Making data structures persistent
Journal of Computer and System Sciences - 18th Annual ACM Symposium on Theory of Computing (STOC), May 28-30, 1986
A singly exponential stratification scheme for real semi-algebraic varieties and its applications
Theoretical Computer Science
Circle shooting in a simple polygon
Journal of Algorithms
Applications of a new space-partitioning technique
Discrete & Computational Geometry
Intersection queries in curved objects
Journal of Algorithms
Counting circular arc intersections
SIAM Journal on Computing
Ray shooting and parametric search
SIAM Journal on Computing
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
Pseudo-triangulations: theory and applications
Proceedings of the twelfth annual symposium on Computational geometry
Handbook of discrete and computational geometry
Handbook of discrete and computational geometry
Ray shooting and lines in space
Handbook of discrete and computational geometry
Hierarchical vertical decompositions, ray shooting, and circular arc queries in simple polygons
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
Online point location in planar arrangements and its applications
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Ray Shooting, Depth Orders and Hidden Surface Removal
Ray Shooting, Depth Orders and Hidden Surface Removal
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We consider segment intersection searching amidst (possibly intersecti ng) algebraic arcs in the plane. We show how to preprocess $n$ arcs in time $O(n^{2+\epsilon})$ into a data structure of size $O(n^{2+\epsilon})$, for any $\epsilon 0$, such that the $k$ arcs intersecting a query segment can be counted in time $O(\log n)$ or reported in time $O(\log n+k)$. This problem was extensively studied in restricted settings (e.g., amidst segments, circles or circular arcs), but no solution with comparable performance was previously presented for the general case of possibly intersecting algebraic arcs. Our data structure for the general case matches or improves (sometimes by an order of magnitude) the size of the best previously presented solutions for the special cases.As an immediate application of this result, we obtain an efficient data structure for the triangular windowing problem, which is a generalization of triangular range searching. As another application, the first substantially sub-quadratic algorithm for a red-blue intersection counting problem is derived. We also describe simple data structures for segment intersection searching among disjoint arcs, and ray shooting among algebraic arcs.