1. XTC: A Practical Topology Control Algorithm for Ad-Hoc Networks
Roger Wattenhofer
Aaron Zollinger

2. Overview What is Topology Control? Context – related work
XTC algorithm
XTC analysis
Worst case
Average case
Conclusions
WMAN 2004

3. Topology Control
Sometimes also clustering, Dominating Set construction Not in this presentation
Drop long-range neighbors: Reduces interference and energy!
But still stay connected (or even spanner)
WMAN 2004

4. Overview What is Topology Control? Context – related work
XTC algorithm
XTC analysis
Worst case
Average case
Conclusions
WMAN 2004

5. Context – Previous Work
Mid-Eighties: randomly distributed nodes [Takagi & Kleinrock 1984, Hou & Li 1986]
Second Wave: constructions from computational geometry, Delaunay Triangulation [Hu 1993], Minimum Spanning Tree [Ramanathan & Rosales-Hain INFOCOM 2000], Gabriel Graph [Rodoplu & Meng J.Sel.Ar.Com 1999]
Cone-Based Topology Control [Wattenhofer et al. INFOCOM 2000]; explicitly prove several properties (energy spanner, sparse graph)
Collecting more and more properties [Li et al. PODC 2001, Jia et al. SPAA 2003, Li et al. INFOCOM 2002] (e.g. local, planar, distance and energy spanner, constant node degree [Wang & Li DIALM-POMC 2003])
Only neighbor direction and relative distance
But: exact node positions known
WMAN 2004

6. K-Neigh (Blough, Leoncini, Resta, Santi @ MobiHoc 2003)
“Connect to k closest neighbors!”
Very simple algorithm.
On average as good as others…
Tough question: What should k be?
[Thanks to P. Santi]
WMAN 2004

7. Percolation
Node density such that the graph is just about to become connected (about 5 nodes per unit disk).
What’s the value for k at percolation?!? (Tough question?)
too sparse
critical density
too dense
WMAN 2004

8. K-Neigh and the Worst Case?
What if the network looks like this:
Does a typical/average network (or parts of an average network) really look like this? Probably not… but…
Still, cool simulation and analysis results by Blough et al.
For example: energy to compute K-Neigh topology is much smaller than CBTC topology (figure right)
k+1 nodes
k+1 nodes
WMAN 2004

9. Overview What is Topology Control? Context – related work
XTC algorithm
XTC analysis
Worst case
Average case
Conclusions
WMAN 2004

10. XTC Algorithm D C G B A E F Each node produces “ranking” of neighbors.
Examples
Distance (closest)
Energy (lowest)
Link quality (best)
Not necessarily depending on explicit positions
Nodes exchange rankings with neighbors A
1. C
2. E
3. B
4. F
5. D
6. G E F
WMAN 2004

11. XTC Algorithm (Part 2) D C G B A E F
Each node locally goes through all neighbors in order of their ranking
If the candidate (current neighbor) ranks any of your already processed neighbors higher than yourself, then you do not need to connect to the candidate. A
1. C
2. E
3. B
4. F
5. D
6. G E F
3. E
7. A
WMAN 2004

12. Overview What is Topology Control? Context – related work
XTC algorithm
XTC analysis
Worst case
Average case
Conclusions
WMAN 2004

13. XTC Analysis (Part 1)
Symmetry: A node u wants a node v as a neighbor if and only if v wants u.
Proof:
Assume 1) u  v and 2) u  v
Assumption 2)  9w: (i) w Áv u and (ii) w Áu v
Contradicts Assumption 1)
WMAN 2004

14. XTC Analysis (Part 1)
Symmetry: A node u wants a node v as a neighbor if and only if v wants u.
Connectivity: If two nodes are connected originally, they will stay so (provided that rankings are based on symmetric link-weights).
If the ranking is energy or link quality based, then XTC will choose a topology that routes around walls and obstacles.
WMAN 2004

15. XTC Analysis (Part 2)
If the given graph is a Unit Disk Graph (no obstacles, nodes homogeneous, but not necessarily uniformly distributed), then …
The degree of each node is at most 6.
The topology is planar.
The graph is a subgraph of the RNG.
Relative Neighborhood Graph RNG(V):
An edge e = (u,v) is in the RNG(V) iff there is no node w with (u,w) < (u,v) and (v,w) < (u,v). u v
WMAN 2004

16. XTC Average-Case
Unit Disk Graph XTC
WMAN 2004

17. XTC Average-Case (Degrees)u
UDG max
UDG avg
GG max
GG avg
XTC max
XTC avg
WMAN 2004

18. XTC Average-Case (Stretch Factor)
XTC vs. UDG – Euclidean
GG vs. UDG – Euclidean
XTC vs. UDG – Energy
GG vs. UDG – Energy
WMAN 2004

19. XTC Average-Case (Geometric Routing)
connectivity rate
worse
GFG/GPSR on GG
GOAFR+ on GXTC
better
GOAFR+ on GG
WMAN 2004

20. No complex assumptions
Conclusion
Even with minimal assumptions, only neighbor ranking, it is possible to construct a topology with provable properties:
Symmetry
Connectivity
Bounded degree
Planarity
Simple algorithm +
XTC lends itself to practical implementation
No complex assumptions
WMAN 2004

21. Really?!? Topology Control
Drop long-range neighbors: Reduces interference and energy!
But still stay connected (or even spanner)
Really?!?
WMAN 2004

22. What Is Interference? Model We want to minimize maximum interference!
Transmitting edge e = (u,v) disturbs all nodes in vicinity
Interference of edge e = # Nodes covered by union of the two circles with center u and v, respectively, and radius |e|
We want to minimize maximum interference!
At the same time topology must be connected or a spanner etc. 8
WMAN 2004

23. Low Node Degree Topology Control?
Most researchers argue that low node degree is sufficient for low interference!
This is not true since you can construct very bad topologies with minimum node degree but huge interference!
WMAN 2004

24. Let’s Study the Following Topology!
…from a worst-case perspective.
WMAN 2004

25. Topology Control Algorithms Produce…
All known topology control algorithms (XTC too!) include the nearest neighbor forest as a subgraph, and produce something like this:
The interference of this graph is (n)!
WMAN 2004

26. But Interference… Interference does not need to be high…
This topology has interference O(1)!!
WMAN 2004

27. Interesting Research Question
We have some preliminary results:
There is no local algorithm that finds a good interference topology
The optimal topology will not be planar, etc.
LISE, LLISE algorithms
[Burkhart, von Rickenbach, Wattenhofer, MobiHoc 2004]
WMAN 2004

28. Simulation UDG, I = 50 XTC, I = 25 LLISE2, I = 23 LLISE10, I = 12
20 Knoten pro Unit Disk
10 Units Seitenlänge
RNG kein Spanner!
LLISE2, I = 23
LLISE10, I = 12
WMAN 2004

29. No complex assumptions
Conclusion
Even with minimal assumptions, only neighbor ranking, it is possible to construct a topology with provable properties:
Symmetry
Connectivity
Bounded degree
Planarity
Simple algorithm
No complex assumptions
XTC lends itself to practical implementation +
But does Topology Control really reduce interference?
WMAN 2004

30. Questions? Comments?
Distributed Computing Group