Network Coding Tomography for Network Failures

Sidharth JaggiMinghua Chen

Hongyi Yao

Tomography (CAT Scan)

Computerized Axial

1

Tomography

Heart

2

Y=TXT?

Network Tomography[V96]…

Objectives:•Topology estimation•Failure localization

Failure type:•Adversarial error: The corrupted packets are carefully chosen by the enemies for specific reasons. •Random error: The network packets are randomly polluted.

001001

@#$%&*

3

Tomography type Active tomography[RMGR04,CAS06]:

All network nodes work cooperatively for 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 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].

S

r1 r2

m1

m1

m2

m2

m2m1

m1+m2am1+bm2

5

Source: Sends packets. Organized as:

Internal Nodes: Random linear coding

Sink gets Y:

Random Linear Network Coding

TX TY=T

X I

v1

v2a1v1+a2v2

Information T: Recover Topology [Sharma08]

6X I =

v1 v2

a1v1+a2v2

Network Coding Aids Tomography

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

Probe messages: M=[1, 2]

su

r

e1

e2

e3

e4

1

2

x=2

3

2

3

2

7

5

YE=[3, 2] YM=[1,2]E=YE-YM=[2,0]x[1,0]x[0,1]

e3e1

3+2

3+2 2.YE=[7, 5] YM=[5,3]E=YE-YM=[2,2]

3

2

e1

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

x[1,1]x[2,1]

x

xx

x

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

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

7

x

0x

2x

back

Summary of Contribution

Topology estimation

Failure localization

Passive tomography for random linear network coding

Adversary error

Random error

Exponential Hardness proof

Exponential[HLCWK05]

Polynomial

No result

Polynomial

Exponential[FM05,HLCWK05]

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

Random linear network coding has great advantages.

Passive = low overhead.

WHY?

No result

8

Failure type

[This work][This work]

[This work] [This work]

Core Concept: IRV

Edge Impulse Response Vector (IRV):The linear transform from the edge to the receiver.Using IRVs we can estimate topology and locate failures.

1

1

1

[1 0 0]

1 00

3

12

3

9

3

2 9 6

[0 3 2]

[2 9 6]

26

2 3

3

1

2 41. Relation between IRVs and network structure:

e1

e2

e3

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

0 0

9

2. Unique mapping from edge to IRV: For random linear network coding, two independent edges has independent IRVs with high probability.

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

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 Errors Stage 1: Collect IRVs

[2,1]

[3,2]

E1=4 , 227 , 1518 , 10

[1,3]

[1,1]

E2=0 , 03 , 36 , 14

<E1> <E2>= < >[0 3 2]

[0 3 2]

10

Stage 2: Recover topology

[0 3 2]

[2 9 6]

[0 0 4]

IRVs from Stage 1: [0 3 2]

[1 0 0]

[0 1 0]

[0 0 1][0 0 2]

[2 9 6]

[0 0 4]

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

e1

e2 e3

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Topology Estimation for Random Errors

Random Failure Localization

Locating random failures:

[2,1]

[3,2]

[0 3 2]

[1 0 0]

[0 1 0]

[0 0 1]

[0 0 2]

[0 3 2]

[2 9 6]

[0 0 4]

IRVs:

[2 9 6]

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

[2 9 6]

[0 3 2]

4 , 227 , 1518 , 10

in < >?[4 27 18]

[2 15 10]

[2 9 6]

[0 3 2]

Preliminaries: The Impulse Response Vector (IRV) of each edge.As long as the topology is given, we can do error localization.

12

Exp

Summary of our contribution

Topology estimation

Failure localization

Adversary error

Random error

Exponential Hardness proof

Exponential[HLCWK05]

Polynomial

No result

Polynomial

Exponential[FM05,HLCWK05]

No result

Failure type

[This work][This work]

[This work] [This work]

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.

Related works

Network Coding Tomography for Network Failures

Thanks!

Questions?

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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