International Journal of Distributed Sensor NetworksVolume 2012 (2012), Article ID 918252, 25 pagesdoi:10.1155/2012/918252

Research Article

The Complexity of the Minimum Sensor Cover Problem with Unit-Disk Sensing Regions over a Connected Monitored Region

Ren-Song Ko

Department of Computer Science and Information Engineering, National Chung Cheng University, 168 University Road, Min-Hsiung Chia-Yi 621, Taiwan

Received 17 June 2011; Revised 12 August 2011; Accepted 16 August 2011

Academic Editor: Yuhang Yang

Copyright © 2012 Ren-Song Ko. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper considers the complexity of the Minimum Unit-Disk Cover (MUDC) problem. This problem has applications in extending the sensor network lifetime by selecting minimum number of nodes to cover each location in a geometric connected region of interest and putting the remaining nodes in power saving mode. MUDC is a restricted version of the well-studied Minimum Set Cover (MSC) problem where the sensing region of each node is a unit-disk and the monitored region is geometric connected, a well-adopted network model in many works of the literature. We first present the formal proof of its NP-completeness. Then we illustrate several related optimum problems under various coverage constraints and show their hardness results as a corollary. Furthermore, we propose an efficient algorithm for reducing MUDC to MSC which has many well-known algorithms for approximated solutions. Finally, we present a decentralized scalable algorithm with a guaranteed performance and a constant approximation factor algorithm if the maximum node density is fixed.

1. Introduction

The research in wireless ad hoc networks has rapidly grown in recent years due to their applications in civil and military domains. Combined with recent developments in micro-electro-mechanical systems and low-cost mass production, various small and low-power devices that integrate sensors with limited on-board processing and wireless communication capabilities begin to emerge. Hence, wireless networks with large numbers of sensors become possible and open up potential of many new applications, such as environment monitoring and surveillance [1].

With the available technology, the sensors are usually battery powered. Due to size and cost constraints, the energy available at each sensor is limited. Therefore, one of the important design considerations in sensor networks is to minimize energy consumption and prolong network lifetime. There is a significant amount of the literature addressing the issue of efficient energy management in generic wireless ad hoc networks from various perspectives, such as medium access control [2, 3], routing [4, 5], broadcasting [6, 7], multicasting [8, 9], and topology control [10, 11]. Of course, similar approaches have been also considered in wireless sensor networks [12–15].

An alternative approach commonly adopted in sensor networks is based on scheduling sensor activity so that some nodes may enter the power saving mode while the remaining active nodes can still provide continuous service [14, 16]. For instance, if all the sensor nodes simultaneously operate in active mode, an excessive amount of energy is wasted and the data collected is highly correlated and redundant. In addition, multiple packet collisions may occur when all the sensors in a certain region try to transmit as a result of a triggering event. Several research results [3, 16] illustrate that a mode of operation alternating active and inactive battery states has a significant reduced energy consumption.

However, such scheduling schemes may face new constraints about sensing coverage introduced by their distributed sensing applications [17]. For example, surveillance applications may require each location of monitored regions to be covered by at least one sensor, while many stronger environmental monitoring, such as military applications, require multiple sensors for fault-tolerant purpose. Besides, triangulation positioning-based tracking applications [18, 19] may require at least three sensors at any locations. Data sampling applications may require a given percentage of monitored regions to be covered.

Therefore, this paper considers the scheduling approach that extends the network lifetime by minimizing the number of active nodes while maintaining coverage constraints. As mentioned earlier, the advantage of this approach is that less packet collisions may occur since less spatially close sensors try to transmit highly correlated and redundant information as a result of a triggering event. Hence, the lifetime of each sensor cover may be extended.

We model a sensor network as a 2D geometric connected region monitored by a set of deployed sensor nodes with unit-disk sensing regions, a realistic assumption that is well adopted in many network models. The coverage constraint is that each location of the 2D region is covered by at least one active node. Note that optimum sensor cover problems may be solved by partitioning monitored regions into disjoint sectors [20, 21]. Here a sector is a maximum region covered by the same set of nodes. Hence the minimum sensor cover problem is transformed to the Minimum Set Cover (MSC) problem which is NP-complete [22] and has been studied extensively in the literature [23–26]. However, with the additional unit-disk sensing region and geometric connected monitored region restrictions, the problem considered here is only a restricted version of MSC, and, to the best of our knowledge, its complexity is still unknown. (That is, it is not trivial to transform each instance of MSC to an instance in the minimum sensor cover problem with unit-disk sensing regions over a connected monitored region in polynomial time.) Thus, we will answer this fundamental question and present the formal proof of its NP-completeness. Furthermore, we illustrate several related optimum problems under different coverage constraints and show their hardness results as a corollary.

Next, we propose the arc sampling algorithm which may effectively and efficiently reduce MUDC to MSC. Consequently, many well-known algorithms can be applied to find approximated solutions. In addition, we present a decentralized scalable algorithm with a guaranteed performance, and a constant approximation factor algorithm if the maximum node density is fixed. Finally, we illustrate simulation results to evaluate the proposed algorithms.

2. Related Work

2.1. Coverage Problems

Meguerdichian et al. [17] defined the coverage problems from several different application domains including deterministic, statistical, worst, and best cases. They also presented optimum polynomial time algorithms to evaluate paths that are the best and least monitored in the sensor network. The work in [27] further defined the exposure problem as measure of how well an object can be observed by the sensor network while it moves along an arbitrary path with an arbitrary velocity. A localized exposure-based coverage algorithm was proposed in [28] for finding the minimal exposure path between two points.

Furthermore, Gui and Mohapatra [29] considered the object tracking applications in which networks operate between surveillance state and tracking state. During surveillance state, they devised a set of metrics for quality of surveillance for detecting moving objects and quantify the trade-off between power conservation and quality. They also proposed an algorithm for each node to determine when to wake up or sleep during the tracking stage.

Tian and Georganas [30] developed a coverage-preserving scheduling scheme to reduce energy consumption by turning off some redundant nodes based on some eligibility rules. Carbunar et al. [31] proposed distributed algorithms for detecting and eliminating redundancy in a sensor network while preserving the network's coverage via Voronoi diagrams, even in cases of sensor failures or insertion of new sensors.

Huang and Tseng [32] considered the -coverage problem to determine whether every point in the monitored region is covered by at least nodes. They reduced this problem to the perimeter-coverage problem which determines the coverage degree of the perimeter of each node's sensing region and presented polynomial-time algorithms in the number of nodes.

In addition to coverage, connectivity also needs to be assured to make sensor networks successfully. It has been shown in [33] that if the communication range of sensors is at least twice as large as their sensing range, then full coverage of a convex region implies connectivity. Wang et al. [34] presented a Coverage Configuration Protocol (CCP) that allows the network to self-configure dynamically to achieve guaranteed degrees of coverage and connectivity.

2.2. Minimum Sensor Cover Problems

In [21], Funke et al. proposed the greedy sector cover algorithm which selects a node that covers the maximum number of uncovered sectors at each iteration step. That is, the problem is reduced to MSC and solved by the greedy algorithm. They proved that the well-known approximation factor remains tight in this restricted version. Here is the maximum number of sectors covered by a single node. To obtain better approximation factors, they also presented a grid placement algorithm and a distributed dominating cover algorithm. These two algorithms have constant approximation factors, but cannot guarantee the full coverage.

Gupta et al. [35] designed an centralized approximation algorithms with the connectivity constraint. Here is the number of sensor nodes. In their definition of the sensor cover problem, the sensing region can take any convex shape. They also mentioned that such a problem is NP-hard as the less general problem of covering discrete points using line segments is known to be NP-hard [36]. On the other hand, the sensing region considered in this paper is restricted to a unit-disk, which is well adopted in many network models. We will prove such a problem remains NP-complete even with the unit-disk restriction. They also proposed a distributed algorithm based on node priorities, but did not provide any guarantee on the solution size.

2.3. Related Optimum Problems

Fowler et al. [37] proved the -completeness of the Box Cover problem which aims at finding the minimum number of identical rectangles to cover a set of given points. Similarly, Megiddo and Supowit [38] considered the Circle Covering problem which is equivalent to the Geometric Disc Covering problem, that is, to find the minimum number of identical disks to cover a set of given points. There are two fundamental differences between MUDC and these two problems. In these two problems, the covered object is a set of discrete points but not a connected region. Hence, in the proofs of the NP-completeness, we have less flexibility in the connected case than in the discrete cases, since we need to ensure the monitored region is connected while constructing a problem instance. Furthermore, the two problems have the flexibility to determine the “good” locations of covering objects (rectangles or disks), which could be anywhere on the plane. On the other hand, in MUDC, the locations of disks are pre-deployed.

Marathe et al. [39] considered several basic optimization problems for unit-disk graphs with hierarchical structures. They presented a general technique to prove the hardness results of several problems. The hardness of these problems, including Box Cover and Circle Covering, was proved via satisfiability problems. The reduction strategy was to use some geometric structures to represent variables. Each clause is represented by a special structure that “glues” the corresponding structures of the variables in the clause.

There are several polynomial approximation algorithms [40–42] for the Geometric Disc Covering problem. Franceschetti et al. pointed out in [43] that the number of possible disk positions can be bounded if any disk that covers at least two points has two of these points on its border. Hence, by performing a search on a subset of the possible disk positions, the running time of these algorithms becomes polynomial and the solution sizes are guaranteed. They also gave a detailed comparison of these algorithms in [44].

3. Preliminaries

In this section, we define the Minimum Unit-Disk Cover (MUDC) problem that aims at finding the least number of nodes with unit-disk sensing regions to fully cover a designated connected region. We prove this problem is intractable; that is, it belongs to the NP-complete class.

Definition 1. Consider a two-dimensional Euclidean metric space , the unit-disk sensing region of a given node is defined as . (In the context of discussing MUDC, we represent a node by its geometric location without any confusion.) Here is the distance in Euclidean metric between and . Furthermore, the unit-disk sensing region of a set of nodes is defined as .

Definition 2. A two-dimensional finite region is said to be unit-disk covered by a set of nodes in a two-dimensional Euclidean metric space if . Furthermore, is called a unit-disk cover (UDC) of .The objective is to find the minimum unit-disk cover (MUDC) of . Note that the optimum problems discussed in this paper could be solved by their associated decision problems in polynomial time. Therefore, we discuss the decision version of MUDC instead, and it can be formally stated in the following.

Problem 3 (MUDC). Given a set of nodes in a two-dimensional Euclidean metric space , a two-dimensional geometric connected finite region , and a positive integer , determine whether there is a subset with such that . Here is the cardinality of .

For simplicity's sake, the geometry of an MUDC problem, that is, the region and the set of nodes, is denoted as .

Thus, we will prove the following theorem.

Theorem 4. MUDC is NP-complete.

The NP-completeness of MUDC will be proved by reduction from the Planar 3-SAT problem, which is known to be NP-complete [45].

Problem 5 (Planar 3-SAT, P3SAT). Given a set of variables and a set of clauses over such that each has (denoted as a boolean formula ) determine whether there is an assignment for the variables so that all clauses are satisfied. (In Lichtenstein's NP-completeness proof of P3SAT [45], an instance of 3SAT is transformed to an instance of P3SAT with being 2 or 3. Thus, the restriction, , does not change the complexity of the problem. This restriction is required to prove Lemma 17.) Furthermore, the bipartite graph (in this NP-completeness proof of MUDC, will be used to construct an equivalent MUDC problem for ) is planar, where . (We remove the edges without any changes in the difficulty of the problem [46].)

That is, let be a boolean formula in P3SAT with clauses and variables. We wish to construct an equivalent MUDC problem with the geometry MUDC where and are the region and the set of nodes transformed from , respectively.

Inspired by Lichtenstein's NP-completeness proof of the Geometric Connected Dominating Set problem [45], MUDC is constructed via structures. Each structure is a geometry containing a polygon and a set of nodes and denoted as . Variables, clauses, and edges of are represented by various structures. Hence, MUDC is constructed from by replacing variables, clauses, and edges with their corresponding structures.

Each structure is constructed in such a way that can be partitioned into two disjoint subsets of equal size, denoted as and , and the MUDC of , except the ones representing clauses, is either or . Thus, a variable is assigned true corresponding to that is the MUDC of ; false corresponds to . For convenience throughout this paper, we assign each node a polarity. The node has positive polarity if or negative polarity if . The property of structures mentioned above can be formally defined in the following.

Definition 6. One calls a structure well aligned if and .

Definition 7. For a structure and , we call partially well behaved on , if the following preconditions hold:(i)(ii)if is a UDC of , (iii)if is an MUDC of , or . (For a given set of nodes , we denote the set of nodes with the same polarity as with superscripts + or − throughout this paper. i.e., and .)

Furthermore, we call well behaved if is partially well behaved on .

Note that, from the above definition, if is well behaved and is an MUDC of , then and only contains the nodes with the same polarity.

The NP-completeness proof of MUDC will proceed as follows.(1)Describe structures representing variables, edges, and clauses. These structures have the properties defined in Definitions 6 and 7.(2)Describe how the above structures may be connected together to represent while preserving the properties defined in Definitions 6 and 7. Here the resulting composite structure is MUDC.(3)We claim that is satisfiable if and only if can be covered by half the nodes of . In the proof of the claim, the properties defined in Definitions 6 and 7 will be used in the forward direction and the backward direction respectively.

4. NP-Completeness Proof of MUDC

We first prove that belongs to the class. This could be done since a nondeterministic algorithm needs only guess a set of nodes, , and verify whether . Besides, as stated in [32], this verification could be done in .

We continue the proof by reduction from the Planar 3-SAT problem. Let be a boolean formula in P3SAT with clauses and variables. We wish to construct an equivalent MUDC problem with the geometry MUDC transformed from the bipartite graph .

4.1. Structures

We encode each variable by the structure, denoted as , shown in Figure 1. represents the shaded region which is a rectangle. represents the set of the nodes which are positioned accordingly and used to cover . Each node has a either positive or negative polarity and is represented by a square or triangle, respectively, in the figure. The may be long enough to prevent unwanted interactions between nearby edge structures.

Figure 1: The structure representing a variable.

Next, we may encode an edge by a strip-like structure, denoted as , which may extend horizontally and vertically. Figure 2 illustrates an example. The shaded region, denoted as , is composed of rectangles. represents the set of the positive and negative polar nodes which are positioned accordingly and used to cover .

Figure 2: The structure representing an edge.

It is not hard to prove the structures and satisfy the following lemma.

Lemma 8. The structures shown in Figure 1 and shown in Figure 2 are well aligned and well behaved.

Proof. The lemma may be proved by induction. For the sake of brevity, the complete proof is given in Appendix A.

Each clause may be represented by a structure, called an -way connector and denoted as . Here and could be the value of 2 and 3. Figure 3 illustrates the possible realization of -way connectors. represents the shaded polygon that will be covered by the set of the nodes, . The geometries of and relative positions of nodes of are shown in Figures 3 and 4 and Tables 1 and 2.

Table 1: The geometry of the 2-way connector shown in Figure 3(a).

Node
Position
Vertex
Position

Table 2: The geometry of the 3-way connector shown in Figure 3(b).

Node
Position
Vertex
Position

(a) 2-way connector

(b) 3-way connector

Figure 3: The structures representing clauses.

(a) 2-way connector

(b) 3-way connector

Figure 4: The labels of .

Furthermore, is divided into disjoint partitions with and we denote . As shown in Figure 3, each node is labeled as an alphabet and a numeral. The nodes with the same numeral belong to the same partition. The alphabets represent the relative polarity of nodes within the same partition. For example, in Figure 3(b), the nodes labeled as C2 and D2 belong to . The two nodes labeled as C2 have the same polarity and have the opposite polarity with the two nodes labeled as D2.

Each partition has a header node which is indicated as a dark circle in Figure 3. We can define the polarity of the partition as the polarity of its header node . The set of header nodes is denoted as . Each partition corresponds to a variable of a clause. What -way connector should represent a clause depends on how the clause is formed from variables. For example, if is composed of two positive literals and one negative literal, then it should be represented by a 3-way connector that has two positive partitions and one negative partition as shown in Figure 7. As described later, an edge and its ends in will be transformed by connecting a variable structure to a partition of an -way connector via an edge structure. Thus, a partition may be viewed as “extended territory” of a variable. (The definition of territory is given in Definition 16. Again, the key point to transform to MUDC is to preserve the properties defined in Definitions 6 and 7 for territories.)

The main reason why an -way connector is constructed in this way is the following lemma.

Lemma 9. Each -way connector, , of Figure 3 has the following properties. (i), is partially well behaved on . (For an MUDC of , the active nodes of each partition have the same polarity. Thus, a variable of can be assigned true or false based on the polarity of the active nodes in its corresponding partition. Here, in the context of discussing a given UDC, we call a node active if it is in the UDC.)(ii)If is a UDC of , then . (At least one header node must be active for covering .)(iii), if . Here contains either all positive polar nodes () or all negative polar nodes (). That is, . (In other words, if the active nodes of each partition have the same polarity and one of them is a header node, then is covered.)

Proof. The idea is to examine each possible case of partition and ensure will not be covered if, for each , less than nodes are active or if exactly nodes but not having the same polarity are active. Furthermore, we need to examine whether will be covered if none of header nodes is active. The complete proof is given in Appendix B.

Note that, from Lemma 9(ii), at least one header node must be active for covering . Thus, together with Lemma 9(i), an MUDC of will allow each variable of a clause to be assigned true or false, based on the polarity of active nodes in its corresponding partition, for the clause being satisfied.

4.2. Composite Structures

Next, we illustrate how structures may be connected together to form a complex structure. As shown in Figure 5, two structures, and , are connected via several squares called connection patches. Each connection patch has two nodes from each structure, for example, and , located at its vertices. (This is formally defined as precondition 25(i). For convenience sake, we use precondition 25(i) for referring to precondition (i) of Definition 25, and will use this labeling throughout this paper.) Besides, the nodes on the same edge of each connection patch have opposite polarities. (This is formally defined as preconditions 25(ii) and 26(ii).) For the sake of brevity, the formal definitions are given in Appendix C. We call the set a port of , and a port is a connected port if there is a connection patch attaching to it. Besides, the nodes from different structures but on the same edge of a connection patch are each other's connection counterpart, for example, and . Obviously, it is easy to derive the following lemma.

Figure 5: Connection patches for connecting two structures.

Lemma 10. A connection patch can be unit-disk covered by the same polar nodes located at its vertices, for example, or in Figure 5.

In this -completeness proof, in order to preserve the partially well-behaved property, we require two structures to be in such a way, that is, least interactively connected, that(1)at least one node from each connected port is active, (this is formally defined as precondition 27(i));(2)nonconnected port nodes do not cover any point, except the vertices, of the connection patches (this is formally defined as precondition 27(ii) which states whether a connection patch can be fully covered only depends on its connected port);(3)nonconnected port nodes of one structure do not cover any region of the other structure, (this is formally defined as precondition 28(i));(4)the connected ports of one structure cannot cover any point, except their connection counterparts, of the other structure (this is formally defined as precondition 28(ii));

The formal definitions about the least interactive connection are also given in Appendix C.

A variable structure can use any two nearby nodes on the side of border as a port, for example, shown in Figure 1. An edge structure uses its endpoints as ports, indicated by the arrows in Figure 2. For an -way connector, each partition contains a port indicated by the arrows in Figure 3. The fact that it is possible to make the above structures least interactively connected via the ports described is proved in Appendix D.

We can define the composite structure in the following definition and derive several lemmas about the least interactive connection. For the sake of brevity, the complete proofs of these lemmas are given in Appendix E.

Definition 11. The structures and are least interactively connected via the connection patches . One calls the composite structure of and . Furthermore, one denotes .

Lemma 12. Suppose and the cardinality of MUDCs of and is and , respectively. If is a UDC of and , then and are MUDCs of and , respectively.

Lemma 13 (Connection Lemma). Consider , , and via one connection patch at the ports and . (We note that this lemma may be extended to more than one connection patches with possible minor modification on Definition 28(ii); however we do not use this property and therefore ignore it.) Furthermore, for any with and with , will be partially well behaved on if the following connection preconditions hold. (i) and are partially well behaved on and , respectively.(ii)There exist an MUDC of , , and an MUDC of , , such that and have the same polarity. (Since and have the same polarity, by Lemma 10, the connection patch is automatically covered without the help of the nodes not in and . Hence, is an MUDC of .)

After describing the properties of the least interactive connection, we describe how these structures may be connected to encode .

Figure 6 illustrates how edge structures are connected to a variable structure with connection patches. The edges may go up or down from the variable structure.

Figure 6: The structure represents a variable with two edges going down and one edge going up. Note that the distance of two nearby ports on the same side must be at least 2 to ensure the least interactive connection.

Figure 7: The variable structure is connected to the positive partition of the 3-way connector, which represents the clause containing two positive literals, and , and one negative literal, . Note that the 3-way connector has two positive partitions and one negative partition. The header nodes are indicated by squares or triangles surrounded by circles.

With -way connectors described above, we can transform an edge and its ends in by connecting a variable structure to a partition of an -way connector via an edge structure. For example, suppose a clause is , then we need a 3-way connector with two positive partitions and one negative partition. As shown in Figure 7, we connect the edge from to one positive partition, the edge from to the negative partition, and the edge from to the other positive partition.

Since all structures are connected via squares, the positions of structures must match, that is, all structures can be placed in a 2D space such that their ports are on a 2D grid with resolution . Obviously, such a placement is possible for variable and edge structures. Furthermore, as indicated in Tables 1 and 2, the relative positions of the ports of each -way connector are integers, so it is also possible to make such a placement for -way connectors.

Note that, in addition to positions, polarities of ports must also match. (Precondition 26(ii) needs to be satisfied.) However, for example, it is possible that the polarities may not match when an edge structure is connected to a connector as shown in Figure 8(a).

(a) The edge's polarities do not match with the connector's

(b) The polarities are inverted to match with the connector's

Figure 8: A polarity inverter structure may be needed to invert the polarity of the edge to connect the edge and 3-way connector.

Therefore, referring to Figure 9, we introduce a structure called a polarity inverter and denoted as to invert the polarity. contains a main structure, rectangles, and two buffers, squares. (The purpose of the buffers is to ensure satisfies the requirement that nonconnected port nodes do not cover any point, except the vertices, of the connection patches, that is, precondition 27(ii).) Similar to edge structures, a polarity inverter use its endpoint pairs, that is, and in Figure 9, as ports.

Figure 9: A structure for inverting the polarity of the edge. After distance 11, the polarities are inverted compared with a normal edge.

The distance between two nearby column-pairs of the main structure is . Thus, the polarities are inverted compared with a normal edge of the same length. With the polarities inverted, the polarity requirements for vertices of a connection patch can be satisfied as shown in Figure 8(b).

Of course, the structure also satisfies the following lemma. The proof is similar to variable structures and is given in Appendix F.

Lemma 14. The structure shown in Figure 9 is well aligned and well behaved.

Definition 15. For a given variable structure , an -way connector is called an associated connector of if is connected to via an edge structure. Furthermore, the partition of where is connected to is called an associated partition of . Similarly, the edges, polarity inverters, and connection patches used to connect and are called associated connection edges, associated polarity inverters, and associated connection patches of , respectively.

For the th variable, , let: be the set of nodes in the corresponding variable structures , : be the set of nodes in all the associated edge structures of , : be the set of nodes in all the associated polarity inverters of , : be the set of nodes in all the associated -way connectors of , : be the set of nodes in all the associated partitions of , : be the shaded region of , : be the union of the shaded region from all the associated edge structures of , : be the union of the shaded region from all the associated polarity inverters of , : be the union of the shaded region from all the associated -way connectors of , : be the union of the shaded region from all the associated connection patches of .

Definition 16. Let , , and . We call the composite structure the territory of the th variable. Furthermore, let . The nodes in are the pieces of the th variable. Note that , and thus .

Note that represents the th variable and all clauses it belongs to as shown in Figure 10. It is not difficult to layout each structure on the plane and make variable structures and connectors far enough to prevent unwanted interactions between nearby structures, that is, structures are least interactively connected. Consequently, we have the following lemma which states that, for a given variable, its territory is partially well behaved on the set of its pieces.

Figure 10: The territory, , of the variable includes variable structure and all associated edge structures, patches, inverters, and connectors. The nodes enclosed in dashed line are pieces of .

Lemma 17. If all the associated structures of the variable structure are least interactively connected, the territory is partially well-behaved on .

Proof. Since structures are least interactively connected, this lemma can be proved by Lemmas 8, 9(i), 14, and Connection Lemma. The fact that for each clause is the key to ensure precondition (ii) of Connection Lemma is satisfied for Connection Lemma being applicable. The complete proof is given in Appendix G. After introducing the structures and their properties, we can define an equivalent MUDC problem with the geometry, MUDC(), for a given boolean formula, , in P3SAT by replacing the variables, clauses, and edges of the bipartite graph with their corresponding structures. Denote that: is the set of nodes in all the variable structures, : is the set of nodes in all the edge structures, : is the set of nodes in all the polarity inverters, : is the set of nodes in all the -way connectors, : is the union of the shaded region from all the variable structures, : is the union of the shaded region from all the edge structures, : is the union of the shaded region from all the polarity inverters, : is the union of the shaded region from all the -way connectors, : is the union of the required connection patches. Let with and , and . Hence, we have the following claim. Note that it is not difficult to prove that the construction from to MUDC() can be done in polynomial time.

Claim B is satisfiable if and only if MUDC() has a UDC with cardinality .

Note that the key to the backward direction of the proof is Lemma 17; that is, for a given variable, its territory is partially well behaved on the set of its pieces. Lemma 17 will be used to derive, that if MUDC() has a UDC with cardinality , then, for each variable, half of its pieces with same polarity needs to be active to unit-disk cover its territory. Therefore, each variable could be assigned true or false based on the polarity of its active pieces.

4.3. Proof of the Claim

Proof. For each , choose the nodes with the polarity which is the same as the assignment of the th variable in a given satisfying instance of . Obviously, only nodes are picked. By Lemmas 8, 14, and 10, , , , and are covered. Furthermore, since is satisfiable, at least one header node of each -way connector is active. By Lemma 9(iii), is covered. Thus MUDC() has a UDC with cardinality . Let MUDC() have a UDC with . We will show that this set must look right.From Lemmas 8, 9(i), and 14, we know that the cardinality of an MUDC of a variable structure, edge structure, -way connector, or polarity inverter is half the number of the nodes in the structure. Since all structures in are least interactively connected and , it is not difficult to derive that, for the th variable structure, is an MUDC of by removing nonassociated edge structures of one by one and Lemma 12.From Lemma 17 and , only contains the nodes with the same polarity. Thus, the th variable could be assigned true or false based on the polarity of the nodes in . Finally, since at least one header node of each -way connector must be active from Lemma 9(ii), the corresponding clause will be true.

5. Extensions of MUDC

MUDC can be easily extended to the following two more general cover problems, which require each location to be unit-disk covered by predefined number of nodes. These problems regard the quality of various services of sensor network applications such as surveillance, object tracking, and fault tolerance.

Problem 18 (Minimum Unit-Disk -Cover, MUDKC). Given a geometry and two positive integers and , determine whether there is a subset with such that ; that is, is unit-disk covered by at least nodes in .

Problem 19 (Minimum Unit-Disk Multicover, MUDM). Given a geometry , a quality of surveillance function , and a positive integer , determine whether there is a subset with such that ; that is, is unit-disk covered by at least nodes in .

We may also consider connectivity and have the following problem.

Problem 20 (Minimum Connected Unit-Disk Cover, MCUDC). Given a geometry , a positive number , and a positive integer , determine whether there is a subset with such that and the graph is connected. Here .

Furthermore, under many environmental data sampling applications, instead of full coverage, a predefined percentage of coverage is required for achieving energy efficiency and preciseness of sampling. The objective of the following problem is to find as few nodes as possible to achieve the coverage requirements.

Problem 21 (Minimum Unit-Disk Partial Cover, MUDPC). Given a geometry , a positive number with , and a positive integer , determine whether there is a subset with such that . Here the function gives the area of a given region. By the NP-completeness of MUDC, it is easy to derive the complexity of the above problems.

Corollary 22. MUDKC, MUDM, MCUDC, and MUDPC are NP-complete.

Proof. Note that every instance of MUDC can be viewed as an instance of MUDKC, MUDM, or MUDPC simply by letting , , or , respectively. Thus MUDC is just a restricted version of these problems, and their NP-completeness follows by trivial transformations from MUDC.For the geometry, MUDC(), described in the NP-completeness proof of MUDC, it is obvious that, for any UDC of , the distance between an active node and its closest active node is less than 2. Thus, it is not difficult to prove that if , is connected. That is, MUDC is just a restricted version of MCUDC with , and the -completeness of MCUDC follows by a trivial transformation from MUDC.

6. Arc Sampling Algorithm for Reducing MUDC to MSC

As stated earlier, MUDC may be solved by partitioning the region into disjoint sectors [20]. Consequently, MUDC is reduced to MSC and many well-known algorithms can be applied, for example, the greedy algorithm is the best approximation algorithm and the approximation factor is well known.

To identify necessary sectors is a key factor to whether the solutions found by the algorithms for the transformed MSC are valid, that is, the solutions are disk covers of the original MUDC. A naive approach for partitioning is to sample at uniform spacings in a grid pattern; then the sampling points covered by the same set of nodes would be grouped into one sector. With enough resolution, all necessary sectors can be successfully identified at the expense of computation time.

However, to determine a good resolution may be difficult. For example, Figure 11 shows that inappropriately increasing resolution may not necessarily find a valid solution, and Figure 13(a) illustrates that the ratio of successfully finding a valid solution decreases as the node density decreases. Therefore, we propose an arc sampling approach which is inspired by the theorem of the paper [32], that is, is covered if and only if the perimeter of each node's sensing region is covered. (Several special cases including boundary are also discussed in [32].)

Figure 11: The solution found at different sampling intervals. Increasing resolution may not necessarily find a valid solution.

Consider the node with its neighbors, that is, the nodes with distance not greater than 2 from . As illustrated in Figure 12, 's perimeter is divided into disjoint arcs by its neighbors' perimeters and the boundary of . It is obvious that all points of each disjoint arc in are covered by the same set of nodes. Thus, we can simply choose a point such as the midpoint from each arc in , for example, from arc , as a sampling point. Note that if 's perimeter cannot be divided, 's perimeter (and thus ) is not covered [32], as indicated in Lines 7~9 of Algorithm 1. From the earlier mentioned theorem of the paper [32], it is easy to derive that is covered if and only if all these sampling points are covered. Thus, the solutions of the transformed by this arc sampling approach are always valid.

Input: ()

Output: a set of sampling points

  (1)

  (2) for all   do

  (3)     

  (4)     for all   such that   do

  (5)        

  (6)       end for

  (7)        if   then

  (8)             exit is not covered by . *

  (9)     end if

(10)     List the points in sorted by their azimuth

            angles on the polar coordinate system

            with reference to

(11)    for all   such that   do

(12)

(13)    end for

(14)

(15)end for

Algorithm 1: Arc sampling algorithm.

Figure 12: Each arc in , that is, , , or , is covered by the same set of nodes. Hence, we can simply use a point, for example, the midpoint marked as a cross, from each arc as a sampling point. Here represents the boundary of .

(a) The ratio of successfully finding a solution with different node density: the greedy algorithm with the arc sampling approach always generates a valid solution

(b) The arc sampling approach is as effective as the grid sampling approach

(c) The arc sampling approach requires less computation time to reduce MUDC to MSC than the grid sampling approach except the densest deployment

Figure 13: The experiment results show that the arc sampling approach is effective and effcient for reducing MUDC to MSC.

The arc sampling algorithm is shown in Algorithm 1. Here is the perimeter of . The outer loop between Lines 2 and 15 will run times. The average time complexity for a node to find all its perimeter intersections with neighbors, that is, Lines 4~6, is . Here is the density of nodes. Note that each node has average neighbors and thus disjoint arcs. Hence, the sorting in Line 10 could be implemented in time. In addition, the time complexity of finding the midpoints, that is, Lines 11~14, is . Thus the overall time complexity of Algorithm 1 is or . On the other hand, it is not difficult to derive that the time complexity of finding all sampling points is for the grid sampling approach. Here is the area of each grid. Together with Figure 13(a), we may conclude that the arc sampling approach will perform more efficiently than the grid sampling approach, particularly with low density of nodes.

We conducted an experiment to compare the grid sampling and the arc sampling approaches. In the experiment, there are 240 nodes deployed uniformly in a square region ranging from to . The radius of the sensing range is 10. The sampling interval of the grid sampling approach is 0.1. After transforming to , the greedy algorithm is used to find the approximated solution. Each result is the average of 100 random deployments.

Figure 13(b) shows the effectiveness of the arc sampling approach. The solutions from both approaches have almost same sizes, that is, same number of nodes. However, Figure 13(a) shows that not every solution obtained from the grid sampling approach is valid. Figure 13(c) illustrates that the arc sampling approach requires less computation time to reduce MUDC to MSC than the grid sampling approach except the densest deployment. Thus, the arc sampling approach is effective and efficient for reducing MUDC to MSC.

7. Decentralized Polynomial Approximation Algorithms

Algorithm 1 and the greedy algorithm may not be suitable for all practical sensor network applications, since it is a centralized algorithm at the cost of potentially excessive communication across the whole network and communication accounts for the majority of energy consumption. The communication power consumption increases with number of nodes and internode distances, so it is not well scalable. Unless the nodes involving in the communication and computation have enough resources, the algorithm may not complete successfully.

Thus, we present a decentralized algorithm in which nodes only require local information by using the divide and conquer technique described in [42] and derive its approximation factor. Furthermore, if the maximum node density is fixed, we may design a constant approximation factor algorithm by using the similar technique. (Note that MUDC remains NP-complete even with fixed maximum node density. It could be easily proved from the fact that the density of MUDC() in the NP-completeness proof of MUDC is bounded by a constant.)

7.1. Decentralized Greedy Approximation Algorithm

The proposed algorithm for the instance proceeds as follows.(1)Use Algorithm 1 to determine sampling points.(2)Divide into vertical strips with width being the diameter of the sensing region, that is, 2. Each strip is left closed and right open. Number strips from left to right. There are total strips.(3)Divide each strip into cells with length being the diameter of the sensing region. Each cell is bottom closed and top open. Number cells from bottom to top. Denote the th cell of the th strip as . There are total cells for each strip.(4)Apply the greedy algorithm to each cell, that is, select a node that covers the maximum number of uncovered sampling points in the cell. Denote the solution of as . (5)Output the solution .

Figure 14(a) illustrates that is divided into cells. Note that each sampling point is located in exactly one cell. This algorithm requires that geometric information of and cells are known a priori to each node and each node's location can be determined after deployment.

(a)

(b)

Figure 14: Divide and conquer: (a) The monitored region (enclosed in heavy line) is divided into cells. (b) (enclosed in heavy line) and its repository (enclosed in thin line)—the solid lines and dashed lines represent closed and open boundaries. Note that the distance between any two points in does not exceed.

For each cell , we define its repository . As illustrated in Figure 14(b), is the region containing all nodes that may cover the sampling points in . Hence, in Step 4, the greedy algorithm is applied to the nodes in each cell's repository and can be implemented in , where is the maximum number of nodes in a repository. Furthermore, Figure 14(b) also illustrates that a node does not need to communicate with others further than times of the sensing radius. Thus, this approach is more scalable than the centralized greedy algorithm.

Theorem 23. The above algorithm has an approximation factor . (Though can be written as by definition, we explicitly write it out to emphasize the approximation factor of the decentralized algorithm is four times the approximation factor of the centralized algorithm.) Here is the maximum number of sampling points covered by a single node.

Proof. The theorem is the result of the shifting lemma in [42]. The proof proceeds as follows.For , denote that is the optimum solution to cover the sampling points in . That is, is the optimum solution for , is the optimum solution for the th strip, and so forth. Thus, from Step 4, . Here is the maximum number of sampling points in covered by a single node.Consider the th strip, and define the following disjoint subsets of : be the set of nodes that only cover the sampling points in , be the set of nodes that cover both the sampling points in and . Note that since the length of cells is the diameter of the sensing region, the union of the above disjoint subsets is . Hence, . Besides, it is obvious that covers all sampling points in . (Here and .) Thus, . Therefore, it can easily be derived that

Similarly, it can easily be derived that

and then

Consequently,

Note that, obviously, .

7.2. Constant Approximation Factor Algorithm with Fixed Maximum Density

When the maximum node density, denoted as , is fixed, the similar divide and conquer technique can be used to derive a constant approximation factor algorithm.

The algorithm is almost the same as the previous one except Step 4, which will be modified as follows:

(4) Apply an exhaustive search for optimum solution to each cell. Denote the solution of as .

Theorem 24. The above algorithm has a constant approximation factor 4.

Proof. Note that the number of nodes in each cell's repository is at most . (Refer to Figure 14(b); the area of each repository is .) Thus, the time complexity of the exhaustive search is at most for each cell. Since is fixed, an optimum solution for each cell can be found with a constant time complexity. Since , from (3), we have

7.3. Performance Evaluation

We conducted various simulations to evaluate the proposed algorithms. In Figure 15, nodes are deployed uniformly within a square region. Figures 15(a) and 15(b) show the solution size and execution time for various sensing ranges in which there are 25 nodes. The optimum solution is found by exhaustive searching. denotes the solution by using Algorithm 1 and the greedy algorithm. and represent the algorithms described in Sections 7.1 and 7.2 respectively. The cover size decreases as the sensing radius increases, since each node can cover a larger region. and have similar performance in terms of cover size and execution time. Furthermore, generates the largest cover size, on average more than , more than , or more than , but requires the least execution time, on average of , of , or of .

(a)

(b)

(c)

(d)

Figure 15: Performance evaluation: the nodes are deployed uniformly within a square region.

Figures 15(c) and 15(d) indicate the solution size and execution time for various number of nodes in which the sensing radius is fixed at 10. The cover size does not change significantly as the number of nodes increases, since the sensing region of each node does not change. and have similar cover size, but requires more execution time than . Similarly, generates the largest cover size, on average more than , more than , or more than , in the least time, on average of , of , or of .

We also considered the scenario in which nodes are deployed in a Gaussian distribution with the peak located at the center of and the variance 15, and the results are illustrated in Figure 16. Here is a square region. In Figures 16(a) and 16(b), the number of nodes is 25 and the sensing radius varies between 8.5 and 12. generates the largest cover size, on average more than , more than , or more than , but requires the least execution time, on average of , of , or of .

(a)

(b)

(c)

(d)

Figure 16: Performance evaluation: the nodes are deployed in a Gaussian distribution with the peak located at the center of A and the variance 15. Here A is a 30×30 square region.

Furthermore, in Figures 16(c) and 16(d), the sensing radius is fixed at 10 and the number of nodes varies between 12 and 30. Similarly, generates the largest cover size, on average more than , more than , or more than , in the least time, on average of , of , or of . For most of cases, has smaller cover sizes than in this scenario.

8. Conclusion

In this paper, we consider the complexity of MUDC, the Minimum Unit-Disk Cover problem. This problem has applications in extending the sensor network lifetime by selecting minimum number of nodes to fully cover a geometric connected region of interest and putting the remaining nodes in power saving mode. MUDC is a restricted version of MSC where the sensing region of each node is a unit-disk and the monitored region is geometric connected, a well-adopted network model in many works of the literature.

To prove the hardness of MUDC, we construct various structures to represent variables and edges of a given P3SAT instance's bipartite graph . With the well-aligned and partially well-behaved properties of these structures, we illustrate that the structures can be unit-disk covered with half of nodes. Furthermore, we introduce the -way connectors to represent clauses, which can be unit-disk covered with half of its nodes if and only if the corresponding clauses have true assignments. Finally, we discuss how complex structures can be constructed by connecting simpler structures while still preserving these properties, that is, via the least interactive connection. Thus, we prove that P3SAT can be directly reduced to MUDC in polynomial time, and obtain the NP-completeness proof of MUDC.

We also discuss several optimum problems with various coverage constraints introduced by different sensing applications. These problems are extensions of MUDC, and their NP-completeness proofs are presented as a corollary.

We propose the arc sampling algorithm which may effectively and efficiently reduce MUDC to MSC, and many well-known algorithms can be applied to find approximated solutions. We also propose a decentralized algorithm with a guaranteed performance. The algorithm requires only local communication, that is, a node does not need to communicate with others further than times of the sensing radius. Thus, this approach is scalable. Furthermore, we present an algorithm with a constant approximation factor 4 if the maximum node density is fixed. Finally, we provide simulation results to evaluate the proposed algorithms and the optimum algorithm in uniform and Gaussian deployment networks. The results show that may have smaller cover size than at the cost of more execution time. In addition, generates the largest cover size in the least time.

Appendices

A. Proof of Lemma 8

In this appendix, we present the proof for Lemma 8.

A.1. Variable Structures

Obviously, the structure shown in Figure 1 has the same number of positive and negative polar nodes and, thus, precondition 7(i) is satisfied.

We call the pair of opposite polar nodes in the th column the column-pair . From Figure 17, it is not difficult to prove that each column-pair must have at least one active node to fully cover . Since there are columns, for a UDC of ; that is, the precondition 7(ii) is satisfied. Besides, if we can pick exactly one node from each column-pair and the resulting set, say , can-unit-disk cover , then is an MUDC since .

Figure 17: The shaded region cannot be covered while none of nodes from the column-pair is active.

It is easy to prove that, if the picked nodes from each column-pair do not have the same polarity, cannot be unit-disk covered by these picked nodes. Suppose that the picked nodes of the th column and the th column have opposite polarities. From the Figure 18, cannot be covered.

Figure 18: The shaded region cannot be covered by two nearby opposite polar nodes.

Thus, the only possibility to cover with nodes is to pick the nodes with the same polarity from each column-pair, which can be easily proved by induction. Figures 19(a) and 19(b) show the base cases of the induction and Figure 19(c) illustrates the induction step. Therefore, precondition 7(iii) is satisfied and is well behaved. Note that the induction step works for both positive and negative cases and also proves that is well aligned.

(a) The base case with only positive polar nodes being active

(b) The base case with only negative polar nodes being active

(c) Induction step for the positive polarity case

Figure 19: Induction proof for that a long variable structure can be unit-disk covered by + 1 nodes with the same polarity.

A.2. Edge Structures

As illustrated in Figure 20, the structure is basically a composite structure from numbers of variable structures connected via connection patches. By Lemma 29 described in Appendix D, the variable structures are least interactively connectable at any two nearby nodes on the side of border. Thus, this lemma can be proved by induction on the composing variable structures with Lemma 10 and Connection Lemma.

Figure 20: An edge is a composite structure of variable structures. Note that a variable structure can be rotated by 90° and still satisfies Lemmas 8 and 29. Here (, ) and (, ) are the connected ports of the connection patch.

B. Proof of Lemma 9

In this appendix, we complete the proof of Lemma 9.

B.1. 2-Way Connectors

Figures 3(a) and 4(a) illustrate the labels of nodes and vertices of for the 2-way connector. Table 1 lists the positions of nodes and vertices relative to . Note that , , , and the header nodes and . Obviously, for and 2, and, thus, precondition 7(i) is satisfied.

As shown in Figure 21(d), only two active nodes and from cannot unit-disk cover even all nodes in are active, which implies that only one node, or , from being active cannot unit-disk cover . Similarly from Figure 21(a), only one node, or , from being active cannot unit-disk cover . Thus, if and , is not a UDC of . That is, if is a UDC of , .

(a) and are active

(b) and are active

(c) and are active

(d) and are active

Figure 21: cannot be fully unit-disk covered by two opposite polar nodes from , even all the nodes in are active. The shaded regions indicate the uncovered regions.

Furthermore, Figure 21 lists all possible cases in which only two opposite polar nodes from being active cannot unit-disk cover even all nodes in are active.

As shown in Figure 22(d), only two active nodes and from cannot unit-disk cover even all nodes in are active, which implies that only one node, or , from being active cannot unit-disk cover . Similarly from Figure 22(a), only one node, or , from being active cannot unit-disk cover . Thus, if and , is not a UDC of . That is, if is a UDC of , . Thus, together with the result from , precondition 7(ii) is satisfied.

(a) and are active

(b) and are active

(c) and are active

(d) and are active

Figure 22: cannot be fully unit-disk covered by two opposite polar nodes from , even if all the nodes in are active. The shaded regions indicate the uncovered regions.

Furthermore, Figure 22 lists all possible cases in which only two opposite polar nodes from being active cannot unit-disk cover even all nodes in are active.

Figure 23 illustrates possible cases in which can be unit-disk covered with two same polar nodes from and two same polar nodes from being active. Together with Figures 21 and 22, precondition 7(iii) is satisfied. With satisfaction of preconditions 7(i), 7(ii), and 7(iii) for and , property (i) is satisfied. Furthermore, in Figure 23, at least one of the header nodes must be active, so property (iii) is also satisfied.

(a) ; and are active

(b) and are active

(c) and are active

Figure 23: With at least one of the header nodes being active, can be fully unit-disk covered by the nodes with the same polarity from each port.

Figure 24 shows that if no header node is active, cannot be unit-disk covered. That is, property (ii) is satisfied. Therefore, we prove that Lemma 9 holds for the 2-way connector shown in Figure 3(a).

Figure 24: Without any header node being active, cannot be fully unit-disk covered, even all the rest of nodes are active.

B.2. 3-Way Connectors

Figures 3(b) and 4(b) illustrate the labels of nodes and vertices of for the 3-way connector. Table 2 lists the positions of nodes and vertices relative to . Note that , , , , and the header nodes , , and . Obviously, for , 2, and 3, and, thus, precondition 7(i) is satisfied.

Figure 25 shows that cannot be fully unit-disk covered without any node in being active, even if all nodes in and are active. Thus, if is a UDC of , .

Figure 25: cannot be fully unit-disk covered without any node in being active, even if all the nodes in and are active. The shaded region indicates the uncovered region.

As shown in Figure 26(d), only two active nodes and from cannot unit-disk cover even if all nodes in and are active, which implies that only one node, or , from being active cannot unit-disk cover . Similarly from Figure 26(a), only one node, or , from being active cannot unit-disk cover . Thus, if and , is not a UDC of . That is, if is a UDC of , .

(a) and are active

(b) and are active

(c) and are active

(d) and are active

Figure 26: cannot be fully unit-disk covered by two opposite polar nodes from , even if all the nodes in and are active. The shaded regions indicate the uncovered regions.

Furthermore, Figure 26 lists all possible cases in which only two opposite polar nodes from being active cannot unit-disk cover even all nodes in and are active.

As shown in Figure 27(d), only two active nodes and from cannot unit-disk cover even all nodes in and are active, which implies that only one node, or , from being active cannot unit-disk cover . Similarly from Figure 27(a), only one node, or , from being active cannot unit-disk cover . Thus, if and , is not a UDC of . That is, if is a UDC of , . Thus, together with the results from and , precondition 7(ii) is satisfied.

(a) and are active

(b) and are active

(c) and are active. Note that the tiny uncovered region is located between and . For example, cannot be covered

(d) and are active

Figure 27: cannot be fully unit-disk covered by two opposite polar nodes from , even if all the nodes in and are active. The shaded regions indicate the uncovered regions.

Furthermore, Figure 27 lists all possible cases in which only two opposite polar nodes from being active cannot unit-disk cover even all nodes in and are active.

Figure 28 illustrates possible cases in which can be unit-disk covered with one node from , two same polar nodes from , and two same polar nodes from being active. Together with Figures 26 and 27, precondition 7(iii) is satisfied. With satisfaction of preconditions 7(i), 7(ii), and 7(iii) for each port, property (i) is satisfied. Furthermore, in Figure 28, one of the header nodes must be active, so property (iii) is also satisfied.

(a) , , , and ; are active

(b) , ; , and ; are active

(c) , ; , and ; are active

(d) , ; , and ; are active

(e) , ; , and ; are active

(f) , ; , and ; are active

(g) , ; , and ; are active

Figure 28: With at least one of the header nodes being active, can be fully unit-disk covered by the nodes with the same polarity from each port

Figure 29 shows that if no header node is active, cannot be unit-disk covered. That is, property (ii) is satisfied. Therefore, we prove that Lemma 9 holds for the 3-way connector shown in Figure 3(b).

Figure 29: Without any header node being active, cannot be fully unit-disk covered, even if all the rest of nodes are active.

C. Formal Definitions about Structure Connection

In this appendix, we define how structures may be connected together to form a complex structure.

Definition 25. The structure, , is connectable, if, refer to Figure 5, there exists a pair of nodes, and , such that(i), (hence, and can be the vertices of a connection patch defined in Definition 26.)(ii) and have opposite polarities. Furthermore, we call connectable at and and the set a port of .

Definition 26. Consider two connectable structure, and . Suppose and are connected together via squares, , called connection patches. Each , , is attached to ports of and of . These ports are positioned at vertices of each connection patch as shown in Figure 5 and called connected ports. and are well connected if the following hold:(i) and (ii)for , if and on the same edge of , and have opposite polarities, for example, and in Figure 5. Furthermore, and are each other's connection counterpart.

In this NP-completeness proof, we would like to show that two structures are connected in such a way that the nodes, except connected ports, of one structure cannot cover any point of the other structure for preserving the partially well-behaved property. Thus, we have the following definitions.

Definition 27. The structure, , is least interactively connectable, if there exists a port, , such that the following preconditions hold. (i)There exists a point that is not at the location of or and can only be unit-disk covered by or . (This precondition requires that at least one node from each connected port needs to be active.)(ii). Here is the connection patch attached to . (Nonconnected port nodes and the connection patch are so far that nonconnected port nodes cannot cover any point, except the vertices, of the connection patch. Thus, whether a connection patch can be fully covered only depends on its connected ports.) Furthermore, we call the least interactively connectable port.

Definition 28. The least interactively connectable structures and are well connected via the connection patches . For , is attached to the least interactively connectable ports of and of . We call and least interactively connected if the following preconditions hold. (i) and . (The distance between any nonconnected port nodes of one structure and any point of the other structure is greater than 1. Thus, the nodes, except connected ports, of one structure cannot cover any point of the other structure.) (ii)for , and . (The connected ports of one structure cannot cover any point, except their connection counterparts, of the other structure.)

D. The Least Interactive Connectability of Structures

Lemma 29. The structure shown in Figure 1 is least interactively connectable at any two nearby nodes on the side of border.

Proof. Suppose and are two nearby nodes as shown in Figure 1. It is not difficult to prove that is a port. Furthermore, as shown in Figure 1, , the midpoint of and , is the point satisfying precondition 27(i). Finally, it is obvious that , and . Hence, it is easy to derive that precondition 27(ii) is satisfied. Therefore, is least interactively connectable at and .

Lemma 30. The structure shown in Figure 2 is least interactively connectable at the endpoint pairs indicated by the arrows.

Proof. It is obvious that the endpoint pairs are also ports, and for each endpoint pair, for example, , . Hence, it is easy to derive that precondition 27(ii) is satisfied at each endpoint pair. Besides, similar to variable structures, the midpoint of each endpoint pair can only be unit-disk covered by the endpoint pair. Thus, is least interactively connectable at the endpoint pairs.

Lemma 31. Each -way connector of Figure 3 is least interactively connectable at the pairs of nodes indicated by the arrows.

Proof. It is obvious that the pairs indicated by the arrows are also ports. Furthermore, for each port, it is obvious that the midpoint of the port is the point satisfying precondition 27(i). Moreover, without loss of generality, it is straightforward to prove from Table 2 that, for the port of the 3-way connector, , and . Hence, it is easy to derive that precondition 27(ii) is satisfied at . Similar argument may apply to other ports and the 2-way connector.

Lemma 32. The structure shown in Figure 9 is least interactively connectable at its endpoint pairs.

Proof. It is obvious that the endpoint pairs are also ports, and for each endpoint pair, for example, . Hence, it is easy to derive that precondition 27(ii) is satisfied at each endpoint pair. Besides, similar to variable structures, the midpoint of each endpoint pair can only be unit-disk covered by the endpoint pair. Thus, is least interactively connectable at the endpoint pairs.

E. Proof of Lemmas about the Least Interactive Connection

E.1. Proof of Lemma 12

First, consider the following lemma.

Lemma 33. Suppose . If is a UDC of , then and are UDCs of and , respectively.

Proof. Suppose is not a UDC of , then there exists a point such that cannot be covered by any node in but some node, say , in .From Definition 28(i), we know the distance between any node in , except connected ports, and any point in is greater than 1. Thus, none of nodes in , except connected ports, can cover any point in . Hence, must belong to some connected port. Without loss of generality, let be of Figure 5. From Definition 28(ii), the only point in which can be covered by is at the location of , and it implies that can only be at the location of .Let be the point that is not at the location of or and can only be covered by or , as stated in Definition 27(i). From the above argument, cannot be covered by any node of . Thus, at least one node from must be active in . Hence, there is a contradiction since either or can cover , that is, the location of . Therefore, is a UDC of and similar argument can also derive that is a UDC of .

With the help of Lemma 33, we can prove Lemma 12.

If is a UDC of , is a UDC of . By Lemma 33, and are UDCs of and , respectively, which implies that and . From Definition 26(i), and, hence, . Thus, and .

E.2. Proof of Connection Lemma

Before proving Connection Lemma, consider the following lemma.

Lemma 34. If via one connection patch, then the following propositions will hold. (We note that this lemma may be extended to more than one connection patches with possible minor modification on Definition 28(ii); however we do not use this property to prove Connection Lemma and therefore ignore it.) (a)If is an MUDC of , then and are MUDCs of and , respectively. (b)If and are MUDCs of and , respectively, then is an MUDC of .

Proof. Let be an MUDC of . By Lemma 33, and are udcs of and respectively.Suppose is not an MUDC of . Then there exists a udc of and . Since is a udc of and is a udc of , is a udc of . Since by Definition 26(i), , which contradicts the assumption that is an MUDC of . Similar argument can apply to . Hence, we have proved proposition (a) of this lemma.Now assume and are MUDCs of and , respectively, and, obviously, is a udc of . Suppose there exists an MUDC of and . By proposition (a), and are MUDC of and , respectively. Again, since , . Hence, or , which contradicts the assumption that and are MUDCs of and , respectively. Hence, is an MUDC of .

With the help of Lemmas 33 and 34, we can prove Lemma 13.

Let be the connection patch. The key point to the proof is precondition (ii). That is, can be automatically covered by the active nodes of ports without the help of nodes not in and , if and have the same polarity. Thus, the MUDC of can only be .

Note that by Definition 26(i). Thus, precondition 7(i) is satisfied since and are partially well behaved on and , respectively.

Moreover, let be a udc of . By Lemma 33, it is easy to derive that and are udcs of and , respectively. Then by precondition (i) and Definition 7(ii), and . Since , . That is, precondition 7(ii) is satisfied.

Now, suppose is an MUDC of and is an MUDC of . Obviously, . Since is a UDC of , . Consequently, by showing there exists an MUDC of such that is a UDC of , we can prove that , which implies that is also an MUDC of .

Note that precondition (ii) states that there exists an MUDC of , , and an MUDC of , , such that and have the same polarity. Let be such a candidate. Hence, by Lemma 34(b), is an MUDC of .

Since and have the same polarity, or . Referring to Figure 5, by Definitions 25(ii) and 26(ii), either the pair (, ) or (, ) is active. Obviously, either pair can unit-disk cover by Lemma 10. Therefore, is a UDC of , which implies that, for any MUDC of , and is also an MUDC of .

Now we need to prove that, for any MUDC of , only contains the nodes with the same polarity. Since is also an MUDC of , by Lemma 34(a), , denoted as , is an MUDC of and , denoted as , is also an MUDC of . By Definition 7(iii), only contains the nodes with the same polarity and so does . Thus, we need to prove that cannot be or . Referring to Figure 5, without loss of generality, let with and being active. Note that it is obvious that . (In other words, cannot be covered by and .) That is, .

Denote . Obviously, , and, hence, . From Definition 27(ii), . Thus, , which implies . Hence, . Similarly, .

Therefore, . That is, . Thus, simply nodes from and cannot unit-disk cover , and at least one more node not in is needed, for example, or . Hence, cannot be an MUDC of .

Therefore, we can conclude that if is an MUDC of , contains only the nodes with the same polarity. That is, precondition 7(iii) is satisfied and, thus, is partially well behaved on .

F. Proof of Lemma 14

Similar to the proof of variable structures in Lemma 8, the well-aligned and well-behaved properties of the main structure can be proved via Figures 30, 31, and 32. Consequently, with the help of Lemma 10 and Connection Lemma, the well-aligned and well-behaved properties of can be proved. Note that, in this case, Connection Lemma is applied to the main structure and both “end”, and . Referring to Figure 33, is the line segment and . Besides, the left buffer is the connection patch for connecting and the main structure. Similar idea can apply to the right end, that is, .

Figure 30: The tiny shaded region as indicated in the middle of the column-pair cannot be covered while none of nodes from the column-pair is active.

Figure 31: The shaded region cannot be covered by two nearby opposite polar nodes.

(a) The base case with only positive polar nodes being active

(b) The base case with only negative polar nodes being active

(c) Induction step for the positive polarity case

Figure 32: Induction proof for that the long main structure can be unit-disk covered by nodes with the same polarity. Note that for the main structure of a polarity inverter, .

Figure 33: Connection Lemma is applied to the main structure and the left end (enclosed in dashed line). is the line segment and .

G. Proof of Lemma 17

For the th variable , consider the composite structure with and . Here is the union of the shaded region from all the associated connection patches, except the ones attached to the associated connectors. An example of is illustrated in Figure 34. In other words, the composite structure of and all associated connectors is the territory . It is obvious that is well behaved by Lemmas 8, 14, and Connection Lemma. After that, we connect each associated -way connector to iteratively. At each iteration, we can prove the partially well behaved property of the composite structure. When all associated connectors are connected, this lemma is proved.

Figure 34: The structure enclosed in dashed line is . The set of nodes enclosed in dashed circle is , and the set of nodes enclosed in solid line is . By proving is partially well-behaved on , this lemma will be proved.

Without loss of generality, consider an MUDC of , . Since for each clause , the clauses are represented by 2-way or 3-way connectors. Note that an instance has a graph structure, , in which there is at most one edge between two nodes. That is, the variables in a clause are different. Thus, for each associated connector of , there is another partition, that is, not enclosed in dashed line in Figure 10, which is not an associated partition. Without loss of generality, suppose is an associated 2-way connector of . Here and . Referring to Figure 34, let be the associated partition and . We want to prove is partially well behaved on .

From Lemma 9(i), Lemma 9(iii), and Definition 7, it is easy to derive that is an MUDC of . Thus, and are the MUDCs of and , respectively, and precondition (ii) of Connection Lemma is satisfied. (In this case, , , , , , and .) Besides, as mentioned earlier, the variables in a clause are different. That is, and are connected via one connection patch. Thus, Connection Lemma is applicable. By Lemma 9(i) and Connection Lemma, is partially well behaved on . This procedure can be iteratively applied to the rest of connectors and, hence, the lemma is proved.

Acknowledgment

This research was supported by National Science Council (NSC), Taiwan, under Grant NSC 99-2221-E-194-021. The author gratefully acknowledges this support.

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