A tight bound for online colouring of disk graphs

Caragiannis, Ioannis; Fishkin, Aleksei V.; Kaklamanis, Christos; Papaioannou, Evi

doi:10.1016/j.tcs.2007.04.025

Cited by 20 publications

(19 citation statements)

References 9 publications

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“…Thus, the time complexity of this algorithm will be much close to the node coloring number of the communication graph G, in which two interfering nodes should be assigned different colors. Notice that, it was proved in [22] that for disk graphs, the tight bound for approximation ratio for online coloring of disk graphs is min{log n, log ψ}. Thus, we know that it is impossible to design distributed algorithm for link scheduling with better asymptotic approximation ratio when no any ordering are allowed among links.…”

Section: D2 G )mentioning

confidence: 99%

Interference-Aware Joint Routing and TDMA Link Scheduling for Static Wireless Networks

Wang

et al. 2008

IEEE Trans. Parallel Distrib. Syst.

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Abstract-We study efficient interference-aware joint routing and TDMA link scheduling for a multihop wireless network to maximize its throughput. Efficient link scheduling can greatly reduce the interference effect of close-by transmissions. Unlike the previous studies that often assume a unit disk graph model, we assume that different terminals could have different transmission ranges and different interference ranges. In our model, it is also possible that a communication link may not exist due to barriers or is not used by a predetermined routing protocol, while the transmission of a node always result interference to all non-intended receivers within its interference range.Using a mathematical formulation, we develop interference aware joint routing and synchronized TDMA link schedulings that optimize the networking throughput subject to various constraints. Our linear programming formulation will find a flow routing whose achieved throughput is at least a constant fraction of the optimum, and the achieved fairness is also a constant fraction of the requirement. Then, by assuming known link capacities and link traffic loads, we study link scheduling under the RTS/CTS interference model and the protocol interference model with fixed transmission power. For both models, we present both efficient centralized and distributed algorithms that use time slots within a constant factor of the optimum. We also present efficient distributed algorithms whose performances are still comparable with optimum, but with much less communications. We prove that the time-slots needed by our faster distributed algorithms are only at most O(min(log n, log ψ)) for RTS/CTS interference model and protocol interference model. Our theoretical results are corroborated by extensive simulation studies.

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Section: D2 G )mentioning

confidence: 99%

Interference-Aware Joint Routing and TDMA Link Scheduling for Static Wireless Networks

Wang

et al. 2008

IEEE Trans. Parallel Distrib. Syst.

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“…Specifically, it first computes the smallest color c min that is not assigned to any neighbor of u (line [10][11]. If c min is smaller than the maximum color in SC(u), c min is assigned to u (line [12][13]. Otherwise, for each u's neighbor v, DC-Local uses mcolor[v.color] to store the the maximal number of the saturation colors of u's neighbors that are assigned with v.color (line [15][16][17].…”

Section: The Existing Solutionmentioning

confidence: 99%

“…OB [56] suggests to assign vertices with the feasible color containing the most vertices. There are considerable studies on online coloring for special graph classes, such as tree [29], interval graph [39], disk graph [13] and bounded treewidth graph [21]. Recent works study scenarios where an online algorithm can query oracle about future information [64,12,10].…”

Section: Static Graph Coloringmentioning

confidence: 99%

Effective and efficient dynamic graph coloring

et al. 2017

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Graph coloring is a fundamental graph problem that is widely applied in a variety of applications. The aim of graph coloring is to minimize the number of colors used to color the vertices in a graph such that no two incident vertices have the same color. Existing solutions for graph coloring mainly focus on computing a good coloring for a static graph. However, since many real-world graphs are highly dynamic, in this paper, we aim to incrementally maintain the graph coloring when the graph is dynamically updated. We target on two goals: high effectiveness and high efficiency. To achieve high effectiveness, we maintain the graph coloring in a way such that the coloring result is consistent with one of the best static graph coloring algorithms for large graphs. To achieve high efficiency, we investigate efficient incremental algorithms to update the graph coloring by exploring a small number of vertices. We design a color-propagation based algorithm which only explores the vertices within the 2-hop neighbors of the update-related and color-changed vertices. We then propose a novel color index to maintain some summary color information and, thus, bound the explored vertices within the neighbors of these vertices. Moreover, we derive some effective pruning rules to further reduce the number of propagated vertices. The experimental results demonstrate the high effectiveness and efficiency of our approach. PVLDB Reference Format:

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“…the ratio of the number of colors used by A to the number of colors in an optimal coloring of G [15]. Notice that F irst − F it combined with any arbitrary vertex ordering computes a 5-approximation coloring in UDGs [2]. By processing the vertices of the graph in a specific order, F irst − F it computes 3-approximate solutions in UDGs.…”

Section: Coloring Sequence Analysismentioning

confidence: 99%

Employing (1 − ∊) Dominating Set Partitions as Backbones in Wireless Sensor Networks

Mahjoub

Matula

2010

2010 Proceedings of the Twelfth Workshop on Algorithm Engineering and Experiments (ALENEX)

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For a random geometric graph G(n, r) of minimum degree δ, we introduce an efficient algorithm for selecting (δ + 1) backbones with disjoint node sets that are each independent (1 − ε) dominating sets of G. The backbone node sets are determined by a graph coloring algorithm employing only the topology (not the geometry) of G(n, r), and the backbone links are selected with link lengths in a narrow window between r and 2r and further to form a planar graph backbone. For large vertex sets (n = 1600, 3200) the resulting backbones are shown to each cover typically over 99% of the vertices of G (i.e. ε < 0.01), with about 30% being fully dominating, which is consistent with the 1 4 constant approximation factor algorithm proposed recently in [23] for the domatic partition problem in Unit Disk Graphs. We establish experimentally by measures of node degrees, link lengths, and interior triangular face counts that each individual backbone has most of the coverage behavior and routing convenience of the triangular "perfect packing" lattice. We further show for each sample G(n, r) that the relatively few vertices not covered by all (δ + 1) backbones are covered by most of the backbones. Hence backbone rotation in a wireless sensor network would reach all sensors (vertices) sufficiently frequently. Our novel backbone generation algorithm confirms experimentally the existence of these (δ + 1) backbones in a random geometric graph, and provides an efficient topologically based centralized algorithm for determining the backbones. We also point out that our novel backbone construction method is flexible such that any efficient coloring algorithm can be plugged into it. In this paper, we experiment with several coloring algorithms: Smallest Last, Largest First, Lexicographic, Radial Sweep and Random and we compare their respective performance. Our emphasis is, however, on SL since it offers robust properties and interesting expected behavior. We also experiment with several random node distributions: uniform, skewed and normal in both unit square and disk of which we also discuss the results. * Boby B. Lyle School of Engineering. Southern Methodist University. Dallas, TX 75275-0122, USA.

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A tight bound for online colouring of disk graphs

Cited by 20 publications

References 9 publications

Interference-Aware Joint Routing and TDMA Link Scheduling for Static Wireless Networks

Interference-Aware Joint Routing and TDMA Link Scheduling for Static Wireless Networks

Effective and efficient dynamic graph coloring

Employing (1 − ∊) Dominating Set Partitions as Backbones in Wireless Sensor Networks

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