Abstract: It has been shown that the maximum distance routing strategy which works well on a Unit Disk Graph (UDG), performs poorly when it is executed on a Non Unit Disk Graph (N-UDG) which reflects the radio irregularity phenomenon. This latter arises from multiple factors, such as antenna and medium type, and is accentuated by environmental factors such as obstacles (e.g., buildings, hills, mountains) and weather conditions. In this paper, we propose a Cross-Layer Greedy Routing algorithm (CL-GR) which enables a correct position-based routing on a N-UDG. It provides two novel greedy routing strategies, termed respectively, Progress towards the sink node through Symmetrical links that experience the lowest Path Loss (PSPL) and progress through symmetrical links, combining the Maximum Distance forwarding strategy and the PSPL (MDPSPL). We compare our CL-GR to an Enhanced version of the Greedy algorithm of the Greedy Perimeter Stateless Routing protocol (GPSR), that we call E-GR and which can be executed on a N-UDG, and to COP _ GARE algorithm. The simulation results show that both PSPL and MDPSPL enable higher Packet Delivery Ratio (PDR) and energy efficiency. In terms of end-to-end delay, while the PSPL strategy significantly increases this metric, the MDPSPL strategy enables a satisfactory end-to-end delay, comparatively to E-GR and COP _ GARE.

Abstract: Over years, many efforts are made for the problem of constructing quality fault-tolerant virtual backbones in wireless network. In case that a wireless network consists of physically equivalent nodes, e.g., with the same communication range, unit disk graph (UDG) is widely used to abstract the wireless network and the problem is formulated as the minimum $k$ -connected $m$ -dominating set problem on the UDG. So far, most results are focused on designing a constant factor approximation algorithm for this NP-hard problem under two positive integers $k$ and $m$ satisfying $m \geq k \geq 1$ and $k \leq 3$ . This paper introduces an approximation algorithm for the problem with $m \geq k \geq 1$ . This algorithm is simple to implement; it connects the components by adding a bounded number of paths, which first computes a 1-connected $m$ -dominating set $D$ and repeats the following steps: (a) search the separators arbitrarily in $(i-1,m)$ -CDS with $i = 2, 3, \cdots, k$ , (b) add a bounded number of paths connecting the components separated by separators in $(i-1,m)$ -CDS to improve the connectivity of $(i-1,m)$ -CDS, until it becomes $k$ -connected, and (c) remove redundant paths if there exist at every iteration. We provide a rigorous theoretical analysis to prove that the proposed algorithm is correct and its approximation ratio is a constant, for any fixed $k$ .

Abstract: Finding the minimum connected dominating set (MCDS) is a key problem in wireless sensor networks, which is crucial for efficient routing and broadcasting. However, the MCDS problem is NP-hard. In this paper, a new approximation algorithm with approximation ratio H(Δ)+3 in time O(n2) is proposed to approach the MCDS problem. The key idea is to divide the sensors in CDS into core sensors and supporting sensors. The core sensors dominate the supporting sensors in CDS, while the supporting sensors dominate other sensors that are not in CDS. To minimize the number of both the cores and the supporters, a three-phased algorithm is proposed. (1) Finding the base-core sensors by constructing independent set (denoted as S1), in which the sensors who have the largest $\frac {|N^{2}(v)|}{|N(v)|}$ (number of two-hop neighbors over the number of one-hop neighbors) will be selected greedily into S1; (2) Connecting all base-core sensors in S1 to form a connected subgraph, the sensors in the subgraph are called cores; (3) Adding the one-hop neighbors of the core sensors to the supporter set S2. This guarantees a small number of sensors can be added into CDS, which is a novel scheme for MCDS construction. Extensive simulation results are shown to validate the performance of our algorithm.

Abstract: Because of its application in the field of security in wireless sensor networks, k-path vertex cover ( $$\hbox {VCP}_k$$ ) has received a lot of attention in recent years. Given a graph $$G=(V,E)$$ , a vertex set $$C\subseteq V$$ is a k-path vertex cover ( $$\hbox {VCP}_k$$ ) of G if every path on k vertices has at least one vertex in C, and C is a connected k-path vertex cover of G ( $$\hbox {CVCP}_k$$ ) if furthermore the subgraph of G induced by C is connected. A homogeneous wireless sensor network can be modeled as a unit disk graph. This paper presents a new PTAS for $$\hbox {MinCVCP}_k$$ on unit disk graphs. Compared with previous PTAS given by Liu et al., our method not only simplifies the algorithm and reduces the time-complexity, but also simplifies the analysis by a large amount.

Abstract: In this article, we study a variant of the dominating set problem known as the liar’s dominating set problem on unit disk graphs. We prove that the liar’s dominating set problem is NP-complete and admits a polynomial time approximation scheme in unit disk graphs.

Abstract: Controlling the manufacturing costs of integrated circuits while increasing their density is of a paramount importance to maintain a certain degree of profitability in the semi-conductor industry. Among various components of a circuit, we are interested in vertical metallic connections known as “vias”. During manufacturing, a complex lithography process is used to form an arrangement of vias on a silicon wafer support, using an optical mask. For manufacturing reasons, a minimum distance between the vias must be respected. Whenever this is not the case, we are talking about a “conflict”. In order to eliminate these conflicts, the industry uses a technique that decomposes an arrangement of vias in several subsets, where minimum distance constraints are respected: the formation of the individual subsets is done, in sequence, on a silicon wafer using one optical mask per subset. This technique is called Multiple Patterning (MP). There are several ways to decompose an arrangement of vias, the goal being to assign the vias to a minimum number of masks, since the masks are expensive. Minimizing the number of masks is equivalent to minimizing the number of colors in a unit disk graph. This is a NP-hard problem however, a number of “good” heuristics exist. A recent and promising technique is based on the direction and self-assembly of the molecules called Directed Self Assembly (DSA), allows to group vias in conflict according to certain conditions. The main challenge is to find the best way of grouping vias to minimize the number of masks while respecting the constraints related to DSA. This problem is a graph coloring problem where the vertices within each color define a set of independent paths of length at most k also called a k-path coloring problem. During the graph modeling, we distinguished two k-path coloring problems: a general problem and an induced problem. Both problems are known to be NP-hard, which explains the use of heuristics in the industry to find a valid decomposition into subsets. In this study, we are interested in exact methods to design optimal solutions and evaluate the quality of heuristics developed in the industry (at Mentor Graphics). We present different methods: an integer linear programming (ILP) approach where we study several formulations, a dynamic programming approach to solve the induced case when k=1 or k=2 and when the graphs have small tree-width; finally, we study a particular case of line graphs. The results of the various numerical studies show that the naive ILP formulations are the best, they list all possible paths of length at most k. Tests on a snippet of industrial instances of at most 2000 vertices (a largest connected component among those constituting an instance) have shown that the two problems, general and induced, are solved in less than 6 seconds, for k=1 and k=2. Dynamic programming, applied to the induced k-path coloring when k=1 and k=2, shows results equivalent to those of the naive ILP formulation, but we expect better results by dynamic programming when the value of k increases. Finally, we show that the particular case of line graphs can be solved in polynomial time by exploiting the properties of Edmonds’ algorithm and bipartite matching.

Abstract: Unit disk graphs are the intersection graphs of unit radius disks in the Euclidean plane. Deciding whether there exists an embedding of a given unit disk graph, i.e. unit disk graph recognition, is an important geometric problem, and has many application areas. In general, this problem is known to be $\exists\mathbb{R}$-complete. In some applications, the objects that correspond to unit disks, have predefined (geometrical) structures to be placed on. Hence, many researchers attacked this problem by restricting the domain of the disk centers. One example to such applications is wireless sensor networks, where each disk corresponds to a wireless sensor node, and a pair of intersecting disks corresponds to a pair of sensors being able to communicate with one another. It is usually assumed that the nodes have identical sensing ranges, and thus a unit disk graph model is used to model problems concerning wireless sensor networks. We consider the unit disk graph realization problem on a restricted domain, by assuming a scenario where the wireless sensor nodes are deployed on the corridors of a building. Based on this scenario, we impose a geometric constraint such that the unit disks must be centered onto given straight lines. In this paper, we first describe a polynomial-time reduction which shows that deciding whether a graph can be realized as unit disks onto given straight lines is NP-hard, when the given lines are parallel to either $x$-axis or $y$-axis. Using the reduction we described, we also show that this problem is NP-complete when the given lines are only parallel to $x$-axis (and one another). We obtain those results using the idea of the logic engine introduced by Bhatt and Cosmadakis in 1987.

Abstract: Since there are no fixed infrastructures in some wireless scenarios, constructing a connected dominating set (CDS) of virtual backbone nodes to reduce communicating overhead and enhance scalability of the network is an effective strategy. However, some virtual backbone nodes may disconnect with each other occasionally, especially in highly mobile scenarios. In the past few years, many CDS construction algorithms try to achieve a minimum CDS (MCDS), and some algorithms are proposed to tackle with the recovery of CDS. Most works assume that each node could easily discover the whole outgoing and incoming neighbors, whether in unit disk graph (UDG) or disk graph (DG). However, it is hard to obtain the global topology information in mobile networks. It is even more challenging in hierarchical wireless networks considering the diverse capabilities of nodes, where some high-level nodes with larger coverages are more prone to be appointed as backbone nodes. In this paper, we propose a CDS-based recovering algorithm for disconnected virtual backbone nodes in mobile hierarchical wireless networks. Numerical results demonstrate that the proposed scheme can achieve a smaller set of nodes to recover the connection of virtual backbone nodes compared with the existing works without increasing communicating overhead.

Abstract: Unit disk graphs are the intersection graphs of unit diameter disks in the Euclidean plane. Recognizing unit disk graph is an important geometric problem, and has many application areas. In general, this problem is shown to be $\exists\mathbb{R}$-complete. However, in some applications, the objects that correspond to unit disks have predefined (geometrical) structures to be placed on. Hence, many scientists attacked this problem by restricting the domain for the centers of the disks. One example to such applications is wireless sensor networks, where each disk corresponds to a wireless sensor node, and a pair of intersecting disks correspond to a pair of sensors being able to communicate with each other. It is usually assumed that the nodes have identical sensing ranges, and thus unit disk graph model is used to model problems concerning wireless sensor networks. In this paper, we also attack the unit disk recognition problem on a restricted domain, by assuming a scenario where the wireless sensor nodes are deployed on the corridors of a building, forming collinear groups. Based on this scenario, we impose a geometric constraint such that the unit disks must be centered onto given straight lines. We show that deciding whether there exists an embedding of unit disk graphs is NP-hard, even if the given straight lines are parallel to either $x$-axis or $y$-axis. Moreover, we remark that if the straight lines are not given, then the problem becomes $\exists\mathbb{R}$-complete.

Abstract: This paper considers the problem of how to efficiently share a wireless medium which is subject to harsh external interference or even jamming. So far, this problem is understood only in simplistic single-hop or unit disk graph models. We in this paper initiate the study of MAC protocols for the SINR interference model (a.k.a. physical model). This paper makes two contributions. First, we introduce a new adversarial SINR model which captures a wide range of interference phenomena. Concretely, we consider a powerful, adaptive adversary which can jam nodes at arbitrary times and which is only limited by some energy budget. Our second contribution is a distributed MAC protocol called Sade which provably achieves a constant competitive throughput in this environment: we show that, with high probability, the protocol ensures that a constant fraction of the non-blocked time periods is used for successful transmissions.

Abstract: In this article, we study the maximum distance-d independent set problem, a variant of the maximum independent set problem, on unit disk graphs. We first show that the problem is NP-hard. Next, we propose a polynomial-time constant-factor approximation algorithm and a PTAS for the problem.

Abstract: A unit disk graph is the intersection graph of disks of equal radii in the plane. The class of unit disk graphs is hereditary, and therefore admits a characterization in terms of minimal forbidden induced subgraphs. In spite of quite active study of unit disk graphs very little is known about minimal forbidden induced subgraphs for this class. We found only finitely many minimal non-unit disk graphs in the literature. In this paper we study in a systematic way forbidden induced subgraphs for the class of unit disk graphs. We develop several structural and geometrical tools, and use them to reveal infinitely many new minimal non-unit disk graphs. Further we use these results to investigate structure of co-bipartite unit disk graphs. In particular, we give structural characterization of those co-bipartite unit disk graphs whose edges between parts form a $$C_4$$ -free bipartite graph, and show that bipartite complements of these graphs are also unit disk graphs. Our results lead us to propose a conjecture that the class of co-bipartite unit disk graphs is closed under bipartite complementation.