Efficient Algorithms for Neighbor Discovery in Wireless Networks

1. Efficient Algorithms for Neighbor Discovery in Wireless Networks Sudarshan Vasudevan (Bell Labs), Micah Adler (FluentMobile), Dennis Goeckel and Don Towsley (UMass Amherst)

2. Problem definition • Wireless nodes dropped over a region • Nodes have very little or no information about the network characteristics • Nodes beginning to power up • Problem: How does each node “discover” the IDs of its neighbors (e.g., nodes in communication range)? • A node i “discovers” node j upon receiving a message from node j

3. One-and-only-one transmit: That ID is discovered (by all) Idle: No discovery Collision: No discovery System model • Each node has a unique ID • Omni-directional transceiver • Node can either transmit or receive at any time • Collision channel model • Bi-directional links between neighboring nodes

4. Motivation • Fundamental problem in large, self-organizing wireless networks • First step in initializing wireless networks • Medium access, routing, topology control depend on knowledge of neighbor IDs • Faster neighbor discovery implies reduced energy consumption

5. Prior work • Aloha-based ND:[MMcGlynn’01, SBorbash’07] • Assume a time-slotted system with nodes synchronized • At each slot: node transmits with probability p • What is the optimal p to maximize discovery rate? • p = 1/n where n is number of neighbors • n known to all nodes

6. Number of questions unresolved… • What is the running time of Aloha-based ND? • Not studied even for single-hop networks! • What happens when nodes do not know n (number of neighbors)? • How to initiate and terminate ND? • Is Aloha-based ND optimal? • Outline: • Single-hop networks • Multi-hop networks

7. Aloha-based ND • What is the time to discover all n neighbors? • Assumptions: • Clique of size n • n known to all nodes • Slotted, synchronous system • Prob. node i is discovered in a time slot: • Prob. of “unsuccessful” slot: • Probability that the slot is idle or collision occurs = 1 – n ps = 1/e

8. Aloha-based ND and Coupon Collection • Given an urn with n coupon types (each corresponding to unique neighbor) • draw a coupon (i.e. discover a neighbor) with probability 1/ne • draw a “blank” coupon (i.e. a collision or an idle slot occurs) with probability 1/e • W: time to discover all n neighbors • Same as waiting time to complete coupon collection • E[W] = ne(log N + Θ(1)) = O(n log n) • Concentration result: W = Θ(n log n) w.h.p

9. Unknown number of neighbors • Algorithm divided into “phases” • Phase k: • Duration slots • Each node transmits with prob p = At most a factor 2 slowdown from when n is known!

10. Receive ~ Exponential(Λ) σ Time Asynchronous Aloha-based ND • Each node alternates between “transmit” and “receive” modes • Analogous to synchronous case, where p = 1/n • factor of two in the denominator due to doubling of collision window in asynchronous operation

11. Asynchronous Aloha-based ND • Exponential “receive” durations implies transmission events are Poisson • Prob. a given transmission is successful is 1/e • Asynchronous algorithm can again be viewed as a coupon collection problem • Prob. of drawing a coupon • Time to discover all neighbors (W) • E[W] = 2σne(log n + Θ(1)) • Two times slower than synchronous version • W = Θ(n log n) w.h.p

12. Initiating ND • Assumption: maximum clock offset of δ • Each node starts ND when its local clock reaches T • Add δ times units to each phase • All nodes in log n-th phase for 2σne(log n + c) time units • In practice: • Mica2 motes 32.768 kHz quartz crystal oscillator • real-time clock accuracy ±10 ppm • δ = 1.7 seconds/day • Temperature-compensated oscillators • accuracy ±1 ppm, δ=160 ms/day

13. Terminate after phase 5, since D4 ≥ 8 and D5 < 16 Terminating ND • Let Dj denote the number of neighbors discovered by node i in the j-th phase • Termination Rule: Stop at the end of j-th phase if Dj-1 ≥ 2j-2 and Dj < 2j-1 • Example: Simulation of clique of size n = 16 • Phase 1: D1 = 0 • Phase 2: D2 = 2 • Phase 3: D3 = 14 • Phase 4: D4 = 15 • Phase 5: D5 = 15

14. Summary of Aloha-based ND • Θ(n log n) Aloha-based ND • A priori knowledge of n not required • At most factor of two slowdown • Asynchronous operation at most two times slower compared to synchronous operation • Allows nodes to start execution at different time instants • Provably correct termination condition • Can we achieve an O(n) ND algorithm?

15. A C A A C C B B B D E D E D E One-and-only-one transmit: That ID discovered (by all) Idle: No discovery Collision: No discovery Collision: No feedback from A,B,C in the 2nd mini-slot D and E know their xmissions failed Success: Only C transmits => all nodes transmit feedback in 2nd mini-slot C no longer transmits and channel contention keeps decreasing! Order-optimal ND in single-hop networks Idea: Divide each slot into two mini-slots to provide feedback to senders Assume nodes detect collisions

16. Order-optimal ND in single-hop networks • Time to discover all neighbors, W = Θ(n) w.h.p • Factor log n improvement over Aloha-based ND • Similar to Aloha-based ND: • No knowledge of n => at most factor 2 slowdown • Asynchronous operation => factor 2 slowdown • Starting ND same as for Aloha-like algorithm • Termination trivial: each node yet to be discovered broadcasts at the end of each phase

17. Order-optimal ND without collision detection • Algorithm divided into “rounds” • Round k lasts ~ O(n/2k) time slots • In Round k, where k < log log n: • Each node transmits with prob. 2k-1/n and includes ID of most recently discovered node • At the end of the round, nodes that hear their IDs back “drop out” • After log log n rounds, run Aloha-based ND

18. Order-optimal ND without collision detection • Insight: • Given n coupons, first n/2 coupons can be collected in linear time, while remaining n/2 coupons require O(n log n) time • n/2k nodes “drop out” in round k • Key result: At most n/log n nodes remain in contention after log log n rounds w.h.p • Remaining n/log n nodes discovered using Aloha-based ND in O(n) time w.h.p • Total running time Θ(n) w.h.p.

19. Multi-hop network analysis • Given a graph G=(V,E) where |V| = n, Δ=max node degree • Aloha-based ND: each node transmits with prob. 1/Δ • Time W until all edges in E are discovered?

20. Multi-hop network analysis(contd.)

21. Lower bound analysis • Given an arbitrary graph G=(V,E) such that |V| = n, max. degree = Δ, and |E| -> ∞ as n -> ∞ • Main result: Any randomized algorithm requires Ω(Δ + ln |E|) time w.h.p • Ω(Δ) lower bound applies trivially • Result follows by proving a lower bound of Ω(ln |E|) when Δ=o(ln |E|) • Assume collision detection at nodes • Does not affect lower bound

22. Defn. of randomized algorithm

23. Lower bound analysis • First establish lower bound for class of uniform randomized algorithms • All nodes run the same algorithm i.e.,

24. g d b a i j h f e c Lower bound analysis • Result: Any graph G=(V,E) with max node degree Δ admits a matching of size at least |E|/2Δ • Proof:At each step, pick arbitrary edge in G; add to matching and remove all adjacent edges from G • At most 2Δ edges removed per step g h a c d i f b e j

25. Lower bound analysis • Why look at matching? • Different edges operate independently • Lower bound for matching yields lower bound for graph G • Histories of neighboring nodes identical until the time T that discovery happens • Assume both nodes discover each other in same slot • log(|E|/2Δ) time to discover all links of matching w.h.p. • When Δ=o(log |E|), this implies a lower bound of Ω(log |E|) and main result follows

26. Lower bound analysis • Non-uniform algorithms • Each node may run a different algorithm • Assume nodes 1..n run A1,..,An respectively • Idea: reduce a non-uniform algorithm to a uniform one • Node i simulates an Ak uniformly at random, independent of any other node • Result: Joint probability of schedules chosen by nodes same under both non-uniform and uniform algorithms • Lower bound of Ω(Δ + log |E|) applies

27. Summary of multi-hop analysis • Aloha-based ND has running time O(Δ log n) w.h.p • Any randomized algorithm requires Ω(Δ+log |E|) w.h.p. • When |E| = Ω(n) (e.g. a connected graph) • Aloha-based ND at most min(Δ, log n) from optimal

28. Other extensions • Analysis extends to randomized directional neighbor discovery • ND algorithms can be adapted for RFID tag identification and counting applications • Unidirectional links can be identified • E.g. nodes broadcast ids of discovered neighbors on termination • Neighbor discovery when nodes have multipacket reception [MobiHoc 2011]

29. Open problems • Order-optimality in multi-hop setting • Can the feedback-based algorithms be extended to a multi-hop network? • Lower bound on deterministic complexity • Exponential gap between randomization and determinism? • More detailed PHY layer models? • Mobility and topology maintenance

30. Comparison with beacon-based ND • Each node transmits beacon once every k time units • Routing protocols: AODV, DSR, GPSR, … • Bluetooth networks, Zigbee, .. • Bluetooth standard recommends k = 14 • When n ~ 100, Beacon-based ND 65 times slower than Aloha-based ND and 300 times slower than feedback-based ND! • GPSR recommends k = 1600 (corresponds to interval of 1s with a slot size of 0.625 ms) • When n ~ 10, Beacon-based ND 60 times slower than Aloha-based ND! • No obvious way to terminate Beacon-based ND

31. Thanks! ?^|/**/

32. References • S. Vasudevan, M. Adler, D. Goeckel, and D. Towsley, “Efficient Algorithms for Neighbor Discovery in Wireless Networks”, In submission to IEEE/ACM Transactions on Networking. • S. Vasudevan, D. Towsley, D. Goeckel, and R. Khalili, “Neighbor Discovery in Wireless Networks and the Coupon Collector’s Problem”, Proceedings of ACM MOBICOM, 2009. • W. Zeng, X. Chen, A. Russell, S. Vasudevan, B. Wang, W. Wei, “Neighbor Discovery in Wireless Networks with Multipacket Reception”, To appear in Proceedings of ACM MOBIHOC, 2011.

33. Backup slide: Order notations