Network Coding Tomography for Network Failures Hongyi Yao Sidharth Jaggi Minghua Chen

Computerized Axial Tomography (CAT Scan) 1

Tomography Heart Y=TX T? 2

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

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]. 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

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

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

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

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

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= E2= 27 , 15 3 , 3 [0 3 2] 18 , 10 6 , 14 [1,1] [3,2] [0 3 2] <E1> <E2>= < > 10

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

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] [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

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]

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