How hard is half-space range searching?

Range Searching Range Searching

Approximate Range Searching Using Binary Space Partitions

We show how any BSP tree TP for the endpoints of a set of n disjoint segments in the plane can be used to obtain a BSP tree of size O(n · depth(TP )) for the segments themselves, such that the range-searching efficiency remains almost the same. We apply this technique to obtain a BSP tree of size O(n log n) such that ε-approximate range searching queries with any constant-complexity convex quer...

Lecture Notes: Range Searching with Linear Space

In this lecture, we will continue our discussion on the range searching problem. Recall that the input set P consists of N points in R. Given an axis-parallel rectangle q, a range query reports all the points of P ∩ q. We want to maintain a fully dynamic structure on P to answer range queries efficiently. We will focus on non-replicating structures [2, 3]. Specifically, consider that each point...

Plane Strain Deformation of a Poroelastic Half-Space Lying Over Another Poroelastic Half-Space

The plane strain deformation of an isotropic, homogeneous, poroelastic medium caused by an inclined line-load is studied using the Biot linearized theory for fluid saturated porous materials. The analytical expressions for the displacements and stresses in the medium are obtained by applying suitable boundary conditions. The solutions are obtained analytically for the limiting case of undrained...

Exploring the Problem Space of Orthogonal Range Searching

Orthogonal range searching is a fundamental problem in computational geometry: preprocess a set of points into a data structure such that we can efficiently answer questions about the points that lie in axis-aligned rectangles called query ranges. There are many interesting variants of orthogonal range searching with differing complexities. We consider classic fundamental variants, new variants...