Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  present new algorithms for approximate range counting, where, for a specified e > 0,  want to count the number of data points in a query range, up to  relative  error of e.  first describe a general framework, adapted from Cohen [10], for this task, and then specialize it to two important instances of range counting: halfspaces in  R  3  and disks in the plane. The technique reduces the approximate range counting problem to that of finding the minimum rank of a data object in the range, with respect to a  random  permutation of the input.A major technical step in our analysis, which  believe to be of independent interest, is a bound of  O ( n  log  n ) on the expected complexity of the overlay of all the Voronoi faces that are generated during a randomized incremental construction of the Voronoi diagram of  n  points in the plane. The same bound holds for the expected complexity of the overlay of all the faces of the minimization diagram of the lor envelope of  n  planes in  R  3 , or for the expected complexity of the overlay of all the normal (or Gaussian) diagram faces of the convex hull of  n  points in  R  3 , that are generated during a randomized incremental construction of the lor envelope or of the hull, respectively. All these bounds are tight in the worst case.The first bound leads to an algorithm that, for a query point  x  ∈  R  2 , efficiently retrieves the sequence of nearest neighbors of  x  in  P , over the random insertion process. A query takes  O (log  n ) expected time, and the expected storage size is  O ( n  log  n ). Similarly, the other bounds lead to an algorithm that, for a query direction ω ∈  S  2 , efficiently retrieves the sequence of the convex hull vertices that are touched by the planes with outward direction ω that support the convex hull during the random insertion process. Again, a query takes  O (log  n ) expected time, and the expected storage size is  O ( n  log  n ). These algorithms are used as the main component in the approximate range counting technique that  present, for ranges that are halfspaces in  R  3  or disks in the plane.
['Haim Kaplan', 'Micha Sharir']
I am writing a research paper on a idea Randomized incremental constructions of three-dimensional convex hulls and planar voronoi diagrams, and approximate range counting
[{"from": "human", "value": "I am writing a research paper on a idea Randomized incremental constructions of three-dimensional convex hulls and planar voronoi diagrams, and approximate range counting"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  present new algorithms for approximate range counting, where, for a specified e > 0,  want to count the number of data points in a query range, up to  relative  error of e.  first describe a general framework, adapted from Cohen [10], for this task, and then specialize it to two important instances of range counting: halfspaces in  R  3  and disks in the plane. The technique reduces the approximate range counting problem to that of finding the minimum rank of a data object in the range, with respect to a  random  permutation of the input.A major technical step in our analysis, which  believe to be of independent interest, is a bound of  O ( n  log  n ) on the expected complexity of the overlay of all the Voronoi faces that are generated during a randomized incremental construction of the Voronoi diagram of  n  points in the plane. The same bound holds for the expected complexity of the overlay of all the faces of the minimization diagram of the lor envelope of  n  planes in  R  3 , or for the expected complexity of the overlay of all the normal (or Gaussian) diagram faces of the convex hull of  n  points in  R  3 , that are generated during a randomized incremental construction of the lor envelope or of the hull, respectively. All these bounds are tight in the worst case.The first bound leads to an algorithm that, for a query point  x  \u2208  R  2 , efficiently retrieves the sequence of nearest neighbors of  x  in  P , over the random insertion process. A query takes  O (log  n ) expected time, and the expected storage size is  O ( n  log  n ). Similarly, the other bounds lead to an algorithm that, for a query direction \u03c9 \u2208  S  2 , efficiently retrieves the sequence of the convex hull vertices that are touched by the planes with outward direction \u03c9 that support the convex hull during the random insertion process. Again, a query takes  O (log  n ) expected time, and the expected storage size is  O ( n  log  n ). These algorithms are used as the main component in the approximate range counting technique that  present, for ranges that are halfspaces in  R  3  or disks in the plane."}]