Abstract: It is a conjecture that in any unit disk graph $$G$$G, $$\alpha (G) \le 3 \cdot \gamma _c(G) + 3$$?(G)≤3·?c(G)+3 where $$\alpha (G)$$?(G) is the size of the maximum independent set in $$G$$G and $$\gamma _c(G)$$?c(G) is the size of minimum connected dominating set in $$G$$G. In this paper, we show that in any unit disk graph $$G$$G, $$\alpha (G) \le 3.399 \cdot \gamma _c(G) + 4.874$$?(G)≤3.399·?c(G)+4.874. Currently, this is the best-known bound.

Abstract: Over years, virtual backbone has attracted lots of attentions as a promising approach to deal with the broadcasting storm problem in wireless networks. One popular way to construct a quality virtual backbone is to solve the minimum connected dominating set problem. However, a virtual backbone computed in this way is not resilient against topology change since the induced graph by the connected dominating set is one-vertex-connected. As a result, the minimum k-connected m-dominating set problem is introduced to construct a fault-tolerant virtual backbone. Currently, the best known approximation algorithm for the problem in unit disk graph assumes k ≤ 3 and m ≥ 1 and its performance ratio is 280 when k = m = 3. In this paper, we use a classical result from graph theory, Tutte decomposition, to design a new approximation algorithm for the problem in unit disk graph for k ≤ 3 and m ≥ 3. In particular, the algorithm features with much simpler structure and much smaller performance ratio, e.g. nearly 66 when k = m = 3. We also conduct simulation to evaluate the performance of our algorithm.

Abstract: In this paper, we study the Antenna Orientation (AO) problem concerning symmetric connectivity in Directional Wireless Sensor Networks. We are given a set of nodes each of which is equipped with one directional antenna with beam-width θ = 2π/3 and is initially assigned a transmission range 1 that yields a connected unit disk graph spanning all nodes. The objective of the problem is to compute an orientation of the antennas and to find a minimum transmission power range r = O(1) such that the induced symmetric communication graph is connected. We propose an algorithm that orients the antennas to yield a symmetric connected graph where the transmission power range is bounded by 6 which is currently the best result for this problem. We also study the performance of our algorithm through simulation.

Abstract: A new energy efficient optimal Connected Dominating Set (CDS) algorithm with activity scheduling for mobile ad hoc networks (MANETs) is proposed This algorithm achieves energy efficiency by minimizing the Broadcast Storm Problem [BSP] and at the same time considering the node's remaining energy The Connected Dominating Set is widely used as a virtual backbone or spine in mobile ad hoc networks [MANETs] or Wireless Sensor Networks [WSN] The CDS of a graph representing a network has a significant impact on an efficient design of routing protocol in wireless networks Here the CDS is a distributed algorithm with activity scheduling based on unit disk graph [UDG] The node's mobility and residual energy (RE) are considered as parameters in the construction of stable optimal energy efficient CDS The performance is evaluated at various node densities, various transmission ranges, and mobility rates The theoretical analysis and simulation results of this algorithm are also presented which yield better results

Abstract: SUMMARY This paper considers the problem of enumerating all maximal cliques in unit disk graphs , which is a plausible setting for applications of finding similar data groups. Our primary interest is to develop a faster algorithm using the geometric structure about the metric space where the input unit disk graph is embedded. Assuming that the distance between any two vertices is available, we propose a new algorithm based on two well-known algorithms called Bron-Kerbosch and Tomita-Tanaka-Takahashi. The key idea of our algorithm is to find a good pivot quickly using geometric proximity. We validate the practical impact of our algo-rithm via experimental evaluations.

Abstract: Trilateration is an effective way to localize a sensor network based on relative distance measures, but the conditions that guarantee the existence of a solution are quite restrictive. If the network topology is a unit disk graph, however, the localization of the network can be achieved also when the standard trilateration fails, using a priori information about “not being connected”. Such an information can be modeled as additional links, namely shadow edges, that can be used to localize also networks that are not localizable via trilateration. In this paper we inspect the applicability of shadow edge localization in the noisy setting, showing some conditions that guarantee the existence of solution and comparing the results of trilateration and shadow edge localization algorithms in a noisy setting, with respect to the error after a post processing done by means of a recursive least square algorithm. The results show that, besides localizing more nodes, the shadow edge approach has better results in terms of localization error.

Abstract: We present a distributed and fully asynchronous algorithm for construction of the partial Delaunay triangulation over quasi unit disk graphs. Provided that the ratio of the maximum to the minimum communication range of nodes is bounded from above by square root of two, our algorithm outputs a connected and planar overlay graph of the input graph, which enables the use of localized geographic routing algorithms that guarantee message delivery. Moreover, under the assumption that the input graph is civilized (i.e., any two network nodes have non-zero minimum Euclidean distance), we show that our algorithm is localized. We show by means of simulation that our approach yields output graphs whose Euclidean spanning ratio is on average significantly smaller compared to those constructed by all other asynchronous approaches.

Abstract: This paper considers a variant of the connected dominating set (CDS) problem in a unit disk graph G = (V, E). The considered problem consists in minimizing the number of CDS vertices that belong to a subset V' ⊆V. As far as we know, this problem has not been treated in the literature. Nevertheless, its resolution would be useful in many communication network applications, such as the selection of relay nodes in heterogenous wireless ad hoc networks where only a subset of powerful nodes (e.g., energy or memory rich nodes) may form the network backbone act as relays, or where it is preferable to select relays from these nodes and minimize the number of non-powerful nodes that act as relays. Replacement of non-powerful nodes might be necessary either at the initialization (after deployment), or during the network lifetime, which justifies the need to minimize their number. The problem is first modeled and reduced to the minimum weighted connected dominating set (WCDS) problem in a vertex weighted graph, and then it is resolved by taking advantage of the simple form of the weight function using integer linear programming (ILP). A heuristic is also proposed for large scale resolution. Simulation results confirms closeness of the proposed heuristic to the optimal solution obtained by the ILP, and scalability of the heuristic.

Abstract: A storage unit stores information on a graph corresponding to a structure data set that represents a three-dimensional structure with a plurality of polygons. An operation unit obtains the information on the graph from the storage unit. The operation unit determines a degree and parameter to be used in the operation of a symmetric polynomial, on the basis of the information on the graph, and then calculates a feature value of the graph corresponding to the structure data set with the symmetric polynomial using the determined degree and parameter.

Abstract: Beaconless topology control algorithms reduce message overhead of local topology constructions compared to conventional (beacon-based) local approaches by avoiding maintenance of neighborhood tables. Moreover, they construct a node's adjacency in the desired topology on demand and only locally, i.e., Do not require network-wide operation. In this work, we present a beaconless topology control algorithm which enables a node to reactively construct a planar backbone graph in its geographic vicinity. This backbone graph is a constant node degree, constant stretch hop-spanner for the input quasi unit disk graph. Our contribution is novel, since all known algorithms with comparable outputs require maintenance of neighborhood tables and are designed for network-wide operation. In addition, it is of significance since there are several applications of it, e.g., In the context of geographic unicast and multicast routing with guaranteed delivery.

Abstract: Finding the maximum flow, or capacity, of wireless multihop networks received a considerable attention by the research community due to its importance from theoretical and practical standpoints. However, since it is np-hard, only bounds can be found using different heuristics. In this poster we find an upper bound on the maximum flow supported by static wireless multihop networks for any load matrix and arbitrary topology in a polynomial time, via a Linear Program, by exploiting only a local interference conflict graph. By noting that interference is local, we replace the optimization problem condition of listing maximal independent sets by listing maximal cliques which improves computational complexity of the solution. Doing this, we had a quadratic programming problem to calculate the exact maximum flow, and not bounds, for any network which is polynomial for some interference models such as the Unit Disk Graph. We applied our model to an example network in [1] and obtained the exact maximum network flow, while [1] suggests only a solution to calculate bounds for the network. The model was also applied to other networks such that the conflict-graph is not perfect, where other models fail to calculate the capacity, and we were able of obtaining the exact capacity.

Abstract: A storage unit stores information on a graph corresponding to a structure data set that represents a three-dimensional structure with a plurality of polygons. An operation unit obtains the information on the graph from the storage unit. The operation unit determines a degree and parameter to be used in the operation of a symmetric polynomial, on the basis of the information on the graph, and then calculates a feature value of the graph corresponding to the structure data set with the symmetric polynomial using the determined degree and parameter.

Abstract: For message delivery in sensor network, greedy forwarding, face routing, and hybrid greedy-face routing have been used extensively. In this paper we present a method for improved face routing by introducing the notion of ‘floating chains’ and ‘external clusters’. The proposed technique removes unnecessary nodes and node clusters from the Gabriel graph extracted from the unit disk graph induced by sensor nodes. We present efficient algorithms for identifying solo-faced chains and external clusters in the network. Removal of such nodes leads to improved performance of face routing methods.

Abstract: The wireless sensor network (WSN) consisting of a large number of autonomous sensors with limited battery. It is a challenging aim to design an energy efficient routing protocol along with original coverage which can save the energy and thereby extend the lifetime of the network. However, in the context of WSN, connected dominating set (CDS) principle has emerged as the most popular approach for energy efficient routing mechanism in WSNs. In this paper, the target coverage is achieved with adjustable sensing range to the proposed distributed CDS based on prime node-ID (P-CDS) modeled in unit disk graph (UDG) (Wan et al. in Distributed construction of CDS in wireless ad hoc networks, 2002) [1]. P-CDS has time complexity O(n) and message complexity of O(n). Theoretical analysis and simulation results are also presented to verify efficiency of our approach.

Abstract: Geographic cluster based routing in ad-hoc wireless sensor networks is a current field of research. Various algorithms to route in wireless ad-hoc networks based on position information already exist. Among them algorithms that use the traditional beaconing approach as well as algorithms that work beaconless (no information about the environment is required besides the own position and the destination). Geographic cluster based routing with guaranteed message delivery can be carried out on overlay graphs as well. Until now the required planar overlay graphs are not being constructed reactively. This thesis proposes a reactive algorithm, the Beaconless Cluster Based Planarization (BCBP) algorithm, which constructs a planar overlay graph and noticeably reduces the number of messages required for that. Based on an algorithm for cluster based planarization it beaconlessly constructs a planar overlay graph in an unit disk graph (UDG). An UDG is a model for a wireless network in which every participant has the same sending radius. Evaluation of the algorithm shows it to be more efficient than the non beaconless variant. Another result of this thesis is the Beaconless LLRAP (BLLRAP) algorithm, for which planarity but not continued connectivity could be proven.

Abstract: Since there is no fixed infrastructure or centralized management in Wireless Sensor Networks (WSNs), a Connected Dominating Set(CDS) has been proposed as a virtual backbone is efficient. A virtual backbone plays a major role in routing, broadcasting, coverage andactivity scheduling. Wireless sensor networks to form a CDS usually by UDG (Unit Disk Graph) models that are used in this model, allnodes have the same message, but this article UDG model instead of a version that is closer to reality called DGB (Disk Graph withBidirectional links) is used in which nodes can adopt different transmission intervals. In many applications, to reduce overhead, increasenetwork lifetime, and so on, to find the MCDS (minimum connected dominating set) is desirable, but the point is that MCDS UDG modelsand DGB, the problem is NP-hard. In addition to the analysis of algorithms, the new algorithm will provide and the efficiency of thealgorithm, especially in terms of energy consumption, through theoretical analysis and simulation algorithms are available to be checked out.

Abstract: Disk contact representations realize graphs by mapping vertices bijectively to interior-disjoint disks in the plane such that two disks touch each other if and only if the corresponding vertices are adjacent in the graph. Deciding whether a vertex-weighted planar graph can be realized such that the disks' radii coincide with the vertex weights is known to be $$\textsf {NP}$$-hard. In this work, we reduce the gap between hardness and tractability by analyzing the problem for special graph classes. We show that it remains $$\textsf {NP}$$-hard for outerplanar graphs with unit weights and for stars with arbitrary weights, strengthening the previous hardness results. On the positive side, we present constructive linear-time recognition algorithms for caterpillars with unit weights and for embedded stars with arbitrary weights.

Abstract: We wish to decide whether a simply connected flexible polygonal structure can be realized in Euclidean space. Two models are considered: polygonal linkages body-and-joint framework and contact graphs of unit disks in the plane. 1 We show that it is strongly NP-hard to decide whether a given polygonal linkage is realizable in the plane when the bodies are convex polygons and their contact graph is a tree; the problem is weakly NP-hard already for a chain of rectangles, but efficiently decidable for a chain of triangles hinged at distinct vertices. 2 We also show that it is strongly NP-hard to decide whether a given tree is the contact graph of interior-disjoint unit disks in the plane.

Abstract: In a wireless sensor network, the virtual backbone plays an important role. Due to accidental damage or energy depletion, it is desirable that the virtual backbone is fault-tolerant. Such a consideration leads to the problem of finding a minimum weight k-connected m-fold dominating set ((k, m)-MWCDS for short). In this paper, we give an (α + 2.5ρ)-approximation for (2, m)-MWCDS with m ≥ 2 in unit disk graph, where α is the performance ratio for the minimum weight m-fold dominating set problem, and ρ is the performance ratio for the {0,1,2}-Steiner Network Design problem. In view of currently best known ratios for α and ρ, (2, m)-MWCDS has a (9 + e)-approximation for m ≥ 3 and a (8 + e)-approximation for m =2, where e is an arbitrary positive real number.

Abstract: In this article, we study approximation algorithms for the problem of computing minimum dominating set for a given set S of n unit disks in ℝ2. We first present a simple O(n log k) time 5-factor approximation algorithm for this problem, where k is the size of the output. The best known 4-factor and 3-factor approximation algorithms for the same problem run in time O(n8 log n) and O(n15 log n) respectively [M. De, G. K. Das, P. Carmi and S. C. Nandy, Approximation algorithms for a variant of discrete piercing set problem for unit disks, Int. J. of Computational Geometry and Appl., 22(6):461–477, 2013]. We show that the time complexity of the in-place 4-factor approximation algorithm for this problem can be improved to O(n6 log n). A minor modification of this algorithm produces a 143-factor approximation algorithm in O(n5 log n) time. The same techniques can be applied to have a 3-factor and a 4513-factor approximation algorithms in time O(n11 log n) and O(n10 log n) respectively. Finally, we propose a very important shifting lemma, which is of independent interest, and it helps to present 52-factor approximation algorithm for the same problem. It also helps to improve the time complexity of the proposed PTAS for the problem.

Abstract: Let G be a unit disk graph in the plane defined by n disks whose positions are known. For the case when G is unweighted, we give a simple algorithm to compute a shortest path tree from a given source in O ( n log ? n ) time. For the case when G is weighted, we show that a shortest path tree from a given source can be computed in O ( n 1 + e ) time, improving the previous best time bound of O ( n 4 / 3 + e ) .

Abstract: Maximum independent set (MIS) is a fundamental problem in graph theory and it has important applications in many areas such as social network analysis, graphical information systems and coding theory. The problem is NP-hard, and there has been numerous studies on its approximate solutions. While successful to a certain degree, the existing methods require memory space at least linear in the size of the input graph. This has become a serious concern in view of the massive volume of today's fast-growing graphs.In this paper, we study the MIS problem under the semi-external setting, which assumes that the main memory can accommodate all vertices of the graph but not all edges. We present a greedy algorithm and a general vertex-swap framework, which swaps vertices to incrementally increase the size of independent sets. Our solutions require only few sequential scans of graphs on the disk file, thus enabling in-memory computation without costly random disk accesses. Experiments on large-scale datasets show that our solutions are able to compute a large independent set for a massive graph with 59 million vertices and 151 million edges using a commodity machine, with a memory cost of 469MB and a time cost of three minutes, while yielding an approximation ratio that is around 99% of the theoretical optimum.