a

‣

a

‣

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

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Asymptotic Error Case

a We show:

‣

a

‣

a

‣

a lim n , 0 I(bi; zi)/n = 1 lim n , 0 I(ai; zi)/n = 1 lim n , 0 I(ai; bi)/n = 1 In summary: lim n , 0 I(zi; zi )/n = 1

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Current Understanding

.. . .. . .. . .. . a1 ak A1 Ak zk z1 z1 zk t x1 y1 yk xk bk b1 B1 Bk N sk s1 tk t1 s .. . .. . N sk s1 tk t1

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.. . .. . .. . .. . a1 ak A1 Ak zk z1 z1 zk t x1 y1 yk xk bk b1 B1 Bk N sk s1 tk t1 s .. . .. . N sk s1 tk t1

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.. . .. . .. . .. . a1 ak A1 Ak zk z1 z1 zk 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

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with zero error feasible

with error

(1,1, . . . ,1) Rate k feasible

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with error

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.

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Current Understanding

with error

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.

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Infeasibility of Multiple Unicast

Theorem

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

N

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a₁ a₂ A₁ A₂ z2 z1 z₁ z₂ x₁ y₁ y₂ x2 s₂ s₁ t₂ t₁ s C D

Infeasibility of Multiple Unicast

Proof (constructive):

Theorem

N

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a₁ a₂ A₁ A₂ z2 z1 z₁ z₂ x₁ y₁ y₂ x2 b₂ b₁ B₁ B₂ s₂ s₁ t₂ t₁ s C D

Infeasibility of Multiple Unicast

Theorem

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:

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Infeasibility of Multiple Unicast

Theorem

N

N N N

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Infeasibility of Multiple Unicast

Theorem

N

N N N

a₁(M) = x1(M) = y1(M) = z1(M) = M1

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Infeasibility of Multiple Unicast

Theorem

N

N N N

a₁(M) = x1(M) = y1(M) = z1(M) = M1

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Infeasibility of Multiple Unicast

Proof (cont.):

Theorem

N

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a

bi reserves one bit to indicate if case 1

or 2 happens a₁ a₂ A₁ A₂ z2 z1 z₁ z₂ x₁ y₁ y₂ x2 s₂ s₁ t₂ t₁ s C D

Infeasibility of Multiple Unicast

Proof (cont.):

Theorem

N

N N N

b_i₍x_i_, x_i_, z_i _{) =} xi if xi = yi (case 1)

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a

bi reserves one bit to indicate if case 1

or 2 happens a₁ a₂ A₁ A₂ z2 z1 z₁ z₂ x₁ y₁ y₂ x2 b₂ b₁ B₁ B₂ s₂ s₁ t₂ t₁ s C D

Infeasibility of Multiple Unicast

Proof (cont.):

Theorem

N

N N N

b_i₍x_i_, x_i_, z_i _{) =} xi if xi = yi (case 1)

z_i if x_i ₌ y_i (case 2)

t is able to decode (M1, M2) correctly at

asymptotic rate 2

Multiple unicast with rate (1, 1) is not feasible

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Take Aways

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

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Take Aways

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