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Network Coding Tomography for Network Failures

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Network Coding Tomography forNetwork Failures

Hongyi Yao

Sidharth Jaggi

Minghua Chen

Computerized Axial

1

Heart

Y=TX

T?

2

[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

- 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

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

- 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

back

e1

x

x

x

x

x=2

.

3+2 2

e1

e3

- 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

- 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

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

- 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.

- 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.

- 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

- 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

[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

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]

- 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.

- 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

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