An Optimal Dynamic Data Structure for Stabbing-Semigroup Queries

Let $S$ be a set of $n$ intervals in $\mathbb{R}$, and let $(\mathbf{S}, +)$ be any commutative semigroup. We assign a weight $\omega(s) \in \mathbf{S}$ to each interval in $S$. For a point $x \in \mathbb{R}$, let $S(x) \subseteq S$ be the set of intervals that contain $x$. Given a point $q \in \mathbb{R}$, the stabbing-semigroup query asks for computing $\sum_{s \in S(q)} \omega(s)$. We propose a linear-size dynamic data structure, under the pointer-machine model, that answers queries in worst-case $O(\log n)$ time and supports both insertions and deletions of intervals in amortized $O(\log n)$ time. It is the first data structure that attains the optimal $O(\log n)$ bound for all three operations. Furthermore, our structure can easily be adapted to external memory, where we obtain a linear-size structure that answers queries and supports updates in $O(\log_B n)$ I/Os, where $B$ is the disk block size. For the restricted case of a nested family of intervals (either every pair of intervals is disjoint or one contains the other), we present a simpler solution based on dynamic trees.