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Network Coding Tomography for Network Failures
Hongyi Yao Sidharth Jaggi Minghua Chen
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Computerized Axial Tomography (CAT Scan) 1
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Tomography Heart Y=TX T? 2
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Network Tomography [V96]… 3 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
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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
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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
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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 Y=T X I = T 6
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Network Coding Aids Tomography
back 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] x x=2 x 3+2 2 . e1 e3 3 1 3 2x x 7 3 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 5 2 x x[2,1] x[0,1] x[1,0] x[1,1] x 3+2 x e2 e4 e3 e1 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. 7
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Summary of Contribution
Passive tomography for random linear network coding WHY? 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. Failure type Topology estimation Failure localization Exponential No result [HLCWK05] Adversary error Exponential Hardness proof [This work] [This work] Exponential No result Random error [FM05,HLCWK05] Polynomial Polynomial [This work] [This work] 8
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Core Concept: IRV 1. Relation between IRVs and network structure: 9
Edge Impulse Response Vector (IRV): The linear transform from the edge to the receiver. Using IRVs we can estimate topology and locate failures. 1 [2 9 6] e1 [0 3 2] 3 1 2 e3 3 1 1 3 1. Relation between IRVs and network structure: 2 3 4 2 1 9 3 IRV(e1) is in the linear space spanned by IRV(e2) and IRV(e3). [1 0 0] 2 6 e2 2 1 9 6 2. Unique mapping from edge to IRV: For random linear network coding, two independent edges has independent IRVs with high probability. 9
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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.
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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.
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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
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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
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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. [ ] [ ] [2 9 6] [0 3 2] [0 0 2] [0 0 4] [0 1 0] [0 0 1] [1 0 0] [2 9 6] [0 3 2] IRVs: in < >? [2 9 6] [2,1] [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
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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 Random error [FM05,HLCWK05] Polynomial Polynomial [This work] [This work]
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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.
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Related works
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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: 14
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