Colorful Strips

Given a planar point set and an integer $k$, we wish to color the points with $k$ colors so that any axis-aligned strip containing enough points contains all colors. The goal is to bound the necessary size of such a strip, as a function of $k$. We show that if the strip size is at least $2k{-}1$, such a coloring can always be found. We prove that the size of the strip is also bounded in any fixed number of dimensions. In contrast to the planar case, we show that deciding whether a 3D point set can be 2-colored so that any strip containing at least three points contains both colors is NP-complete. We also consider the problem of coloring a given set of axis-aligned strips, so that any sufficiently covered point in the plane is covered by $k$ colors. We show that in $d$ dimensions the required coverage is at most $d(k{-}1)+1$. Lower bounds are given for the two problems. This complements recent impossibility results on decomposition of strip coverings with arbitrary orientations. Finally, we study a variant where strips are replaced by wedges

Coloring and guarding arrangements

Given a simple arrangement of lines in the plane, what is the minimum number c of colors required so that we can color all lines in a way that no cell of the arrangement is monochromatic? In this paper we give worst-case bounds on the number c for both the above question and for some of its variations. Line coloring problems can be redefined as geometric hypergraph coloring problems as follows: if we define Hline-cell as the hypergraph whose vertices are lines and edges are cells of the arrangement, then c is equal to the chromatic number of this hypergraph. Specifically, we prove that this chromatic number is between Ω(log n= log log n) and O( √n). Furthermore, we give bounds on the minimum size of a subset S of the intersection points between pairs of lines in A such that every cell contains at least a vertex of S. This may be seen as the problem of guarding cells with vertices when the lines act as obstacles. The problem can also be defined as the minimum vertex cover problem in the hypergraph Hvertex-cell, the vertices of which are the line intersections, and the hyperedges are vertices of a cell. Analogously, we consider the problem of touching the lines with a minimum subset of the cells of the arrangement, which we identify as the minimum vertex cover problem in the Hcell-zone hypergraph

Necklaces, convolutions, and X + Y

SCOPUS: cp.kAlgorithms - ESA 2006, 14th Annual European Symposium, Zurich, Switzerland, September 11-13, 2006, Proceedingsinfo:eu-repo/semantics/publishe

Necklaces, convolutions, and X+Y

We give subquadratic algorithms that, given two necklaces each with n beads at arbitrary positions, compute the optimal rotation of the necklaces to best align the beads. Here alignment is measured according to the p norm of the vector of distances between pairs of beads from opposite necklaces in the best perfect matching. We show surprisingly different results for p = 1, p even, and p = \infty. For p even, we reduce the problem to standard convolution, while for p = \infty and p = 1, we reduce the problem to (min, +) convolution and (median, +) convolution. Then we solve the latter two convolution problems in subquadratic time, which are interesting results in their own right. These results shed some light on the classic sorting X + Y problem, because the convolutions can be viewed as computing order statistics on the antidiagonals of the X + Y matrix. All of our algorithms run in o(n^2) time, whereas the obvious algorithms for these problems run in \Theta(n^2) time.info:eu-repo/semantics/publishe

Continuous Yao graphs

In this paper, we introduce a variation of the well-studied Yao graphs. Given a set of points S⊂R2 and an angle 0π, but on the other hand is always connected for θ⩽π. Furthermore, we show that cY(θ) is a region-fault-tolerant geometric spanner for convex fault regions when θ<π/3. For half-plane faults, cY(θ) remains connected if θ⩽π. Finally, we show that cY(θ) is not always self-approaching for any value of θ