Computational geometry: an introduction
Computational geometry: an introduction
Functional approach to data structures and its use in multidimensional searching
SIAM Journal on Computing
Algorithms (2nd ed.)
Partition trees for triangle counting and other range searching problems
SCG '88 Proceedings of the fourth annual symposium on Computational geometry
Quasi-optimal range searching in spaces of finite VC-dimension
Discrete & Computational Geometry - Selected papers from the fourth ACM symposium on computational geometry, Univ. of Illinois, Urbana-Champaign, June 6 8, 1988
Discrete & Computational Geometry - Selected papers from the fifth annual ACM symposium on computational geometry, Saarbrücken, Germany, June 5-11, 1989
Higher-dimensional Voronoi diagrams in linear expected time
Discrete & Computational Geometry
Computational geometry in C
ACM Computing Surveys (CSUR)
Generating random polygons with given vertices
Computational Geometry: Theory and Applications
Primitives for the manipulation of general subdivisions and the computation of Voronoi
ACM Transactions on Graphics (TOG)
Selected papers from the 12th annual symposium on Computational Geometry
On the Difficulty of Range Searching
WADS '95 Proceedings of the 4th International Workshop on Algorithms and Data Structures
Heuristics for the Generation of Random Polygons
Proceedings of the 8th Canadian Conference on Computational Geometry
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This paper studies the idea of answering range searching queries using simple data structures. The only data structure we need is the Delaunay Triangulation of the input points. The idea is to first locate a vertex of the (arbitrary) query polygon {\cal q} and walk along the boundary of the polygon in the Delaunay Triangulation and report all the points enclosed by the query polygon. For a set of uniformly distributed random points in 2-D and a query polygon the expected query time of this algorithm is O(n^{1/3} + Q + {\bf E}K + L_{r}n^{1/2}), where Q is the size of the query polygon {\cal q}, {\bf E}K = O(n\bcdot area({\cal q})) is the expected number of output points, L_{r} is a parameter related to the shape of the query polygon {\cal q} and n, and L_{r} is always bounded by the sum of the edge lengths of {\cal q}. Theoretically, when L_{r} = O(1/n^{1/6}) the expected query time is O(n^{1/3} + Q + {\bf E}K), which improves the best known average query time for general range searching. Besides the theoretical meaning, the good property of this algorithm is that once the Delaunay Triangulation is given, no additional preprocessing is needed. In order to obtain empirical results, we design a new algorithm for generating random simple polygons within a given domain. Our empirical results show that the constant coefficient of the algorithm is small, at least for the special (practical) cases when the query polygon is either a triangle (simplex range searching) or an axis-parallel box (orthogonal range searching) and for the general case when the query polygons are generated by our new polygon-generating algorithms and their sizes are relatively small.