  Relay selection for cooperative communications promises significant
performance improvements, and is, therefore, attracting considerable attention.
While several criteria have been proposed for selecting one or more relays,
distributed mechanisms that perform the selection have received relatively less
attention. In this paper, we develop a novel, yet simple, asymptotic analysis
of a splitting-based multiple access selection algorithm to find the single
best relay. The analysis leads to simpler and alternate expressions for the
average number of slots required to find the best user. By introducing a new
`contention load' parameter, the analysis shows that the parameter settings
used in the existing literature can be improved upon. New and simple bounds are
also derived. Furthermore, we propose a new algorithm that addresses the
general problem of selecting the best $Q \ge 1$ relays, and analyze and
optimize it. Even for a large number of relays, the algorithm selects the best
two relays within 4.406 slots and the best three within 6.491 slots, on
average. We also propose a new and simple scheme for the practically relevant
case of discrete metrics. Altogether, our results develop a unifying
perspective about the general problem of distributed selection in cooperative
systems and several other multi-node systems.
