Wireless Communication using Directional Antennas

Directional Antennas Focus the energy in a desired direction: Antenna Array Desired Communication Point Geometric representation:  r

Communication Links A

Requirement #1 A B Strong Connectivity

Requirement #2 B A Hop Spanner |AB| ≤ 1: minpath(A, B) has a constant #hops.

Requirements #3 B A Minimum Radius

How can A reach B? B A

Problem Statement Given: –plane point set S, fixed angle  –one  -antenna per point Find: –orientations of  -antennas –a minimum radius r such that the induced communication graph is a strongly connected hop spanner.

B A Case  = 180 o R Euclidean MST

180 o Antennas - Clustering R BFS Traversal Select non-adjacent edges of the MST.

180 o Antennas - Clustering R Select non-adjacent edges of the MST. BFS Traversal

180 o Antenna Orientations R

Basic Observation R

R B A

180 o Antennas of Radius 2 R B A

How can A reach B now? B A

(180 o, 2)-Communication Graph R Communication Graph for:  = 180 o Radius = 2 Strongly connected Hop factor = 3

What about < 180 o ? Next:  ≥ 120 o

K 2 K o Antennas – 3 Point Connectivity A C B Want: Strong Connectivity Plane Coverage

120 o Antennas – 3 Point Connectivity A C B Want: Strong Connectivity Plane Coverage A C B Radius = Maximum Pairwise Distance

120 o Antennas – 3 Point Connectivity A C B Radius = Maximum Pairwise Distance ≤ 2 A C B Observation: Radius = 3 also covers the unit disk around each point

120 o Antennas – 3 Point Connectivity A C B Radius = Maximum Pairwise Distance ≤ 2 A C B Observation: Radius = 3 also covers the unit disk around each point

120 o Antennas – 3 Point Connectivity A C B Radius = Maximum Pairwise Distance ≤ 2 A C B Observation: Radius = 3 also covers the unit disk around each point

Connecting 3-Point Clusters A C B Assumption: Clusters at unit distance. X Z Y X Z Y A C B

Connecting 3-Point Clusters A C B Assumption: Clusters at unit distance. X Z Y X Z Y A C B

Connecting 3-Point Clusters A C B Assumption: Clusters at unit distance. X Z Y X Z Y A C B

120 o Antennas - Clustering R Euclidean MST(S) Partition S into clusters of ≥ 3 nodes

(120 o, 5)-Communication Graph R r = 5: Each cluster is strongly connected and covers the enclosing unit halo

(120 o, 5)-Communication Graph R r = 5: Each cluster is strongly connected and covers the enclosing unit halo B A

120 o Antennas - Summary of Results Given: –plane point set S –fixed angle  ≥ 120 o There exist –orientations of  -antennas Such that –radius r = 5 establishes a communication graph that is a strongly connected, 5-hop spanner. Lower bound: r = 2

K 1 K 2 K 3  ≥ 90 o : Similar Approach 4 Point Connectivity D K 4 A B C

K 1 K 2 K 3  ≥ 90 o : Similar Approach 4 Point Connectivity D K 4 A B C D A B C Communication Graph Radius = Second longest pairwise distance

 ≥ 90 o : Similar Approach Radius + 1: Unit halo coverage D A B C Allows us to connect clusters at unit distance

90 o Antennas - Clustering R Euclidean MST Partition S into clusters of ≥ 4 nodes

r = 7: Each cluster is strongly connected and covers the enclosing unit halo (90 o, 7)-Communication Graph R B A

90 o Antennas - Summary of Results Given: –plane point set S –fixed angle  ≥ 90 o There exist –orientations of  -antennas Such that –radius r = 7 establishes a communication graph that is a strongly connected, 6-hop spanner. Lower bound: r = 2

OPEN What about  < 90 ? This approach does not work: Strong connectivity: –each antenna must cover at least one point Plane coverage: –some antennas cover no points Conflicting criteria! Restriction too strong!