Broadcasting in UDG Radio Networks with Unknown Topology

1. Broadcasting in UDG Radio Networks with Unknown Topology WeizmannLiverpoolWeizmannQuébecWeizmannLiverpool Yuval Emek, Leszek Gąsieniec, Erez Kantor, Andrzej Pelc,David Peleg, Chang Su,

2. stations = points in UDG radio networks • transmitting range = 1 • unit disk graph – UDG • (nodes, edges, paths, …) • distributed synchronous model • in every round: transmit or receive • message heard iff exactly one neighbor transmits • else: silence or collision (same effect)

3. distributed synchronous model (1) no transmission (silence) (2) single transmission (3) multiple transmission v can receive the message from u v u w vcannotreceive the message

4. distributed synchronous model (1) no transmission (silence) (3) multiple transmission • collisions cannot be distinguished from silence v u w vcannotreceive the message

5. Unknown topology (ad hoc) • a unique coordinate system • each node knows its own coordinates • does not know the: • coordinates of any other node • the number of nodes • the diameter

6. Unknown topology (ad hoc) • known granularity g = • inverse of minimum Euclidean distance • , for every pair of nodes • typically:d is much smaller then 1 and g is much larger than 1

7. Broadcasting • a distinguished source node • source’s message should be heard by all nodes • remote nodes – use graph’s paths • connected graphs

8. Broadcasting • two models are considered: • conditional wake up: - nodes are initially idle • wakes up upon hearing a message • spontaneous wake up: • – all nodes are awake from the beginning • execution time = • #rounds until all nodes hear the source’s message

9. Deterministic model • decisions of a node on round tdepends only on: • own coordinates • t itself • messages heard so far

10. = diameter of the UDG network (in hops) • = granularity: inverse of min Euclidean distance • s • v This work • execution time depends on two parameters: • not Euclidean diameter

11. This work • upper bound • lower bound • conditionalwake up • spontaneouswake up

12. Previous results • roughly divided into 2 subareas: • centralized: complete knowledge, designing fast schedulers • distributed: local knowledge, designing fast protocols (this work)

13. from • to • Alon, Bar-Noy, Linial, Peleg ’91: constant D Centralized model • Chlamtac, Kutten ’85: formulating the model of radio networks • Chlamtac, Weinstein ‘91 • Gaber, Mansour ‘95 • Elkin, Kortsarz ‘05 • Gasieniec, Peleg, Xin ‘05 • Kowalski, Pelc (to appear)

14. Bar-Yehuda, Goldreich, Itai ’92: • Kushilevitz, Mansour ’98: • Czumaj, Rytter ’03: Distributed model • unknown topology, no labels, randomized: • first to study distributed broadcasting (also deterministic) • (tight!)

15. Chlebus, Gasieniec, Gibbons, Pelc, Rytter ’02: • Kowalski, Pelc ’05: Distributed model • unknown topology, knowing own labels, spontaneous wake up, deterministic: • Kowalski, Pelc ’05: unknown topology, knowing own labels, conditional wake up, deterministic

16. Spontaneous wake up – lower bound • Theorem. deterministic broadcasting algorithm A, and  choice of parameters D,g,  UDG network N of diameter D and granularity g s.t. A requires • rounds to broadcast in N under the spontaneous wake up model.

17. clusters • k consists of • cells Chain networks • each cell may be occupied with a node or empty • each cluster contain at least one occupied cell • source cell (always occupied) in source cluster 0

18. clusters • form a clique Chain networks • there is no edge between any and any for |k-i|>1

19. the message go from directly to Chain networks • from to when only one node from transmit the message

20. the message go from directly to Chain networks • from to when only one node from transmit the message

21. Chain networks • if there exists a node in that heard the message • then all the nodes of must being heard the source message

22. The broadcasting algorithm A • knows the coordinates of the cells • does not know which cells are occupied and which are empty (except the source) • knows that there is at least one occupied cell in every cluster • a typical instruction: “transmit if occupied” • St = cells scheduled to transmit on round t by A

23. decisions are made separately for every k and online based on The adversary • goal: slow down the broadcasting algorithm • decides for every cell whether occupied or empty

24. = number of occupied cells in Game between the algorithm and the adversary • u • St schedule to transmit • adversary decide: • (1) single transmission • (2) silence / collision • algorithm can learn? • what u can learn?

25. u Game between the algorithm and the adversary • adversary: • (1) reveal these cells (occupied/empty) • (2) report silence / collision • must be consists with previous reports

26. u Game between the algorithm and the adversary • v • algorithm knows v • St schedule to transmit by the algorithm • algorithm can learn whether: • (1) • (u hear v) • (2) • (u did not hear v)

27. u Game between the algorithm and the adversary • adversary: • (1) reveal these cells • (2) report silence / collision • (2) report that collision occur • must be consists with previous reports

28. Lower bound • ti = first round on which the nodes of ireceive the message • , number of round for delivering the message from ito i+1

29. execution time: Lower bound • adversary guarantees : • for ti<cg2 • , for i<cg2/log (g)

30. Conditional wake up – lower bound

31. execution time: • rounds • rounds • rounds • rounds Conditional wake up – lower bound • chain network • N1 • N2 • N3 • ND/2 • diameter 2

32. Conditional wake up – lower bound • Theorem. deterministic broadcasting algorithm A, and  g,  UDG network N of diameter 2 and granularity g s.t. A requires • rounds to broadcast in N under the conditional wake up model.

33. blocks • in each block: • 1> • auxiliary cells • opposite each block: • a target cell • 1> The network N • exactly 1 target cell is occupied • g auxiliary cells • target

34. The network N • target cell is outside of the • transmitting range of any other blocks • there is at least oneoccupied cell in the block that opposite to the occupied target cell • the network is connected • auxiliary • target

35. Adversary • can no longer guarantee that no messages are being heard • distinguish silence from collision (stronger model)

36. Game between the algorithm and the adversary • st • Adversary: • (1) reveal some cells • (2) report: collision occur • (3) report: silence / collision

37. execution continues for • rounds Adversarial policy • dead blocks– all cells are revealed, target cell is empty • on every round we “kill” at most 1 block and reveal at most 1 cell in each “live” block

38. The concatenate network • the auxiliary cells of Niis outside the transmitting range of the next source node si+1 • the target cell of Niis inside of the transmitting range of the next source node si+1

39. The concatenate network • the message must be delivered via target nodes and auxiliary nodes

40. The concatenate network • execution time:

41. Summary • upper bound • lower bound • conditionalwake up • spontaneouswake up

42. END • Thank You!!!