Online Conflict‐Free Coloring for Intervals

Chen, Ke; Fiat, Amos; Kaplan, Haim; Levy, Meital; Matoušek, Jiřı́; Mossel, Elchanan; Pach, János; Sharir, Micha; Smorodinsky, Shakhar; Wagner, Uli; Welzl, Emo

doi:10.1137/s0097539704446682

Cited by 52 publications

(74 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Hoffman et al [20] give tight bounds for the conflict-free chromatic art gallery problem under rectangular visibility in orthogonal polygons: Θ(log log n) are sometimes necessary and always sufficient. Chen et al [13] consider the online version of the conflict-free coloring of a set of points on the line, where each newly inserted point must be assigned a color upon insertion, and at all times the coloring has to be conflict-free. Also in the online scenario, Bar-Nov et al [9] consider a certain class of k-degenerate hypergraphs which sometimes arise as intersection graphs of geometric objects, presenting an online algorithm using O(k log n) colors.…”

Section: Related Workmentioning

confidence: 99%

Three Colors Suffice: Conflict-Free Coloring of Planar Graphs

Abel¹,

Álvarez²,

Gour³

et al. 2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

View full text Add to dashboard Cite

A conflict-free k-coloring of a graph assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and v's neighbors. Such colorings have applications in wireless networking, robotics, and geometry, and are well-studied in graph theory. Here we study the natural problem of the conflict-free chromatic number χ CF (G) (the smallest k for which conflict-free k-colorings exist), with a focus on planar graphs.For general graphs, we prove the conflict-free variant of the famous Hadwiger Conjecture: If G does not contain K k+1 as a minor, then χ CF (G) ≤ k. For planar graphs, we obtain a tight worst-case bound: three colors are sometimes necessary and always sufficient. In addition, we give a complete characterization of the algorithmic/computational complexity of conflict-free coloring. It is NP-complete to decide whether a planar graph has a conflict-free coloring with one color, while for outerplanar graphs, this can be decided in polynomial time. Furthermore, it is NP-complete to decide whether a planar graph has a conflict-free coloring with two colors, while for outerplanar graphs, two colors always suffice. For the bicriteria problem of minimizing the number of colored vertices subject to a given bound k on the number of colors, we give a full algorithmic characterization in terms of complexity and approximation for outerplanar and planar graphs.

show abstract

Section: Related Workmentioning

confidence: 99%

Three Colors Suffice: Conflict-Free Coloring of Planar Graphs

Abel¹,

Álvarez²,

Gour³

et al. 2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

View full text Add to dashboard Cite

show abstract

“…It was shown in [9] that the static version of this problem can be solved using 1 + log 2 n colors and this is also the best that we can do. For the online case, Chen et al [6] gave an algorithm that uses O(log 2 n) colors. In between the static and the online models, Bar-Noy et al [4] studied two semionline models, namely the dynamic offline model where the entire sequence of the vertices is given but the vertices have to be colored one-by-one according to the order of the sequence and the colors cannot be changed later, and the online absolute position model where the vertices are presented in the online fashion but the positions of all vertices on the line are known, i.e, for every vertex we know how many other vertices are on the left and right of this vertex.…”

Section: Definition 3 (Dual Cf-coloring) Let (X R) Be a Range Spacementioning

confidence: 99%

“…Randomized algorithms have been proposed for the online dual CF-coloring for intervals [6], unit disks [5,7], and hypergraphs (i.e., general range space) [3].…”

Section: Definition 3 (Dual Cf-coloring) Let (X R) Be a Range Spacementioning

confidence: 99%

Dynamic Offline Conflict-Free Coloring for Unit Disks

Chan

Chin

Hong

et al. 2009

Approximation and Online Algorithms

View full text Add to dashboard Cite

Abstract.A conflict-free coloring for a given set of disks is a coloring of the disks such that for any point p on the plane there is a disk among the disks covering p having a color different from that of the rest of the disks that covers p. In the dynamic offline setting, a sequence of disks is given, we have to color the disks one-by-one according to the order of the sequence and maintain the conflict-free property at any time for the disks that are colored. This paper focuses on unit disks, i.e., disks with radius one. We give an algorithm that colors a sequence of n unit disks in the dynamic offline setting using O(log n) colors. The algorithm is asymptotically optimal because Ω(log n) colors is necessary to color some set of n unit disks for any value of n [9].

show abstract

“…In addition to this practical motivation, this new coloring model has drawn much attention of researchers through its own theoretical interest and such colorings have been the focus of several recent papers [1,4,6,8,9,10,13].…”

Section: Theorem 14 Let R Be a Family Of N Axis-parallel Rectanglesmentioning

confidence: 99%

On The Chromatic Number of Geometric Hypergraphs

Smorodinsky¹

2007

SIAM J. Discrete Math.

Self Cite

View full text Add to dashboard Cite

Abstract.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 and obtain the following results: (i) Any hypergraph that is induced by a family of n simple Jordan regions such that the maximum union complexity of any k of them (for 1 ≤ k ≤ m) is bounded by U (m) andThus, for example, we prove that any finite family of pseudo-discs 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(log 2 (n)) colors.

show abstract

Online Conflict‐Free Coloring for Intervals

Cited by 52 publications

References 7 publications

Three Colors Suffice: Conflict-Free Coloring of Planar Graphs

Three Colors Suffice: Conflict-Free Coloring of Planar Graphs

Dynamic Offline Conflict-Free Coloring for Unit Disks

On The Chromatic Number of Geometric Hypergraphs

Contact Info

Product

Resources

About