Network Coding Tomography for
Network Failures
Sidharth Jaggi
Minghua Chen
Hongyi Yao
Computerized Axial
Tomography (CAT Scan)
1
Tomography
Heart
Y=TX
T?
2
Network Tomography [V96]…
@#$%&*
001001
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.
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
the optimal throughput for
multicast[RNSY00].
m1
m2
m1
m2
am
m11+bm
+m2 2

Random linear network coding suffices,
in addition to its distributed feature and
low design complexity[TMJMD03].
m1
m2
r1
r2
5
Random Linear Network Coding

Source: Sends packets. Organized as:
X
I
v1

Internal Nodes: Random linear coding
v1
Information T: Recover
Topology [Sharma08]
Sink gets Y:
Y=T
a1v1+a2v2
a1v1+a2v2
v2

v2
X
I
=
TX
T
6
back
Network Coding Aids Tomography


Network
coding scheme
is used
u:x(e1),
3)=x(e
1)+2x(e
Routing scheme
is used by
u: x(eby
3)=x(e
x(e4)=x(e
2). 2),
x(e4)=x(e1)+x(e2).
Probe messages:
M=[1, 2]
s
1
2
xx=2
e1
x
e2
x
.
3+2 2
e3
3
3
2
u
2
3+2 x
2x
7
x3
5
x2
0
YE=[7,
=[3, 5]
2]
YM=[5,3]
=[1,2]
r
x[0,1]
x[1,0]
x[2,1]
x[1,1]
e4
E=YE-YM=[2,2]
=[2,0]
e1

Network coding scheme is enough for r to locate error edge e1.

Routing scheme is not enough for r to locate error edge e1.
e3
7
Summary of Contribution
Passive tomography for random linear network coding
WHY?

Topology
estimation
Failure the
localization
It Failure
turnstype
out that
the idea
underlying
example
holds even the coding is done in a random fashion.
Adversary


No result
Exponential
[HLCWK05]
Exponential
Hardness proof
Random
errorlinear network coding has great advantages.
[This work]
Passive
= low
Random
error
No result
overhead.
Polynomial
[This work]
[This work]
Exponential
[FM05,HLCWK05]
Polynomial
[This work]
8
Core Concept: IRV
0
2 1
9e1
6
[
Edge Impulse Response Vector (IRV):
The linear transform from the edge to the receiver.
]
[
Using IRVs we can estimate topology and locate
failures.
0
3
2
0
1 3
e33 2
]
1
1. Relation between IRVs and network
structure:
2
1
3
]
1
0
0
2
3
9
[
IRV(e1) is in the linear space spanned by IRV(e2) and IRV(e3).
1
3
4
6
2
e2
1 9
2
0
0
6
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
4 , 2
27 , 15
18 , 10
0 , 0
E2=
3 , 3
6 , 14
[1,3]
[
E1=
[2,1]
]
0
3
[1,1]
[3,2]
2
[
<E1>
<E2>= <
>
]
0
3
2
10
Topology Estimation for Random Errors
9
6
0
4
0
]
]
[
0
3
2
[
0
0
4
2
9
6
]
[
]
]
0
3
2
[
[
IRVs from Stage 1:
2
[
Stage 2: Recover topology
[

]
0
0
2
]
[
0
[
[
spanned by IRV(e2) and IRV(e3).
0
0
1
0
0
1
0
]
]
1
]
According to: IRV(e1) is in the linear space
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.
]
]
9
6
[2,1] +
3
2
4 , 2
[3,2] =
27 , 15
18 , 10
]
]
0
3
2
]
[
[
E=
2
9
6
0
3
2
]
Locating random failures:
[3,2]
]
[
0
2
9
6
[
[
[
in <
]
]
1
0
0
[
[
[
]
[
0
0
1
]
[
0
1
0
]
[
0
0
4
]
[
0
0
2
]
[
0
3
2
]
[
IRVs:
2
9
6
4 2
27 15
2 [2,1]
18 10 >?
12
Summary of our contribution
Failure type
Adversary
error
Random
error
Topology estimation
Failure localization
No result
Exponential
[HLCWK05]
Exponential
[This work]
Hardness proof
[This work]
No result
Exponential
[FM05,HLCWK05]
Polynomial
[This work]
Polynomial
[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