This paper is an extended version of our paper published in Lin, Y.; Wu, Q. Approximate Algorithms for Sensor Deployment with k-coverage in Constrained 3D Space. In Proceedings of the 16th International Conference on Parallel and Distributed Systems, Shanghai, China, 8–10 December 2010.

Sensor networks have been used in a rapidly increasing number of applications in many fields. This work generalizes a sensor deployment problem to place a minimum set of wireless sensors at candidate locations in constrained 3D space to

Sensor networks have been widely used in many agricultural, military, and industrial applications. The technological advances in both sensing and communication have significantly improved the quality of sensors at a reduced cost, making it possible to deploy more sensors than before to achieve quality through quantity. The redundancy in the deployment of sensors is often explored to cover targets of interest with multiple sensors for fault tolerance and sustained operation.

Depending on the nature of the environment and the type of the application, sensors could be deployed in either a deterministic or a stochastic manner. In the former, sensors are typically mounted manually at some predetermined locations to meet a certain deployment objective; while in the latter, sensors are not bound to any specific location and are oftentimes dropped randomly by vehicles or airplanes to cover a large geographical region. In the past decade, a significant number of research efforts have been made in both of the deployment scenarios for various purposes, e.g., to localize potential targets in the former [

In this paper, we consider a specific type of wireless sensor networks (WSNs) that are deployed in challenged environments for microclimate monitoring, such as cultural heritage sites with historical frescos, sculpture paintings, or religionary statues, which are typically located on the 3D surface of constrained space [

This paper generalizes and investigates a sensor deployment problem in constrained 3D space to achieve

For each of these four problems, we further consider a connected version where all sensors in the deployment region must be connected for communication, referred to as C-DLDT, C-DLCT, C-CLDT, and C-CLCT. We first study a subproblem, i.e., Steiner Tree Problem with Minimum number of Steiner Points on Constrained Locations (STP-MSPCL), and design an approximate algorithm based on minimum spanning tree with an approximation ratio of 5 in a 2D plane and 12 in 3D space. We solve C-DLDT using an approximate algorithm with an approximation ratio of

We implement the proposed deployment algorithms and evaluate their performance in a simulated setting. For performance comparison, we also design and implement an optimal solution based on an integer linear programming (ILP) formulation, which is only meant for small-scale problem instances, and a genetic algorithm (GA)-based approach. Extensive simulations show that these algorithms consistently outperform the GA-based heuristic and achieve a close-to-optimal performance in small-scale problem instances and a significantly superior overall performance than the theoretical upper bound. These results shed light on the efficiency of the proposed deployment schemes and their great potential for practical sensor network applications. For example, many cultural relics in the world such as the ancient Great Wall and Terracotta Army in China are suffering from serious environmental pollution and degradation. Deploying a network of sensors in the constrained surroundings to measure environmental parameters such as humidity, temperature, and lighting is a key component of a viable solution for protection from environment-related threats. The proposed algorithms could be used to produce cost-effective sensor deployment schemes in such real-life scenarios.

Compared to the conference version [

We developed a method based on integer linear programming (ILP) to find the lower bound of the optimal solution to the DLDT problem by formulating it as an ILP task.

We derived a tighter approximation ratio of Algorithm 1 for DLDT based on the lower bound

We provided a detailed rigorous proof on the upper bound on the number of divisions intersected by

We derived a more accurate approximation ratio of the algorithm designed for DLCT with target object discretization.

We designed a completely new approach to CLCT, which first divides the CLCT problem into two subproblems, and then converts the two subproblems to CLDT and DLCT, respectively.

We implemented the proposed algorithms and evaluated the performance against the optimal solution and a genetic algorithm-based approach through extensive simulations.

The rest of the paper is organized as follows.

Sensor deployment for coverage and connectivity in 2D space has been investigated in-depth in the literature. Many efficient techniques have been proposed, including several recent efforts based on integer linear programming (ILP) models and artificial intelligence such as randomization-based genetic algorithm [

as well as bee colony algorithm and particle swarm optimization [

The traditional 3D

There also exist a number of efforts on sensor deployment for

In [

The problems we study in this paper differ from the aforementioned research efforts in several aspects: (i) We consider a set of constrained sensor deployment problems in 3D space; (ii) We exhaust the combinations of various deployment constraints posed on candidate sensor locations and target objects, which can be either discrete points or continuous areas; (iii) We require the deployed sensors to

Our network model considers sensors with a certain sensing radius

We study the constrained sensor deployment problem in 3D space, which is formulated as an optimization problem: Given a 3D convex region

Discrete

Discrete

Continuous

Continuous

Note that in the above problems, the convex 2D areas for continuous target objects and feasible sensor locations could be positioned anywhere in 3D space (i.e., they may not be located on the same plane). At each discrete point for sensor deployment, we can deploy at most one sensor. When a feasible sensor location is a continuous 2D area, we can deploy a sensor at any position within that area. Furthermore, we consider a connected version of each of the four sensor deployment problems where all deployed sensors must be connected, referred to as C-DLDT, C-DLCT, C-CLDT, and C-CLCT.

We first prove the NP-completeness of DLDT and propose an approximate algorithm with an approximation ratio of

We show that the known NP-complete Discrete Unit Disk Cover (DUDC) problem [

We can restrict DLDT to DUDC by allowing only those instances where

We develop a method based on integer linear programming (ILP) to find the lower bound of the optimal solution to the DLDT problem by formulating it as an ILP task. Let variable

In the above ILP task, the objective is to minimize the total number of sensors to be deployed. The first constraint ensures that each target point is

Let

Andersen et al. modified the greedy algorithm originally designed for the Set Cover problem [

Considering that each of

We use

After the

We use

By rearranging

We consider a function

In Algorithm 1, after

The inequality in Equation (

We would like to point out that DLDT with

Since

According to the constraint defined in Equation (

In DLCT, the candidate sensor locations are given as a set of discrete points, and the surfaces of the target objects are composited by a finite set of continuous 2D areas, in which, any point must be

The first step for intersected circle calculation is to compute the sensing region within a continuous target area

The second step for division calculation is to determine all divisions formed by the intersected circles calculated at the previous step. This problem appears to be computationally challenging in geometry, but it turns out that tracing all possible divisions can be done in polynomial time. In fact, Huang et al. pointed out that the total number of intersected divisions could be as many as

We prove this lemma by reduction to absurdity. Assume that a newly added circle passes an intersection point

We prove this theorem using mathematical induction. Let

Suppose that

According to Lemma 1,

In this case, an arc of

Subcase (1):

Subcase (2):

Subcase (3):

Subcase (4):

In this case, two arcs of

In sum, the average increment of divisions for any existing circle

According to Lemma 1,

We can represent a division using a sequence of component intersection points and arcs. To facilitate division calculation, we provide below several definitions.

We design an algorithm in Algorithm 2 to determine all divisions in a continuous area

Calculate all the intersection points on

Add all the arcs in

Add

Add all arcs on

Select

Add

Add

Select

Add

Add

Add

Incorporate

We use Algorithm 2 to determine all divisions in each continuous target area

The third step for covering set calculation is to determine the set of divisions a discrete sensor location

If a division contains a non-arc boundary segment of a target area, then the segment is covered by a sensor if and only if the distance from each endpoint of the segment to the sensor location is less than or equal to

The complexity of the entire discretization process is dominated by the complexity of the last two steps, i.e.,

We compute the maximum total number of divisions in all continuous target areas. Since every continuous target object is a convex area, Lemma 2 shows that the maximum number of divisions in one continuous target area is

Hence, the approximation ratio of GreedyDLCT is

In CLDT, the feasible sensor locations are given as a set of 2D continuous areas, and the target objects are given as a set of discrete points. We may place sensors at any point within these continuous areas for sensor deployment. The key idea for solving CLDT is to use a similar three-step discretization process to discretize a continuous sensor location into divisions in order to convert CLDT to DLDT without loss of precision, referred to as GreedyCLDT.

Similar to DLCT, during discretization in CLDT, the first step for intersected circle calculation is to compute the valid sensor region within each continuous feasible sensor location

The second step for division calculation is to use Algorithm 2 to determine all possible divisions at every continuous sensor location

The second step for division calculation is to use Algorithm 2 to determine all possible divisions at every continuous sensor location

We choose

The third step for covering set calculation is to compute the set of discrete target points that every discrete point resulted from the previous step can cover. The above three-step discretization process converts CLDT to DLDT, and the time complexity of this transformation is

In CLCT, both sensor locations and target objects are given as continuous 2D areas. This problem is more challenging to tackle because we can not directly convert it to DLDT using the same discretization method for DLCT and CLDT. We propose to first divide CLCT into two subproblems, and then convert the two subproblems to CLDT and DLCT, respectively. Based on this approach, we design an approximate algorithm for CLCT.

Let

There are four cases in the CLCT problem.

Case 1: At least a target point exists outside

Case 2: No target point is outside

Case 3: No target point is outside

Case 4: All the target areas are in the inner side of

Let

Subproblem 1: Given

Subproblem 2: Given

Note that in Case 4, the CLCT problem is the same as Subproblem 1.

We first tessellate 3D region

Problem 1: Given

Problem 2: Given

Problem 2 is a CLDT problem with an approximation algorithm, so that we can use the approximate solution

In Problem 2, if selected location divisions contain a point in

Obviously, a set of sensors

Since the approximation ratio of GreedyCLDT is

Since we can deploy

We provide the upper bound

According to Equations (

We construct a DLCT problem as follows: Given a set

Consider the DLCT problem. For each sensor location

Based on the definition of

By converting Subproblem 2 to DLCT that is solved by an approximation algorithm, we can obtain an approximation solution

There are only two existing forms for the points intersected by

Case 1: an isolated point in

Case 2: a set of concyclic points, all of which only one or two sensor location points can cover.

If all the target areas can be

Assume that in a sensor location area, there exist two location points

Let

Based on Equation (

We consider another set of four connected deployment problems by imposing a requirement on the sensor network connectivity: C-DLDT, C-DLCT, C-CLDT, and C-CLCT. Obviously, all these problems are still NP-complete since their corresponding version without the connectivity requirement is a special case where every sensor has a sufficiently large communication radius.

We design an approximate algorithm for each of these connected sensor deployment problems. We use C-DLDT as an example to explain the algorithm design, which is based on the result of DLDT returned by Algorithm 1. Suppose that the set of sensor locations from Algorithm 1 for DLDT is

We propose an approximate algorithm for STP-MSPCL based on the minimum spanning tree algorithm in Algorithm 3, which solves the node-weighted Steiner tree problem by reducing it to the edge-weighted Steiner tree problem. In lines 1–9, it assigns a weight to every link. If both of the end points of a link belong to

Compute the shortest path

Compute a minimum spanning tree

Let

For STP-MSPCL in 2D space where

We first show that the degree of a Steiner point is at most 5. Assume that a Steiner point

We provide a tight example of Theorem 6 in

Before further analyzing STP-MSPCL in 3D space, we would like to describe a long-debated historical “thirteen spheres” problem, which asks if 13 non-overlapping spheres of an identical size can touch the surface of another (central) sphere in 3D space. In a symmetrical configuration, we can place 12 spheres at those positions corresponding to the vertices of a regular icosahedron concentric with the central sphere. As these 12 spheres do not touch each other, there was a conjecture that a 13-th sphere may be added, which was the subject of the famous discussion between Isaac Newton and David Gregory in 1694, and it was finally proved in 1953 that at most 12 spheres can touch the central sphere [

We set the radius of each sphere to be

The algorithm for C-DLDT is a two-stage algorithm: (i) use the approximation algorithm with an approximation ratio of

We conduct simulation-based performance evaluation on the proposed algorithms for constrained sensor deployment in 3D space using a set of randomly generated problem instances.

We would like to make an emphasis that the proposed methods for sensor deployment are approximate algorithms in nature, which are essentially different from heuristic approaches without any performance bound. In particular, we focus on the performance evaluation of the algorithm designed for DLDT (Algorithm 1) because it serves as the base of the solutions to the other problems, and the performance of this algorithm is a direct reflection of the performance of the other proposed algorithms.

In the simulation, we consider a 3D cube of dimensions 100 m × 100 m × 100 m as the region of interest, where a given number of discrete sensor locations and target points are randomly placed. Each sensor has a sensing diameter of 25 m (

For the evaluation of an approximate algorithm with an approximation ratio, one important aspect is to examine how it actually performs within the proven performance bound in different scenarios. For that purpose, we compare GreedyDLDT with an optimal solution, which is obtained by using the GNU Linear Programming Kit (GLPK) package to solve DLDT under the ILP formulation. Also, for a more practical evaluation of the performance, we compare GreedyDLDT with a genetic algorithm (GA)-based heuristic approach for sensor deployment, which mainly follows the design and implementation of the genetic operators (i.e., crossover, translocation, and mutation) in [

The surveillance region is extended from a 2D plane to a 3D space, and divided into a number of uniform contiguous voxels of unit size.

The sensing model is changed from probabilistic sensing to Definite Range Law Approximation (Cookie Cutter) [

The sensor type is changed from heterogeneous to homogeneous, and accordingly, no priority is given to any sensor during the population initialization.

The cost function considers the number of target points that have been sufficiently covered and is penalized against the total number of target points multiplied by

The fitness function is redesigned to minimize the number of deployed sensors under the constraint imposed from the cost function.

In the first set of experiments, we set the number of discrete sensor locations to 100 and set

Based on the number of deployed sensors computed by GreedyDLDT and the optimal solution, we are able to measure the actual approximation ratio of GreedyDLDT. We plot in

In the second set of experiments, we set the number of discrete sensor locations to 200 and the number of target points to 100, and compare GreedyDLDT with the GA-based approach and the optimal solution with different

Similarly, based on the number of deployed sensors computed by GreedyDLDT and the optimal solution, we are able to measure the actual approximation ratio in comparison with the theoretical one, as plotted in

In order to evaluate the performance of GreedyDLDT with realistic deployment constraints, we consider an exhibition hall at Zhejiang Provincial Museum, Hangzhou, China [

We run the proposed GreedyDLDT algorithm, the GA-based approach, and the optimal solution to determine the sensor deployment scheme in this exhibition hall as the number of exhibition items are increased from 5 to 50 at an interval of 5 items. The performance measurements in

Similarly, we focus on the performance evaluation of C-DLDT because it serves as the base of the solutions to the connected version of the other three problems. We also run the proposed two-stage (first coverage and then connectivity) Greedy algorithm, the GA-based approach, and the optimal solution for C-DLDT to solve the two sets of problem instances created in

We generalized and investigated a class of problems to deploy a minimum set of wireless sensors at candidate locations in constrained 3D space to achieve

We designed an approximate algorithm for DLDT with an approximation ratio of

The simulation results show that the proposed deployment algorithms consistently outperform a genetic algorithm-based approach and achieve a close-to-optimal performance in small-scale problem instances. The measured approximation ratios in the simulations are generally much less than the theoretical upper bounds derived for the worst cases, which illustrates the efficacy of these algorithms in practical applications.

This research is sponsored by National Science Foundation under Grant No. CNS-1560698 and Oak Ridge National Laboratory, U.S. Department of Energy, under Contract No. DE-AC05-00OR22725/4000141963 with New Jersey Institute of Technology. We would also like to thank two anonymous reviewers whose comments have greatly helped us improve the presentation and technical contents of this manuscript.

C.Q. Wu formulated the problems and designed the algorithms and the experiments; L. Wang made contributions to the implementation and evaluation of the proposed algorithms.

The authors declare no conflict of interest.

Illustration of two intersection cases between a new circle and the existing circles: (

An example of Algorithm 2.

Intersected circles and divisions.

A tight example of Theorem 6.

Comparison of the average performance with the standard deviation between GreedyDLDT (Greedy), Genetic Algorithm (GA), and Optimal Algorithm (OPT) for DLDT in response to a varying number of target points.

Measured average approximation ratio of GreedyDLDT for DLDT in response to a varying number of target points.

Comparison of the average performance with the standard deviation between GreedyDLDT (Greedy), Genetic Algorithm (GA), and Optimal Algorithm (OPT) for DLDT in response to varying

Measured average approximation ratio of GreedyDLDT for DLDT in response to varying

Performance comparison of GreedyDLDT (Greedy), Genetic Algorithm (GA), and Optimal Algorithm (OPT) for a museum exhibition hall with sensor mounting constraints in response to a varying number of exhibition items.

Comparison of the average performance with the standard deviation between the 2-stage Greedy Algorithm, Genetic Algorithm (GA), and the Optimal Algorithm (OPT) for C-DLDT in response to a varying number of target points.

Comparison of the average performance with the standard deviation between the 2-stage Greedy Algorithm, Genetic Algorithm (GA), and the Optimal Algorithm (OPT) for C-DLDT in response to varying