Conflict-free colorings of simple geometric regions with applications to frequency assignment in cellular networks

Even, Guy; Lotker, Zvi; Ron, Dana; Smorodinsky, Shakhar

doi:10.1109/sfcs.2002.1181994

Cited by 109 publications

(231 citation statements)

References 14 publications

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“…The study of the above problem was initiated in [9] and continued in a series of recent papers [3-6, 8, 11, 12, 14, 15]. For a recent survey on the problem and its applications, we refer to [16].…”

Section: Introductionmentioning

confidence: 99%

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Conflict-Free Coloring for Rectangle Ranges Using O(n .382) Colors

Ajwani

Elbassioni

Govindarajan

et al. 2012

Discrete Comput Geom

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Given a set of points P ⊆ R 2 , a conflict-free coloring of P is an assignment of colors to points of P , such that each non-empty axis-parallel rectangle T in the plane contains a point whose color is distinct from all other points in P ∩ T. This notion has been the subject of recent interest, and is motivated by frequency assignment in wireless cellular networks: one naturally would like to minimize the number of frequencies (colors) assigned to bases stations (points), such that within any range (for instance, rectangle), there is no interference. We show that any set of n points in R 2 can be conflict-free colored with˜Owith˜ with˜O(n .382+) colors in expected polynomial time, for any arbitrarily small > 0. This improves upon the previously known bound of O(p n log log n/ log n).

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Section: Introductionmentioning

confidence: 99%

“…The work in [9] presented a general framework for computing a conflict-free coloring for several types of ranges. In particular, for the case where the ranges are discs in the plane, they present a polynomial-time coloring algorithm that uses O(log n) colors for conflict-free coloring, and this bound is shown to be tight.…”

Section: Introductionmentioning

confidence: 99%

Conflict-Free Coloring for Rectangle Ranges Using O(n .382) Colors

Ajwani

Elbassioni

Govindarajan

et al. 2012

Discrete Comput Geom

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“…Earlier, conflictfree colorings were mainly considered for some concrete hypergraphs, usually defined by geometric means [6]. János Pach suggested to us that it would be worth while to study the conflict free colorings of almost disjoint transfinite set systems.…”

Section: Introductionmentioning

confidence: 99%

Conflict free colorings of (strongly) almost disjoint set-systems

et al. 2010

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f : ∪A → ρ is called a conflict free coloring of the set-system A (with ρ colors) ifThe conflict free chromatic number χCF (A) of A is the smallest ρ for which A admits a conflict free coloring with ρ colors.A is a (λ, κ, μ)-system if |A| = λ, |A| = κ for all A ∈ A, and A is μ-almost disjoint, i.e. |A ∩ A | < μ for distinct A, A ∈ A. Our aim here is to studyfor λ κ μ, actually restricting ourselves to λ ω and μ ω.For instance, we prove that • for any limit cardinal κ (or κ = ω) and integers n 0, k > 0, GCH implies χCF (κ +n , t, k + 1) = κ +(n+1−i) if i · k < t (i + 1) · k, i = 1, . . . , n; κ if (n + 1) · k < t;• if λ κ ω > d > 1, then λ < κ +ω implies χCF (λ, κ, d) < ω and λ ω (κ) implies χCF (λ, κ, d) = ω;• GCH implies χCF (λ, κ, ω) ω2 for λ κ ω2 and V = L implies χCF (λ, κ, ω) ω1 for λ κ ω1;• the existence of a supercompact cardinal implies the consistency of GCH plus χCF (ℵω+1, ω1, ω) = ℵω+1 and χCF (ℵω+1, ωn, ω) = ω2 for 2 n ω;• CH implies χCF (ω1, ω, ω) = χCF (ω1, ω1, ω) = ω1, while MAω 1 implies χCF (ω1, ω, ω) = χCF (ω1, ω1, ω) = ω. * Corresponding author.† The preparation of this paper was partially supported by OTKA grants K 61600 and K 68262.

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“…The model of conflict-free coloring was introduced in [1] and further studied in [2][3][4]. This model arises from frequency assignment problems in cellular networks.…”

Section: Introductionmentioning

confidence: 99%

“…Algorithms that use O(log n) colors (where n is the number of regions) are given in [1] for the problems of conflict-free coloring of disks, axis-parallel rectangles (with constant ratio between the largest and smallest rectangle), regular hexagons (with constant ratio between the largest and smallest hexagon) and general congruent centrally symmetric convex regions in the plane. In [3] it is shown that for general axis parallel rectangles a CF-coloring that uses only O(log 2 n) colors is possible.…”

Section: Introductionmentioning

confidence: 99%

Conflict-free coloring of unit disks

Lev-Tov

Peleg

2009

Discrete Applied Mathematics

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a b s t r a c tThe paper considers the geometric conflict-free coloring problem, introduced in [G. Even, Z. Lotker, D. Ron, S. Smorodinsky, Conflict-free colorings of simple geometric regions with applications to frequency assignment in cellular networks, SIAM J. Comput. 33 (2003) 94-133]. The input of this problem is a set of regions in the plane and the output is an assignment of colors to the regions, such that for every point p in the total coverage area, there exist a color i and a region D such that p ∈ D, the region D is colored i, and every other region D that contains p is not colored i. The target is to minimize the number of colors used. This problem arises from issues of frequency assignment in radio networks. The paper presents an O(1) approximation algorithm for the conflict-free coloring problem where the regions are unit disks.

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Conflict-free colorings of simple geometric regions with applications to frequency assignment in cellular networks

Cited by 109 publications

References 14 publications

Conflict-Free Coloring for Rectangle Ranges Using O(n .382) Colors

Conflict-Free Coloring for Rectangle Ranges Using O(n .382) Colors

Conflict free colorings of (strongly) almost disjoint set-systems

Conflict-free coloring of unit disks

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