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Simple Network Codes for Instantaneous Recovery from Edge Failures in Unicast Connections

Salim Yaacoub El Rouayheb,Alex Sprintson

Costas Georghiades

Department of Electrical Engineering

Texas A&M University

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Information vs. Commodity Flow

b1

b1 b1

Replication

b1b2

b1+ b2

Encoding

Replication

Encoding

3

Network Coding

tt11tt22

aabb

ss11 ss22

aa bb

demands b demands a

Network Coding increases the throughput!

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Recovery from Failures

Input A graph G(V,E) Link capacities c(e) for e E

Number of packets that can be transmitted by e per time unit

source sV, destination tV. h packets need to be sent reliably

from s to t. Goal:

Ensure the destination node t receives h packets even if a link fails

Link capacities

tt

ss

uu vv2 2

1

11

11

a b

5

Coding for restoration

Standard approach: Rerouting upon a failure

tttt

ss

uu vv

2 2

1

11

1

1

tt

ss

uu vv

2 2

1

11

1

1

ss

2 2

1

11

1

1

ss

2 2

1

11

1

1

m1 m2

6

Coding for restoration

With network coding: Instantaneous recovery Do not need to change

coding/routings

t

a,b

a b

abb a

s

t

0,0 0,0

0 b

b0 a

a,b

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Coding Advantage

tt

ss

2 2

uu vvww

cutFailure!

How many packets can be sent reliably from s to t? With instantaneous recovery No rerouting

Traditional approach One packet

Network Coding Two packets

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Resilient Capacity Definition: Resilient capacityResilient capacity,

Cr, of a unicast network G(V,E) is the maximum number of packets that can be sent reliably from s to t.

Necessary Condition: G(V,E) must have Cr paths

between s and t in G/e, for any eE.

Min-Cut Max-Flow:Cr = minC(s,t)

X

e2E (C )

c(e) ¡ maxe2E (C )

c(e)

s

t

1 1 2

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Achieving Capacity

Resilient Capacity can be achieved by linear network coding [Koetter and Medard 03]

Robust network codes can be found in polynomial time For multicast and unicast Jaggi et al. [04] Required field size O(k|E|)

k is the number of terminals

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Results: Focus on h=2 Introduce the concept of a simple

unicast network Simple networks are minimal - every

link is essential for achieving capacity Show that minimal network have a

certain structure Use this structure to design robust

network codes over GF(2)

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Simple Unicast Network

tt

ss

A simple network.

Lemma: There exist a robust linear network code over F for a unicast network iff there exists one for the corresponding simple network over the same field!

First, we build a corresponding simple simple networknetwork N is feasible N is minimal Every node of degree 3 No multiple edges

2 2

1 1

1 1

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Structure of minimal networks

We prove that minimal unicast networks have a special structure Facilitates efficient

network codes Small field size

tt

ss

13

Network Structure

Type BType A

1

1 1

1 1

2

Type C

2

1 1

Type D

1 1

2

Lemma: Simple networks can only include nodes of the following types

Block Structure

s t

tt

ss

Block A

Block C

Block B

Block A

Block B

tt

ss

Block A

Block C

Block B

Block A

Block B

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Structure of Simple Networks

Lemma: For any cut of size 3 holds One node must be of type A Two others must be of type

C These nodes must be

connected

1

A

1

1 1

B1 1

1

C

2

1 1

D1

2

s

t

A

1

1 1A

1

1 1

Cannot occur – contradicts minimality

s

t

A

1

1

C

Can occur

1

1 1

22

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Proof Techniques

Proof techniques Use minimality property of simple networks Use the concept of network flows A cycle in the residual graph - the network is

not minimal

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Proof Techniques

2 2

tt

ss

22 1.5 1.5

Unicast NetworkCorresponding Flow Network

Unicast Network is feasible <==> Corresponding Flow Network admits a flow b of value 3.

11 1

.5 .5

1.5 1.5

.5 .5

11 1

tt

ss

Theorem:

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Proof Techniques

tt

ss

tt

1 11

1 1

.5

.5 .5 .5 .5

.5

11

1

.5

.5.5

.5

.5

.5

.5

.5

.5

.5 1 1 .5.5

Unicast Network

1

0 1 0

10

Residual Graph

ssFlow of value 3

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Code description

ss a,bGF(2)

a

a+b

b

Coding at the source

Coding for block A

m1 m2 m3m1 m2

m2

m2

m2

m3

m1 m2 m2m3

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Code Description(3)

Coding for block C

m1 m2m3m1

m1

m2

m2

m3

m3m3

m4

m4

m4

m1m3

m4m2

m1 m2m4m4m3

m1 +m2

m2

m2 +m4

m3

Coding for block B

m3

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Code Robustness Suppose the Network

does not have block C Transfer matrix T:

T is of full rank!

Block AB

I 2I 3I 1

O1 O2 O3

0

@O1O2O3

1

A =

0

@1 1 00 1 10 1 0

1

A

| {z }T

0

@I 1I 2I 3

1

A

A

B

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Code Robustness(2)

Even when one edge fails in block AB, rank {O1,O2,O3}≥2.

The source will be always able to decode the two original packets even if one edge fails.

ss

AB

AB

AB

AB

AB

Failure

rank=3

rank=3

rank=3

rank ≥ 2

rank ≥ 2

rank ≥ 2

a a+b b

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Summary

We studied instantaneous recovery from link failures in communication networks

We showed that the minimality implies very simple and elegant structure

We built robust linear network code for instantaneous recovery over GF(2) (for h=2).

Our bound on the field size does not depend on the network size; Compared to the previous bound of O(|E|).

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Open Problems Extend our results for h>2

Study the structure of the network Find the required field size

Derive an efficient algorithm for finding robust network codes for Unicast networks.

We conjecture that the required field size only depends on the number of packets

Does not depend on the network size