Given sorted $0\leq x_1 \leq x_2 \leq ... \leq x_n$ and $y_1 \geq y_2 \geq ... \geq y_n \geq 0$ non negative integers accessible through oracles, with the additional constraints $x_{i+1}-x_i \leq 1$ and $y_i - y_{i+1} \leq 1$. Can we approximate the minimum of $x_i + y_i$ with $o(n)$ oracle queries to $x_i, y_i$ values, or is $\Omega(n)$ required?
For the exact case, the following example shows we need to read all indices to determine the minimum: $x_i=i$ and $y_i=n+1-i$. If the algorithm doesn't read one index (say index $k$), then the adversary can set $x_k=k, y_k=n+1-k$ so that the overall minimum is $n+1$, or is can set $x_k=k-1=x_{k-1}$ and $y_k=n+1-i$, at which case the overall minimum is $n$. So the algorithm needs to read all $n$ bits to differentiate inputs.
A $2-$approximation can be taken by returning the index $k$ that minimizes $|x_i-y_i|$. This can be done using binary search using $O(\log n)$ queries.
Can we do better (either approximation constant or number of queries)?
