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Network Coding Tomography for Network Failures. Hongyi Yao. Sidharth Jaggi Minghua Chen. Tomography (CAT Scan). Computerized Axial. 1. Tomography. Heart. Y=TX T?. 2. Network Tomography. [V96]…. Objectives : Topology estimation Failure localization. @#$%&*. 001001.

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Network coding tomography for network failures

Network Coding Tomography forNetwork Failures

Hongyi Yao

Sidharth Jaggi

Minghua Chen


Tomography cat scan
Tomography (CAT Scan)

Computerized Axial

1


Tomography
Tomography

Heart

Y=TX

T?

2


Network tomography
Network Tomography

[V96]…

  • Objectives:

    • Topology estimation

    • Failure localization

@#$%&*

001001

  • Failure type:

    • Adversarial error: The corrupted packets are carefully chosen by the enemies for specific reasons.

    • Random error: The network packets are randomly polluted.

3


Tomography type
Tomography type

  • Active tomography[RMGR04,CAS06]:

    • All network nodes work cooperativelyfor tomography.

    • Probe packets from the sources are required.

    • Heavy overhead on computation & throughput.

  • Passive tomography [RMGR04, CA05, Ho05, This work]:

    • Tomography is done during normal communications.

    • Zero overhead on computation & throughput.

4


Network coding
Network coding

S

  • Network coding suffices to achieve to the optimal throughput for multicast[RNSY00].

  • Random linear network coding suffices, in addition to its distributed feature and low design complexity[TMJMD03].

m1

m2

m1

m2

am1+bm2

m1+m2

m1

m2

r1

r2

5


Random linear network coding
Random Linear Network Coding

  • Source: Sends packets. Organized as:

  • Internal Nodes: Random linear coding

  • Sink gets Y:

X

I

v1

v2

v1

a1v1+a2v2

a1v1+a2v2

v2

Information T: Recover Topology [Sharma08]

TX

X

I

T

Y=T

=

6


Network coding aids tomography

back

e1

x

x

x

x

x=2

.

3+2 2

e1

e3

Network Coding Aids Tomography

  • Network coding scheme is used by u:x(e3)=x(e1)+2x(e2), x(e4)=x(e1)+x(e2).

  • Routing scheme is used by u:x(e3)=x(e1), x(e4)=x(e2).

Probe messages:

M=[1, 2]

e1

e3

3

1

3

2x

7

3

x

YE=[3, 2]

YM=[1,2]

E=YE-YM=[2,0]

YE=[7, 5]

YM=[5,3]

E=YE-YM=[2,2]

s

r

2

2

u

2

2

x

5

0

x[1,1]

x[2,1]

x[0,1]

x[1,0]

3+2

e2

e4

  • Network coding scheme is enough for r to locate error edge e1.

  • Routing scheme is not enough for r to locate error edge e1.

7


Summary of contribution
Summary of Contribution

  • It turns out that the idea underlying the exampleholds even the coding is done in a random fashion.

  • Random linear network coding has great advantages.

  • Passive = low overhead.

Passive tomography for random linear network coding

WHY?

Failure type

Topology estimation

Failure localization

Exponential

No result

[HLCWK05]

Adversary

error

Exponential

Hardness proof

[This work]

[This work]

Exponential

No result

[FM05,HLCWK05]

Random

error

Polynomial

Polynomial

[This work]

[This work]

8


Network coding tomography for network failures

Core Concept: IRV

0

0

Edge Impulse Response Vector (IRV):

The linear transform from the edge to the receiver.

UsingIRVswe can estimate topology and locate failures.

1

[2 9 6]

e1

[0 3 2]

3

1

2

e3

3

1

3

1

1. Relation between IRVs and network structure:

2

3

4

2

1

3

9

IRV(e1) is in the linear space spanned by IRV(e2) and IRV(e3).

[1 0 0]

6

2

e2

2

1

0

9

6

0

2. Unique mapping from edge to IRV:

For random linear network coding, two independent edges has independentIRVs with high probability.

9


Network tomography by irvs
Network tomography by IRVs

  • The concept of IRV significantly aids network tomography:

    • The relations between IRVs and network structure is used to estimate network topology.

    • The unique mapping between network edge and its IRV is used to locate network failures.


Topology estimation for random errors
Topology Estimation for Random Errors

  • Why study random failures:

    • For network without errors, the only information about the network is the transform matrix T. Thus recovering network topology is hard [SS08].

    • Surprisingly, for network with random failures (errors, or packet loss), the IRV of the failure edge will be exposed, letting us recovering network topology efficiently.


Topology estimation for random errors1
Topology Estimation for Random Errors

  • Stage 1: Collect IRVs

[2,1]

4 , 2

0 , 0

[1,3]

E1=

E2=

27 , 15

3 , 3

[0 3 2]

18 , 10

6 , 14

[1,1]

[3,2]

[0 3 2]

<E1> <E2>= < >

10


Topology estimation for random errors2
Topology Estimation for Random Errors

  • Stage 2: Recover topology

[2 9 6]

[0 0 4]

[0 3 2]

[2 9 6]

[0 0 4]

IRVs from Stage 1:

[0 3 2]

[0 0 2]

According to: IRV(e1) is in the linear space

spanned by IRV(e2) and IRV(e3).

[1 0 0]

[0 1 0]

[0 0 1]

e1

e2

e3

11


Network coding tomography for network failures

[2 9 6]

[0 3 2]

Random Failure Localization

Exp

Preliminaries: The Impulse Response Vector (IRV) of each edge.

As long as the topology is given, we can do error localization.

[4 27 18]

[2 15 10]

[1 0 0]

[2 9 6]

[0 3 2]

[0 0 2]

[0 0 4]

[0 1 0]

[0 0 1]

[2 9 6]

in < >?

[2,1]

IRVs:

[0 3 2]

[3,2]

Locating random failures:

[2 9 6]

[0 3 2]

4 , 2

E= [2,1] + [3,2] =

27 , 15

18 , 10

12


Summary of our contribution
Summary of our contribution

Failure type

Topology estimation

Failure localization

Exponential

No result

[HLCWK05]

Adversary

error

Exponential

Hardness proof

[This work]

[This work]

Exponential

No result

[FM05,HLCWK05]

Random

error

Polynomial

Polynomial

[This work]

[This work]


Future direction
Future direction

  • Current work: From existing good network codes to tomography algorithms.

  • Another direction: From some criteria to new network codes.

  • For instance, network Reed-Solomon code[HS10], satisfies:

    • Optimal multicast throughput

    • Low complexity and distributed designing.

    • Significantly aids tomography:

      • Failure localization without centralized topology information.

      • Adversary localization can be done in polynomial time.



Network coding tomography for network failures
Network Coding Tomography forNetwork Failures

  • Thanks!

  • Questions?

Details in: Hongyi Yao and Sidharth Jaggi and Minghua Chen, Network Tomography for Network Failures, under submission to IEEE Trans. on Information Theory, and arxiv: 0908-0711

14