1. 1 Network Coding and its Applications in Communication Networks Alex Sprintson Computer Engineering Group Department of Electrical and Computer Engineering Texas A&M University

2. 2 Information Course webpage http://cegroup.ece.tamu.edu/spalex/netcod Office Hours TBA, WERC 333D Or by appointment spalex@ece.tamu.eduspalex@ece.tamu.edu Will use neo email

3. 3 Information vs. Commodity Flow b1b1 b1b1 b1b1 Replication b1b1 b2b2 b 1 + b 2 Encoding Replication Encoding

4. 4 Network coding New research area (Ahlswede et. al. 2000) Benefits many areas Networking, Communication, Distributed computing Uses tools from Information Theory, Algebra, Combinatorics Has a potential for improving: throughput, robustness, reliability and security of networking and distributed systems.

5. 5 Communication Networks Directed graph G=(V,E). Source nodes S. Terminal nodes T. Requirement: deliver data from S to T.

6. 6 The multicast setting Multicast: One source transmits data to all terminals.

7. 7 Standard approach Each node forwards and possibly duplicates messages Paths for unicast Trees for multicast

8. 8 Standard approach Steiner Tree Given a set T of points (terminals), interconnect them by a tree of shortest length.

9. 9 Standard tree packing Steiner Tree Packing Given a source node s and a set T of terminals, interconnect them by a tree of shortest length.

10. 10 Network coding Introduce nodes that do more than forwarding + duplication. Each outgoing packets is a function of incoming packets. m1m1 m2m2 m3m3 F 1 (m 1,m 2,m 3 ) F 2 (m 1,m 2,m 3 ) encoding

11. 11 Network coding Linear Coding over finite fields Examples

12. 12 Traditional Approach Steiner Tree Packing Integer reservation - 1 message per time unit Fractional/time sharing solution 1.5 messages per time unit

13. 13 Network coding helps! One message per time unit 1.5 messages per time unit 2 messages per time unit

14. 14 Delay Minimization Jain and Chou (2004) Delay =3 Delay =2

15. 15 Energy Minimization Wu et al. (2003); Wu, Chou, Kung (2004) Lun, Médard, Ho, Koetter (2004) Without network coding - 4 messages With network coding - 3 messages

16. 16 Cost minimization Cost without network coding 4 with network coding 3 - 33% reduction

17. 17 Research Issues How useful is network coding? Multicast Practical Implementation Beyond Multicast Networks with cycles What operations should be performed at each node? Linear vs. non-linear operations The minimum size of a packet Under what conditions is a given network coding problem solvable How to find suitable edge functions? How many encoding nodes are needed?

18. 18 Syllabus In this class: Overview of the main results Overview of the current research Applications in communication networks Directions for future research

19. 19 Syllabus Introduction Introduction to network coding Introduction to network algorithms Benefits and coding advantage Diversity coding Linear Network coding

20. 20 Course Outline Network optimization and Linear Programming (LP) Basics of LP LP duality Primal-dual algorithms Approximation of NP-hard problem The probabilistic method

21. 21 Course Outline Network flows and disjoint path algorithms (3) Theory of network flows Disjoint paths algorithms Increasing network connectivity Network reliability

22. 22 Course Outline Network coding (12) Algebraic framework Polynomial – time algorithms for network code desing Randomized algorithms Distributed network coding Information-flow decomposition Network coding for cyclic networks Encoding complexity Practical network coding

23. 23 Course Outline Design of robust communication networks Bandwidth allocation Network coding-based methods Connection to convolution codes Knots and special cases Information-Theoretic approach for network management

24. 24 Course Outline Network Error Correction (4) Robust networks Correcting adversarial errors Secure network coding

25. 25 Course Outline Encoding Complexity Bounds on the number of encoding nodes Hardness results

26. 26 Course Outline Coding for non-multicast networks Insufficiency of linear network codes Non-linear network codes

27. 27 Course Outline Network coding applications Applications in distributed computing Applications in sensor networks Network planning in wireless and ad-hoc networks Applications for network storage

28. 28 Course Outline Web models and information retrieval algorithms Modeling the web Taxonomy of information retrieval models Retrieval evaluation

29. 29 Course Outline Textbook: No textbook will be used. Notes and research papers will be available on the course website. A good reference site: www.networkcoding.info www.networkcoding.info

30. 30 Course Outline Tentative Grading Policy: Assignments 40% Project 40% Student Presentations 20%

31. 31 Network Capacity In general: the maximum number of packets that can be sent throughout the network. For a given list of source-destination pairs Each link has a limited capacity

32. 32 Network Capacity Multicast connections: The maximum number of packets that can be sent from s to T. Each link has a limited capacity

33. 33 Main Theorem for Multicast The network capacity is equal to the minimum size of a cut that separates s and a terminal t i  T Can be achieved by linear network codes

34. 34 Cut Definition: A cut (V 1,V 2 ) is a partition of the nodes of the graph into two subsets V 1 and V 2 =V\V 1 Size of the cut: The total capacity of links between nodes in V 1 and V 2.

35. 35 Cut (cont.) We say that a cut (V 1,V 2 ) separates s and t if s  V 1 and t  V 2.

36. 36 Flow interpretation Menger’s theorem: The minimum size of a cut between s and t is h  There are h disjoint paths between s and each t A special case of min-cut – max-flow theorem

37. 37 Flow interpretation

38. 38 Upper bound Let h be the size of the min- cut between s and a terminal t  T We cannot send more than h packets to this terminal

39. 39 Challenge Let h be the size of the min-cut between s and a terminal t  T Can send h packets to each terminal t  T separatly e.g., by using h disjoint paths Problem: How to send them to all terminals simultaneously ? Solution: Network coding

40. 40 Challenge Path clashing is resolved by using NC