Network Coding Tomography for Network Failures

1. Network Coding Tomography forNetwork Failures Hongyi Yao Sidharth Jaggi Minghua Chen

2. Tomography (CAT Scan) Computerized Axial 1

3. Tomography Heart Y=TX T? 2

4. Network Tomography [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

5. Tomography type • 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

6. 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

7. 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 X I T Y=T = 6

8. back e1 x x x x x=2 . 3+2 2 e1 e3 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] 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

9. Summary of Contribution • 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

10. 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

11. 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.

12. 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.

13. 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

14. 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

15. [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

16. 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 [FM05,HLCWK05] Random error Polynomial Polynomial [This work] [This work]

17. 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.

18. Related works

19. Network Coding Tomography forNetwork 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