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49
ConflictFree Colorings of Simple Geometric Regions with Applications to Frequency Assignment in Cellular Networks
, 2002
"... Motivated by a frequency assignment problem in cellular networks, we introduce and study a new coloring problem that we call Minimum ConflictFree Coloring (MinCFColoring). In its general form, the input of the MinCFcoloring problem is a set system (X, S), where each S 2 S is a subset of X . The ..."
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Cited by 60 (8 self)
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Motivated by a frequency assignment problem in cellular networks, we introduce and study a new coloring problem that we call Minimum ConflictFree Coloring (MinCFColoring). In its general form, the input of the MinCFcoloring problem is a set system (X, S), where each S 2 S is a subset of X . The output is a coloring of the sets in S that satisfies the following constraint: for every x 2 X there exists a color i and a unique set S 2 S, such that x 2 S and (S) = i. The goal is to minimize the number of colors used by the coloring .
Online ConflictFree Coloring for Intervals
, 2006
"... We consider an online version of the conflictfree 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 conflictfree, in the sense that in every interval I there is a color that appears exactly once ..."
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Cited by 26 (6 self)
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We consider an online version of the conflictfree 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 conflictfree, in the sense that in every interval I there is a color that appears exactly once in I. We present deterministic and randomized algorithms for achieving this goal, and analyze their performance, that is, the maximum number of colors that they need to use, as a function of the number n of inserted points. We first show that a natural and simple (deterministic) approach may perform rather poorly, requiring Ω ( √ n) colors in the worst case. We then derive two efficient variants of this simple algorithm. The first is deterministic and uses O(log 2 n) colors, and the second is randomized and uses O(log n) colors with high probability. We also show that the O(log 2 n) bound on the number of colors used by our deterministic algorithm is tight on the worst case. We also analyze the performance of the simplest proposed algorithm when the points are inserted in a random order, and present an incomplete analysis that indicates that, with high
Monadic Secondorder Logic for Parameterized Verification
 in: Proc. 19th Symp. on Parallelism in Algorithms and Architectures (SPAA
, 1994
"... Given a set of points P ⊆ R 2, a conflictfree coloring of P is an assignment of colors to points of P, such that each nonempty axisparallel 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 ..."
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Cited by 25 (0 self)
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Given a set of points P ⊆ R 2, a conflictfree coloring of P is an assignment of colors to points of P, such that each nonempty axisparallel 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 conflictfree colored with Õ(n.382+ɛ) colors in expected polynomial time, for any arbitrarily small ɛ> 0. This improves upon the previously known bound of O ( p n log log n/log n).
Conflictfree colorings of shallow discs
 In Proc. 22nd Annual ACM Symposium on Computational Geometry (SoCG
, 2006
"... We prove that any collection of n discs in which each one intersects at most k others, can be colored with at most O(log 3 k) colors so that for each point p in the union of all discs there is at least one disc in the collection containing p whose color differs from that of all other members of the ..."
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Cited by 23 (3 self)
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We prove that any collection of n discs in which each one intersects at most k others, can be colored with at most O(log 3 k) colors so that for each point p in the union of all discs there is at least one disc in the collection containing p whose color differs from that of all other members of the collection that contain p. This is motivated by a problem on frequency assignments in cellular networks, and improves the best previously known upper bound of O(log n) when k is much smaller than n. 1
ConflictFree Colorings of Rectangles Ranges
 In Proc. 23rd International Symposium on Theoretical Aspects of Computer Science (STACS 2006
, 2006
"... Abstract. Given the range space (P, R), where P is a set of n points in IR 2 and R is the family of subsets of P induced by all axisparallel rectangles, the conflictfree coloring problem asks for a coloring of P with the minimum number of colors such that (P, R) is conflictfree. We study the foll ..."
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Cited by 19 (1 self)
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Abstract. Given the range space (P, R), where P is a set of n points in IR 2 and R is the family of subsets of P induced by all axisparallel rectangles, the conflictfree coloring problem asks for a coloring of P with the minimum number of colors such that (P, R) is conflictfree. We study the following question: Given P, is it possible to add a small set of points Q such that (P ∪ Q, R) can be colored with fewer colors than (P, R)? Our main result is the following: given P, and any ǫ ≥ 0, one can always add a set Q of O(n 1−ǫ) points such that P ∪ Q can be conflictfree colored using Õ(n3 8 (1+ǫ) ) 1 colors. Moreover, the set Q and the conflictfree coloring can be computed in polynomial time, with high probability. Our result is obtained by introducing a general probabilistic recoloring technique, which we call quasiconflictfree coloring, and which may be of independent interest. A further application of this technique is also given. 1
Slotted Scheduled Tag Access in MultiReader RFID Systems
"... Abstract—Radio frequency identification (RFID) is a technology where a reader device can “sense ” the presence of a closeby object by reading a tag device attached to the object. To improve coverage, multiple RFID readers can be deployed in the given region. In this paper, we consider the problem of ..."
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Cited by 13 (1 self)
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Abstract—Radio frequency identification (RFID) is a technology where a reader device can “sense ” the presence of a closeby object by reading a tag device attached to the object. To improve coverage, multiple RFID readers can be deployed in the given region. In this paper, we consider the problem of slotted scheduled access of RFID tags in a multiple reader environment. In particular, we develop centralized algorithms in a slotted time model to read all the tags using nearoptimal number of time slots. We consider two scenarios – one wherein the tag distribution in the physical space is unknown, and the other where tag distribution is known or can be estimated a priori. For each of these scenarios, we consider two cases depending on whether a single channel or multiple channels are available. All the above version of the problem are NPhard. We design approximation algorithms for the single channel and heuristic algorithms for the multiple channel cases. Through extensive simulations, we show that for the single channel case, our heuristics perform close to the approximation algorithms. In general, our simulations show that our algorithms significantly outperform Colorwave, an existing algorithm for similar problems. I.
How to play a coloring game against a colorblind adversary
 In Proc. 22nd Annual ACM Symposium on Computational Geometry (SoCG 2006
, 2006
"... We study the problem of conflictfree (CF) coloring of a set of points in the plane, in an online fashion, with respect to halfplanes, nearlyequal axisparallel rectangles, and congruent disks. As a warmup exercise, the online CF coloring of points on the line with respect to intervals is also con ..."
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Cited by 12 (2 self)
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We study the problem of conflictfree (CF) coloring of a set of points in the plane, in an online fashion, with respect to halfplanes, nearlyequal axisparallel rectangles, and congruent disks. As a warmup exercise, the online CF coloring of points on the line with respect to intervals is also considered. We present randomized algorithms in the oblivious adversary model, where the adversary does not see the colors used. For the problems considered, the algorithms always produce valid CF colorings, and use O(log n) colors with high probability (these bounds are optimal in the worst case). Our randomized online algorithms are considerably simpler than previous algorithms for this problem and use fewer colors. We also present a deterministic algorithm for the CF coloring of points in the plane with respect to nearlyequal axisparallel rectangles, using O(polylog(n)) colors. This is the first efficient deterministic online CF coloring algorithm for this problem. 1
State of the Union (of Geometric Objects)
 CONTEMPORARY MATHEMATICS
"... Let C be a set of geometric objects in R d. The combinatorial complexity of the union of C is the total number of faces of all dimensions on its boundary. We survey the known upper bounds on the complexity of the union of n geometric objects satisfying various natural conditions. These bounds play ..."
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Cited by 11 (7 self)
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Let C be a set of geometric objects in R d. The combinatorial complexity of the union of C is the total number of faces of all dimensions on its boundary. We survey the known upper bounds on the complexity of the union of n geometric objects satisfying various natural conditions. These bounds play a central role in the analysis of many geometric algorithms, and the techniques used to attain these bounds are interesting in their own right.
Online Conflictfree Colorings for Hypergraphs
, 2007
"... We provide a framework for online conflictfree coloring (CFcoloring) of any hypergraph. We use this framework to obtain an efficient randomized online algorithm for CFcoloring any kdegenerate hypergraph. Our algorithm uses O(k log n) colors with high probability and this bound is asymptotically ..."
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Cited by 11 (2 self)
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We provide a framework for online conflictfree coloring (CFcoloring) of any hypergraph. We use this framework to obtain an efficient randomized online algorithm for CFcoloring any kdegenerate hypergraph. Our algorithm uses O(k log n) colors with high probability and this bound is asymptotically optimal for any constant k. Moreover, our algorithm uses O(k log k log n) random bits with high probability. As a corollary, we obtain asymptotically optimal randomized algorithms for online CFcoloring some hypergraphs that arise in geometry. Our algorithm uses exponentially fewer random bits compared to previous results. We introduce deterministic online CFcoloring algorithms for points on the line with respect to intervals and for points on the plane with respect to halfplanes (or unit discs) that use Θ(log n) colors and recolor O(n) points in total.
Online ConflictFree Coloring for Intervals
, 2004
"... We consider an online version of the conflictfree 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 conflictfree, in the sense that in every interval I there is a color that appears exactly once ..."
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Cited by 10 (2 self)
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We consider an online version of the conflictfree 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 conflictfree, in the sense that in every interval I there is a color that appears exactly once in I. We present several deterministic and randomized algorithms for achieving this goal, and analyze their performance, that is, the maximum number of colors that they need to use, as a function of the number n of inserted points. We first show that a natural and simple (deterministic) approach may perform rather poorly, requiring Ω ( √ n) colors in the worst case. We then derive several efficient algorithms. The first algorithm is randomized and simple to analyze; it requires an expected number of at most O(log² n) colors, and produces a coloring which is valid with high probability. The second algorithm is deterministic, and is a variant of the initial simple algorithm; it uses a maximum of Θ(log 2 n) colors. The third algorithm is a randomized variant of the second algorithm; it requires an expected number of at most O(log n log log n) colors and always produces a valid coloring. We also analyze the performance of the simplest proposed algorithm when the points are inserted in a random order, and present