Given data consisting of $n$ coordinates $\left((x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)\right)$ sorted by their $x$-values, and $m$ sorted query points $(q_1, q_2, \ldots, q_m)$, find the linearly interpolated values of the query points according to the data. We assume $q_i \in (\min_j x_j, \max_j x_j)$
I heard off-hand that this problem could be solved in $O(m+n)$ time but can only think of an $O(m \log n)$ algorithm. I can't seem to find this particular problem in any of the algorithm textbooks.
interpolated =  for q in qs: find x[i] such that x[i] <= q <= x[i+1] with binary search t = (q - x[i]) / (x[i+1] - x[i]) interpolated.append(y[i] * (1-t) + y[i+1] * t)
This gives us a runtime of $O(m \log n)$, it's unclear to me how to get this down to $O(m + n)$ as the search for $x_i$ must be done for every query point.