Interval tree: find all intervals containing a given interval

Question

Given an interval tree $T$ and an interval $I$, I need to find an algorithm that returns all intervals in $T$ that contain $I$. The asymptotic running time should be $O(\min(n,(k + 1) \log n))$ where $k$ is the number of intervals returned. The only solution I can think of is bruteforcing in $O(n)$ time, but that's clearly too simple.
While looking for an answer I came across stabbing queries and the problem seemed very similar to this one. Is this something I could base my algorithm on or am I on the wrong track?

Edit: I might have an idea, but I'm still missing something.
Let $I=[I_s,I_e]$ and $x$ a node. If $x.s \leq I.s$ then there is an interval that contains $I$ in the left subtree of $x$ if and only if $\mathrm{left}[x].\max\geq I.e$. Suppose you had a similar statement for the right subtree of $x$, then you could do the following:
If the condition for the left or/and subtree is satisfied, recurse on those trees. Since there must be an interval covering $I$ in these subtrees this will result in $k$ paths with a maximum length of $\log n$.
If $x.s > I.s$ there can't be an interval in the right subtree that covers $I$, but there might be one in the left, so recurse on left. If this happens repeatedly until you find a leaf, you get a path of length $logn$ without finding anything.

My question is: given $x.s \leq I.s$, how can I know for sure whether the right subtree of $x$ contains an interval covering $I$?

Answer 1

Hint: A (non-root) interval can only contain I if its parent contains I.

For the algorithm in the pseudo-code below, TestInterval is not called for the same vertex twice, so the complexity is $O(n)$.

Consider the subtree S consisting of all nodes which TestInterval is called on. At every node in S which is not a leaf node, an interval must have been added to intervalContainingI (and k incremented). A binary tree with k non-leaf nodes can have at most $k+1$ leaf nodes. Therefore the complexity is in fact $O(k)$.

vector<int> intervalsContainingI;

void TestInterval(interval x)
{
    if (x.start <= I.start && x.end >= I.end)
    {
        intervalsContainingI.push_back(x);
        if (Exists(x.leftChild))
        {
            TestInterval(x.leftChild);
        }
        if (Exists(x.rightChild)
        {
             TestInterval(x.rightChild);
        }
    }
}

TestInterval(root_node);

Answer 2

If the nodes contain a max_end attribute in addition to the interval, you can use the max_end to stop looking further down the tree.

def find_containing_intervals(interval, root_node):
    stack = [root_node]
    containing_intervals = []
    while stack:
        node = stack.pop()
        if node is None:
            continue
        if node.max_end < interval.end:
            continue
        if node.interval.contains(interval):
            containing_intervals.append(interval)
        if node.interval.start <= interval.start:
            stack.append(node.right_child)
        stack.append(node.left_child)

    return containing_intervals

Recursive solution:

class Node:
    def find_containing(self, interval):
        intervals = []
        if self.max_end < interval.end:
            return intervals
        if self.interval.contains(interval):
            intervals.append(interval)
        if self.right_child is not None and self.interval.start <= interval.start:
            intervals.extend(self.right_child.find_containing(interval))
        if self.left_child is not None:
            intervals.extend(self.left_child.find_containing(interval))
        return intervals