1. Localized Topology Control Algorithms for Heterogeneous Wireless Networks
Ning Li and Jennifer C. Hou
University of Illinois at Urbana-Champaign
Presented by Andrew Tzakis

2. The Goal
Increase the energy efficiency and Network capacity in wireless ad hoc networks and wireless sensor networks.

3. How to Reach this Goal
Create a topology control algorithm to improve energy efficiency and network connectivity.
Modify each node to use the lowest transmission power that will maintain the same level of network connectivity.
Benefits:
Energy efficiency will obtained by not transmitting at max power.
Spatial reuse and mitigate MAC-level contention will both be improved.
Connectivity preserved.

4. Related work
Most of the current algorithms assume Homogeneous Networks.
This is not realistic, even identical radios can have different ranges.
Rodoplu and Meng (R&M) can work on Heterogeneous network
But is later shown to not be as efficient.
Also the resulting topology is sensitive to the model used in computation.

5. Proposed Solutions
Li and Hou propose two localized topology control algorithms for heterogeneous wireless mutli-hop networks.
Directed Relative Neighborhood Graph (DRNG)
Directed Local Spanning Subgraph (DLSS)

6. Characteristics of DRNG and DLSS
DRNG and DLSS derive topologies with Smaller average node degress (both logical and physical.
Reduces MAC level contection (for better throughput).
They also produce smaller average link lengths while maintaining network connectivity.
Smaller link lengths implies smaller transmission power needed (creating better efficiency).

7. What needs to be proved Li and Hou prove the following:
The topology derived under DRNG or DLSS preserves network connectivity.
The out-degree of any node in the topology by DLSS or DRNG is bounded by a constant.
The topology generated by DRNG or DLSS preserves network bi-directionality.

8. How DRNG and DLSS work Information Collection Topology Construction
Each node locally collects the information of the neighborhood.
Topology Construction
Each node creates a proper set of neighbors for the final topology using information from step 1.
Construction of topology with only bi-directional links (optional)
Each node adjusts its set of neighbors to make sure all edges are reachable.

9. Information Collection
Each node needs to know all of the edges in its neighborhood.

10. Information Collection (2)
This can be found by having all nodes broadcast using is maximal power a hello message that has a unique node id and its max transmit power.
After each node collects its reachable neighborhood, it can broadcast this information so that each node has E(GR) for the whole graph.
This is needed for the next step, topology construction.

11. Weight Function 1 2 u1 5 v2 5 3 v1 u2

12. Topology Construction DRNG

13. Topology Construction DLSS
3:If there is no path then add it
6:Once all paths have been reconnected, stop.
An example will be shown in a few slides.

14. Topology with Only Bi-Directional Links
Some links can be uni-directional in GDLSS and GDRNG.
Can apply either Addition or removal to obtain bi-directional topologies.
v2  v3
v3  v1

15. Addition And Removal Addition Removal
Add an extra edge (v,u) into GA if (u,v)  E(GA), (v,u)  E(GA), and d(v,u)  RV.
Removal
Delete any edge (u,v)  E(GA) if (v,u)  E(GA)

16. The Proof From earlier we need to prove Assumptions:
The topology derived under DRNG or DLSS preserves network connectivity.
The out-degree of any node in the topology by DLSS or DRNG is bounded by a constant.
The topology generated by DRNG or DLSS preserves network bi-directionality.
Assumptions:
G is always strongly connected.

17. Connectivity

18. Connectivity (2) Example of DLSSu p u 10 v 5
ADD p 6 v 5 6
ADD
Not Added
: Stop p reaches all nodes u 10 v p

19. Connectivity (3)
GDRNG says p must exist for (u,v) not to be in the graph. However, if p exists then (u,v) would not be in GDLSS

20. Bi-directionality There exist graphs which are not bi-direction
This can occur when maximum transmission ranges are non-uniform.
Prove that DLSS or DRNG preserves bi-directionality if it exists from the beginning

21. Bi-directionality(2)

22. Degree Bound
The out-degree of a node u under algorithm A is denoted as degAout(u) is the number of out-neighbors.
The in-degree of u under algorithm A is denoted as degAin(u) is the number of in-neighbors.

23. Out Degree Bound
This is true because we know v has to at least transmit at rmin.
If there was a node inside the range then an out neighbor of u to that node would not exist since v would have a shorter weight to that node.

24. Out Degree Bound (2)
To pack the nodes as densely as we can, we will assume the worst case when all other nodes transmit at rmin.
For links a and b to exist in DLSS or DRNG a  b  c.
The most dense packing would make them all equal yielding an equilateral triangle.
Angle wiuwj must be  /3
By induction we can prove there can only be 6 such slices

25. Out Degree Bound (3) Worst case outside the disk of rmin
All nodes transmit at rmin and are rmin/2 away from each other
u is at rmax

26. Out Degree Bound (4)
By taking the area defineed by area(rmax + rmin/2) minus the inter circle, and divide that by the rmin/2 circle area we get the max number outside the rmin/2 circle.
Note the 6 is from the inter circle

27. Out Degree Bound (5)
We can get a tighter bound on the area using hexagons.

28. In Degree Bound
In degree is similar to the inter circle, if you try to add a node p anywhere around the outside,the outside node would become the in node for p.

29. Simulation Study 1
50 Nodes uniformly distributed in 1000m x 1000m area with transmission range of 200m,250m

30. Simulation Study 2
Nodes varied from 100 to 300

31. Simulation Study 3

32. Conclusion DRNG and DLSS do the following for heterogeneous networks
Preserve network connectivity
Preserve network bi-directionality
The out-degree of any node is bounded
Simulation shows that DRNG and DLSS work much better that R&M.