1. Analysis of a Cone-Based Distributed Topology Control Algorithm for Wireless Multi-hop Networks
L. Li, J. Y. Halpern
Cornell University
P. Bahl, Y. M. Wang, and R. Wattenhofer
Microsoft Research, Redmond
Presented by Lu-chuan Kung
For CS598hou Sp2006
University of Illinois at Urbana-Champaign

2. Outline Motivation Bigger Picture and Related Work
Basic Cone-Based Algorithm
Summary of Two Main Results
Properties of the Basic Algorithm
Optimizations
Properties of Asymmetric Edge Removal
Performance Evaluation

3. Motivation for Topology Control
Example of No Topology Control with maximum transmission radius R (maximum connected node set)
High energy consumption
High interference
Low throughput

4. Example of No Topology Control with Smaller Transmission Radius
Network may partition

5. Example of Topology Control
Global connectivity
Low energy consumption
Low interference
High throughput

6. Basic Cone-Based Algorithm
Assumption: receiver can determine the direction of sender
Directional antenna community: Angle of Arrival problem
Each node u broadcasts “Hello” with increasing power (radius)
Each discovered neighbor v replies with “Ack”.
Each node u increases power until each cone of degree αcontains a node, or u transmits with maximum power
Who should be my neighbor in the graph?
What is my transmission power?

7. Notation E = { (u,v)  V x V: v is a discovered neighbor by node u}
G = (V, E)
E may not be symmetric
(B,A) in E but (A,B) not in E

8. Notation: Two Symmetric Sets
E+ = { (u,v): (u,v)  E or (v,u)  E }
Symmetric closure of E
G+ = (V, E+ )
E- = { (u,v): (u,v)  E and (v,u)  E }
Asymmetric edge removal
G- = (V, E- )

9. Summary of Two Main Results
Let GR = (V, ER), ER = { (u,v): d(u,v)  R }
Connectivity Theorem
If   150, then G+ preserves the connectivity of GR and the bound is tight.
Asymmetric Edge Theorem
If   120, then G- preserves the connectivity of GR and the bound is tight.

10. Properties of the Basic Algorithm
Counter-example for  = 

11. Counter-example for  = 150 + 

12. Counter-example for  = 150 + 

13. Connectivity Lemma
For   150 ( 5/6 )
if d(A,B) = d  R and (A,B)  E+, there must be a pair of nodes W,Y, one red and one green, with distance d(W,Y) less than d(A,B).

14. Connectivity Lemma Sketch of ProofB A z
z is in Nα(B) with minimal Angle(z,B,A)
Case 1:
Angle(z,B,A) < 60° Then d(A,z) < d(A,B), therefore the Lemma holds ( W=A, Y=z )

15. B A z y w x
Case 2:
Angle(z,B,A) > 60° Must exist y such that Angle(z,B,y) <= α
Similarly there exists w and x st Angle(w,A,x) <= α
Then either d(w,z) < d(A,B) or d(x,y) < d(A,B)
Q.E.D

16. Connectivity Theorem
Order the edges in ER by length and induction on the rank in the ordering
For every edge in ER, there’s a corresponding path in G+ .
If   150, then G+ preserves the connectivity of GR and the bound is tight.

17. Optimizations Shrink-back operation Asymmetric edge removal
“Boundary nodes” can shrink radius as long as not reducing cone coverage
Asymmetric edge removal
If   120, remove all asymmetric edges
Pairwise edge removal
If  < 60, remove longer edge e2 B e1  A e2 C

18. Properties of Asymmetric Edge Removal
Counterexample for  = 

19. For   120 ( 2/3 ) Asymmetric Edge Lemma
if d(A,B)  R and (A,B)  E, there must be a pair of nodes, W or X and node B, with distance less than d(A,B).

20. Asymmetric Edge Theorem
Two-step inductions on ER and then on E
For every edge in ER , if it becomes an asymmetric edge in G , then there’s a corresponding path consisting of only symmetric edges.
If   120, then G- preserves the connectivity of GR and the bound is tight.

21. Performance Evaluation
Simulation Setup
100 nodes randomly placed on a 1500m-by-1500m grid. Each node has a maximum transmission radius 500m.
Performance Metrics
Average Radius
Average Node Degree

22. Average Radius

23. Average Node Degree

24. Comparison with Other TC
SMECN: small minimum-energy communication network (requires location information)

25. Reconfiguration In response to mobility, failures, and node additions
Based on Neighbor Discovery Protocol (NDP) beacons
Joinu(v) event: may allow shrink-back
Leaveu(v) event: may resume “Hello” protocol
AngleChangeu(v) event: may allow shrink-back or resume “Hello” protocol
Careful selection of beacon power

26. Summary
Distributed cone-based topology control algorithm that achieves maximum connected node set
If we treat all edges as bi-directional
150-degree tight upper bound
If we remove all unidirectional edges
120-degree tight upper bound
Simulation results show that average radius and node degree can be significantly reduced

27. Comments The requirement to measure angle-of-arrival is not practical
CBTC doesn’t work better than distance-based topology control

28. The Aladdin Home Networking System
Phoneline
Ethernet
LAN
Powerline
Network
Home
Gateway
Wireless
Sensor
Network
Alert
Router IM

29. Bigger Picture and Related Work
Routing
Topology
Control
Selective
Node
Shutdown
[Hu 1993]
[Ramanathan & Rosales-Hain 2000]
[Rodoplu & Meng 1999]
[Wattenhofer et al. 2001]
[GAF]
[Span]
MAC / Power-controlled MAC
[MBH 01]
[WTS 00]
Relative Neighborhood Graphs, Gabriel graphs, Sphere-of-Influence graphs, -graphs, etc.
Computational
Geometry

30. The Why-150 Lemma
150 =