Network Optimization

1. Network Optimization Lecture 14 - 1 Minimum Energy Communication TC

2. Minimum Energy Communication TC • How to find minimum energy routes between two communication nodes? • If global information is available, can be easily found by some shortest path algorithm (e.g. Dijkstra’s algorithm) • If global information is notavailable, should be found by some protocols based on the flooding approach (e.g. AODV)

3. Minimum Energy Communication TC • Desire a topology that contains the minimum-energy paths between any pair of nodes, at the same time, keeps as sparer as possible to alleviate the redundant overhead caused by flooding messages Sparer topology Original topology Minimum-energy topology

4. Minimum Energy Communication TC • Minimum-energy property • Given a network G, a topology (subgraph) G’ of G satisfies this property i.f.f. it has a minimum energy path between any two nodes that are connected in G G G’ G’ Original graph Subgraph with minimum-energy property Subgraph without minimum-energy property

5. Minimum Energy Communication TC • Given a subgraph G’=(V, E’), an edge (u,v)E’ is redundant if there is a path r from u to v in G such that | r | > 1 and c(r) ≤ c(u,v) v v r = (u, h, k, v) | r | = 3 c(r) ≤ c(u,v) (u , v) is redundant r = (u, h, v) | r | = 2 c(r) ≤ c(u,v) (u , v) is redundant k h h u u

6. Minimum Energy Communication TC • Let Gmin=(V, Emin) be the subgraph G such that (u, v)Emin i.f.f. there is no path r such that |r|>1 and C(r) ≤ C(u,v) • Theorem: • Gmin is the smallest subgraph of G with minimum-energy property • A subgraph G’ of G has the minimum-energy property i.f.f. it contains Gmin as a subgraph

7. pmax Minimum Energy Communication TC • Constructing Gmin needs a global view of the network • Instead, it is practical to construct a spare topology that containsGmin with local information G’=(V, E’) Gmin = (V, Emin)

8. Minimum Energy Communication TC • Edge (u, v) E’ is k-redundant if there is a path r in G’ such that | r | = k and C(r) ≤ C(u, v) • E2 consist of all and only edges in E that are not 2-redundant • E2  Emin G2 = (V, E2) G=(V, E)

9. Minimum Energy Communication TC • Widely used energy consumption models in theoretic analyses : • P(u, v) = ||u, v||α, α= 2 • P(u, v) = c + ||u, v||α, α= 2, c > 0 • P(u, v) = ||u, v||α, α> 2 • P(u, v) = c + ||u, v||α, α> 2, c > 0

10. Relay Region • Given a transmitteru and a receiverv, a node r could be used asa relay node i.f.f. • P(u, v) > P(u, r) + P(r, v) • ||u, v|| + c > ||u, r|| + c + ||r, v|| + c • ||u, v|| > ||u, r|| + ||r, v|| + c Transmitter u Relay node r Receiver v

11. Mathematic representation Relay Region • Given a transmitter u and a node r, loc of any possible receiver v such that relaying through r consumes less power than directly transmitting from u to v, is called the relay region of r for u Transmitter u Relay node r Possible Receiver located on (x, y)

12. Enclosure Region u Pmax N(u): neighbors of u in Pmax

13. Enclosure Graph • Definition: Given a set of node V, an edge (u,v) is in the enclosure graph EG,c(V) i.f.f. v N,c(u) • Theorem: EG,c(V)  Gmin(V) • Minimum energy path p = v1v2…vm-1vm • (v1,v2)  Gmin(V) • There is no other node relaying from v1 to v2 using less energy • Node v2 is in the enclosure region of v1 • (v1,v2)  EG,c(V) • Same reason for any vivi+1 in p vi

14. Enclosed Region • Preserving the links to all neighbors in each enclosed region ensures that the minimum energy topology Gmin is contained • Two challenges • How to reduce the number of neighbors in each enclosed region? (e.g. only nodes in G2 = (V, E2)) • How to reduce the broadcasting energy required to identify the enclosed region?

15. u MECN (Rodoplu and Meng, 1999)

16. u MECN (Rodoplu and Meng, 1999)

17. N u MECN (Rodoplu and Meng, 1999)

18. N u MECN (Rodoplu and Meng, 1999)

19. N u MECN (Rodoplu and Meng, 1999)

20. N u SMECN (Li and Halpern, 2001)

21. N u SMECN (Li and Halpern, 2001)

22. N N u SMECN (Li and Halpern, 2001)

23. Theorem • Communication graph constructed by SMECN has the minimum-energy property, i.e. containing Gmin • Theorem • If the search regions are circular, the communication graph constructed by SMECN is a subgraph of the communication graph constructed by MECN

24. Simulation Results • 200 nodes are uniformly placed in a rectangular region of 1500 by 1500 meters, each with transmission range of 500 meters • 1/d4 transmit power roll-off • Omni-directional antenna • Ignore reception power consumption • All nodes periodically send UDP traffic to a sink node situated at the boundary of the network

25. SMECN MECN • Average node degree: • MECN: 3.64 • SMECN: 2.80 • MECN consumed roughly 49% power more than SMECN.

26. Average number of neighbors over time

28. Dynamic Distributed Networks • A node moves from one position to other position can be viewed as two events: • One node is deactivated at the old position • One node is activated at the new position • When adding a new node v: • Node r broadcasts its position to nearby nodes • Only some node u whose enclosure region contains r need to be updated • Remove each neighbor v of u if v Rur

29. Dynamic Distributed Networks • Remove an node r: • Node r broadcasts its position to nearby nodes • Only some node u whose enclosure region contains r need to be updated • Revive each non-neighbor v of u if v Rur