Dynamic Trees and Dynamic Point Location

Abstract

This paper describes new methods for maintaining a point-location data structure for a dynamically changing monotone subdivision $\cal S$. The main approach is based on the maintenance of two interlaced spanning trees, one for $\cal S$ and one for the graph-theoretic planar dual of $\cal S$. Queries are answered by using a centroid decomposition of the dual tree to drive searches in the primal tree. These trees are maintained via the link-cut trees structure of Sleator and Tarjan [J. Comput. System Sci., 26 (1983), pp. 362--381], leading to a scheme that achieves vertex insertion/deletion in O(log n) time, insertion/deletion of k-edge monotone chains in O(log n + k) time, and answers queries in O(log2n ) time, with O(n) space, where n is the current size of subdivision $\cal S$. The techniques described also allow for the dual operations expand and contract to be implemented in O(log n) time, leading to an improved method for spatial point location in a 3-dimensional convex subdivision. In addition, the interlaced-tree approach is applied to on-line point location (where one builds $\cal S$ incrementally), improving the query bound to $O(\log n\log\log n)$ time and the update bounds to O(1)amortized time in this case. This appears to be the first on-line method to achieve a polylogarithmic query time and constant update time.