Dynamic trees and dynamic point location

TLDR: The main approach is based on the maintenance of two interlaced spanning trees, one for S and one for the graph-theoretic planar dual of S, which allows for the dual operations expand and contract to be implemented in O(logn) time, leading to an improved method for spatial point location in a 3-dimensional convex subdivision.

Abstract: This paper describes new methods for maintaining a point-location data structure for a dynamically changing monotone subdivisionS. The main approach is based on the maintenance of two interlaced spanning trees, one forS and one for the graph-theoretic planar dual ofS. 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(logn) time, insertion/deletion of k-edge monotone chains in O(logn+k) time, and answers queries in O(log 2 n) time, with O(n) space, where n is the current size of subdivision S. The techniques described also allow for the dual operations expand and contract to be implemented in O(logn) 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 S incrementally), improving the query bound to O(logn log logn) time and the update bounds to O(1) amortized time in this case. This appears to be the rst on-line method to achieve a polylogarithmic query time and constant update time.