On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles
Discrete & Computational Geometry
Applications of random sampling in computational geometry, II
Discrete & Computational Geometry - Selected papers from the fourth ACM symposium on computational geometry, Univ. of Illinois, Urbana-Champaign, June 6 8, 1988
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
Online conflict-free coloring for intervals
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Conflict-Free Coloring of Points and Simple Regions in the Plane
Discrete & Computational Geometry
Conflict-free colorings of shallow discs
Proceedings of the twenty-second annual symposium on Computational geometry
How to play a coloring game against a color-blind adversary
Proceedings of the twenty-second annual symposium on Computational geometry
Conflict-free coloring for intervals: from offline to online
Proceedings of the eighteenth annual ACM symposium on Parallelism in algorithms and architectures
Conflict-free coloring for rectangle ranges using O(n.382) colors
Proceedings of the nineteenth annual ACM symposium on Parallel algorithms and architectures
Delaunay graphs of point sets in the plane with respect to axis-parallel rectangles
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Proceedings of the twenty-fourth annual symposium on Computational geometry
Coloring Axis-Parallel Rectangles
Computational Geometry and Graph Theory
Dynamic Offline Conflict-Free Coloring for Unit Disks
Approximation and Online Algorithms
Online conflict-free colorings for hypergraphs
ICALP'07 Proceedings of the 34th international conference on Automata, Languages and Programming
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A finite family R of simple Jordan regions in the plane defines a hypergraph H = H(R) where the vertex set of H is R and the hyperedges are all subsets S ⊂ R for which there is a point p such that S = {r ∈ R|p ∈ r. The chromatic number of H(R) is the minimum number of colors needed to color the members of R such that no hyperedge is monochromatic. In this paper we initiate the study of the chromatic number of such hypergraphs. We obtain the following results:(i) any hypergraph that is induced by a family of n simple Jordan regions (not necessarily convex) such that the union complexity of any m of them is given by u(m) and u(m)/m is non-decreasing is O(u(n)/n)-colorable. Thus, for example we prove that any finite family of pseudodiscs can be colored with a constant number of colors.(ii) any hypergraph induced by a finite family of planar discs is four-colorable. This bound is tight. In fact, we prove that this statement is equivalent to the Four-Color Theorem.(iii) any hypergraph induced by n axis-parallel rectangles is O(log n)-colorable. This bound is asymptotically tight.Our proofs are constructive. Namely, we provide deterministic polynomial-time algorithms for coloring such hypergraphs with only "few" colors (that is, the number of colors used by these algorithms is upper bounded by the same bounds we obtain on the chromatic number of the given hypergraphs)As an application of (i) and (ii) we obtain simple constructive proofs for the following:(iv) Any set of n Jordan regions with near linear union complexity admits a conflict-free (CF) coloring with polylogarithmic number of colors.(v) Any set of n axis-parallel rectangles admits a CF-coloring with O(log2(n)) colors.