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1. Wireless Network Information Flow: From Cell Phones to Matroids David Tse Wireless Foundations EECS U.C. Berkeley EECS Colloquium April 14, 2010 Joint work with Salman Avestimehr and Suhas Diggavi. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAAAAAAAAA

2. Cooperative Wireless Systems Cooperative gain: power diversity spatial multiplexing Gain scales with density of nodes.

3. Communication System Design Step 1: Use information theory to derive fundamental limits Step 2: Derive architectural insights and efficient schemes.

4. Wireless Information Flow … … Basic question: What is the maximum information rate deliverable by the wireless relay network from s to d?

5. Some History Enc Dec Shannon 48 DecR:EncR Enc Dec Open problem for more than 30 years.

6. Why is the Wireless Problem Hard? Wired max-flow = min-cut (Ford-Fulkerson 56) 5/26

7. Why is the Wireless Problem Hard? Signals interact via: broadcast superposition together with background noise. … …

8. Approach: Deterministic Bridge Wired Wireless Deterministic max-flow = min-cut Idea: focus on signal Interaction, not random noise.

9. Deterministic Approximation Transmit a real number Gaussian channel Least significant bits are truncated at noise level. Deterministic channel

10. Broadcast and Superposition

11. Algebraic Representation Received Signal: D S B2 B1 b1 A1 b2 b3 b4 b5 c1 A2 c2 c3 c4 c5 • S: 5 by 5 shift matrix • All operations are over binary field.

12. Example 1: Single Relay Network DecR:EncR Enc Dec x x

13. Example 2: A Two-Staged Network A1 B1 D S A2 B2 12/26

14. General Networks: Min-Cut Upper bound D S A1 A2 B1 B2

15. Max-Flow = Min-Cut • Theorem: (Avestimehr, Diggavi & T 07) • Result holds for general linear transformations between nodes, not just shift matrices. • In wired networks, is just summation of the capacity of links from W to Wc • This is a generalized max-flow min-cut theorem.

16. D S A1 A2 B1 B2 One Way to Achieve Capacity • Source randomly codes information bits. • Each relay forwards random linear combinations of received bits. • Destination solves received equations to decode information bits.

17. Architecture for Cooperative Systems • Performance: • better than conventional relaying schemes • captures power, diversity, spatial multiplexing gains • provably close to optimal for Gaussian channels. • Source encodes information as in point-to-point communication. • Relays extract received bits above noise level and re-encode randomly. • Destination receives equations from source and relays and decodes.

18. D S A1 A2 B1 B2 Back to Deterministic Networks • We showed that random coding at relays is optimal. • Turns out that routing is in fact sufficient (Amaudruz & Fragouli 08). • Interesting connections with matroid theory identified. (Yazdi & Savari 09, Goemans, Iwata & Zenklusen 09)

19. Information Flow Revisited Wired networks: flow = number of edge disjoint paths Wireless networks: flow = number of linear independent equations that can be pushed thru the network. A1 B1 D S A2 B2

20. Propagation of Linear Independence A1 B1 D S A2 B2 Goal: find a sequence of maximal-sized linear independent sets that are “linked”.

21. Linking over the Air Subsets of nodes that can be “linked over the air” are related by an invertible matrix. A1 B1 D S A2 B2

22. Linking Over the Wire A1 B1 D S A2 B2 Subset of nodes that can be “linked over the wire” are perfectly matched in the bipartite graph.

23. General Linking Systems (Schjiver 78) • ¤ satisfies certain axiomatic properties. • Linking function: • maximum flow from A to B.

24. Linking Networks (Goemans et al 09) A1 B1 D S A2 B2

25. Convolution of Linking Systems Cut

26. Max-Flow Min-Cut Again

27. Algorithms • There is a one-to-one correspondence between linking systems and matroids. • Evaluating the linking function is equivalent to rank optimization on matroids. • Can use existing algorithms to compute the max-flow.

28. Conclusion • Characterizing information flow on wireless networks is a long-standing open problem • We make progress by modeling it as a linear deterministic network flow problem. • A max-flow min-cut theorem is proved. • Architectural insights on design of cooperative systems. • Strong connections with matroid theory.