SINR Diagram with Interference Cancellation Merav Parter Joint

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

Wireless Radio Networks d S 1 Ø Stations with radio device Ø Synchronous operation S 2 S 3 Ø Wireless channel Ø No centralized control S 4 S 5

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

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)

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

Fundamental Rule of the SINR model Station si is heard at point p ∈ d - S iff Reception Threshold (>1)

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

Successive Interference Cancelation (SIC) Signal Received Signal Interference

Decode the Strongest Received Signal Decode (Rounding) Interference

and subtract (cancel) it … Received Signal Amplify Interference

Can you hear me? (Uniform Powers) Without SIC: With SIC: To ``hear” s 1, receiver p should cancel signals of closer stations in order s 3 p s 2 s 1 s 4

Can you hear me? (with IC) s 3 No Yes p s 2 No Yes s 1 s 4

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

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

SINR Diagram with Uniform Power All stations transmit with power 1 (Ψi=1 for every i) Theorem: q Every reception zone H(si) is convex and fat. q H(si) ⊆ Vor(i). [Avin et al. PODC 09] [Avin, Emek, Kantor, Lotker, Peleg, Roditty, PODC 2009]

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

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

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

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

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)

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

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

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

High-Order Voronoi Diagram and SIC-SINR Diagram

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

Drawing Map: Introducing Hyperplane Arrangement

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

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

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

Bounding #Cells

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

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

Questions? Thanks for listening!