1. Topology Control –power control
Outline
introduction
History Review
K-neighbor graph

2. Power control
Adjust transmission power of nodes such that the resulting network is connected and energy consumption is optimized

3. Motivation Limited energy in wireless network
Energy can be saved if the topology itself is energy efficient

4. Power saving
Network layer
MAC layer
Physical layer
Power control
Routing
Awake-sleep

5. History review
Energy Model
Metrics
Main Methods

6. Energy Model
Omni-directional antennas + Uniform power detection thresholds(t)
Signal power falls inversely proportional to dk
1<K<5
P=t
P=t* dk

7. Observation 1
Transmission through small hops is more power efficient than through big hops. d1 d2 d3
d1+d2+d3

8. Interference Model Transmit /Receive mode
Transmission area: a disk centered at the node with radii equal to it’s transmission range
Transmit /Receive mode
Sleep /Idle mode
y is not interfered if X is in transmit mode and all other y’s neighbors
is in sleep/indle mode. x y

9. Observation 2
Because there could be more simultaneous transmission with small hops than big hops, using small hops can improve throughput.

10. History review
Energy Model
Metrics
Main Methods

11. Metrics Energy efficiency Throughput Average Degree Delay Small hops
Big hops
Small hops

12. Small hop VS Big hop
Minimum transmission range obtain optimal performance?

13. History review
Energy Model
Metrics
Main Methods

14. Main Methods Homogeneous transmission range
-a common value for all nodes
Node-based transmission range
-each node has a different transmission range

15. Homogeneous transmission range
Assumption: every node knows the positions of other nodes (GPS)
Basic Idea: take the longest edge in the minimum spanning tree(MST)
weakness: centralized

16. Node-based transmission range
Feature: fully distributed, localized
Well-known Proximity graphs:
Relative neighborhood graph(RNG)
Gabriel graph(GG)
Yao graph(YG)
Common: all these graphs are well- known sparse spanners. In addition, they all contain the Euclidean Minimum Spanning Tree (EMST) as a subgraph. However, all of these graphs have no constant degree.

17. Relative neighborhood graph(RNG)
RNG has an edge between u and v, if there is no node w such that

18. Gabriel graph(GG)
GG graph has an edge between two nodes u and v such that there is no node w

19. The disk can be broken arbitrarily
Yao Graph
Given a set of nodes in 2-dimensional space, suppose we partition the space around each node into k(k>=6) sectors of a fixed angle and connect the node to the nearest neighbor in each sector.
The disk can be broken arbitrarily

20. Pros & Cons Pros simple and easy to implement
average node degree is bounded by a constant
Cons
The maximum degree can be as large as n-1 V1 V2 Vi u
Vi-1 Vi

21. Question! Less->increase transmission range
Can we keep the number of neighbors of a node around an optimal (minimum) value k?
Less->increase transmission range
More->decrease transmission range
What’s the minimum number k than can ensure connectivity?

22. K-Neighbors Graph

23. Asymmetric Connectivity1 1 e 1 d f 1 c g 3 e d f 1 b c g
Range radii 2 b a
Strongly connected
Message from “a” to “b” has
multi-hop acknowledgement route a 2 3 1 b d g f e c a
Nodes transmit messages within a range depending on their battery power, e.g., agb cgb,d ggf,e,d,a

24. Symmetric Connectivity
Two nodes are symmetrically connected iff they are within transmission range of each other
Increase range of “b” by 1 and decrease “g” by 2 a 2 1 b d g f e c
Symmetric Connectivity
Node “a” cannot get acknowledgement directly from “b” a 2 3 1 b d g f e c
Asymmetric Connectivity

25. Symmetric K-Neighbors Graph
Definition 1. The symmetric super-graph of G is defined as the undirected graph G+ obtained from G by adding the undirected edge (i, j) whenever edge [i, j] or [j, i] is in G. Formally, G+ = (N,E+), where E+ = {(i, j)|([i, j] ∈ E) or ( [j, i] ∈ E)}.
Definition 2. The symmetric sub-graph of G- is defined
as the undirected graph G- obtained from G by removing
All the non-symmetric edges. Formally, G- = (N,E-), where
E-={(i, j)|([i, j] ∈ E) and ( [j, i] ∈ E)}.

26. Theorem k???

27. K-Neighbors Protocol Assumption: Nodes are stationary
The maximum transmission power is the same for all the nodes
Given n, P is chosen in such a way that the communication graph that results is connected with w.h.p
A distance estimation mechanism, possibly error prone, is available to every node
The nodes initiate the k-Neigh protocol at different time. However, the difference between nodes wake up time is upper bounded by a known constant

28. More……
Node i wakes up at time ti, with ti ∈ [0, ]. At random time t1,i chosen in the interval [ti + ,ti + +d], node i announces its ID at maximum power.
For every message received from other nodes, i stores the identity and the estimated distance of the sender
At time ti d, i orders the list of its neighbors (i.e.,of the nodes from which it has received the announcement message) based on the estimated distance; let Li be the list of the k nearest neighbors of node i (if i has less than k neighbors, Li is the list of all its neighbors). ex

29. Simple Example La: f d b e Lb: c d a f Lc: b Ld: b a Le: a Lf: a b Lsa
LSb c d a
LSc b
LSd b a
LSe a
LSf a

30. More….
4. At random time t2 i chosen in the interval [ti d +τ, ti d+τ] (τ is an upper bound on the duration of step 3), node i announces its ID and the list Li at maximum power.
5. At time ti d +τ node i, based on the lists Lj received from its neighbors, calculates the set of symmetric neighbors in Li. Let LSi be the list of symmetric neighbors of node i, and let j be the farthest node in LSi .
6. Node i sets its transmitting power Pi to the power needed to transmit at distance δe(ij), where δe(ij) is the estimated distance
between nodes i and j. ex

31. Some results

32. Future Work
Adapt k-neighbor to mobility?