Carmen Cortés

Lower bounds for computing statistical depth

Given a finite set of points S, two measures of the depth of a query point θ with respect to S are the Simplicial depth of Liu and the Halfspace depth of Tukey (also known as Location depth). We show that computing these depths requires Ω(nlog n) time, which matches the upper bound complexities of the algorithms of Rousseeuw and Ruts. Our lower bound proofs may also be applied to two bivariate sign tests: that of Hodges, and that of Oja and Nyblom.

Bichromatic separability with two boxes: A general approach

Let S be a set of n points on the plane in general position such that its elements are colored red or blue. We study the following problem: Find a largest subset of S which can be enclosed by the union of two, not necessarily disjoint, axis-aligned rectanglesRandBsuch thatR (resp.B) contains only red (resp. blue) points. We prove that this problem can be solved in O(n^2logn) time and O(n) space. Our approach is based on solving some instances of Bentleys maximum-sum consecutive subsequence problem. We introduce the first known data structure to dynamically maintain the optimal solution of this problem. We show that our techniques can be used to efficiently solve a more general class of problems in data analysis.

Diagonal flips in outer-triangulations on closed surfaces

We show that any two outer-triangulations on the same closed surface can be transformed into each other by a sequence of diagonal flips, up to isotopy, if they have a sufficiently large and equal number of vertices.

Diagonal flips in outer-torus triangulations

Abstract An outer-torus triangulation G with n vertices is a fixed embedding of a simple graph on the torus such that there is one face bounded by a cycle of length n and other faces are all triangular. We show that any two outer-torus triangulations with the same number of vertices can be transformed into each other by a sequence of diagonal flips, through outer-torus triangulations, up to homeomorphism.

Diagonal flips in outer-Klein-bottle triangulations

Abstract We determine the complete list of the irreducible outer-Klein-bottle triangulations, and we prove that any two outer-Klein-bottle triangulations with the same number of vertices can be transformed into each other by a sequence of diagonal flips, up to homeomorphism.

Compact Grid Representation of Graphs

A graph G is said to be grid locatable if it admits a representation such that vertices are mapped to grid points and edges to line segments that avoid grid points but the extremes. Additionally G is said to be properly embeddable in the grid if it is grid locatable and the segments representing edges do not cross each other. We study the area needed to obtain those representations for some graph families.

Transforming Triangulations on Nonplanar Surfaces

We consider whether any two triangulations of a polygon or a point set on a nonplanar surface with a given metric can be transformed into each other by a sequence of edge flips. The answer is negative in general with some remarkable exceptions, such as polygons on the cylinder, and on the flat torus, and certain configurations of points on the cylinder.

Reporting Bichromatic Segment Intersections from Point Sets

In this paper, we introduce a natural variation of the problem of computing all bichromatic intersections between two sets of segments. Given two sets R and B of n points in the plane defining two sets of segments, say red and blue, we present an O(n2) time and space algorithm for solving the problem of reporting the set of segments of each color intersected by segments of the other color. We also prove that this problem is 3-Sum hard and provide some illustrative examples of several point configurations.

Cover contact graphs

We study problems that arise in the context of covering certain geometric objects (so-called seeds, e.g., points or disks) by a set of other geometric objects (a so-called cover, e.g., a set of disks or homothetic triangles). We insist that the interiors of the seeds and the cover elements are pairwise disjoint, but they can touch. We call the contact graph of a cover a cover contact graph (CCG). We are interested in two types of tasks: (a) deciding whether a given seed set has a connected CCG, and (b) deciding whether a given graph has a realization as a CCG on a given seed set. Concerning task (a) we give efficient algorithms for the case that seeds are points and covers are disks or triangles. We show that the problem becomes NP-hard if seeds and covers are disks. Concerning task (b) we show that it is even NP-hard for point seeds and disk covers (given a fixed correspondence between vertices and seeds).