Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version

Square and rectangle covering with outliers

Proceedings of the 3rd Frontiers of Algorithmics Workshop (FAW’09)info:eu-repo/semantics/publishe

Square and rectangle covering with outliers

DecretOANoAutActifinfo:eu-repo/semantics/inPres

Square and rectangle covering with outliers

info:eu-repo/semantics/publishe

Square and Rectangle Covering with Outliers

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Square and Rectangle Covering with Outliers

For a set of n points in the plane, we consider the axis-aligned (p; k)-Box COVERING problem: Find p axis-aligned, pairwise disjoint. boxes that together contain exactly n-k points. Here, our boxes are either squares or rectangles, and we want to minimize the area of the largest box. For squares, we present algorithms that find the solution in O(n + k log k) time for p = 1. and in O(n log n + k(p) log(p) k) time for p = 2; 3. For rectangles we have running times of O(n + k(3)) for p = 1 and O(n log n + k(2+p) log(p-1) k) time for p = 2; 3. In all cases; our algorithms use O(n) space.11sciescopu