Topology Control – power control

1. Topology Control –powercontrol 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. x Interference Model • 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. 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. Small hops Big hops Metrics • Energy efficiency • Throughput • Average Degree • Delay

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

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

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 +2 +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. d e b a c f Simple Example Lb: c d a f Lc: b Ld: b a Le: a Lf: a b La: f d b e Lsa f d e 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 +2 +d +τ, ti +2 +2d+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 + 3 +2d +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?