Balanced Islands in Two Colored Point Sets in the Plane

Let $S$ be a set of $n$ points in general position in the plane, $r$ of which are red and $b$ of which are blue. In this paper we prove that there exist: for every $\alpha \in \left [ 0,\frac{1}{2} \right ]$, a convex set containing exactly $\lceil \alpha r\rceil$ red points and exactly $\lceil \alpha b \rceil$ blue points of $S$; a convex set containing exactly $\left \lceil \frac{r+1}{2}\right \rceil$ red points and exactly $\left \lceil \frac{b+1}{2}\right \rceil$ blue points of $S$. Furthermore, we present polynomial time algorithms to find these convex sets. In the first case we provide an $O(n^4)$ time algorithm and an $O(n^2\log n)$ time algorithm in the second case. Finally, if $\lceil \alpha r\rceil+\lceil \alpha b\rceil$ is small, that is, not much larger than $\frac{1}{3}n$, we improve the running time to $O(n \log n)$

Matching random colored points with rectangles

Let S[0,1]2 be a set of n points, randomly and uniformly selected. Let RB be a random partition, or coloring, of S in which each point of S is included in R uniformly at random with probability 1/2. We study the random variable M(n) equal to the number of points of S that are covered by the rectangles of a maximum strong matching of S with axis-aligned rectangles. The matching consists of closed rectangles that cover exactly two points of S of the same color. A matching is strong if all its rectangles are pairwise disjoint. We prove that almost surely M(n)=0.83n for n large enough. Our approach is based on modeling a deterministic greedy matching algorithm, that runs over the random point set, as a Markov chain.Research supported by projects MTM2015-63791-R MINECO/FEDER and Gen. Cat. DGR 2017SGR1640Postprint (author's final draft

Rainbow polygons for colored point sets in the plane

Given a colored point set in the plane, a perfect rainbow polygon is a simple polygon that contains exactly one point of each color, either in its interior or on its boundary. Let $\operatorname{rb-index}(S)$ denote the smallest size of a perfect rainbow polygon for a colored point set $S$, and let $\operatorname{rb-index}(k)$ be the maximum of $\operatorname{rb-index}(S)$ over all $k$-colored point sets in general position; that is, every $k$-colored point set $S$ has a perfect rainbow polygon with at most $\operatorname{rb-index}(k)$ vertices. In this paper, we determine the values of $\operatorname{rb-index}(k)$ up to $k=7$, which is the first case where $\operatorname{rb-index}(k)\neq k$, and we prove that for $k\ge 5$, $$\frac{40\lfloor (k-1)/2 \rfloor -8}{19} \leq\operatorname{rb-index}(k)\leq 10 \bigg\lfloor\frac{k}{7}\bigg\rfloor + 11.$$ Furthermore, for a $k$-colored set of $n$ points in the plane in general position, a perfect rainbow polygon with at most $10 \lfloor\frac{k}{7}\rfloor + 11$ vertices can be computed in $O(n\log n)$ time.Comment: 23 pages, 11 figures, to appear at Discrete Mathematic

Edge-Removal and Non-Crossing Configurations in Geometric Graphs

A geometric graph is a graph G = (V;E) drawn in the plane, such that V is a point set in general position and E is a set of straight-line segments whose endpoints belong to V . We study the following extremal problem for geometric graphs: How many arbitrary edges can be removed from a complete geometric graph with n vertices such that the remaining graph still contains a certain non-crossing subgraph. The non-crossing subgraphs that we consider are perfect matchings, subtrees of a given size, and triangulations. In each case, we obtain tight bounds on the maximum number of removable edges.Postprint (published version

Drawing the double circle on a grid of minimum size

In 1926, Jarník introduced the problem of drawing a convex n-gon with vertices having integer coordinates. He constructed such a drawing in the grid [1, c ·n 3/2]2 for some constant c > 0, and showed that this grid size is optimal up to a constant factor. We consider the analogous problem of drawing the double circle, and prove that it can be done within the same grid size. Moreover, we give an O(n log n)-time algorithm to construct such a point set.Consejo Nacional de Ciencia y Tecnologia (México)Programa de Apoyo a Proyectos de Investigación e Innovación Tecnológica (Universidad Nacional Autónoma de México)Comisión Nacional de Investigación Científica y Tecnológica (Chile)Fondo Nacional de Desarrollo Científico y Tecnológico (Chile

Linguistic and maternal genetic diversity are not correlated in Native Mexicans

Mesoamerica, defined as the broad linguistic and cultural area from middle southern Mexico to Costa Rica, might have played a pivotal role during the colonization of the American continent. The Mesoamerican isthmus has constituted an important geographic barrier that has severely restricted gene flow between North and South America in pre-historical times. Although the Native American component has been already described in admixed Mexican populations, few studies have been carried out in native Mexican populations. In this study, we present mitochondrial DNA (mtDNA) sequence data for the first hypervariable region (HVR-I) in 477 unrelated individuals belonging to 11 different native populations from Mexico. Almost all of the Native Mexican mtDNAs could be classified into the four pan-Amerindian haplogroups (A2, B2, C1, and D1); only two of them could be allocated to the rare Native American lineage D4h3. Their haplogroup phylogenies are clearly star-like, as expected from relatively young populations that have experienced diverse episodes of genetic drift (e.g., extensive isolation, genetic drift, and founder effects) and posterior population expansions. In agreement with this observation, Native Mexican populations show a high degree of heterogeneity in their patterns of haplogroup frequencies. Haplogroup X2a was absent in our samples, supporting previous observations where this clade was only detected in the American northernmost areas. The search for identical sequences in the American continent shows that, although Native Mexican populations seem to show a closer relationship to North American populations, they cannot be related to a single geographical region within the continent. Finally, we did not find significant population structure in the maternal lineages when considering the four main and distinct linguistic groups represented in our Mexican samples (Oto-Manguean, Uto-Aztecan, Tarascan, and Mayan), suggesting that genetic divergence predates linguistic diversification in Mexico