Algorithms for Colourful Simplicial Depth and Medians in the Plane

The colourful simplicial depth of a point x in the plane relative to a configuration of n points in k colour classes is exactly the number of closed simplices (triangles) with vertices from 3 different colour classes that contain x in their convex hull. We consider the problems of efficiently computing the colourful simplicial depth of a point x, and of finding a point, called a median, that maximizes colourful simplicial depth. For computing the colourful simplicial depth of x, our algorithm runs in time O(n log(n) + k n) in general, and O(kn) if the points are sorted around x. For finding the colourful median, we get a time of O(n^4). For comparison, the running times of the best known algorithm for the monochrome version of these problems are O(n log(n)) in general, improving to O(n) if the points are sorted around x for monochrome depth, and O(n^4) for finding a monochrome median.Comment: 17 pages, 8 figure

Orthogonal weighted linear L1 and L∞ approximation and applications

AbstractLet S={s1,s2,...,sn} be a set of sites in Ed, where every site si has a positive real weight ωi. This paper gives algorithms to find weighted orthogonal L∞ and L1 approximating hyperplanes for S. The algorithm for the weighted orthogonal L1 approximation is shown to require O(nd) worst-case time and O(n) space for d ≥ 2. The algorithm for the weighted orthogonal L∞ approximation is shown to require O(n log n) worst-case time and O(n) space for d = 2, and O(n⌊dl2 + 1⌋) worst-case time and O(n⌊(d+1)/2⌋) space for d > 2. In the latter case, the expected time complexity may be reduced to O(n⌊(d+1)/2⌋). The L∞ approximation algorithm can be modified to solve the problem of finding the width of a set of n points in Ed, and the problem of finding a stabbing hyperplane for a set of n hyperspheres in Ed with varying radii. The time and space complexities of the width and stabbing algorithms are seen to be the same as those of the L∞ approximation algorithm

Algorithms for Computing Closest Points for Segments

Given a set $P$ of $n$ points and a set $S$ of $n$ segments in the plane, we consider the problem of computing for each segment of $S$ its closest point in $P$. The previously best algorithm solves the problem in $n^{4/3}2^{O(\log^*n)}$ time [Bespamyatnikh, 2003] and a lower bound (under a somewhat restricted model) $\Omega(n^{4/3})$ has also been proved. In this paper, we present an $O(n^{4/3})$ time algorithm and thus solve the problem optimally (under the restricted model). In addition, we also present data structures for solving the online version of the problem, i.e., given a query segment (or a line as a special case), find its closest point in $P$. Our new results improve the previous work.Comment: Accepted to STACS 202

A linear-time algorithm for constructing a circular visibility diagram

To computer circular visibility inside a simple polygon, circular arcs that emanate from a given interior point are classified with respect to the edges of the polygon they first intersect. Representing these sets of circular arcs by their centers results in a planar partition called the circular visibility diagram. An O(n) algorithm is given for constructing the circular visibility diagram for a simple polygon with n vertices.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41346/1/453_2005_Article_BF01206329.pd