  We take a unifying approach to single selection optimal stopping problems
with random arrival order and independent sampling of items. In the problem we
consider, a decision maker (DM) initially gets to sample each of $N$ items
independently with probability $p$, and can observe the relative rankings of
these sampled items. Then, the DM faces the remaining items in an online
fashion, observing the relative rankings of all revealed items. While scanning
the sequence the DM makes irrevocable stop/continue decisions and her reward
for stopping the sequence facing the item with rank $i$ is $Y_i$. The goal of
the DM is to maximize her reward. We start by studying the case in which the
values $Y_i$ are known to the DM, and then move to the case in which these
values are adversarial.
  For the former case, we write the natural linear program that captures the
performance of an algorithm, and take its continuous limit. We prove a
structural result about this continuous limit, which allows us to reduce the
problem to a relatively simple real optimization problem. We establish that the
optimal algorithm is given by a sequence of thresholds $t_1\le t_2\le\cdots$
such that the DM should stop if seeing an item with current ranking $i$ after
time $t_i$. Additionally we are able to recover several classic results in the
area such as those for secretary problem and the minimum ranking problem. For
the adversarial case, we obtain a similar linear program with an additional
stochastic dominance constraint. Using the same machinery we are able to pin
down the optimal competitive ratios for all values of $p$. Notably, we prove
that as $p$ approaches 1, our guarantee converges linearly to 0.745, matching
that of the i.i.d.~prophet inequality. Also interesting is the case $p=1/2$,
where our bound evaluates to $0.671$, which improves upon the state of the art.
