# Variations on the Theme of Network Coding

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```					   Network Coding Tomography for
Network Failures

Sidharth Jaggi       Hongyi Yao
Minghua Chen
Computerized Axial
Tomography (CAT Scan)

1
Tomography

Heart

Y=TX
T?

2
Network Tomography [V96]…

Objectives:
•Topology estimation
@#\$%&*
001001                                •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.

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
S

   Network coding suffices to achieve to         m1            m2

the optimal throughput for
m1   m2
multicast[RNSY00].

m1+m2
am1+bm2

m1              m2
   Random linear network coding suffices,
in addition to its distributed feature and
low design complexity[TMJMD03].                   r1         r2

5
Random Linear Network Coding
   Source: Sends packets. Organized as:

X        I
v1     v2
   Internal Nodes: Random linear coding
v1
a1v1+a2v2
a1v1+a2v2
v2

Information T: Recover
   Sink gets Y:                           Topology [Sharma08]

Y=T            X        I       =   TX               T
6
back

Network Coding Aids Tomography

     Networkscheme scheme by used by u:x(e), x(e4)=x(e2). 2),
Routing coding is used is u: x(e3)=x(e1 3)=x(e1)+2x(e
x(e4)=x(e1)+x(e2).
Probe messages:          x=2
x
3+2 . 2x
M=[1, 2]
1
e1       3       3    e3
x
3
7
2x                    =[7, 2]
YE=[3, 5]
s                                            r               =[5,3]
YM=[1,2]
2             2   u   2            x
0
2
5
x                 3+2 x             x[2,1]
x[1,0]
x[0,1]
x[1,1]           =[2,2]
E=YE-YM=[2,0]

e2                     e4                        e1          e3

   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
Passive tomography for random linear network coding

WHY?
       turns out Topology estimation Failure localization
It Failure type that the idea underlying the example
holds even the coding is done in a random fashion.
Exponential
No result        [HLCWK05]
Random linear network coding has greatproof
error
Exponential
[This work]        [This work]
Exponential
Passive = low      No result
      Random                          [FM05,HLCWK05]
error        Polynomial         Polynomial
[This work]         [This work]

8
Core Concept: IRV
0           0

Edge Impulse Response Vector (IRV):                            2 1

[
9e1
The linear transform from the edge to the receiver.              6

]
Using IRVs we can estimate topology and locate

[
0
failures.                                                            3
2               1 3
e3 2
3               1

]
1
3
1. Relation between IRVs and network                             2                3                   4
structure:                                                       1                    3
9
2
IRV(e1) is in the linear space spanned by IRV(e2) and IRV(e3).                                    2
6

[
1
0   e2
0
2 9
1 0         6
0

]
2. Unique mapping from edge to IRV:
For random linear network coding, two independent edges
has independent IRVs with high probability.
9
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]
4 , 2                   0 , 0
[1,3]
E1=   27 , 15       E2=       3 , 3

[
18 , 10                 6 , 14        0
3
[3,2]
[1,1]
2

]
[

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

10
Topology Estimation for Random Errors

   Stage 2: Recover topology

[

[
2           0
9           0
6           4

[

[

[
IRVs from Stage 1:         0    2   0

]

]
3    9   0

[
0
2    6   4          3

[
]

]

]
0
2
0

]
2

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

[

[

[
1   0       0
0   1       0
spanned by IRV(e2) and IRV(e3).                    0   0       1

]

]

]
e1

e2    e3

11
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 2
27 15

[
[

[

[

[

[

[
2   0     0    0   0    0   1

[

[
in <   18 10 >?

[
IRVs:                                                           2 [2,1]           2   0
9   3     0    0   1    0   0

]
]
9               9   3
6   2     2    4   0    1   0                    6
6   2
]
]

]

]

]

]

]

]

[
]

]
0
3
[3,2]
Locating random failures:                                             2

]
[

[

2             0             4 , 2
E=   9   [2,1] +   3   [3,2] =   27 , 15
6             2             18 , 10
]

]

12
Summary of our contribution

Failure type   Topology estimation   Failure localization

Exponential
No result            [HLCWK05]
Exponential         Hardness proof
error
[This work]          [This work]
Exponential
No result
Random                             [FM05,HLCWK05]
error         Polynomial            Polynomial
[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?

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

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Lingjuan Ma