  Ranking and selection (R&S) aims to select the best alternative with the
largest mean performance from a finite set of alternatives. Recently,
considerable attention has turned towards the large-scale R&S problem which
involves a large number of alternatives. Ideal large-scale R&S procedures
should be sample optimal, i.e., the total sample size required to deliver an
asymptotically non-zero probability of correct selection (PCS) grows at the
minimal order (linear order) in the number of alternatives, $k$. Surprisingly,
we discover that the na\"ive greedy procedure, which keeps sampling the
alternative with the largest running average, performs strikingly well and
appears sample optimal. To understand this discovery, we develop a new
boundary-crossing perspective and prove that the greedy procedure is sample
optimal for the scenarios where the best mean maintains at least a positive
constant away from all other means as $k$ increases. We further show that the
derived PCS lower bound is asymptotically tight for the slippage configuration
of means with a common variance. For other scenarios, we consider the
probability of good selection and find that the result depends on the growth
behavior of the number of good alternatives: if it remains bounded as $k$
increases, the sample optimality still holds; otherwise, the result may change.
Moreover, we propose the explore-first greedy procedures by adding an
exploration phase to the greedy procedure. The procedures are proven to be
sample optimal and consistent under the same assumptions. Last, we numerically
investigate the performance of our greedy procedures in solving large-scale R&S
problems.
