Searching in Dynamic Catalogs on a Tree

In this paper we consider the following modification of the iterative search problem. We are given a tree $T$, so that a dynamic catalog $C(v)$ is associated with every tree node $v$. For any $x$ and for any node-to-root path $\pi$ in $T$, we must find the predecessor of $x$ in ${\cup }_{v\in \pi }C(v)$. We present a linear space dynamic data structure that supports such queries in $O(t(n)+\mathrm{\mid }\pi \mathrm{\mid })$ time, where $t(n)$ is the time needed to search in one catalog and $\mathrm{\mid }\pi \mathrm{\mid }$ denotes the number of nodes on path $\pi$. We also consider the reporting variant of this problem, in which for any $x_1$, $x_2$ and for any path ${\pi }^{\mathrm{\prime }}$ all elements of ${\cup }_{v\in {\pi }^{\mathrm{\prime }}}(C(v)\cap [x_1,x_2])$ must be reported; here ${\pi }^{\mathrm{\prime }}$ denotes a path between an arbitrary node $v_0$ and its ancestor $v_1$. We show that such queries can be answered in $O(t(n)+|\pi’|+ k)$time,where$k$ is the number of elements in the answer. To illustrate applications of our technique, we describe the first dynamic data structures for the stabbing-max problem, the horizontal point location problem, and the orthogonal line-segment intersection problem with optimal $O(\log n/\log \log n)$ query time and poly-logarithmic update time.