Crossed Cube Ring: A k-connected virtual backbone for wireless sensor networks

Abstract

Connected dominating set is widely used to construct a virtual backbone for effective routing in wireless sensor networks. The k-connected dominating set, where at least k disjoint paths between any pair of nodes exist, is used to meet the fault tolerance requirement of wireless sensor networks. It is an open problem of finding a minimum approximation ratio, measured by the ratio between the number of nodes in the constructed virtual backbone and the number of nodes in the optimal minimum connected dominating set. In this paper, we propose Crossed Cube Ring Algorithm (CCRA) to make the k-connected dominating set virtual backbone with a lower approximation ratio than any existing algorithms. Both theoretical analysis and simulation experiments demonstrate that CCRA can reduce the approximation ratio by 20% than any other algorithms. Therefore, CCRA can be used to construct more compact k-connected dominating set than the conventional algorithms.

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 $\alpha \le 3\gamma +2$, 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 ($k,m$)-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 ($k,m$)-CDS is an important indicator of the robustness and efficiency of a VB in WSNs. Therefore, reducing the size of the ($k,m$)-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 ($k,m$)-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 $\alpha \le 3.2833\gamma +4.5590$ 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 $(12+14k+\frac{34k}{2^k})\gamma$, 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.