Dynamic ordered sets with exponential search trees
Journal of the ACM (JACM)
Tight bounds for dynamic convex hull queries (again)
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Voronoi diagrams in n · 2o(√lg lg n) time
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Optimal dynamic vertical ray shooting in rectilinear planar subdivisions
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Distribution-sensitive point location in convex subdivisions
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Well-separated pair decomposition in linear time?
Information Processing Letters
Localized Spanner Construction for Ad Hoc Networks with Variable Transmission Range
ADHOC-NOW '08 Proceedings of the 7th international conference on Ad-hoc, Mobile and Wireless Networks
Optimal halfspace range reporting in three dimensions
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Succinct geometric indexes supporting point location queries
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Optimal dynamic vertical ray shooting in rectilinear planar subdivisions
ACM Transactions on Algorithms (TALG)
Optimal in-place algorithms for 3-D convex hulls and 2-D segment intersection
Proceedings of the twenty-fifth annual symposium on Computational geometry
Compact oracles for approximate distances around obstacles in the plane
ESA'07 Proceedings of the 15th annual European conference on Algorithms
External memory range reporting on a grid
ISAAC'07 Proceedings of the 18th international conference on Algorithms and computation
A fast algorithm for three-dimensional layers of maxima problem
WADS'11 Proceedings of the 12th international conference on Algorithms and data structures
Succinct geometric indexes supporting point location queries
ACM Transactions on Algorithms (TALG)
Entropy, triangulation, and point location in planar subdivisions
ACM Transactions on Algorithms (TALG)
| Hi-index | 0.00 |
We consider the static planar point location problem in an arbitrary polygonal subdivision given by n segments. We assume points come from the [u]^2 grid, and consider algorithms for the RAM with words of O(lg u) bits. We give the first solution to the problem which can surpass the traditional query time of O(lg n). Specifically, we can obtain a query time of O(\sqrt {\lg u}). Though computational geometry on a grid has been investigated for a long time (including for this problem), it is generally not known how to make good use of a bounded universe in problems of such nonorthogonal flavor. Our result shows this limitation can be surpassed, at least for planar point location. A result by Timothy Chan, appearing independently in FOCS'06, also achieves sublogarithmic query times. Combining the two results, we obtain the following bound. For any S \geqslant2, the exists a data structure using space O(n · S) which supports queries in time: {\text{O}}\left( {{\text{min}}\left\{ {\frac{{{\text{lg n}}}} {{{\text{lg lg n}}}},\sqrt {\frac{{\lg u}} {{\lg \lg u}},} \frac{{\lg u}} {{\lg S}}} \right\}} \right)