Comment by anderskaseorg

You have ignored the much larger space and time cost of constructing the range minimum query tree in the first place. Furthermore, nobody would arbitrarily cut off a problem at “billions” and then consider O(√billions) or O(∜billions) to be “effectively O(1)”: a theoretician would complain that O() by definition measures the limiting behavior as the problem size tends to ∞, an engineer would complain that 65536 or 256 is very a important factor in practice, and both types are better served by leaving O(√n) or O(∜n) as-is instead of trying to hand-wave it away.