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 π in T , we must find the predecessor of x in ∪v∈πC(v). We present a linear space dynamic data structure that supports such queries in O(t(n) + |π|) time, where t(n) is the time needed to search in one catalog and |π| denotes the number of nodes on path π. We also consider the reporting variant of this problem, in which for any x1, x2 and for any path π, all elements of ∪v∈π′(C(v) ∩ [x1, x2]) must be reported; here π denotes a path between an arbitrary node v0 and its ancestor v1. We show that such queries can be answered in O(t(n) + |π′|+ 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 linesegment intersection problem with optimal O(log n/ log logn) query time and poly-logarithmic update time. ∗Department of Computer Science, University of Bonn. Email [email protected]