Crossed Cube Ring: A k-connected virtual backbone for wireless sensor networks
Introduction
Wireless Sensor Networks (WSNs) is widely used in many civilian applications such as long-term and low-cost geographical monitoring, traffic control, healthcare, and home automation (Lu and Tang, 2014). One of the challenges of using WSNs is how to prolong the network lifetime, especially with a wide distribution of sensors under extreme environments. One such example is the environmental monitoring in Daiyun Mountain National Nature Reserve, China (Ren, 2011). The challenges in this example include a wide distribution of sensors, long-term data collection, and dense data collection, which cause the energy efficiency problem. The ultimate goal of such WSNs deployed in the crucial environments is often to deliver the sensing data from sensor nodes to sink node and then conduct further analysis at the sink node. Data collection becomes an important factor in determining the performance of such WSNs.
In most WSNs, battery is the sole energy source of the sensor nodes, we expect WSNs to work on batteries for several months (Amir et al., 2015; Bagchi et al., 2015). One efficient approach to prolong WSN lifetime is topology control (Zhao et al., 2012, Guo et al., 2012, Jiang et al., 2016; Shan et al., 2014), which aims to save energy consumption by optimizing the network topology. It is widely acceptable to construct a robust Virtual Backbone (VB) (Chen et al., 2013; Gnawali et al., 2013) to meet this target. In this way, each sensor forwards data to the sink node through VB, which is a good choice for the convergence of data flows, since VB can cover the network topology efficiently.
WSNs often use a Connected Dominating Set (CDS) to construct a VB for efficient routing and broadcasting operations (Kim et al., 2009). The nodes within CDS are called dominators. The nodes that are adjacent to a dominator are called dominatees. As shown in Fig. 1, dominatees can forward messages to the dominators, which constitute a CDS. Once a CDS-based VB is used in WSNs, the routing path search space will be restricted to the CDS instead of the whole network. This can lead to shorter search time of routing path, smaller routing table size and simpler routing maintenance. In addition, due to the limited number of sensor nodes in the CDS-based VB, WSNs with the CDS feature can adapt to topology change quickly.
The size of CDS is the primary concern to measure the quality of a VB (Kim et al., 2009). Since all the nodes in a VB share the communication channel, a smaller VB suffers less from the interference problem. Finding the Minimum CDS (MCDS) was proven to be an NP-hard problem, but it has polynomial-time approximation scheme in unit disk graphs (Cheng et al., 2003). The size is often measured by the approximation ratio, which is defined as the ratio between the number of nodes in the constructed VB and the number of nodes in the optimal minimum CDS. A conjecture viewed as the following open problem was given in Wu et al. (2010).
Open problem (conjecture): In a unit disk graph, the minimum size of CDS by any approximate optimal algorithm is , where γ is the size of any optimal minimum CDS, and α is the size of CDS by any approximate optimal algorithm.
Many efficient algorithms for constructing CDS have been proposed (Kim et al., 2009, Cheng et al., 2003, Du et al., 2011, Wan et al., 2002, Wan et al., 2008, Wu et al., 2006, Wu et al., 2010, Funke et al., 2006, Gao et al., 2009, Li et al., 2011, Li et al., 2012, Liu et al., 2013Kim et al., 2009, Cheng et al., 2003, Du et al., 2011, Wan et al., 2002, Wan et al., 2008, Wu et al., 2006, Wu et al., 2010, Funke et al., 2006, Gao et al., 2009, Li et al., 2011, Li et al., 2012, Liu et al., 2013 Cheng et al., 2003, Du et al., 2011, Funke et al., 2006, Gao et al., 2009, Kim et al., 2009, Li et al., 2011, Li and Bartos, 2014, Liu et al., 2013, Wan et al., 2002, Wan et al., 2008, Wu et al., 2006, Wu et al., 2010 He et al., 2013; Ding et al., 2011, Ding et al., 2012, Do et al., 2013). It is a consensus that the VB containing only one CDS is not robust. As shown in Fig. 1, if one of the vertexes in CDS-1 fails, the VB is broken. However, if there is another VB such as CDS-2, the network is still functional when CDS-1 fails. It is a challenge to construct a VB with multiple CDSs enabled for fault tolerance. k-Connected m-Dominating Set that referred to as ()-CDS can serve as a VB for solving this challenge, where k-Connectivity requires that there exist at least k disjoint paths between any pair of nodes in a VB, and m-Domination ensures that there exist at least m Dominator neighbors for each dominatee. The size of the ()-CDS is an important indicator of the robustness and efficiency of a VB in WSNs. Therefore, reducing the size of the ()-CDS is the main aim of this paper.
To the best of our knowledge, there is a lack of effective methods to construct CDS by rule based networks, although some work using the properties of Crossed Cube in WSNs has been done (Cheng et al., 2013). Crossed Cube, which is proven to be superior to the hypercube counterpart in many aspects, has shorter diameter and higher connectivity. Because time complexity can be reduced by decreasing the diameter of the network, this is of great importance for route searching in a large-scale network. On the basis of the simple topology of Ring and high connectivity of Crossed Cube (Wang and Lin, 2005; Cheng et al., 2013), a novel network topology named Crossed Cube Ring (CCR) is proposed in this paper. Based on the CCR, a distributed Crossed Cube Ring Algorithm (CCRA) is proposed to construct the ()-CDS as a VB. The main contributions of this paper are summarized as follows:
1. The dominating set is constructed by a new algorithm based on Induced Tree of the Crossed Cube. The smallest dominating set size with can be obtained, which is the closest to the conjecture of a minimum CDS.
2. Based on the simple topology of Ring and high connectivity of Crossed Cube, a novel network topology named CCR is proposed, which can be used to construct the backbone to meet the requirements of k-Connected set with a smaller size.
3. A distributed CCRA algorithm is developed to construct a fault tolerant VB with the CCR formation. It is verified by the theoretical analysis that CCRA can generate a k-CDS with the approximation ratio of , which is smaller than the ratio achieved in the existing works.
The rest of this paper is organized as follows: Section 2 briefly summarizes the related works on CDS. Section 3 describes the problem statement of our work. Section 4 presents the proposed CCRA algorithm for VB construction. Theoretical analysis and simulation results are shown in Sections 5 and 6, respectively. Finally, Section 7 concludes this paper.
Section snippets
Related work
It has been widely accepted that a CDS of a graph representing a wireless network is the best candidate to serve as a virtual backbone. A Dominating Set (DS) of a graph is a set of vertexes such that for every edge , or . A CDS of a graph is a DS of G such that the subgraph of G induced by the vertexes in this DS is connected. The necessary and sufficient conditions for a network topology to have a CDS are that the network topology is connected. CDS
Notations and definitions
The important notations and definitions are introduced in this section. The network topology is represented as a Unit Disk Graph.
The network is denoted as a graph , where V(G) is the vertex set and E(G) is the edge set. Each vertex represents a sensor node and each edge denotes a communications link between two nodes. If a node is within the transmission range of the other, the communications link between these two nodes is represented by a directed edge from one node to the other.
The CCRA algorithm for K-CDS construction
Theorem 1 illumines a distributed local decision way to make the whole graph k-connected by constructing Crossed Cubes CQn only using the local graph information. Based on the novel network topology, Crossed Cube Ring, a distributed algorithm CCRA is proposed, which has three phases:
Phase 1. Construct a local graph .
Phase 2. Construct a 1-connected dominating set C1 using ITCC algorithm.
Phase 3. Construct a k-connected dominating set C using CCRA algorithm which is based on Crossed Cube Ring.
Theoretical analysis
This section will elaborate on the correctness of the CCRA for a k-CDS that is shown in Theorem 2 and Theorem 3. The computation of approximation ratio of the k-CDS is shown in Lemma 3, 4 and Theorem 4. The message and time complexity is shown in Theorem 5.
An example of CCRA
At first, this section will evaluate CCRA by constructing a 2-CDS as an example. Because the existing simulation software does not integrate the CDS algorithm module, we have developed the simulator based on our laboratory VC++ platform that was adopted in our previous related studies. There are 100 nodes randomly placed in a 100*100 region with a transmission range of 25, where the original network is connected. Figs. 7 (a)–(e) illustrate the execution stages of our algorithm. After all
Conclusion
In this paper, the k-connected dominating set problem is studied in the context of WSNs. First, a new Induced Tree based on the Crossed Cube is defined, which can be constructed recursively. Based on the Induced Tree in the , a novel 1-CDS algorithm called ITCC is presented. Second, based on the simple topology of Ring and high connectivity of the Crossed Cube, a novel network topology, Crossed Cube Ring (CCR), is proposed. The CCR structure is better than Crossed Cube in extension. For
Acknowledgments
The authors wish to thank National Natural Science Foundation of China (Grant NO: 61072080, 61572010, U1405255). Fujian Normal University Innovative Research Team (No. IRTL1207). The Project of the Education Department of Fujian Province (No. JAT160328) and the scientific research project in Fujian University of Technology (GY-Z160066, GY-Z160130, GY-Z160138). Natural Science Foundation of Fujian Province of China (2017J05098). Young and Key Project of Fujian Education Department Funds (JZ160461
References (43)
- et al.
Dimension-adjacent trees and parallel construction of independent spanning trees on crossed cubes
J. Parallel Distrib. Comput.
(2013) - et al.
Approximation algorithms for load-balanced virtual backbone construction in wireless sensor networks
Theor. Comput. Sci.
(2013) - et al.
On approximation algorithms of k-connected m-dominating sets in disk graphs
Theor. Comput. Sci.
(2007) - et al.
Minimum connected dominating sets and maximal independent sets in unit disk graphs
Theor. Comput. Sci.
(2006) - et al.
Efficient data collecting and target parameter estimation in wireless sensor networks
J. Netw. Comput. Appl.
(2015) - et al.
Optimal radius for connectivity in duty-cycled wireless sensor networks
ACM Trans. Sens. Netw. (TOSN)
(2015) - et al.
Dominating set and network coding-based routing in wireless mesh networks
IEEE Trans. Parallel Distrib. Syst.
(2013) - et al.
A polynomial-time approximation scheme for the minimum-connected dominating set in ad hoc wireless networks
Networks
(2003) - Dai, F., Wu, J., 2005. On constructing k-connected k-dominating set in wireless networks. In: Proceeding of the IEEE of...
- et al.
Efficient algorithms for topology control problem with routing cost constraints in wireless networks
IEEE Trans. Parallel Distrib. Syst.
(2011)
Efficient virtual backbone construction with routing cost constraint in wireless networks using directional antennas
IEEE Trans. Mob. Comput.
Formulations for the minimum 2-connected dominating set problem
Electron. Notes Discret. Math.
CDS-based virtual backbone construction with guaranteed routing cost in wireless sensor networks
IEEE Trans. Parallel Distrib. Syst.
The crossed cube architecture for parallel computing
IEEE Trans. Parallel Distrib. Syst.
Hamilton-connectivity of crossed cubes
J. QingDao Univ.
A simple improved distributed algorithm for minimum CDS in unit disk graphs
ACM Trans. Sens. Netw. (TOSN)
Analysis on theoretical bounds for approximating dominating set problems
Discret. Math., Algorithms Appl.
CTPan efficient, robust, and reliable collection tree protocol for wireless sensor networks
ACM Trans. Sens. Netw. (TOSN)
Sleep scheduling for critical event monitoring in wireless sensor networks
IEEE Trans. Parallel Distrib. Syst.
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