Related papers: Algorithms for Colourful Simplicial Depth and Medi…
The colourful simplicial depth conjecture states that any point in the convex hull of each of d+1 sets, or colours, of d+1 points in general position in R^d is contained in at least d^2+1 simplices with one vertex from each set. We verify…
The colourful simplicial depth problem in dimension d is to find a configuration of (d+1) sets of (d+1) points such that the origin is contained in the convex hull of each set (colour) but contained in a minimal number of colourful…
Let $P$ be a set of $n$ points in $d$-dimensions. The simplicial depth, $\sigma_P(q)$ of a point $q$ is the number of $d$-simplices with vertices in $P$ that contain $q$ in their convex hulls. The simplicial depth is a notion of data depth…
The colorful simplicial depth of a collection of d+1 finite sets of points in Euclidean d-space is the number of choices of a point from each set such that the origin is contained in their convex hull. We use methods from combinatorial…
Let $P$ be a $k$-colored set of $n$ points in the plane, $4 \leq k \leq n$. We study the problem of deciding if $P$ contains a subset of four points of different colors such that its Rectilinear Convex Hull has positive area. We show this…
Given $d+1$ sets of points, or colours, $S_1,\ldots,S_{d+1}$ in $\mathbb R^d$, a colourful simplex is a set $T\subseteq\bigcup_{i=1}^{d+1}S_i$ such that $|T\cap S_i|\leq 1$, for all $i\in\{1,\ldots,d+1\}$. The colourful Carath\'eodory…
Inspired by Barany's colourful Caratheodory theorem, we introduce a colourful generalization of Liu's simplicial depth. We prove a parity property and conjecture that the minimum colourful simplicial depth of any core point in any…
Let $S$ be a $k$-colored (finite) set of $n$ points in $\mathbb{R}^d$, $d\geq 3$, in general position, that is, no {$(d + 1)$} points of $S$ lie in a common $(d - 1)$}-dimensional hyperplane. We count the number of empty monochromatic…
We consider a bichromatic two-center problem for pairs of points. Given a set $S$ of $n$ pairs of points in the plane, for every pair, we want to assign a red color to one point and a blue color to the other, in such a way that the value…
We show that any point in the convex hull of each of (d+1) sets of (d+1) points in R^d is contained in at least floor((d+2)^2/4) simplices with one vertex from each set.
Consider a pair of plane straight-line graphs, whose edges are colored red and blue, respectively, and let n be the total complexity of both graphs. We present a O(n log n)-time O(n)-space technique to preprocess such pair of graphs, that…
We study colored coverage and clustering problems. Here, we are given a colored point set where the points are covered by (unknown) $k$ clusters, which are monochromatic (i.e., all the points covered by the same cluster, have the same…
We study a geometric facility location problem under imprecision. Given $n$ unit intervals in the real line, each with one of $k$ colors, the goal is to place one point in each interval such that the resulting \emph{minimum color-spanning…
We study the problem of colouring the vertices of a polygon, such that every viewer in it can see a unique colour. The goal is to minimise the number of colours used. This is also known as the conflict-free chromatic guarding problem with…
The investigation of colour symmetries for periodic and aperiodic systems consists of two steps. The first concerns the computation of the possible numbers of colours and is mainly combinatorial in nature. The second is algebraic and…
An algorithm is demonstrated that finds an ordinary intersection in an arrangement of $n$ lines in $\mathbb{R}^2$, not all parallel and not all passing through a common point, in time $O(n \log{n})$. The algorithm is then extended to find…
In this article, we consider a collection of geometric problems involving points colored by two colors (red and blue), referred to as bichromatic problems. The motivation behind studying these problems is two fold; (i) these problems appear…
The degree of a map between orientable manifolds is a fundamental concept in topology that aids in understanding the structure and properties of the manifolds and the maps between them. Numerous studies have been conducted on the degree of…
Constructing partitions of colored points is a well-studied problem in discrete and computational geometry. We study the problem of creating a minimum-cardinality partition into monochromatic islands. Our input is a set $S$ of $n$ points in…
We consider the problem of coloring a 3-colorable graph in polynomial time using as few colors as possible. This is one of the most challenging problems in graph algorithms. In this paper using Blum's notion of ``progress'', we develop a…