19 research outputs found
Point Location in Incremental Planar Subdivisions
We study the point location problem in incremental (possibly disconnected) planar subdivisions, that is, dynamic subdivisions allowing insertions of edges and vertices only. Specifically, we present an O(n log n)-space data structure for this problem that supports queries in O(log^2 n) time and updates in O(log n log log n) amortized time. This is the first result that achieves polylogarithmic query and update times simultaneously in incremental planar subdivisions. Its update time is significantly faster than the update time of the best known data structure for fully-dynamic (possibly disconnected) planar subdivisions
Dynamic Distribution-Sensitive Point Location
We propose a dynamic data structure for the distribution-sensitive point
location problem. Suppose that there is a fixed query distribution in
, and we are given an oracle that can return in time the
probability of a query point falling into a polygonal region of constant
complexity. We can maintain a convex subdivision with vertices
such that each query is answered in expected time, where OPT
is the minimum expected time of the best linear decision tree for point
location in . The space and construction time are . An
update of as a mixed sequence of edge insertions and deletions
takes amortized time. As a corollary, the randomized incremental
construction of the Voronoi diagram of sites can be performed in expected time so that, during the incremental construction, a nearest
neighbor query at any time can be answered optimally with respect to the
intermediate Voronoi diagram at that time.Comment: To appear in Proceedings of the International Symposium of
Computational Geometry, 202
Dynamic Planar Point Location in External Memory
In this paper we describe a fully-dynamic data structure for the planar point location problem in the external memory model. Our data structure supports queries in O(log_B n(log log_B n)^3)) I/Os and updates in O(log_B n(log log_B n)^2)) amortized I/Os, where n is the number of segments in the subdivision and B is the block size. This is the first dynamic data structure with almost-optimal query cost. For comparison all previously known results for this problem require O(log_B^2 n) I/Os to answer queries. Our result almost matches the best known upper bound in the internal-memory model
External Memory Planar Point Location with Fast Updates
We study dynamic planar point location in the External Memory Model or Disk Access Model (DAM). Previous work in this model achieves polylog query and polylog amortized update time. We present a data structure with O(log_B^2 N) query time and O(1/B^(1-epsilon) log_B N) amortized update time, where N is the number of segments, B the block size and epsilon is a small positive constant, under the assumption that all faces have constant size. This is a B^(1-epsilon) factor faster for updates than the fastest previous structure, and brings the cost of insertion and deletion down to subconstant amortized time for reasonable choices of N and B. Our structure solves the problem of vertical ray-shooting queries among a dynamic set of interior-disjoint line segments; this is well-known to solve dynamic planar point location for a connected subdivision of the plane with faces of constant size