On Approximating the Depth and Related Problems

Abstract

We study the question of finding a deepest point in an arrangement of regions, and provide a fast algorithm for this problem using random sampling, showing it sufficient to solve this problem when the deepest point is shallow. This implies, among other results, a fast algorithm for solving linear programming with violations approximately. We also use this technique to approximate the disk covering the largest number of red points, while avoiding all the blue points, given two such sets in the plane. Using similar techniques imply that approximate range counting queries have roughly the same time and space complexity as emptiness range queries.

Relative (p,ε)-Approximations in Geometry

“…As in the planar case, Theorem 5.5 improves over the previous results [6,25], which require Ω( 1 ε 2 log 2 n) time to answer a query. However, in a subsequent work, Afshani and Chan [2] managed to obtain an improved solution.…”

“…Observe that Theorems 5.3 and 5.4 improve over the previous results in [6,25], which have query time Ω( 1 ε 2 log 2 n). It would also be interesting to compare these results to the recent technique of Aronov and Sharir [7]; as presented, this technique caters only to range searching in four and higher dimensions, but it can be adapted to two or three dimensions too.…”

“…Here, for a set P of n points in R d , for d ≥ 2, the best known algorithm for exact halfspace range counting with nearlinear storage takes O(n 1−1/d ) time [33]. As shown in several recent papers, if we only want to approximate the count, as in (1), there exist faster solutions, in which the query time is close to O(n 1−1/ d/2 ) (and is polylogarithmic in two and three dimensions) [6,7,[25][26][27].…”

Bichromatic separability with two boxes: A general approach

“…For example, the solution to the EIEB-problem for the point set illustrated in Figure 1 is n − 2 (being n the cardinal of the original set), since by removing the points r 1 and b 1 we can obtain two rectangles, R and B, each of them containing only red and blue points, respectively. It is easy to check that it is not possible to remove only one point in order to improve this number.…”

“…The clustering can be obtained by using simple geometric shapes such as circles or boxes. In [1,7], circles and parallel-axis boxes respectively, are considered for the selection. In [1], the following problem is studied: given a bicolored point set, find a ball that contains the maximum number of red points without containing any blue point inside it.…”