Distributed Coordinate-free Hole Recovery

Transcription

1 Distributed Coordinate-free Hole Recovery Xiaoyun Li Department of Computing and Electronic Systems University of Essex Colchester, UK CO4 3SQ Abstract This paper proposes a distributed coordinate-free non-triangulated hole recovery algorithm for sensor networks called 3MeSH-DR (Triangular Mesh Distributed hole Recovery), which recovers coverage holes with the minimum number of redundant nodes using only connectivity information. If this hole cannot be triangulated by edges between pairs of nodes closer that 2 units from each other, the boundary of the hole is detected accurately with the minimum number of edges. It is shown to be error-free by thousands of simulations with different random topologies and node densities. Because of its high reliability and modest requirements for computing power and memory, 3MeSH- DR is an efficient solution for hole detection and recovery without location information in real sensor network applications. Index Terms hole detection; hole recovery; coverage; triangulation; graph theory; wireless sensor network I I. INTRODUCTION n this paper, an algorithm for distributed coverage hole recovery in sensor networks is proposed which activates redundant nodes adjacent to a hole which cannot be covered by existing active nodes. If this cannot be done, then an accurate hole boundary is detected. Location awareness is not necessary, since only connectivity information is required. The algorithm is an enhanced version of 3MeSH [1-2]. A. Related work There has been much related research on the coverage problem for both mobile and static sensor networks. Most such proposals use computational geometry with geometric tools such as Voronoi diagrams to detect holes. Solutions exist for single-level coverage, in static sensor networks [3-9], and in mobile sensor networks [10-12]. Also, solutions exist for multiple-level coverage in dense static sensor networks [13-15]. All these techniques require coordinate information about the sensor nodes in the target sensing area. An algebraic topological method utilizing homology theory can detect single-level coverage holes without coordinates [16-17]. It employs a central control algorithm that requires connectivity information for all nodes in the sensing area, but cannot guarantee to detect the boundary of a hole accurately. If there are N nodes, the calculation time is O(N 5 ). A boundary detection algorithm using a communication graph has also been proposed [18-19]. It selects one or more nodes as seeds, to build a shortest path tree by flooding. With no coverage holes in a continuous target area, nodes which lie the same number of hops away from the seed node should form an unbroken circle. Therefore each node can examine those David K. Hunter Department of Computing and Electronic Systems University of Essex Colchester, UK CO4 3SQ dkhunter@essex.ac.uk neighbors which are an equal number of hops away from it, to determine whether or not such a circle can be formed. In this way, the boundary nodes of a coverage hole can be detected. This algorithm is suitable for a coordinate-free environment, but it requires the node density to be high enough to guarantee sufficient accuracy, which is not suitable for randomly deployed sensor networks, because the node density of the area adjacent to a hole is generally lower than in a hole-free area. Also, it cannot detect multiple adjacent holes. Another problem is the high communication overhead of frequent flooding when the network topology changes due to node failure or mobility. As a node failure can disrupt full coverage, self-recovery must be implemented. GS3 [20] and SoRCA [21] employ selfhealing algorithms to recover a hole with adjacent redundant nodes, but they require coordinate location information for all sensor nodes. Their algorithms partition the target area into fixed hexagonal cells, and assume a cell is covered if at least one node is inside it, hence they cannot maintain critical full coverage without redundancy. To the best of our knowledge, no hole recovery algorithm for coordinate-free scenarios has yet been proposed. B. A brief introduction to the 3MeSH hole detection algorithm 3MeSH (Triangular Mesh Self-healing Hole detection algorithm) is a distributed coordinate-free hole detection algorithm. Assume each node has a circular sensing range with radius R. A subset of active nodes is elected initially so that any two active nodes are more than R units away from each other. A node lying closer than R units to any active node is considered as a redundant node because it is covered by the active node. After active node election, each node becomes either active or redundant. Then the algorithm monitors the network to detect boundary nodes among all the active nodes. A boundary node is an active node adjacent to a coverage hole. 3MeSH requires only the connectivity information about whether two nodes are R units apart for active node election, and whether two nodes are 2R units apart for boundary node detection. Hence 3MeSH is coordinate-free. It can detect any large hole (non-triangulated hole) accurately (Figure 1b), but cannot detect trivial holes (triangulated holes) as shown in Figure 2. Trivial holes can only be detected accurately if accurate distances between nodes are known. Given a set of sensor nodes V in a coordinate-free environment, a graph G(V, E) can be defined, where the set E contains all links between each node in the set V and its 1

2 neighbors. The graph hence indicates connectivity between neighbors by means of edges, but need not directly relate to the geographical or spatial organization of the nodes. Figure 1a. 3MeSH ring. Figure 1b. A large hole. Figure 1 shows how G(V N, E), a sub-graph of G(V, E), is defined by a node and its neighbors, in order to facilitate discussion of the algorithm. In Figure 1a, node A is a nonboundary node (since it is not on the boundary of any large hole), but in Figure 1b, a large hole exists, and node B is a boundary node. Each active node N defines a graph G(V N, E), where V N is the set of active nodes neighboring N, and E is the set of links between neighboring pairs of nodes in V N, and N itself is not included in V N. In Figure 1a, all the nodes in V A define a closed polygon, known as a ring. If the nodes in set V A cannot by themselves define links which partition the area within the ring into triangles, it is known as a 3MeSH ring. By definition, all the links terminating on node A, in addition to the links in set E, partition the area within the 3MeSH ring into triangles, therefore node A is not adjacent to a large hole. However in Figure 1b, all the neighbors of node B, namely nodes B 1, B 2, B 3 and B 4, do not form a ring encircling B, and nodes C 1 through to C n must be added to do this. Therefore a large hole exists, enclosed by a ring with at least four edges, which cannot be partitioned into triangles by links. Proof of the algorithm can be found in references [1-2]. II. Figure 2. Trivial hole. 3MESH-DR: THE DISTRIBUTED HOLE RECOVERY ALGORITHM 3MeSH-DR is an improved version of the 3MeSH hole detection and recovery algorithm. The boundary node and hole detection algorithm is exactly the same as before [1-2], but the hole recovery algorithm becomes distributed and more efficient. A. Distributed hole recovery in 3MeSH-DR After a hole is detected by 3MeSH, one of the boundary nodes adjacent to it broadcasts a hole message containing the IDs of all boundary nodes. Assume each node is aware of its neighbors during self-organization of the network. Distributed hole recovery for 3MeSH-DR is described below: 1. If a redundant node is adjacent to all the boundary nodes of the hole, it elects itself as an active node in order to recover the hole, and the algorithm terminates. This process is unchanged as described in [1-2]. 2. Each of the N boundary nodes adjacent to the hole broadcasts a neighbor message containing node IDs of all its neighbors (either active or redundant) lying within 2R units. N is the number of boundary nodes adjacent to the hole. 3. Each redundant node covered by any boundary node starts to compare received neighbor messages with its own neighbor set. If its neighbors are a subset of a boundary node s neighbor set, then it can make no more contribution towards recovering the hole than the boundary node. So it quits hole recovery. This is a distributed algorithm to filter out redundant nodes. 4. A redundant node elects itself as an active node and broadcasts a hole recovery message if it is connected to two boundary nodes no less than N / 3 hops away (Fig. 3a), providing another node has not already done so. If no single redundant node is activated, and a boundary node connected to redundant node D and another boundary node connected to redundant node E are no less than N /3 hops away (Fig. 3b), either node D or E broadcasts a hole recovery message and elects both D and E as active nodes, providing another node has not already done so. Hole recovery then restarts from step 1 for the reduced hole(s). If no redundant node is activated, an unrecoverable hole is detected. The algorithm terminates. N / 3 is the smallest integer no less than N/3 (see Necessary Condition A). Two boundary nodes are x hops apart if they are connected via x 1 other boundary nodes which are each adjacent to the hole. 5. If hole recovery has been repeated for more than M iterations without convergence (M = 5 is suggested by simulation), further distributed hole detection is performed by the redundant nodes. A redundant node that did not quit in step 3 and which is connected to any two connected boundary nodes broadcasts a boundary neighbor message, providing another node has not already done so. If any two connected boundary nodes cannot find a redundant node connected to both of them, an unrecoverable hole is detected (see proof of condition B), and the algorithm terminates. Fig. 3a 1 bridge redundant node A 1 Fig. 3b 2 bridge redundant nodes A 1, A 2 This algorithm is distributed and error-free. Any redundant node which can recover the hole elects itself as an active node, until this takes place. A hole is regarded as unrecoverable only if all possible redundant nodes become active and the hole has still not been recovered. This does not mean that the algorithm 2

3 has difficulty converging, because in step 3, the distributed algorithm running on the redundant nodes themselves has filtered out most unnecessary nodes. Further hole detection in step 5 is important to limit the convergence of the algorithm to just a few iterations, especially for high density networks. 3MeSH-DR can also accurately detect the boundary of an unrecoverable hole. Fig. 4a shows an example of a detected unrecoverable hole where the recovery algorithm does not employ step 5. It terminates after 30 iterations when all adjacent redundant nodes are activated. This could easily deplete the energy of sensor nodes by transmitting many neighbor messages, and could cause extremely long hole recovery delay. The bold polygon is the boundary of the detected hole, and the shaded area in the center of the polygon is the uncovered area of the hole. The small circles are activated nodes, and each large circle is the coverage area with one activated node in the centre. Fig. 4b shows the result when using further hole detection as specified in step 5. It terminates at the sixth iteration and detects the hole boundary accurately, with six redundant nodes being activated. This process can detect more than 95% of unrecoverable holes, with an accurate boundary, in six iterations, and other unrecoverable holes (5%) are detected in 12 iterations at most according to the simulations of Section 3. Fig. 4a. Hole detection without step 5 Fig. 4b. Hole detection with step 5 This distributed hole recovery algorithm has been shown to be efficient and error free through thousands of simulations with different random node deployments. It can elect the minimum number of nodes to recover the hole without redundancy, because in each step only one or two redundant nodes are elected. If the hole is recovered by the elected redundant node(s), they must be necessary. If the hole is not recovered, then it has been reduced in size. Hence the elected node(s) are necessary for further hole recovery, and to detect the boundary of a unrecoverable hole accurately. When a hole is detected i.e. not triangulated by the boundary nodes, using the minimum number of nodes to recover this hole means that all the recovery nodes are necessary. For example, if a hole can be recovered by N nodes, and is not recovered after discarding any one of them, then they are all necessary to recover this hole. Although there may be more than one way to recover a hole with different numbers of necessary nodes, all these options are said to have used the minimum number of nodes. Therefore the minimum number of nodes to recover a hole does not necessarily mean the optimal number of nodes. To select the optimal number of nodes required to recover a hole could be NP-hard even in coordinate-aware scenarios, therefore it is not discussed here. In coordinate-free scenarios, hole boundary detection is accurate if the hole is enclosed by a polygon with the minimum number of edges. For example, a hole is detected as being enclosed by a polygon with five edges (a pentagon), if it could not be enclosed by any four nodes (a quadrangle), so the hole boundary detection is accurate. Although there could be more than one way to enclose the hole by a pentagon with different subset of nodes, only one option has the smallest enclosed area. However, it is impossible to determine which option encloses the smallest area in coordinate-free scenarios. B. Necessary conditions for hole recovery In 3MeSH-DR, the hole cannot be recovered (triangulated) if it cannot be reduced in size by one or two connected redundant nodes which are connected to two boundary nodes no less than N /3 hops apart. This is a necessary condition to triangulate a hole as described below. For the 3MeSH hole detection algorithm, a hole is detected which is enclosed by more than three boundary nodes. All the redundant nodes adjacent to the hole are within R units of one or more of the boundary nodes. In the best case, the coverage area for any boundary node B, which includes all the redundant nodes lying closer than R from B is a circle of radius 2R (large circles in Figure 5a). If the hole can be covered by circles of radii 2R centred on each of the boundary nodes, it can possibly be recovered by redundant nodes covered by these boundary nodes. In this case, the hole can have been triangulated by links connecting two boundary nodes within 4R units of one another. Otherwise the hole cannot be recovered by redundant nodes. 3MeSH-DR can also recover a hole detected by a hole detection algorithm other than 3MeSH, if all the available adjacent redundant nodes are within 2R units of one or more boundary nodes. Hence the area covered by a boundary node, plus the area covered by all the redundant nodes lying within 2R units of it, lies within a circle with radius 3R. Based on the analysis above, a necessary condition to recover (triangulate) a hole is summarized below. Necessary condition A: if the hole can be triangulated by links between pairs of boundary nodes up to 6R (4R for 3MeSH) units apart, this hole may possibly be triangulated (i.e. recovered) by redundant nodes lying up to 2R (R for 3MeSH) units from any of the boundary nodes. Otherwise the hole cannot be recovered by one or more redundant nodes. Necessary condition for Necessary Condition A: Assume a hole is enclosed by N boundary nodes and cannot be triangulated by links of length 2R or shorter. However it can be triangulated by longer links (for example, of length 4R or 6R). In this case, at least one longer link should exist which connects at least one pair of boundary nodes separated by no fewer than N / 3 hops. Proof: this necessary condition is proved by contradiction. If no such link connects any two boundary nodes separated by no fewer than N / 3 hops, then consider a planar graph defined by links between pairs of nodes less than N / 3 hops apart. It must be possible to define a polygon in this way with 3

4 more than three edges, which cannot be triangulated. Hence the hole cannot be triangulated, proving the condition. The necessary condition for Necessary Condition A motivates the hole recovery process in step 4 of 3MeSH-DR (activating one or two connected redundant nodes which are connected to boundary nodes no less than N / 3 hops away). If this condition is not met, hole recovery terminates early because the hole is unrecoverable. Figure 5a. A hole triangulated by links up to 4R units long. Figure 5b. A hole triangulated by links up to 2R units long. Figure 5a shows a hole enclosed by a polygon with nine edges. Boundary nodes B0, B3 and B6 are 3 (= N/3) hops apart, and they are connected with links no more than 4R units long. This hole may be triangulated by redundant nodes lying up to R units away from one or more of the boundary nodes according to Condition A. Figure 5b shows a possible triangulation of this hole using redundant nodes. C0, C3 and C6 are redundant nodes lying within R units of boundary nodes B0, B3 and B6 respectively. They are fully connected with links up to 2R units long. Hence B0, B3 and B6 are fully connected, with two formerly redundant nodes connecting each pair. This hole is recovered by redundant nodes. Figure 6. Proof for condition B Necessary condition B: Any two neighboring boundary nodes should have at least one common redundant node connected to both of them. Otherwise the hole cannot be recovered. Proof: If two adjacent boundary nodes B 1 and B 2 have no common redundant nodes connected to them, for any path between B 1 and B 2 {B 1 R 1 R n B 2 } where n 2 (Figure 6), the polygon P m {B 1 R 1 R n B 2 B 1 } cannot be triangulated by links between vertices on the polygon. This is because a triangle with B 1 and B 2 as vertices does not exist, therefore P m cannot be partitioned into triangles. Hence the hole with B 1 and B 2 as boundary nodes cannot be recovered, proving necessary condition B. C. Other details of the distributed algorithm 1) Accurate hole boundary detection When a hole is reduced in size by activating a redundant node, how is the new boundary determined accurately? The activated node(s) may connect to nodes other than the two boundary nodes which were no less than N/3 hops apart, and thus triangulate part of the hole. In this case, all the boundary nodes connected to the activated node(s) should be discarded for the next hole recovery iteration, except those (e.g. B 1 and B 5 in Fig. 3a) which are connected to boundary node(s) (e.g. B 6 ) which are not connected to the bridge node(s) (e.g. A 1 ). Therefore only nodes {B 1, B 6, B 5, A 1 } are adjacent to the reduced hole in Fig. 3a, yielding a more accurate hole boundary for the next hole recovery iteration. 2) Loop avoidance The elected redundant nodes have become active nodes, and should not be considered in further node election for hole recovery. Otherwise there is a possibility of the recovery process repeatedly activating the same node indefinitely. D. Advantages in resource constrained sensor networks 3MeSH-DR is the first practical hole detection and recovery algorithm for sensor network applications. It has the following advantages over a centralized algorithm. 1) Computation time Each node only needs to compare its connectivity information to the boundary nodes in order to filter out unnecessary redundant nodes. Assume the computation is running on the PIC 18f252 microchip [22], for which the execution time of a 16-bit add operation is 2 µs. For a hole enclosed by N = 10 boundary nodes, with high node density λ = 20, the calculation time for each node is 10 4λ 2µs = 1.6 ms, whereas in the centralized algorithm [1-2], the computational time could be several tens of seconds with node density λ > 6. 2) Communication time Only the N boundary nodes adjacent to the hole need to broadcast their connectivity information. Moreover, one or at most two nodes broadcast their connectivity information at each hole recovery iteration. In the worst case, for N = 10, the convergence time is 13 iterations according to the simulation. If each iteration activates two redundant nodes, the total communication time is (N ) 800 bits / bps = 1.5 seconds. The mean communication time is much shorter than in the worst case, because the mean number of iterations for convergence is less than three, and for more than 94% of hole recoveries, only one redundant node is activated in each iteration. Therefore the average communication time is: (N + 3) 800bits / 19200bps = 0.54 seconds. It is almost constant, not being sensitive to increased node density, according to simulation with various node densities. Another communication overhead for 3MeSH-DR is the broadcast at step 5 for at most N redundant nodes which are each connected to any two adjacent boundary nodes. Because the broadcast message contains only the neighboring boundary nodes ID, which is very short, and only under 10% of hole recoveries need this process (as demonstrated by simulation), this communication overhead can be neglected. 3) Accurate hole boundary detection The hole boundary detected using 3MeSH-DR is more accurate than with existing coordinate-free centralized algorithms including 3MeSH [1-2], and that using the 4

5 homology algorithm [16-17]. In 3MeSH-DR, all the activated redundant nodes are involved in detecting the hole precisely; see the comparison of Figure 7a and 7b with the same node deployment. The small black circles are activated redundant nodes for hole recovery, and the bold polygons are detected unrecoverable holes. The centralized hole recovery algorithm of 3MeSH (Figure 7a) can only detect the hole enclosed by the active boundary nodes, whereas the distributed hole recovery algorithm (3MeSH-DR in Figure 7b) not only detects the hole enclosed by the active boundary nodes, but also activates the minimum number of redundant nodes to recover the holes so that unrecoverable holes are either partly recovered or divided into smaller holes. Therefore more accurate boundaries of unrecoverable holes are detected. holes/simulation on average for λ = 14). The number of recoverable holes using more than one iteration is largest for 5 λ 7 (around 3 holes/simulation on average), and drops slowly to less than one hole per simulation on average for λ 10. The number of unrecoverable holes drops when node density increases. For λ = 3 and 4, the number of unrecoverable holes is largest (around 4 holes/simulation on average); for higher node densities, the average number of unrecoverable holes drops quickly to less than one hole per simulation (λ 7), and drops to less than 0.01 holes/simulation (λ 10). For λ = 2 the average number of unrecoverable holes is 2.9 holes/simulation, which should be larger, but some holes cannot be detected due to the boundary effect in sparse sensor networks. For high node densities (λ > 10), most detected holes are recoverable using one redundant node. Figure 7a. Centralized hole detection. Figure 7b. Distributed hole detection. III. SIMULATION RESULTS For each node density 2 λ 14, simulations are performed in a target area of 64 square units (1 square unit = πr 2 ) within 100 different random node deployments for each node density. During the 1300 simulations, holes are detected including 9464 recoverable holes and 1628 unrecoverable holes holes could be recovered using one redundant node in the first iteration, and 1876 holes are recovered using more than one redundant node and more than one iteration. Figure 8 shows that the mean number of recoverable holes is largest for node densities 6 λ 8 (around 10 holes on average for one simulation). For λ = 8, the number of holes recovered in the first iteration is largest (8 holes/simulation on average), then it drops slowly for higher node densities (5 Figure 8. Number of holes. Because holes recovered using one redundant node require zero communication and calculation overheads for both centralized and distributed algorithms, those 7770 holes are not discussed below. Only the 1628 unrecoverable holes and 1871 recoverable holes requiring more than one iteration are discussed below. Figure 9 compares the number of hole recovery iterations with and without further hole detection (step 5 of 3MeSH-DR). During 1300 simulations (100 simulations per node density), the 1876 recoverable holes recovered by more than one redundant node do not benefit from the further hole detection process because they could be recovered previously. For the 1628 unrecoverable holes, if iterations are permitted without further hole detection, there are 15 unrecoverable holes detected after more than 10 iterations, including one terminating at the 30 th iteration (λ = 13), and one terminating at the 23 rd iteration (λ = 12), which is the worst case for convergence of hole recovery. If further hole detection is performed at each iteration after the fifth recovery iteration, which is suggested by 3MeSH-DR, the number of unrecoverable holes detected after no more than five iterations is exactly the same as that without the further detection process (the worst case). However the number of unrecoverable holes detected at the sixth iteration almost doubles (from 98 to 192) compared to that without further 5

6 detection. Furthermore the convergence of hole detection improves significantly for unrecoverable holes detected after more than seven iterations, which is exactly the same as when performing further hole detection at each iteration (the best case). Simulation shows that performing further hole detection for each iteration after the fifth iteration can achieve similar convergence as when performing further detection at each iteration. But it causes much less communication overhead, because 94% of the 1876 recoverable holes are recovered and 87% of the 1628 unrecoverable holes are detected before six iterations, so the communication overhead is not necessary for the further hole detection process. Figure 9. Unrecoverable holes using different detection processes. IV. CONCLUSIONS This paper proposed an innovative coverage hole detection and recovery algorithm for coordinate-free sensor networks. Both the hole detection and hole recovery algorithms are distributed without any central control, which is very convenient for randomly deployed sensor networks with resource constrained nodes, especially in those scenarios with frequent topology change due to node mobility or unpredictable node failures. Simulation results with thousands of different random node deployments and node densities show that the algorithm is accurate and reliable when detecting the precise boundary of any unrecoverable holes, recovering all recoverable holes using the minimum number of redundant nodes, with an average convergence time of less than three iterations. Its high reliability and efficiency, as well as its distributed character, indicate that this is a practical hole detection and recovery algorithm for real resource constrained sensor networks with limited energy, limited memory and computational capability, and lack of location information etc. This algorithm may also be used in detection of routing holes, jamming holes, and most sink and worm holes in sensor networks, and is more efficient and reliable than existing solutions for these problems. REFERENCES [1] X. Li, D. K. Hunter, K. Yang, Distributed Coordinate-Free Hole Detection and Recovery, IEEE GLOBECOM, San Francisco, USA, November 27 December 1, [2] X. Li, D. K. Hunter, Distributed Coordinate-free Algorithm for Full Sensing Coverage, International Journal of Sensor Networks, [3] B. Carbunar, A. Grama, J. Vitek, O. Carbunar, Coverage Preserving Redundancy Elimination in Sensor Networks, IEEE SECON, October [4] H. Zhang, J. C. 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