I think you can apply a Merge Sort Tree.
I solved a similar problem a few days ago where I had to find the value closest to
k in a given range [L, R]. The problem was LRQUER.
In your case you can do the following:
Instead of storing a single value in any node, you can store a range. This range will be the sorted list of the values stored in its children. Then, while querying, you will query in this range. For a single query
[L, R], you can find the
upper_bound() and then return the minimum of all the values found like this.
Edit: To handle point updates, we’ll have to update the corresponding parent nodes too. We can update a node in O(log(n)) time by searching for the previous element and then inserting a new element, using something like
std::set (because we have to store the elements in sorted order) in C++. If duplicate elements are allowed, then we can use
std::multiset in C++.
Now, there can be at most O(log(n)) updates in a single update operation because the depth of the tree will be of the order of log(n), making the complexity O(log^2(n)).