Annual Conference of ITAAnnual Conference of ITAACITA 2009ACITA 2009

Neighbor Discovery in Wireless Networks and the Coupon Collector’s Problem

Sudarshan Vasudevan (UMass), Don Towsley (UMass), Dennis Goeckel (UMass) and Ramin Khalili (EPFL)*

Problem Definition

• Nodes have just been placed/thrown/dropped• Every node has a unique ID• Nodes are beginning to power up• How does each node determine the IDs of its

neighbors to begin ad hoc network formation?• Challenge: Nothing is known at the time of deployment

What is missing?

• No apriori transmission scheduling among nodes can cause interference

• No prior knowledge of node density• Lack of synchronization among nodes• Detecting when to start and terminate neighbor

discovery phase non-trivial

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

Typical Approach

• ALOHA – each node transmits with probability p

Prior Work

• Aloha-like ND Algorithms [MMcGlynn01,SVasudevan05,SBorbash07] • Focus on determining optimal p

• Time required to discover all neighbors under optimal settings unknown

• Assume prior information about node density• ND initiation/termination not handled

Aloha-like Discovery

• Assume node density known and perfect synchronization among nodes

• Nodes cannot distinguish between collision and idle time slot (hence, a node does not know when it has been discovered)

• Each node transmits with p = 1/n• Our Key Observation: Reduces to Coupon

Collector’s Problem• Time to discover all neighbors )ln ( nn

Removing Assumptions

• Unknown n: Algorithm execution in phases• In phase j, transmit with probability • Only a factor 2 slowdown from knowing n

• Asynchronism: Collision duration doubled• Each phase 2 times longer• Factor of 2 slowdown from synchronous execution

• Account for clock skews to allow different start times• Termination condition based on number of nodes

discovered in each phase

j21

Collision Detection-based ND

• Each node sends feedback of reception status• Once a node has been discovered by its neighbors, it

stops transmitting• A factor of ln n improvement in time to discover

neighbors, which is • Order optimal

• Unknown node density, asynchronous execution and initiation/termination• Handled similar to Aloha-like neighbor discovery

Conclusions

• Result: a ND algorithm running in time when there is no feedback and time when nodes provide feedback of reception status, with no assumptions on:• Knowledge of number of neighbors• Synchronization among nodes• Initial starting time• Knowledge of when to terminate

)(n

)ln ( nn)(n

* Portion of work done when the author was a post-doctoral researcher at UMass Amherst