1. 1 Simple Network Codes for Instantaneous Recovery from Edge Failures in Unicast Connections Salim Yaacoub El Rouayheb, Alex Sprintson Costas Georghiades Department of Electrical Engineering Texas A&M University

2. 2 Information vs. Commodity Flow b1b1 b1b1 b1b1 Replication b1b1 b2b2 b 1 + b 2 Encoding Replication Encoding

3. 3 Network Coding t1t1t1t1 t2t2t2t2 abababab s1s1s1s1 s2s2s2s2 a b demands bdemands a Network Coding increases the throughput!

4. 4 Recovery from Failures Input A graph G(V,E) Link capacities c(e) for  e  E Number of packets that can be transmitted by e per time unit source s  V, destination t  V. h packets need to be sent reliably from s to t. Goal: Ensure the destination node t receives h packets even if a link fails Link capacities tsu v 2 2 1 1 1 1 1 a b

5. 5 Coding for restoration Standard approach: Rerouting upon a failure t

6. 6 Coding for restoration With network coding: Instantaneous recovery Do not need to change coding/routing

7. 7 Coding Advantage t s 2 2 u v w cut Failure! How many packets can be sent reliably from s to t? With instantaneous recovery No rerouting Traditional approach One packet Network Coding Two packets

8. 8 Resilient Capacity Resilient capacity Definition: Resilient capacity, C r, of a unicast network G(V,E) is the maximum number of packets that can be sent reliably from s to t. Necessary Condition: G(V,E) must have C r paths between s and t in G/e, for any e  E. Min-Cut Max-Flow: C r = m i n C ( s, t ) X e 2 E ( C ) c ( e ) ¡ max e 2 E ( C ) c ( e )

9. 9 Achieving Capacity Resilient Capacity can be achieved by linear network coding [Koetter and Medard 03] Robust network codes can be found in polynomial time For multicast and unicast Jaggi et al. [04] Required field size O(k|E|) k is the number of terminals

10. 10 Results: Focus on h=2 Introduce the concept of a simple unicast network Simple networks are minimal - every link is essential for achieving capacity Show that minimal network have a certain structure Use this structure to design robust network codes over GF(2)

11. 11 Simple Unicast Network ts A simple network.  Lemma: There exist a robust linear network code over F for a unicast network iff there exists one for the corresponding simple network over the same field! simple network First, we build a corresponding simple network N is feasible N is minimal Every node of degree 3 No multiple edges 2 2 1 1 1 1

12. 12 Structure of minimal networks We prove that minimal unicast networks have a special structure Facilitates efficient network codes Small field size ts

13. 13 Network Structure Type B Type A 1 1 1 11 2 Type C 2 1 1 Type D 1 1 2 Lemma: Simple networks can only include nodes of the following types

14. Block Structure

15. 15 Structure of Simple Networks Lemma: For any cut of size 3 holds One node must be of type A Two others must be of type C These nodes must be connected 1 A 1 1 1 B 11 1 C 2 1 1 D 1 2

16. 16 Proof Techniques Proof techniques Use minimality property of simple networks Use the concept of network flows A cycle in the residual graph - the network is not minimal

17. 17 Proof Techniques 2 2 ts 2 2 1.5 Unicast Network Corresponding Flow Network Unicast Network is feasible Corresponding Flow Network admits a flow b of value 3. 1 1 1.5 1.5.5 1 1 1 ts Theorem:

18. 18 Proof Techniques t s t 1 1 1 11.5 1 1 1 11 Unicast Network 1 01 0 1 0 Residual Graph s Flow of value 3

19. 19 Code description s a,b  GF(2) a a+b b Coding at the source Coding for block A m1m1 m2m2 m3m3 m1m1 m2m2 m2m2 m2m2 m2m2 m3m3 m1m1 m2m2 m2m2 m3m3

20. 20 Code Description(3) Coding for block C m1m1 m2m2 m3m3 m1m1 m1m1 m2m2 m2m2 m3m3 m3m3 m3m3 m4m4 m4m4 m4m4 m1m1 m3m3 m4m4 m2m2 m1m1 m2m2 m4m4 m4m4 m3m3 m 1 +m 2 m2m2 m 2 +m 4 m3m3 Coding for block B m3m3

21. 21 Code Robustness Suppose the Network does not have block C Transfer matrix T: T is of full rank! Block AB I 2 I 3 I 1 O 1 O 2 O 3 0 @ O 1 O 2 O 3 1 A = 0 @ 110 011 010 1 A |{z} T 0 @ I 1 I 2 I 3 1 A A B

22. 22 Code Robustness(2) Even when one edge fails in block AB, rank {O 1,O 2,O 3 }≥2. The source will be always able to decode the two original packets even if one edge fails. s AB Failure rank=3 rank ≥ 2 a a+b b

23. 23 Summary We studied instantaneous recovery from link failures in communication networks We showed that the minimality implies very simple and elegant structure We built robust linear network code for instantaneous recovery over GF(2) (for h=2). Our bound on the field size does not depend on the network size; Compared to the previous bound of O(|E|).

24. 24 Open Problems Extend our results for h>2 Study the structure of the network Find the required field size Derive an efficient algorithm for finding robust network codes for Unicast networks. We conjecture that the required field size only depends on the number of packets Does not depend on the network size