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Sudarshan Vasudevan (Bell Labs), Micah Adler (FluentMobile), Dennis Goeckel and Don Towsley

(UMass Amherst)

Efficient Algorithms for Neighbor Discovery in Wireless Networks

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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

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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

Idle: No discovery Collision: No discovery One-and-only-one transmit: That ID is discovered (by all)

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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

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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

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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

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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

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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

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Unknown number of neighbors

• Algorithm divided into “phases”– Phase k:

• Duration slots• Each node transmits with prob p =

c)2 e(ln kk 2k1/2

c) n e(log n Duration 1/n, : n log Phase

c) 4 (ln e 4 Duration 1/4, :2 Phase

c) 2 (ln e 2 Duration 1/2, :1 Phase

2

p

p

p

c) n e(log n 2 c) 2 e(ln2 W i

nlog

1i

i2

At most a factor 2 slowdown from when n is known!

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Asynchronous Aloha-based ND

• Each node alternates between “transmit” and “receive” modes

Time

Receive ~ Exponential(Λ)

σ

• Analogous to synchronous case, where p = 1/n

• factor of two in the denominator due to doubling of collision window in asynchronous operation

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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

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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

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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

Terminate after phase 5, since D4 ≥ 8 and D5 < 16

w.h.p nodes all discovers

and phase st-1 n log in terminates node Each :Result 2

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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?

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Order-optimal ND in single-hop networks

Idle: No discovery

AB

C

D E

Collision: No discovery

AB

C

D E

Collision: No feedback from A,B,C in the 2nd mini-slotD and E know their xmissions failed

One-and-only-one transmit: That ID discovered (by all)

AB

C

D E

Success: Only C transmits => all nodes transmit feedback in 2nd mini-slotC no longer transmits and channel contention keeps decreasing!

Idea: Divide each slot into two mini-slots to provide feedback to senders Assume nodes detect collisions

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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

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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

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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.

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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?

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Multi-hop network analysis (contd.)

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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

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Defn. of randomized algorithm

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Lower bound analysis

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

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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

a

b

c

e

dg

f

j

i

ha c

b e

d f

g h

i j

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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

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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

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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

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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]

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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

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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

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Thanks!

?^|/**/

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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.

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Backup slide: Order notations