1. 1 Constant Density Spanners for Wireless Ad-Hoc Networks Discrete Mathematics and Algorithms Seminar Melih Onus April 5 2005

2. 2 Ad-Hoc Networks Mobile Devices communicating via radio Network without centralized control The wireless units, or nodes, are represented by a graph, and two nodes are connected by an edge if and only if they are within transmission range of each other Transmissions of messages interfere at a node if at least two of its neighbors transmit a message at the same time. A node can only receive a message if it does not interfere with any other message.

3. 3 Unit Disk Graph Model In theory, its assumed that nodes form a unit disk graph Two nodes can communicate if they are within Euclidean distance 1 (equal transmission ranges) Problems: In reality Signal propagation of real antennas not clear-cut disk Thus, algorithms designed for unit disk graph model may not work well in practice The transmission range of a message is not the same as its interference range

4. 4 Our communication model The transmission range of a message is not the same as its interference range The transmission and interference areas of a node are not necessarily disk-shaped Provides a realistic model for physical carrier sensing

5. 5 Our communication model A set V of nodes are distributed in an arbitrary way in a 2-dimensional Euclidean space For a given cost function c and given transmission range r t, transmission area of u is { v  V | c(u,v)  r t } For given interference range r i, interference area of v is { u  V | c(u,v)  r i }

6. 6. u v w.. ok! Transmission & Interference Area Node u is guaranteed receive a message from a node v in its transmission area as long as there is no other node w  V in its interference area that transmits a message at the same time

7. 7 Transmission Range. u r t /(1+  ) r t /(1-  ) Nodes can communicate if distance  r t /(1+  ) Nodes cannot communicate if distance > r t /(1-  ) In range (r t /(1-  ), r t /(1+  )), it is unspecified whether massage arrives Cost Function: c(v,w)  [(1-  )d(v,w), (1+  )d(v,w)]   [0,1), fixed constant d(v,w): the Euclidean distance between v and w

8. 8. u v w.. ok! Physical carrier sensing Nodes cannot only send and receive messages but they can also perform physical carrier sensing Nodes can set their sensing threshold T Sensing range grows monotonically with T

9. 9 Carrier Sense Transmission & Interference Areas For a given carrier sensing threshold T, carrier sensing transmission area of u is { v  V | c(u,v)  r st (T) } For a given carrier sensing threshold T, carrier sensing interference area of u is { v  V | c(u,v)  r si (T) } r st (T): carrier sensing transmission(CST) range r si (T): carrier sensing interference(CSI) range

10. 10 If node v transmits a message and v is in the CST range of node u, then u senses the message transmission If node u senses a message transmission, then there is at least one node w in the CSI area of u that transmitted a message Carrier Sensing v. u.. w ok!

11. 11 Dominating set A dominating set (DS) is a subset of nodes such that either a node is in DS or has a neighbor in DS. A minimum dominating set (MDS) is a DS with smallest possible number of nodes AB C D E F GGBD E F A C

12. 12 Our Results The nodes do not know the total number of nodes The dominating set protocol generates a constant approximation of a MDS in O(log 4 n) communication rounds, with high probability If physical carrier sensing is not available and the nodes have no estimate of the size of the network, then  (n) are necessary for obtaining a constant approximation of MDS. (Jurdzinski, Stachowiak 2002)

13. 13 Preliminary Scenario r st =r t, so CST area is equal to transmission area r si =r i, so CSI area is equal to interference area

14. 14 Preliminary DS Algorithm Nodes can either be active or inactive The active nodes are the candidates for the dominating set Algorithm: –If v is active, then v sends out an ACTIVE signal. If v is inactive and v did not sense any ACTIVE signal, it becomes active again. –If v is active, then v sends out a LEADER signal with probability ½. If v decides not to send out a LEADER signal, but senses a LEADER signal from at least one other node, then v becomes inactive.

15. 15 Example I A B C D E A B C D E Active Inactive Active signal Leader signal Interference range Transmission range Dominating Set {B, C}

16. 16 Example II A B C D E A B C D E Active Inactive Active signal Leader signal Interference range Transmission range C will sense leader signal of B {B} is not a dominating set

17. 17 Ideas There may be active nodes within range r t at the end of the algorithm, but at most constant number of them Distributed Coloring: Each node divides the time into time frames of k slots for a given constant k There is no active node with same time slot within range r i of an active node Two different sensing threshold k is number of active nodes in CSI area of a node

18. 18 Sensing Thresholds The nodes use two different sensing thresholds, T a and T i, depending on their state –The sensing threshold T a has a CSI range of r t –The sensing threshold T i has a CST range of r i rsrs

19. 19 DS Algorithm Time Step I: –If v is active and in its active slot, then v sends out an ACTIVE signal –If v is inactive and v did not sense any ACTIVE signal for the last k slots using a sensing threshold of T a, v senses with threshold T i, and if it does not sense anything, it becomes active and declares the current slot number as its active slot –If v did sense some ACTIVE signal in one of the last k slots, it just performs sensing with threshold T a and records the outcome

20. 20 DS Algorithm Time Step II –If v is active and is in its active slot, then v sends out a LEADER message containing its ID with some fixed probability p –If v decides not to send out a LEADER message but it either senses a LEADER message with threshold T a or receives a LEADER message, v becomes inactive.

21. 21 Why k slots? If an inactive node v sensed an active signal, there is at least one active node u in its carrier sense interference area There is at most constant number of active nodes in carrier sense interference area of a node, say k’ Choose k as k > k’ Then, if there is an active node in carrier sense interference area of u, but there is no active node in its transmission area, then v will be active at this slot

22. 22 Analysis If there is no active node in transmission area of an active node u, then u will stay as active forever, since inactive nodes cannot be active in its slot. If u become active after v, then c(u, v) > r s, since u will sense all k slots before becoming active. r s is the CST range when CSI range is equal to r t

23. 23 Analysis A node u is called leader if it is active and there is no other active node v of same color with c(u,v)  r t Lemma: Every connected component of active nodes needs a most O( log n) steps, w.h.p., until every node in it either becomes inactive or becomes a leader

24. 24 Analysis Lemma: At any time, if active, nonleading nodes cover an area A=  (log 3 n), the number of leaders emerging from these nodes is  (A/log 2 n), w.h.p. Theorem: If all nodes are initially inactive, then after O(log 4 n) rounds of the algorithm, the leaders form a static dominating set of constant density, with high probability.

25. 25 Assumptions Fixed identification numbers of any form are not required The nodes do not know the total number of nodes We only require that the mobile hosts can synchronize up to some reasonably small time difference, which can be done, for example, with the help of GPS signals

26. 26 Constant density spanner Constant density spanner: Given a graph G find subgraph G’ of G such that distance of two nodes in G’ is less than a constant factor of original distance –Dominating Set –Distributed Coloring –Gateway Selection

27. 27 Conclusion More realistic transmission and interference model New communication model that considers physical sensing Polylogarithmic constant approximation DS algorithm under the realistic wireless model

28. 28 References K. Kothapalli, C Scheideler, M. Onus, A. Richa. Constant Density Spanners For Wireless Ad-hoc Networks, submitted to SPAA 05 T. Jurdzinski, G. G. Stachowiak. Probabilistic algorithms for the wakeup problem in single hop radio networks, ISAAC 535-549, 2002 Fabian Kuhn, Thomas Moscibroda, and Roger Wattenhofer, Initializing Newly Deployed Ad Hoc and Sensor Networks, MOBICOM, Philadelphia, USA, September 2004.