1. SINR Diagram with Interference Cancellation Merav Parter Joint work with Chen Avin, Asaf Cohen, Yoram Haddad, Erez Kantor, Zvi Lotker and David Peleg SODA 2012, Kyoto

2. Stations with radio device Synchronous operation Wireless channel No centralized control Stations with radio device Synchronous operation Wireless channel No centralized control S1S1 S2S2 S3S3 S4S4 S5S5 Wireless Radio Networks d

3. Reception Map - Who Hears Me? Vor(i) := Voronoi Cell of station s i S. Each point ``hears the nearest station Voronoi Diagram M. Shamos and D. Hoey, FOCS '75

4. Reception Map - Physical Model Point ``hears a station if it can decode its massage Attempting to model signal attenuation and interference explicitly Signal to Interference plus Noise Ratio (SINR)

5. Physical Models: Signal to interference & noise ratio Transmission Energy Transmission Energy Path-loss exponent Received signal strength of station s i Receiver point Station s i Noise Interference

6. Station s i is heard at point p d - S iff Fundamental Rule of the SINR model Reception Threshold (>1)

7. Interference Cancellation Receivers decode interfering signals Cancel them from received message Decode their intended message Receivers decode interfering signals Cancel them from received message Decode their intended message optimal The optimal strategy in several scenarios (strong interference, spread spectrum communication, etc.) optimal The optimal strategy in several scenarios (strong interference, spread spectrum communication, etc.) J.G. Andrews. Wireless Communications, IEEE, 2005

8. Successive Interference Cancelation (SIC) Received Signal Interference Signal

9. Decode the Strongest Interference Received Signal Decode (Rounding)

10. and subtract (cancel) it … Signal Interference Received Signal Amplify

11. Can you hear me? (Uniform Powers) s3s3 s2s2 s1s1 s4s4 Without SIC: p With SIC:

12. Can you hear me? (with IC) s3s3 s2s2 s1s1 s4s4 p No Yes

13. The Power of SIC: Exponential Node Chain Q: How many slots are required to schedule all links? n Without SIC: n slots are required for any power assignment single With SIC: A single slot is required using uniform powers Can receive multiple messages at once! Can receive multiple messages at once!

14. SINR Diagram A map characterizing the reception zones of the network stations Reception Zone of Station s i Null Zone Cell := Maximal connected component within a zone. Cell

15. All stations transmit with power 1 (Ψ i =1 for every i) SINR Diagram with Uniform Power Theorem: Every reception zone H(s i ) is convex and fat. H(s i ) Vor(i). [Avin, Emek, Kantor, Lotker, Peleg, Roditty, PODC 2009] [Avin et al. PODC09]

16. Reception Map for Physical Model with Interference Cancellation Can I receive s 1 ?? How hears me? Goal: Uniform Power with SIC

17. SIC-SINR Diagram No Cancellation 1- Cancellation 2- Cancellation A map characterizing the reception zone of stations in the SINR where receivers employ IC Reception Map for Station s 1

18. Complexity of SIC-SINR Diagram The polynomial depends on cancellation ordering. Aprioi there are exponentially many. The polynomial depends on cancellation ordering. Aprioi there are exponentially many. SIC-SINR reception zones are not convex in d nor hyperbolically convex in d+1. SIC-SINR reception zones are not convex in d nor hyperbolically convex in d+1.

19. The Map Drawing Challenge Are there exponentially many cells? reception zone How to draw the reception zone of IC a given station under IC setting? reception zone How to draw the reception zone of IC a given station under IC setting? What is the characteristic polynomial of the zone? What is the characteristic polynomial of the zone? Depends on cancellation ordering, but a priori there are exponentially many Depends on cancellation ordering, but a priori there are exponentially many

20. Challenges Nice Properties of Zones Bounding #Cells (how many?) Bounding #Cells (how many?) Map Drawing (which are they?) Point Location Task (how to store them?) Point Location Task (how to store them?) Algorithms Topology

21. Reception Region of Cancellation Ordering H(s 1 ) H(s 2, s 1 ) H(s 2, s 3, s 1 ) H(s 3, s 2, s 1 ) H(s 3, s 1 )

22. Reception Region of Cancellation Ordering (CO) H(s 2 )H(s 1 |S-{s 2 })

23. High-Order Voronoi Diagram Extension of the ordinary Voronoi diagram Cells are generated by more than one point in S. Every region consists of locations having the same closest points in S. Extension of the ordinary Voronoi diagram Cells are generated by more than one point in S. Every region consists of locations having the same closest points in S. Order k=2 Cells generators M. Shamos and D. Hoey, FOCS '75

24. Ordered order-k Voronoi Diagram Or, inductively Sorted list of generators For k=2

25. High-Order Voronoi Diagram and SIC-SINR Diagram

26. The Topology of Reception Zone with SIC The Reception zone of every networks station is a collection of O(n 2d ) shapes. Each shape is: Convex cancellation ordering Correspond to distinct cancellation ordering high-order Voronoi cell Fully contained in the high-order Voronoi cell

27. Drawing Map: Introducing Hyperplane Arrangement

28. Labelled Arrangement (1,2,3)(1,2,3) (1, 3,2) Stations ordering is required only once Observe: (3, 1, 2) (3, 2, 1) (2, 3, 1) (2, 1, 3 ) p

29. Drawing SIC-SINR Reception Map For station s 1 (1,2,3)(1, 3,2) ( 3,1,2) ( 3,2,1) ( 2,3,1) ( 2,1,3) 1 1 2,1 3,1 3,2,1 2,3,1

30. Bounding #Cells – The Compactness Parameter Reception Threshold (>1) Path-loss exponent

31. Bounding #Cells

32. Challenges Nice Properties of Zones Bounding #Cells (how many?) Bounding #Cells (how many?) Map Drawing (which are they?) Point Location Task (how to store them?) Point Location Task (how to store them?) Algorithms Topology

33. Challenges – What We Know #Cells General- O(n 2d ) Compact network- O(1) #Cells General- O(n 2d ) Compact network- O(1) Map Drawing Efficient construction of labeled arrangement O(n 2d+1 ) Map Drawing Efficient construction of labeled arrangement O(n 2d+1 ) Point Location Task Poly-time construction. Logarithmic time per query Point Location Task Poly-time construction. Logarithmic time per query Algorithms Topology Nice Properties of Zones Collection of convex cells Each related to high-order Voronoi cell.

34. Questions? Thanks for listening!