Experimental Study of Independent and Dominating Sets in Wireless ...
Experimental Study of Independent and Dominating Sets in Wireless Sensor Networks Using Graph Coloring Algorithms Dhia Mahjoub and David W. Matula Lyle School of Engineering Southern Methodist University Dallas, TX 75275-0122, USA [email protected], [email protected]
Abstract. The domatic partition problem seeks to maximize the partitioning of the nodes of the network into disjoint dominating sets. These sets represent a series of virtual backbones for wireless sensor networks to be activated successively, resulting in more balanced energy consumption and increased network robustness. In this study, we address the domatic partition problem in random geometric graphs by investigating several vertex coloring algorithms both topology and geometry-aware, color-adaptive and randomized. Graph coloring produces color classes with each class representing an independent set of vertices. The disjoint maximal independent sets constitute a collection of disjoint dominating sets that offer good network coverage. Furthermore, if we relax the full domination constraint then we obtain a partitioning of the network into disjoint dominating and nearly-dominating sets of nearly equal size, providing better redundancy and a near-perfect node coverage yield. In addition, these independent sets can be the basis for clustering a very large sensor network with minimal overlap between the clusters leading to increased efficiency in routing, wireless transmission scheduling and data-aggregation. We also observe that in dense random deployments, certain coloring algorithms yield a packing of the nodes into independent sets each of which is relatively close to the perfect placement in the triangular lattice. Keywords: Wireless Sensor Networks, random geometric graphs, independent sets, dominating sets, domatic partition problem, graph coloring, triangular lattice.
1 Introduction and Related Work Connectivity and coverage are two fundamental quality of service objectives to optimize in Wireless Sensor Network (WSN) applications subject to the constraints that energy dissipation is reduced at the sensor level and the network lifetime is prolonged [9]. If we assume large-scale static Wireless Sensor Networks randomly deployed at a high density for applications like area-monitoring, then the area coverage can be approximated by the coverage of the sensor locations. In other words, owing to the large and dense sensor population, we can approximate the coverage of each point in the given area by covering each sensor location [4]. If we model the sensor network as a B. Liu et al. (Eds.): WASA 2009, LNCS 5682, pp. 32–42, 2009. © Springer-Verlag Berlin Heidelberg 2009
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geometric graph, then we can use strong concepts like coloring, independence, cover, and domination to solve the connected coverage problem. Extensive research has been carried out in the past years using the geometric graph model and the consensus has been to restructure or reorganize the network into a hierarchy so as to satisfy the connectivity and coverage objectives but also improve network capacity, reduce energy consumption and interference, and provide a robust, efficient infrastructure for routing and higher-level applications. The hierarchy restructuring can take two forms: build a collection of virtual backbones that are duty-cycled [4,9,16]; or cluster the network, where any node should have at least one cluster-head in its neighborhood [6,21]. The virtual backbone and cluster-heads should constitute a dominating set that covers all nodes in the network. Moreover, in the case of clustering, two cluster-heads should have minimum or no overlap, hence the independent set relevance. The partitioning of points into sets that are both dominating and independent is illustrated by considering the traditional cellular lattice. In this idealized case, large hexagonal regions of the triangular lattice are each partitioned into a fixed number k of smaller hexagons. The design effectively partitions the totality of small hexagons into k disjoint sets, where members of each disjoint set denote local regions which may broadcast with common frequencies sufficiently separated in distance not to interfere, but repetitively placed to broadcast to the whole geographic region. In the graph model, these k disjoint sets form a k-coloring of the implicit regular geometric graph with these sets being both independent and dominating, and all of the same size. On the other hand, constructing a virtual backbone can reduce to building a Minimum Dominating Set (MDS), itself approximable by building a Maximal Independent Set (MIS) [19]. The elements of the MDS play the role of coordinators for purposes of sensing coverage and/or network connectivity. A single MDS however puts heavy burden on its elements to sustain the network services. To maximize the lifetime of the network, the role of coordinators should be rotated among the nodes in the network, so that every node gets a chance to sleep and prolong its operational lifetime. Hence, the relevance of finding the maximum collection of disjoint dominating sets known as the Maximum Domatic Partition (MDP) problem [10], an NP-hard problem, as well as its variants: the k-domatic partition [17] and maximum disjoint set cover [4] problems. The MDP problem can address the area coverage; where it is known as the Maximum Lifetime Target Coverage problem [18], as well as the node coverage or clustering; where it’s called the Maximum Clustering Lifetime problem [16]. In both cases, the MDP problem has useful applications in ensuring a sustained quality of coverage in area monitoring applications, as well maintaining the network connectivity at all times for the benefit of efficient routing while prolonging the network lifetime. 1.1 Contributions Other than the application of one coloring algorithm in [5], we have not found, in the literature, any systematic investigation of various coloring algorithms to optimize the partitioning of nodes of a random geometric graph into disjoint sets that are both independent and dominating. Random geometric graphs (RGG) can closely represent the topological structure of sensor networks [9]. A random geometric graph, G(n,r), is formed by placing n points uniformly at random in the unit square and connecting two points if their Euclidian distance is at most r. Another widely used communication
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model when all nodes have the same transmission range r, is the Unit Disk Graph (UDG) where the vertices of the graph are embedded in the plane and two vertices are adjacent if their Euclidian distance is less or equal to 1 (r is normalized to one unit). RGGs and UDGs are related in the sense that a random geometric graph simply induces a probability distribution on the unit disk graph. Both models assume a regular disk communication model. In practice, however, the communication range is nonisotropic and irregular: the radio signal is disrupted by large scale fading effects like reflection, diffraction and scattering; small scale fading effects such as interference, heterogeneous sending powers, and transmission errors [19]. Therefore, the quasiUDG (qUDG) model was introduced in the literature to account for these irregularities [19]. In our work, we use the RGG model with the perfect disk range, and even though the disk model is oversimplified, we believe our results are still applicable in practice. For instance, when the communication range is irregular but has a lower bound, it can be regarded as a regular disk with radius equal to the lower bound [2]. With this approach, our results can provide conservative bounds on the properties we studied in RGGs using graph coloring. In this work, our goal is to show that a large subset V* ⊂ V of a random geometric graph, e.g. over 80% of the vertices, can be partitioned by an appropriate coloring algorithm into disjoint fully and (1-ε) dominating sets for ε quite small, e.g. (1-ε)>90%. For a random geometric graph in the unit square, there will be regions such as the four corners where the average degree is atypically low, and similarly there may be regions where the density of vertices is atypically high. If we can exclude a small portion of the vertices with atypically high or low degree, we are interested to find out if there is a subset of regularly distributed vertices V* ⊂ V containing most of V that can be covered by disjoint (1-ε) dominating sets for ε quite small, e.g. (1-ε)>0.9. In this paper, our contributions are two-fold. First, we present the results of an experimental investigation of the comparative performance of known and proposed coloring algorithms in random geometric graphs that are categorized as: topology-based, geometry-aware, randomized and color-adaptive. Graph coloring yields a partition into disjoint color classes of vertices where each color class forms an independent set of vertices. Our interest is not in minimizing the number of colors used but in evaluating the properties of the color classes produced. These properties are analyzed with particular regard for obtaining roughly equal sized independent sets that are also maximal independent, hence disjoint dominating. Certain coloring algorithms are shown to generate a collection of similar sized disjoint and verifiably dominating sets that have equally good performance in covering the network; hence these sets constitute good candidates for duty-cycled virtual backbones or minimal overlap clustering. Moreover, by relaxing the full domination constraint on the resulting disjoint independent sets, we are able to obtain a near-optimal node coverage yield with a much higher redundancy in the number of independent sets. For instance, using smallest-last (SL) coloring [15] on a random geometric graph G(1600, 0.12), Table 1 shows that 25.9% of the network nodes can be packed in 7 independent sets (color classes) of average size 59.3 each covering on average 99.97% of the entire network. Furthermore, if we consider the first 14 colors, then 50.97% of the nodes partition into 14 sets, where each color of the second 7 colors covers on average 99.85% of the network. Similarly, the first 21 colors incorporate 73.83% of the nodes, where each one of the last 7 colors covers on average 99.30% of the network. We argue that by using selected coloring algorithms on
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random geometric graphs that model wireless sensor networks, we can incrementally pack a core portion of the network (25, 50 and 75%) into dominating and nearlydominating independent sets of roughly equal size which offer a high node coverage yield. Our second contribution consists in observing that in very dense random geometric deployments of nodes, the packing of the nodes in the independent sets produced by our preferred colorings is nearly as regular and dense as the perfect packing in the triangular lattice. This property is somewhat remarkable, given the randomness of the deployment, and that some of the coloring algorithms are topology based without any reference to the geometrical positioning of the points. Table 1. Coverage results obtained with SL coloring algorithm (10 case average) Color groups
1-7 8-14 15-21 22-28 29-35 36-42
Cumulative number of vertices in color groups (%) 25.9 50.97 73.83 91.59 99.41 100.00
Average node coverage over 7 colors (%) 99.97 99.85 99.30 94.82 60.69 6.05
Average independent set size 59.3 57.21 52.25 40.58 17.85 1.35
Standard deviation on independent set size 0.92 1.54 2.22 5.85 8.03 1.61
1.2 Related Work To ensure connected coverage of a sensing field, two main techniques are used: the first is to structure the network into a regular pattern close to the perfect triangular lattice, and the second is to use sleep scheduling where multiple subsets of the nodes constitute virtual backbones and are activated one at a time to provide a connected coverage [9]. Adopting the first technique, the ACE algorithm [6] distributively produces clusters with regular separation close to the hexagonal lattice. Layered Diffusion based Coverage Control Protocol (LDCC) [20] applies triangular tessellation to cover the sensor field with the minimum number of active nodes. GS³ [21], is a location-aware scheme that produces an approximate hexagonal close packing of the sensors, and Optimal Geographical Density Control (OGDC) protocol [22] exploits nodes’ location coordinates to create a triangle tessellation of the nodes to be active at a given time. Using the second approach, Cardei et al. [5] propose an original centralized algorithm, based on sequential vertex coloring, which maximizes the number of disjoint dominating sets for the purpose of increasing the network lifetime, whereas, Moscibroda and Wattenhofer [16] and Thai et al. [18] provide randomized distributed algorithms which approximate the optimal solution of the maximum domatic problem within O(log n). The rest of the paper is organized as follows. Section 2 reviews the graph coloring algorithms employed and discusses the domatic partition problem. Section 3 provides bounds on the size of the maximal independent sets produced by the coloring algorithms in random geometric graphs. The experimental results are demonstrated in Section 4. Finally, Section 5 concludes the paper, and outlines our current and future work.
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2 Background 2.1 Vertex Domination and Independence in Graphs Dominating set: In a graph G = (V, E), a dominating set S ⊆ V of G is a subset of vertices such that every vertex of G is either in S or has at least one neighbor in S [19]. We define a (1-ε) dominating set S as having at most ε|V| vertices that are not adjacent to members of S, i.e. the (1-ε) dominating set S covers nearly all vertices of G as ε→0. Independent set: An independent set S ⊆ V of a graph G = (V, E) is a subset of vertices such that no two vertices ∀u , v ∈ S are adjacent in G. S is a maximal independent set (MIS) if no vertex can be added to it without violating its independence, and equivalently if any vertex not in S has a neighbor in S, i.e., if S forms a dominating set. Vertex coloring (independent set partition): Given an undirected graph G = (V, E), a vertex coloring of G is a mapping f:V→{1,…,k} such that f(x) ≠ f(y) if (x,y) ∈ E. The minimum number k to color G is called the chromatic number of G and denoted χ(G) [14]. Each set of vertices with the same color forms an independent set; hence a k coloring is a partition into k independent sets of the graph G. Domatic partition: A domatic partition is a partition of the vertices so that each part is a dominating set of the graph. The domatic number of a graph G is the maximum number of dominating sets in a domatic partition of G, or equivalently, the maximum number of disjoint dominating sets [8]. The domatic number of a graph G, D(G), is at most δ(G)+1, where δ is the minimum degree, since every vertex can be dominated by at most δ(G)+1 disjoint dominating sets. 2.2 Taxonomy of Vertex Coloring Algorithms A sequential coloring algorithm of a graph G is an algorithm operating in the following two stages: (a) Determine a vertex ordering K (for sequential coloring) [13] of the vertices of G, and (b) Determine a color selection strategy to color the vertices in the ordering K. a. Classification of vertex orderings The coloring sequence or vertex ordering is the arrangement of the vertex set V into a specific sequence K = (v1, v2,…, vn) that will next be colored according to a specific color selection strategy. Table 2 summarizes the coloring sequences that we studied. For further details on existing orderings, we invite the reader to consult the references cited. We also introduce geometry-aware vertex orderings which we briefly present below. Center First method (CF): The vertices are sorted in non-decreasing order according to their Euclidian distance from the center c (0.5, 0.5) of the unit square. Boundary First method (BF): The vertices are sorted in non-increasing order according to their distance from the center c of the unit square. This tends to build larger independent sets since we start from the boundary where vertices have lower degrees than in the interior of the unit square.
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Spiral Center First method (SCF): This ordering is built by spiraling from the center out in a clockwise continuous sweep. Top Down Sweep method (TDS): This ordering is built by sweeping the vertices in the unit square from left to right, top to bottom. Table 2. Classification of vertex orderings from the literature Coloring algorithm Identifier ordering Largest First (LF) [13] Smallest First (SF) Smallest Last (SL) [14,15] Smallest First Recursive (SFR) Random Sequence (RS) [13] Saturation LF (SLF) or DSATUR [13]
Class Labeling-based Topology-based Topology-based Topology-based Topology-based Random Dynamic or color-adaptive
Vertex ordering characteristic Lexicographic order on identifier Non-increasing order on degree Non-decreasing order on degree Recursive minimum degree last Recursive minimum degree first Uniform random order on vertex identifier Dynamic ordering subject to degree of saturation of the vertex to color
b. Color selection strategy Greedy-Color also known as First-Fit or Grundy function [13] is often used as the coloring strategy where to color a vertex v; we pick the smallest color not used by any adjacent vertex of v. With Greedy-Color the first color class is always a maximal independent set hence also a dominating set. Also, the vertices colored with the ith color dominate the induced subgraph determined by all vertices of color at least i.
3 Bounds on MIS Size in Random Geometric Graphs Considering the unit square and as n→∞, we can provide an upper bound on the average number of nodes that can be packed in a maximal independent set (MIS). We start from the fact that placing disks on the vertices of a triangular lattice (or equivalently, at the centers of regular hexagons in the dual lattice) is asymptotically optimal, in terms of the number of disks needed to achieve full coverage of a plane [4]. The nodes of a maximum size independent set are the centers of disks of radius r, where no two disks can mutually cover their respective centers (there is no edge between two disk centers). As n→∞, the coloring algorithm produces the pattern where three disk centers are close enough to form the vertices of an equilateral triangle, hence the relevance of the triangular lattice model. Based on the area of an equilateral triangle of side r, the unit square can pack on average 1 / r
2
3 triangles. Notice that a triangle 4
is incident to 3 vertices and a vertex is incident to 6 triangles, therefore we can obtain essentially 1 / r
2
3 vertices in the unit square. Moreover, we can use the proof pro2
vided by J. Diaz et al. in [7] on the independence number β n (r ) (size of the maximum independent set) in random geometric graphs. The authors prove that 2 βn (r ) ≤ (1 + (1/ r ) ) which provides us with another absolute upper bound on the size of
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the maximal independent sets obtained by coloring random geometric graphs. Similarly, a lower bound on the size of the maximal independent set is obtained with r ' = r 3 and the minimum size of the MIS which still covers all vertices is 1 /
r '2 3 . 2
4 Experimental Results and Analysis We experiment with a much broader choice of polynomial-time coloring algorithms than the related work in the literature [5]. We also consider much larger graph instances than [5] which allows us to bring far better properties, in terms of redundant coverage yield, of the maximal disjoint dominating and (1-ε) dominating sets in random geometric graphs. Our experiments consisted in running the collection of 11 coloring algorithms on a random geometric graph G(n,r) deployed in a unit square where n=1600 and r belongs to {0.06, 0.12. 0.18, 0.24, 0.3, 0.36}, which yields average node degrees in {17, 65, 138, 232, 342, 463}. We run each coloring algorithm on 10 instances of the same graph G(n,r). Given the random placement of the points in the plane, each graph instance is a different combinatorial instance of the same graph G(n,r). Considering the unit square and normalizing the nominal transmission range r to the interval [0,1], we define the network density μ(r) as the number of nodes per nominal transmission area: μ(r)=n.π.r², which also estimates the average node degree. Large scale and highly dense WSNs are prevalent in theory and practice. For instance, the ExScal project is the largest real life sensor network assembled to date [1] consisting of 10000 sensor devices placed in a tiered grid-like fashion on a 1.3km by 300m remote area in Florida. Although not randomly deployed, this initiative sets the ground for very large scale wireless sensor deployments. Moreover, the recent literature abounds with WSN experimental settings of uniform random deployments with high network sizes and densities. Table 3 shows a selection of references that we investigated. In practice, very dense networks suffer from interference and collision; therefore, as the network size grows, the nodes have to scale down their transmission range to keep the network density to a desirable level or a more involved MAC protocol becomes warranted to alleviate the increased channel contention. Our work applies to random geometric graphs in general, and to Wireless Sensor Networks in particular. Therefore, regardless of the application, we provide good experimental results on the asymptotic upper bounds of the properties that we have studied. Table 3. Network sizes and densities from the literature Paper Cardei et al. [3] Chan et al. [6] He et al. [11] Iyengar et al. [12] Wang et al. [20] Zhou et al. [23]
Deployment region 100x100 Unit square 160x160 1000x1000 Disk of radius 60 400x400
n 100 2500 300 200-10000 1200,1500 1500
r 50 50 100 15,17 80
Avg. degree 47.39 10,20,50,100 67.69 5,14,28,57,114,172,287 75,96 156
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4.1 Nearly-Equal Sized Independent Sets Figure 1 shows the cardinality of the color classes (independent sets size, 10 case average) for n=1600 and r=0.12. SL and DSATUR produce the minimal coloring and tend to have roughly equal sized lower color classes. We verified that those color classes are also dominating sets of the whole graph. SFR and BF tend to use many more colors but build a few larger independent sets and show a sharper decrease in the independent sets sizes. For the same network instance, Figure 2 plots the node coverage of the individual color classes obtained with multiple colorings. We observe that nearly half the color classes have node coverage of more than 90%. We also notice that the size of the largest color class is nearly double the size of the smallest color class that still offers more than 90% node coverage. This corroborates the benefit of graph coloring in yielding a substantial collection of disjoint dominating and (1ε) dominating sets with equal coverage that can be duty cycled. Notice that the points where SFR and BF intersect SL and DSATUR represent the number of disjoint dominating sets that we describe later.
Fig. 1. Evolution of the color classes’ sizes
Fig. 2. Node coverage of the color classes
4.2 Collection of Disjoint Dominating Sets Figure 3 shows the number of disjoint dominating sets (domatic number) obtained with 5 of the 11 coloring algorithms we experimented. For lack of space, we only present the most relevant results. Notice that DSATUR and SL offer similar performances. SFR and BF which consider low degree nodes first tend to build larger independent sets therefore they yield a much higher domatic partition size. Random which exploits no intelligence in the coloring produces the lowest results. In their seminal paper on domatic partitions [8], Feige et al. prove that every arbitrary graph with maximum degree Δ and minimum degree δ contains a domatic partition of size ((1-o(1))(δ-1))/ln Δ, where any o(1) term denotes a function of Δ alone that goes to zero as Δ increases. They turn this proof into a polynomial time, centralized algorithm that produces a domatic partition of size Ω(δ/lnΔ). In our experiments, notice that lowerb (i.e. lower bound) is the asymptotic bound of the domatic partition number obtained by the algorithm of [8]. mindeg+1 is the upper bound of the domatic number. We are basically obtaining a much higher domatic partition size than [8] which is
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an approximation algorithm for arbitrary graphs. However, our coloring algorithms are applied on random geometric graphs which can explain the advantages taken from the geometry to produce better results. 4.3 Node Packing in the Independent Sets We consider r=0.12 with several n values as depicted in Table 4. Table 4. Cardinality of the MIS obtained with different colorings n 400 800 1600 3200 6400 10000
Avg. degree 16 32 65 128 259 405
SF 52 57 62 63 64 65
SL 49 57 61 65 68 71
SFR 56 61 69 71 74 77
Random 46 50 54 54 56 57
BF 55 61 68 71 74 77
DSATUR 50 56 63 67 71 72
2 The triangular packing upper bound is 1/ (0.12) 3 = 80.18 .
2
The βn (r ) upper bound is (1+(1/0.12))² = 87.11. As n→∞, SL and DSATUR which
both are topology-based efficient coloring algorithms are within 89% and 82% of both bounds, which is quite good. Our best results are given by BF which is geometry-aware and SFR which recursively colors smaller degrees first. They are within 96% and 88% of both bounds. Figure 4 shows a sample layout of the vertices in a MIS obtained with SL on G(1600, 0.12). Notice that for the display, we increase r by 40% to join the vertices of the MIS so we can observe the close to triangular lattice packing of the nodes. Furthermore, by increasing the transmission radius r only between the independent set vertices, we transform the independent set into a nearly-regular connected dominating set with full connected coverage.
Fig. 3. Evolution of the domatic partition number
Fig. 4. MIS instance, size=75, r=0.12
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5 Conclusion and Future Work In this paper, we have given a comparative study of the performance of several graph coloring algorithms in addressing the domatic partition problem in random geometric graphs. We experimentally show good results on the node coverage yield of the disjoint fully dominating and (1-ε) dominating independent sets produced by the coloring algorithms. Owing to the large scale graphs and broad choice of coloring algorithms that we considered, we uncovered better properties on the graph models employed than in the literature [5] and revealed the quasi-triangular lattice of the placement of points in the maximal independent sets produced. Our future work is to study distributed graph coloring algorithms to address the domatic partition problem. Performance in a distributed setting is certainly not as good as in a centralized environment where the global topology of the network is available for study. However, Wireless Sensor Networks are often randomly deployed; therefore distributed and localized solutions have been the main focus of research in the recent years, since they offer better scalability and robustness. In fact, the work we presented in this paper constitutes a basis for comparison and evaluation of localized solutions and an upper bound on the cardinality of the domatic partition and packing of the nodes into maximal independent sets. These two properties are very useful in establishing redundant virtual backbones and regular clustering patterns for a better routing. We also intend to exploit the observed properties to design a holistic and robust routing approach that self-configures and adapts to the state of the environment in a hostile setting.
Acknowledgements We thank both Dr. Mihaela Iridon and Dr. Saeed Abu-Nimeh for several valuable discussions and Ilteris Murat Derici for his initial work on the SL algorithm and RGG.
References [1] Arora, A., et al.: ExScal: Elements of an Extreme Scale Wireless Sensor Network. In: Proc. of 11th IEEE RTCSA 2005 (2005) [2] Bai, X., et al.: Deploying Wireless Sensors to Achieve Both Coverage and Connectivity. In: Proc. of 7th ACM MOBIHOC 2006, pp. 131–142 (2006) [3] Cardei, M., et al.: Connected Domination in Multihop Ad Hoc Wireless Networks. In: Proc. of 6th Int. Conf. on Computer Science and Informatics (CS&I 2002), pp. 251–255 (2002) [4] Cardei, M., Wu, J.: Energy-efficient coverage problems in wireless ad-hoc sensor networks. J. of Computer Communications 29(4), 413–420 (2006) [5] Cardei, M., et al.: Wireless Sensor Networks with Energy Efficient Organization. J. of Interconnection Networks 3(3-4), 213–229 (2002) [6] Chan, H., Perrig, A.: ACE: An Emergent Algorithm for Highly Uniform Cluster Formation. In: Karl, H., Wolisz, A., Willig, A. (eds.) EWSN 2004. LNCS, vol. 2920, pp. 154– 171. Springer, Heidelberg (2004) [7] Díaz, J., Petit, J., Serna, M.: Random geometric problems on [0,1]2. In: Rolim, J.D.P., Serna, M., Luby, M. (eds.) RANDOM 1998. LNCS, vol. 1518, pp. 294–306. Springer, Heidelberg (1998)
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[8] Feige, U., Halldórsson, M.M., Kortsarz, G., Srinivasan, A.: Approximating the domatic number. SIAM J. of Computing 32(1), 172–195 (2003) [9] Ghosh, A., Das, S.K.: Coverage and connectivity issues in wireless sensor networks: A survey. J. of Pervasive and Mobile Computing 4(3), 303–334 (2008) [10] Ha, R.W., et al.: Sleep scheduling for wireless sensor networks via network flow model. J. of Computer Communications 29(13-14), 2469–2481 (2006) [11] He, Y., et al.: Energy Efficient Connectivity Maintenance in Wireless Sensor Networks. In: Proc. of International Conference on Intelligent Computing (ICIC 2006), pp. 95–105 (2006) [12] Iyengar, R.: Low-coordination Topologies For Redundancy In Sensor Networks. In: Proc. of 6th ACM MOBIHOC 2005, pp. 332–342 (2005) [13] Kubale, M.: Graph Colorings. American Mathematical Society (2004) [14] Matula, D., Marble, G., Isaacson, J.D.: Graph Coloring Algorithms. In: Read, R. (ed.) Graph Theory and Computing, pp. 109–122. Academic Press, New York (1972) [15] Matula, D., Beck, L.L.: Smallest-last ordering and clustering and graph coloring algorithms. J. of the ACM 30(3), 417–427 (1983) [16] Moscibroda, T., Wattenhofer, R.: Maximizing the Lifetime of Dominating Sets. In: Proc. of 5th IEEE WMAN 2005 (2005) [17] Pemmaraju, S.V., Pirwani, I.A.: Energy Conservation via Domatic Partitions. In: Proc. of 7th ACM MOBIHOC 2006, pp. 143–154 (2006) [18] Thai, M.T., et al.: O(log n)-Localized Algorithms on the Coverage Problem in Heterogeneous Sensor Networks. In: Proc. of 26th IEEE IPCCC 2007, pp. 85–92 (2007) [19] Wagner, D., Wattenhofer, R.: Algorithms for Sensor and Ad Hoc Networks, Advanced Lectures. Springer, Heidelberg (2007) [20] Wang, B., Fu, C., Lim, H.B.: Layered Diffusion based Coverage Control in Wireless Sensor Networks. In: Proc. Of 32nd IEEE Conference on Local Computer Networks, pp. 504–511 (2007) [21] Zhang, H., Arora, A.: GS3: Scalable Self-configuration and Self-healing in Wireless Sensor Networks. J. of Computer Networks 43(4), 459–480 (2003) [22] Zhang, H., Hou, J.C.: Maintaining Sensing Coverage and Connectivity in Large Sensor Networks. J. of Ad Hoc and Sensor Wireless Networks 1(1-2), 89–124 (2005) [23] Zhou, Y., et al.: A Point-Distribution Index and Its Application to Sensor-Grouping in Wireless Sensor Networks. In: Proc. of IWCMC 2006, pp. 1171–1176 (2006)
Abstract. The domatic partition problem seeks to maximize the partitioning of the nodes of the network into disjoint dominating sets. These sets represent a series of virtual backbones for wireless sensor networks to be activated successively, resulting in more balanced energy consumption and increased network robustness. In this study, we address the domatic partition problem in random geometric graphs by investigating several vertex coloring algorithms both topology and geometry-aware, color-adaptive and randomized. Graph coloring produces color classes with each class representing an independent set of vertices. The disjoint maximal independent sets constitute a collection of disjoint dominating sets that offer good network coverage. Furthermore, if we relax the full domination constraint then we obtain a partitioning of the network into disjoint dominating and nearly-dominating sets of nearly equal size, providing better redundancy and a near-perfect node coverage yield. In addition, these independent sets can be the basis for clustering a very large sensor network with minimal overlap between the clusters leading to increased efficiency in routing, wireless transmission scheduling and data-aggregation. We also observe that in dense random deployments, certain coloring algorithms yield a packing of the nodes into independent sets each of which is relatively close to the perfect placement in the triangular lattice. Keywords: Wireless Sensor Networks, random geometric graphs, independent sets, dominating sets, domatic partition problem, graph coloring, triangular lattice.
1 Introduction and Related Work Connectivity and coverage are two fundamental quality of service objectives to optimize in Wireless Sensor Network (WSN) applications subject to the constraints that energy dissipation is reduced at the sensor level and the network lifetime is prolonged [9]. If we assume large-scale static Wireless Sensor Networks randomly deployed at a high density for applications like area-monitoring, then the area coverage can be approximated by the coverage of the sensor locations. In other words, owing to the large and dense sensor population, we can approximate the coverage of each point in the given area by covering each sensor location [4]. If we model the sensor network as a B. Liu et al. (Eds.): WASA 2009, LNCS 5682, pp. 32–42, 2009. © Springer-Verlag Berlin Heidelberg 2009
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geometric graph, then we can use strong concepts like coloring, independence, cover, and domination to solve the connected coverage problem. Extensive research has been carried out in the past years using the geometric graph model and the consensus has been to restructure or reorganize the network into a hierarchy so as to satisfy the connectivity and coverage objectives but also improve network capacity, reduce energy consumption and interference, and provide a robust, efficient infrastructure for routing and higher-level applications. The hierarchy restructuring can take two forms: build a collection of virtual backbones that are duty-cycled [4,9,16]; or cluster the network, where any node should have at least one cluster-head in its neighborhood [6,21]. The virtual backbone and cluster-heads should constitute a dominating set that covers all nodes in the network. Moreover, in the case of clustering, two cluster-heads should have minimum or no overlap, hence the independent set relevance. The partitioning of points into sets that are both dominating and independent is illustrated by considering the traditional cellular lattice. In this idealized case, large hexagonal regions of the triangular lattice are each partitioned into a fixed number k of smaller hexagons. The design effectively partitions the totality of small hexagons into k disjoint sets, where members of each disjoint set denote local regions which may broadcast with common frequencies sufficiently separated in distance not to interfere, but repetitively placed to broadcast to the whole geographic region. In the graph model, these k disjoint sets form a k-coloring of the implicit regular geometric graph with these sets being both independent and dominating, and all of the same size. On the other hand, constructing a virtual backbone can reduce to building a Minimum Dominating Set (MDS), itself approximable by building a Maximal Independent Set (MIS) [19]. The elements of the MDS play the role of coordinators for purposes of sensing coverage and/or network connectivity. A single MDS however puts heavy burden on its elements to sustain the network services. To maximize the lifetime of the network, the role of coordinators should be rotated among the nodes in the network, so that every node gets a chance to sleep and prolong its operational lifetime. Hence, the relevance of finding the maximum collection of disjoint dominating sets known as the Maximum Domatic Partition (MDP) problem [10], an NP-hard problem, as well as its variants: the k-domatic partition [17] and maximum disjoint set cover [4] problems. The MDP problem can address the area coverage; where it is known as the Maximum Lifetime Target Coverage problem [18], as well as the node coverage or clustering; where it’s called the Maximum Clustering Lifetime problem [16]. In both cases, the MDP problem has useful applications in ensuring a sustained quality of coverage in area monitoring applications, as well maintaining the network connectivity at all times for the benefit of efficient routing while prolonging the network lifetime. 1.1 Contributions Other than the application of one coloring algorithm in [5], we have not found, in the literature, any systematic investigation of various coloring algorithms to optimize the partitioning of nodes of a random geometric graph into disjoint sets that are both independent and dominating. Random geometric graphs (RGG) can closely represent the topological structure of sensor networks [9]. A random geometric graph, G(n,r), is formed by placing n points uniformly at random in the unit square and connecting two points if their Euclidian distance is at most r. Another widely used communication
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model when all nodes have the same transmission range r, is the Unit Disk Graph (UDG) where the vertices of the graph are embedded in the plane and two vertices are adjacent if their Euclidian distance is less or equal to 1 (r is normalized to one unit). RGGs and UDGs are related in the sense that a random geometric graph simply induces a probability distribution on the unit disk graph. Both models assume a regular disk communication model. In practice, however, the communication range is nonisotropic and irregular: the radio signal is disrupted by large scale fading effects like reflection, diffraction and scattering; small scale fading effects such as interference, heterogeneous sending powers, and transmission errors [19]. Therefore, the quasiUDG (qUDG) model was introduced in the literature to account for these irregularities [19]. In our work, we use the RGG model with the perfect disk range, and even though the disk model is oversimplified, we believe our results are still applicable in practice. For instance, when the communication range is irregular but has a lower bound, it can be regarded as a regular disk with radius equal to the lower bound [2]. With this approach, our results can provide conservative bounds on the properties we studied in RGGs using graph coloring. In this work, our goal is to show that a large subset V* ⊂ V of a random geometric graph, e.g. over 80% of the vertices, can be partitioned by an appropriate coloring algorithm into disjoint fully and (1-ε) dominating sets for ε quite small, e.g. (1-ε)>90%. For a random geometric graph in the unit square, there will be regions such as the four corners where the average degree is atypically low, and similarly there may be regions where the density of vertices is atypically high. If we can exclude a small portion of the vertices with atypically high or low degree, we are interested to find out if there is a subset of regularly distributed vertices V* ⊂ V containing most of V that can be covered by disjoint (1-ε) dominating sets for ε quite small, e.g. (1-ε)>0.9. In this paper, our contributions are two-fold. First, we present the results of an experimental investigation of the comparative performance of known and proposed coloring algorithms in random geometric graphs that are categorized as: topology-based, geometry-aware, randomized and color-adaptive. Graph coloring yields a partition into disjoint color classes of vertices where each color class forms an independent set of vertices. Our interest is not in minimizing the number of colors used but in evaluating the properties of the color classes produced. These properties are analyzed with particular regard for obtaining roughly equal sized independent sets that are also maximal independent, hence disjoint dominating. Certain coloring algorithms are shown to generate a collection of similar sized disjoint and verifiably dominating sets that have equally good performance in covering the network; hence these sets constitute good candidates for duty-cycled virtual backbones or minimal overlap clustering. Moreover, by relaxing the full domination constraint on the resulting disjoint independent sets, we are able to obtain a near-optimal node coverage yield with a much higher redundancy in the number of independent sets. For instance, using smallest-last (SL) coloring [15] on a random geometric graph G(1600, 0.12), Table 1 shows that 25.9% of the network nodes can be packed in 7 independent sets (color classes) of average size 59.3 each covering on average 99.97% of the entire network. Furthermore, if we consider the first 14 colors, then 50.97% of the nodes partition into 14 sets, where each color of the second 7 colors covers on average 99.85% of the network. Similarly, the first 21 colors incorporate 73.83% of the nodes, where each one of the last 7 colors covers on average 99.30% of the network. We argue that by using selected coloring algorithms on
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random geometric graphs that model wireless sensor networks, we can incrementally pack a core portion of the network (25, 50 and 75%) into dominating and nearlydominating independent sets of roughly equal size which offer a high node coverage yield. Our second contribution consists in observing that in very dense random geometric deployments of nodes, the packing of the nodes in the independent sets produced by our preferred colorings is nearly as regular and dense as the perfect packing in the triangular lattice. This property is somewhat remarkable, given the randomness of the deployment, and that some of the coloring algorithms are topology based without any reference to the geometrical positioning of the points. Table 1. Coverage results obtained with SL coloring algorithm (10 case average) Color groups
1-7 8-14 15-21 22-28 29-35 36-42
Cumulative number of vertices in color groups (%) 25.9 50.97 73.83 91.59 99.41 100.00
Average node coverage over 7 colors (%) 99.97 99.85 99.30 94.82 60.69 6.05
Average independent set size 59.3 57.21 52.25 40.58 17.85 1.35
Standard deviation on independent set size 0.92 1.54 2.22 5.85 8.03 1.61
1.2 Related Work To ensure connected coverage of a sensing field, two main techniques are used: the first is to structure the network into a regular pattern close to the perfect triangular lattice, and the second is to use sleep scheduling where multiple subsets of the nodes constitute virtual backbones and are activated one at a time to provide a connected coverage [9]. Adopting the first technique, the ACE algorithm [6] distributively produces clusters with regular separation close to the hexagonal lattice. Layered Diffusion based Coverage Control Protocol (LDCC) [20] applies triangular tessellation to cover the sensor field with the minimum number of active nodes. GS³ [21], is a location-aware scheme that produces an approximate hexagonal close packing of the sensors, and Optimal Geographical Density Control (OGDC) protocol [22] exploits nodes’ location coordinates to create a triangle tessellation of the nodes to be active at a given time. Using the second approach, Cardei et al. [5] propose an original centralized algorithm, based on sequential vertex coloring, which maximizes the number of disjoint dominating sets for the purpose of increasing the network lifetime, whereas, Moscibroda and Wattenhofer [16] and Thai et al. [18] provide randomized distributed algorithms which approximate the optimal solution of the maximum domatic problem within O(log n). The rest of the paper is organized as follows. Section 2 reviews the graph coloring algorithms employed and discusses the domatic partition problem. Section 3 provides bounds on the size of the maximal independent sets produced by the coloring algorithms in random geometric graphs. The experimental results are demonstrated in Section 4. Finally, Section 5 concludes the paper, and outlines our current and future work.
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2 Background 2.1 Vertex Domination and Independence in Graphs Dominating set: In a graph G = (V, E), a dominating set S ⊆ V of G is a subset of vertices such that every vertex of G is either in S or has at least one neighbor in S [19]. We define a (1-ε) dominating set S as having at most ε|V| vertices that are not adjacent to members of S, i.e. the (1-ε) dominating set S covers nearly all vertices of G as ε→0. Independent set: An independent set S ⊆ V of a graph G = (V, E) is a subset of vertices such that no two vertices ∀u , v ∈ S are adjacent in G. S is a maximal independent set (MIS) if no vertex can be added to it without violating its independence, and equivalently if any vertex not in S has a neighbor in S, i.e., if S forms a dominating set. Vertex coloring (independent set partition): Given an undirected graph G = (V, E), a vertex coloring of G is a mapping f:V→{1,…,k} such that f(x) ≠ f(y) if (x,y) ∈ E. The minimum number k to color G is called the chromatic number of G and denoted χ(G) [14]. Each set of vertices with the same color forms an independent set; hence a k coloring is a partition into k independent sets of the graph G. Domatic partition: A domatic partition is a partition of the vertices so that each part is a dominating set of the graph. The domatic number of a graph G is the maximum number of dominating sets in a domatic partition of G, or equivalently, the maximum number of disjoint dominating sets [8]. The domatic number of a graph G, D(G), is at most δ(G)+1, where δ is the minimum degree, since every vertex can be dominated by at most δ(G)+1 disjoint dominating sets. 2.2 Taxonomy of Vertex Coloring Algorithms A sequential coloring algorithm of a graph G is an algorithm operating in the following two stages: (a) Determine a vertex ordering K (for sequential coloring) [13] of the vertices of G, and (b) Determine a color selection strategy to color the vertices in the ordering K. a. Classification of vertex orderings The coloring sequence or vertex ordering is the arrangement of the vertex set V into a specific sequence K = (v1, v2,…, vn) that will next be colored according to a specific color selection strategy. Table 2 summarizes the coloring sequences that we studied. For further details on existing orderings, we invite the reader to consult the references cited. We also introduce geometry-aware vertex orderings which we briefly present below. Center First method (CF): The vertices are sorted in non-decreasing order according to their Euclidian distance from the center c (0.5, 0.5) of the unit square. Boundary First method (BF): The vertices are sorted in non-increasing order according to their distance from the center c of the unit square. This tends to build larger independent sets since we start from the boundary where vertices have lower degrees than in the interior of the unit square.
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Spiral Center First method (SCF): This ordering is built by spiraling from the center out in a clockwise continuous sweep. Top Down Sweep method (TDS): This ordering is built by sweeping the vertices in the unit square from left to right, top to bottom. Table 2. Classification of vertex orderings from the literature Coloring algorithm Identifier ordering Largest First (LF) [13] Smallest First (SF) Smallest Last (SL) [14,15] Smallest First Recursive (SFR) Random Sequence (RS) [13] Saturation LF (SLF) or DSATUR [13]
Class Labeling-based Topology-based Topology-based Topology-based Topology-based Random Dynamic or color-adaptive
Vertex ordering characteristic Lexicographic order on identifier Non-increasing order on degree Non-decreasing order on degree Recursive minimum degree last Recursive minimum degree first Uniform random order on vertex identifier Dynamic ordering subject to degree of saturation of the vertex to color
b. Color selection strategy Greedy-Color also known as First-Fit or Grundy function [13] is often used as the coloring strategy where to color a vertex v; we pick the smallest color not used by any adjacent vertex of v. With Greedy-Color the first color class is always a maximal independent set hence also a dominating set. Also, the vertices colored with the ith color dominate the induced subgraph determined by all vertices of color at least i.
3 Bounds on MIS Size in Random Geometric Graphs Considering the unit square and as n→∞, we can provide an upper bound on the average number of nodes that can be packed in a maximal independent set (MIS). We start from the fact that placing disks on the vertices of a triangular lattice (or equivalently, at the centers of regular hexagons in the dual lattice) is asymptotically optimal, in terms of the number of disks needed to achieve full coverage of a plane [4]. The nodes of a maximum size independent set are the centers of disks of radius r, where no two disks can mutually cover their respective centers (there is no edge between two disk centers). As n→∞, the coloring algorithm produces the pattern where three disk centers are close enough to form the vertices of an equilateral triangle, hence the relevance of the triangular lattice model. Based on the area of an equilateral triangle of side r, the unit square can pack on average 1 / r
2
3 triangles. Notice that a triangle 4
is incident to 3 vertices and a vertex is incident to 6 triangles, therefore we can obtain essentially 1 / r
2
3 vertices in the unit square. Moreover, we can use the proof pro2
vided by J. Diaz et al. in [7] on the independence number β n (r ) (size of the maximum independent set) in random geometric graphs. The authors prove that 2 βn (r ) ≤ (1 + (1/ r ) ) which provides us with another absolute upper bound on the size of
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the maximal independent sets obtained by coloring random geometric graphs. Similarly, a lower bound on the size of the maximal independent set is obtained with r ' = r 3 and the minimum size of the MIS which still covers all vertices is 1 /
r '2 3 . 2
4 Experimental Results and Analysis We experiment with a much broader choice of polynomial-time coloring algorithms than the related work in the literature [5]. We also consider much larger graph instances than [5] which allows us to bring far better properties, in terms of redundant coverage yield, of the maximal disjoint dominating and (1-ε) dominating sets in random geometric graphs. Our experiments consisted in running the collection of 11 coloring algorithms on a random geometric graph G(n,r) deployed in a unit square where n=1600 and r belongs to {0.06, 0.12. 0.18, 0.24, 0.3, 0.36}, which yields average node degrees in {17, 65, 138, 232, 342, 463}. We run each coloring algorithm on 10 instances of the same graph G(n,r). Given the random placement of the points in the plane, each graph instance is a different combinatorial instance of the same graph G(n,r). Considering the unit square and normalizing the nominal transmission range r to the interval [0,1], we define the network density μ(r) as the number of nodes per nominal transmission area: μ(r)=n.π.r², which also estimates the average node degree. Large scale and highly dense WSNs are prevalent in theory and practice. For instance, the ExScal project is the largest real life sensor network assembled to date [1] consisting of 10000 sensor devices placed in a tiered grid-like fashion on a 1.3km by 300m remote area in Florida. Although not randomly deployed, this initiative sets the ground for very large scale wireless sensor deployments. Moreover, the recent literature abounds with WSN experimental settings of uniform random deployments with high network sizes and densities. Table 3 shows a selection of references that we investigated. In practice, very dense networks suffer from interference and collision; therefore, as the network size grows, the nodes have to scale down their transmission range to keep the network density to a desirable level or a more involved MAC protocol becomes warranted to alleviate the increased channel contention. Our work applies to random geometric graphs in general, and to Wireless Sensor Networks in particular. Therefore, regardless of the application, we provide good experimental results on the asymptotic upper bounds of the properties that we have studied. Table 3. Network sizes and densities from the literature Paper Cardei et al. [3] Chan et al. [6] He et al. [11] Iyengar et al. [12] Wang et al. [20] Zhou et al. [23]
Deployment region 100x100 Unit square 160x160 1000x1000 Disk of radius 60 400x400
n 100 2500 300 200-10000 1200,1500 1500
r 50 50 100 15,17 80
Avg. degree 47.39 10,20,50,100 67.69 5,14,28,57,114,172,287 75,96 156
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4.1 Nearly-Equal Sized Independent Sets Figure 1 shows the cardinality of the color classes (independent sets size, 10 case average) for n=1600 and r=0.12. SL and DSATUR produce the minimal coloring and tend to have roughly equal sized lower color classes. We verified that those color classes are also dominating sets of the whole graph. SFR and BF tend to use many more colors but build a few larger independent sets and show a sharper decrease in the independent sets sizes. For the same network instance, Figure 2 plots the node coverage of the individual color classes obtained with multiple colorings. We observe that nearly half the color classes have node coverage of more than 90%. We also notice that the size of the largest color class is nearly double the size of the smallest color class that still offers more than 90% node coverage. This corroborates the benefit of graph coloring in yielding a substantial collection of disjoint dominating and (1ε) dominating sets with equal coverage that can be duty cycled. Notice that the points where SFR and BF intersect SL and DSATUR represent the number of disjoint dominating sets that we describe later.
Fig. 1. Evolution of the color classes’ sizes
Fig. 2. Node coverage of the color classes
4.2 Collection of Disjoint Dominating Sets Figure 3 shows the number of disjoint dominating sets (domatic number) obtained with 5 of the 11 coloring algorithms we experimented. For lack of space, we only present the most relevant results. Notice that DSATUR and SL offer similar performances. SFR and BF which consider low degree nodes first tend to build larger independent sets therefore they yield a much higher domatic partition size. Random which exploits no intelligence in the coloring produces the lowest results. In their seminal paper on domatic partitions [8], Feige et al. prove that every arbitrary graph with maximum degree Δ and minimum degree δ contains a domatic partition of size ((1-o(1))(δ-1))/ln Δ, where any o(1) term denotes a function of Δ alone that goes to zero as Δ increases. They turn this proof into a polynomial time, centralized algorithm that produces a domatic partition of size Ω(δ/lnΔ). In our experiments, notice that lowerb (i.e. lower bound) is the asymptotic bound of the domatic partition number obtained by the algorithm of [8]. mindeg+1 is the upper bound of the domatic number. We are basically obtaining a much higher domatic partition size than [8] which is
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an approximation algorithm for arbitrary graphs. However, our coloring algorithms are applied on random geometric graphs which can explain the advantages taken from the geometry to produce better results. 4.3 Node Packing in the Independent Sets We consider r=0.12 with several n values as depicted in Table 4. Table 4. Cardinality of the MIS obtained with different colorings n 400 800 1600 3200 6400 10000
Avg. degree 16 32 65 128 259 405
SF 52 57 62 63 64 65
SL 49 57 61 65 68 71
SFR 56 61 69 71 74 77
Random 46 50 54 54 56 57
BF 55 61 68 71 74 77
DSATUR 50 56 63 67 71 72
2 The triangular packing upper bound is 1/ (0.12) 3 = 80.18 .
2
The βn (r ) upper bound is (1+(1/0.12))² = 87.11. As n→∞, SL and DSATUR which
both are topology-based efficient coloring algorithms are within 89% and 82% of both bounds, which is quite good. Our best results are given by BF which is geometry-aware and SFR which recursively colors smaller degrees first. They are within 96% and 88% of both bounds. Figure 4 shows a sample layout of the vertices in a MIS obtained with SL on G(1600, 0.12). Notice that for the display, we increase r by 40% to join the vertices of the MIS so we can observe the close to triangular lattice packing of the nodes. Furthermore, by increasing the transmission radius r only between the independent set vertices, we transform the independent set into a nearly-regular connected dominating set with full connected coverage.
Fig. 3. Evolution of the domatic partition number
Fig. 4. MIS instance, size=75, r=0.12
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5 Conclusion and Future Work In this paper, we have given a comparative study of the performance of several graph coloring algorithms in addressing the domatic partition problem in random geometric graphs. We experimentally show good results on the node coverage yield of the disjoint fully dominating and (1-ε) dominating independent sets produced by the coloring algorithms. Owing to the large scale graphs and broad choice of coloring algorithms that we considered, we uncovered better properties on the graph models employed than in the literature [5] and revealed the quasi-triangular lattice of the placement of points in the maximal independent sets produced. Our future work is to study distributed graph coloring algorithms to address the domatic partition problem. Performance in a distributed setting is certainly not as good as in a centralized environment where the global topology of the network is available for study. However, Wireless Sensor Networks are often randomly deployed; therefore distributed and localized solutions have been the main focus of research in the recent years, since they offer better scalability and robustness. In fact, the work we presented in this paper constitutes a basis for comparison and evaluation of localized solutions and an upper bound on the cardinality of the domatic partition and packing of the nodes into maximal independent sets. These two properties are very useful in establishing redundant virtual backbones and regular clustering patterns for a better routing. We also intend to exploit the observed properties to design a holistic and robust routing approach that self-configures and adapts to the state of the environment in a hostile setting.
Acknowledgements We thank both Dr. Mihaela Iridon and Dr. Saeed Abu-Nimeh for several valuable discussions and Ilteris Murat Derici for his initial work on the SL algorithm and RGG.
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