> knowing the position and value of two elements in a sorted list
replyThat's a prior about the distribution, if a relatively weak one (in some sense, at least).
This relies on knowledge of the distribution, just querying in the middle of
A = [1, 2, 4, 8, 16, ..., 2^(n-1)]
is slower than binary search
reply> just querying in the middle
replyIt's an interpolation search. You interpolate the values you evaluated by whatever method you'd like. No one forces you to do linear interpolation. You can very easily fit a quadratic polynomial with the last 3 points, for example.
Interpolation search seems to have a convergence rate of log log n. That's pretty efficient.