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Journal of Computer Systems, Networks, and Communications
Volume 2008 (2008), Article ID 160453, 10 pages
http://dx.doi.org/10.1155/2008/160453
Research Article

Constrained Delaunay Triangulation for Ad Hoc Networks

Department of Computer Science and Engineering, Indian Institute of Technology Guwahati, Guwahati 781039, Assam, India

Received 4 September 2007; Revised 12 January 2008; Accepted 22 February 2008

Academic Editor: Bechir Hamdaoui

Copyright © 2008 D. Satyanarayana and S. V. Rao. 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

Geometric spanners can be used for efficient routing in wireless ad hoc networks. Computation of existing spanners for ad hoc networks primarily focused on geometric properties without considering network requirements. In this paper, we propose a new spanner called constrained Delaunay triangulation (CDT) which considers both geometric properties and network requirements. The CDT is formed by introducing a small set of constraint edges into local Delaunay triangulation (LDel) to reduce the number of hops between nodes in the network graph. We have simulated the CDT using network simulator (ns-2.28) and compared with Gabriel graph (GG), relative neighborhood graph (RNG), local Delaunay triangulation (LDel), and planarized local Delaunay triangulation (PLDel). The simulation results show that the minimum number of hops from source to destination is less than other spanners. We also observed the decrease in delay, jitter, and improvement in throughput.

1. Introduction

An ad hoc network is a collection of wireless nodes with no fixed infrastructure. In wireless ad hoc networks, routing is one of the important challenging tasks because of its distributed nature, mobility, limited transmission range, shared communication media, and limited node capabilities. Many researchers proposed various network topologies for efficient routing. These topologies are used to route the packets from source to destination. These network topologies can be broadly divided into three categories: flat, hierarchical, and hybrid network structures. The flat network graphs form connected graph with all nodes in the network, for example, minimum spanning tree (MST), nearest neighbor graph (NNG), relative neighborhood graph (RNG), Gabriel graph (GG), Yao graph (Yao), and Delaunay triangulation (Del) [13]. In the hierarchical structures, only some subset of nodes is existed in the network graph based on the hierarchies created by the network. Mainly, hierarchical structures are based on dominating set, connected dominating set, and 𝑑-dominating set [48]. On the other hand, hybrid structures [9, 10] combine both hierarchical structures and flat structures, and the graph contains the subset of nodes of the original network graph.

Another type of network graph is unit disk graph (UDG) [11] which can be defined as follows. Assume each node has the same and fixed transmission range of one unit, then there exists an edge between two nodes and if and only if their Euclidean distance is not more than one unit. The number of edges in UDG is in , where is the number of nodes. The communication cost in this graph is high due to the large number of edges.

To reduce the communication cost over UDG, many researchers derived various connected geometric subgraphs of UDG, called spanner, with linear number of edges. The formal definition of spanners are as follows. A spanning subgraph is a -spanner of a graph if and only if the number of vertices in graph 𝐺1 is equivalent to the number of vertices in graph 𝐺, and the length of the shortest path connecting any two vertices in graph 𝐺1 is bounded by a value 𝑡 to the length of the shortest path connecting the same vertices in the graph 𝐺. In other words, a graph 𝐺1 is a 𝑡-spanner of graph if and , , where is the vertex set of graph and is the length of the shortest path between two vertices and in the graph .

Computation of any geometric spanner requires position of nodes. So, each node knows its position using some positioning techniques such as global positioning system (GPS). Some of the important spanners are GG [2], RNG [2], Del [12], Yao [3, 13], and variations of Delaunay triangulation [1, 9, 10, 1417].

These spanners are used as underlying network graph by various position-based routing protocols like greedy perimeter stateless routing (GPSR) [18], FACE routing [19], adaptive FACE routing (AFR) [20], greedy FACE greedy (GFG) routing [19], GOAF [21], and Bose's localized routing protocols [22].

1.1. Survey

Recently, many researchers have shown interest in geometric spanners because of its wide range of applications. Some of the spanners often used in wireless networks are briefly discussed below.

The edge exists between two nodes and in GG if there is no node inside the circle with as the diameter as shown in Figure 1. In other words, the edge if and only if the circle centered at with the radius does not contain any other node inside, where denotes distance between and . Bose et al. have given a localized algorithm [19] to construct GG for ad hoc networks. The spanning ratio of GG is bounded by [2, 23].

60453.fig.001
Figure 1: Rules of GG, RNG, and Yao graphs.

Two nodes are connected in RNG if and only if their lune does not contain any other node . The lune is defined as the intersection of the two circles of radius centered at and , respectively, see Figure 1. The algorithm to construct RNG for wireless networks is given in [18]. Bose et al. [2] have proved that the spanning ratio of RNG is in . RNG is a subgraph of GG.

In Yao graph [3, 13], the transmission range circle of a node is divided into equal sectors, where , as shown in Figure 1. In each sector, closest node is connected with by an edge if exists. Ties are broken arbitrarily. The spanning ratio of Yao graph is bounded by a constant. Unlike GG and RNG, Yao graph is not a planar graph.

We can define Delaunay triangulation (Del) [12] of as the triangulation of all the nodes such that the circumcircle of each triangle should not contain any other node , see Figure 2. Delaunay triangulation is a planar graph. The spanning ratio of Del is 2.42 [24, 25]. Hence, it is a -spanner of UDG. RNG and GG are subgraphs of Del, but still it maintains linear number of edges in the graph. Del is not suitable for ad hoc networks because it contains longer edges than the transmission range. The unit Delaunay triangulation UDel [15] can be obtained from the Delaunay triangulation Del by removing all the edges longer than one unit. UDel is a -spanner of UDG [9, 15].

60453.fig.002
Figure 2: Rules of Del and LDel.

Though the unit Delaunay triangulation (UDel) is suitable for ad hoc networks, it is not known how to construct UDel locally. Li et al. [1] gave a localized algorithm to construct Delaunay triangulation for ad hoc networks called localized Delaunay triangulation (LDel). In LDe, as shown in Figure 2, the circumcircle of each triangle does not contain any other node such that , , and where denotes the set of 1-hop neighbors for node . In this algorithm, first it constructs LDe with one hop neighbors. LDe is a t-spanner of UDG [15], but LDe is not a planar graph. A planarization algorithm is applied on LDe to make it a planar graph, and it is called planarized local Delaunay triangulation (PLDel) or simply localized Delaunay triangulation (LDel). LDel is a planar -spanner of UDG [1, 15]. It contains all the edges that are in the both UDG and Del.

Gao et al. [9, 14] proposed a spanner called restricted Delaunay graph (RDG) which combines the node clustering algorithm with Delaunay triangulation graph. Alzoubi et al. [10, 17] proposed another spanner which first constructs the backbone using clustering algorithm followed by the Delaunay triangulation.

LDel contains many desirable geometric properties which are suitable for wireless ad hoc networks, but at the same time some of the problems in LDel motivated us to propose a new spanner called constrained Delaunay triangulation (CDT). The problems include high hop count between the nodes and large packet delays.

Some times, in LDel, packet traversal may take more than one hop even though the nodes are in the communication range of each other. This can happen due to the geometric property of LDel. For example, consider the LDel shown in Figure 3(a), the outer circle indicates the transmission range of node . Nodes and are neighbor nodes of . Even though the node is in the transmission range of node , the packet traversal from node to node takes more than one hop. Thus, the edge cannot be placed in the LDel. The same thing can happen to other spanners also. For example, Gabriel graph (see Figure 3(b)) does not include the edge in the graph due to its property. In this paper, we propose a new spanner called constrained Delaunay triangulation (CDT) by introducing longer edges to reduce the number of hops and packet delay between the nodes. We call these longer edges as constraints of localized Delaunay triangulation (LDel). The main objectives of the constrained Delaunay triangulation are reducing the number of hops between the nodes in LDel and reducing the end-to-end packet delay. We have simulated CDT in ns-2.28 and compared with PLDel, LDe, GG, and RNG using various greedy routing protocols. The simulation results show that CDT give better performance in terms of number of hops, packet delay, jitter, and throughput.

fig3
Figure 3: More hops in LDel and GG.
60453.fig.004
Figure 4: Constraint edges selection.
fig5
Figure 5: Constraint edge is added to PLDel.
60453.fig.006
Figure 6: Non-Delaunay triangles , , , and are created in CDT.

The remaining part of this paper is arranged as fallows. In Section 2, we define constrained Delaunay triangulation followed by the algorithm to construct CDT. Section 3 presents the results of ns2 [26] simulation. We conclude the paper with the future research direction in Section 4.

2. Constrained Delaunay Triangulation (CDT)

Localized Delaunay triangulation (LDel) [1, 15] is popular because of its useful geometric properties and utility in wireless networks. In LDel, even though two nodes exist in the communication range of each other, they do not have a direct edge between the nodes because of the geometric properties. The same things happen in other geometric graphs such as GG and RNG. This leads to longer path from source to destination, hence more number of transmissions and high latency. We are motivated by this problem and proposed CDT which reduces the number of transmissions and latency.

If we have an edge between the nodes and in Figure 3(a) or and in Figure 3(b), it reduces the number of hops between the nodes and in turn leads to many other benefits such as reduction in the delay, jitter, and energy consumption; but, these type of edges violate the geometric properties of the spanner. So, in order to improve the various network parameters, we introduce the so-called constraint edges into the network graph, but most of other edges satisfy the Delaunay property. This gives us improvement in network parameters and at the same time satisfying various geometric properties.

Constrained Delaunay triangulation is well-studied problem in the area of computational geometry. Chew proposed a divide-and-conquer algorithm for CDT [27]. This algorithm is not suitable for wireless ad hoc networks because it is a centralized algorithm. In this section, we give a localized algorithm to construct CDT for wireless networks.

Constrained Delaunay triangulation is a triangulation of the given set of nodes with the following properties: (1) CDT is a subgraph of UDG; (2) constraint edges exist at two hop away; (3) CDT is a planar graph; (4) the total number of edges in CDT is in ; and (5) CDT is a spanner of UDG.

The algorithm to construct CDT is a localized algorithm, that is, it uses only the constant number of neighborhood hop information. In this algorithm, we assume that each node is static and knows its location information. We can broadly divide the CDT algorithm into three parts. First, we construct the planarized local Delaunay triangulation (PLDel) [1, 15] with one hop neighborhood information. For the sake of completeness, we briefly describe the algorithm. Each node computes Delaunay triangles Del with its 1-hop neighbors. The Gabriel edges are computed from Delaunay triangles which will never be deleted from local Delaunay triangulation (LDel). Remove all the inconsistent triangles. A triangle is called consistent if the nodes , , and contain the triangle . The resulting graph is called LDel1 which is not planar graph. A separate planarization algorithm is applied on LDel1 to induce a planarized local Delaunay triangulation (PLDel).

In the second part of CDT algorithm, we find BLACK nodes which are at least two-hop apart. These Black nodes place nonintersecting constraint edges on top of PLDel and mark the other side of the constraint edge to BROWN. The resulting graph is not planar graph. The third part planarizes this graph and forms new triangles with the constraint edges. The resulting graph is called constrained Delaunay triangulation (CDT). Please note that the new edges added belong to UDG. Formally, the algorithm is described as shown in Algorithm 1.

alg1
Algorithm 1

Constrained Delaunay triangulation can be maintained efficiently in a distributed fashion. CDT has nice theoretical guarantees. The simulation results show that the performance of CDT is better than PLDel, GG, and RNG.

Theorem 1. The expected number of BLACK nodes in any 2-hop disk is in 𝑂(1).

Theorem 2. The expected total number of BLACK and BROWN nodes in a unit disk is in 𝑂(1).

Proof. From the definition of BROWN node, for each BLACK node, the other side of each constraint edge is a BROWN node. Since the maximum number of constraint edges for each BLACK node is 5, the total number of BROWN nodes for a single BLACK node is in 𝑂(1). We know that the total number of BLACK nodes in unit disk is in 𝑂(1), from Theorem 1. Thus, the total number of BROWN nodes in unit disk is in 𝑂(1). Hence, the total number of BLACK and BROWN nodes are in 𝑂(1).

From theoretical point of view, the constraint edges in CDT do not affect the link quality, as the definition of UDG says that every node in its vicinity receives the transmitted packet [1, 911, 14, 15, 16, 17]. On the other hand, the quality of signal strength decreases with the increase in distance to the receiving node [2830]. In this case, the constraint edges may affect the link quality, as the constraint edges are considerably long. In addition to the distance, link quality also depends on walls, buildings, mountains, and weather conditions, which might obstruct signal propagation. We can use the following models for link quality [31, 32]. In our proposed method, we can consider distance parameter for link quality, while selecting the constrained edges. In other words, consider the link quality as a parameter for choosing the constraint edges.

In the remaining part of the section, we analyze the communication cost in terms of the number of transmissions. In addition to the messages of planarized local Delaunay triangulation [1, 15], CDT uses the following control messages:

(1)neighbor_packet. This packet is used to broadcast the 1-hop neighborhood information;(2)constraint_packet. This packet is used to broadcast the constraint edge information;(3)edgecross_packet. This packet is used to broadcast the edgecross information.

The message neighbor_packet is broadcasted at most once by each node in the network graph. So, the communication cost for this message is in 𝑂(𝑛) bits, where n is the total number of nodes. Only BLACK and BROWN nodes use the message constraint_packet. Each BLACK and BROWN nodes broadcast the message constraint_packet at most once. The communication complexity for the message constraint_packet is in 𝑂(𝑛). The message edgecross_packet is used by only WHITE and BROWN nodes. Each WHITE and BROWN node uses edgecross_packet at most once in CDT. So, the message complexity for this packet is in 𝑂(𝑛) bits. Hence follows the theorems.

Theorem 3. The communication cost of constructing CDT from PLDel is in 𝑂(𝑛).

Theorem 4. The communication complexity to construct CDT is in 𝑂(𝑛log𝑛).

Proof. We know that the message complexity for constructing PLDel is in 𝑂(𝑛log𝑛) [1]. The total number of message exchanges by CDT, excluding PLDel, is in 𝑂(𝑛). Thus, the total communication cost of constructing the CDT is in 𝑂(𝑛log𝑛). Hence, Theorem 4 the is proved.

3. Simulation and Performance

In order to evaluate the performance of the proposed spanner, we have done various experimental work using networks simulator (nanoseconds 2.28). We have simulated GG, RNG, LDel1, and PLDel, and compared with our CDT spanner. The following routing protocols are simulated to run on these spanners.

(1)Greedy routing (Grdy). As shown in Figure 7(a), let the node 𝑏 be the destination node, then the current node 𝑎 selects a node 𝑐 as the next relay node in its transmission range such that the distance 𝑐𝑏 is the smallest among its neighbors. Here, the circle with the center at 𝑎 represents the transmission range of node 𝑎. Please note that if there exists a node in the shaded area, then the node 𝑐 cannot be next relay node.(2)Compass routing (Cmp). Assume that the node 𝑏 is the destination node. The current node 𝑎 sends packet to the next relay node 𝑐 which makes the smallest angle, that is, 𝑐𝑎𝑏 is the smallest among all 𝑎's neighbors, see Figure 7(b).(3)Random compass routing (RandCmp). It is similar to the compass routing. As shown in Figure 7(c), let 𝑎 and 𝑏 be the current and destination nodes, respectively. We compute the two nodes 𝑐 and 𝑑, above and below the line 𝑎𝑏, respectively. Here, the angle 𝑐𝑎𝑏 and 𝑑𝑎𝑏 are the minimum angles above and below the line 𝑎𝑏, respectively, among 𝑎's neighbors. The current node 𝑎 sends packet to the destination node, by randomly selecting, either node 𝑐 or node 𝑑 as the next relay node.(4)Most forward routing (MFR). That is shown in Figure 7(d), the current node 𝑎 selects the next relay node 𝑐 to transmit the packet to the destination node 𝑏 such that the distance 𝑑𝑏‖ is the smallest among 's neighbors.(5) Nearest neighbor routing (NN). Let and be the current and destination nodes, respectively, as shown in Figure 7(e). For a particular angle , node selects the next relay node such that , and is the nearest neighbor of .(6) Farthest neighbor routing (FN). In this routing protocol, for the given angle , as shown in Figure 7(f), the current node selects the next relay node such that , and is the farthest neighbor of node . Here, is the destination node.

fig7
Figure 7: Greedy routing algorithms.

The greedy routing, compass routing, and random compass routing provide guarantee delivery of packets if the underlying network topology is a Delaunay triangulation [18, 19].

3.1. Performance Metrics

In this paper, we have considered the following metrics to evaluate the performance of different spanners.

(1)Hop count. This metric represents the total number of hops traveled by a data packet from source to destination on the underlying network graph.(2)Delivery ratio. The delivery ratio (DR) is the ratio of the total number of packets received successfully to the total packets sent by the source.(3)Delay. Delay metric is the total time spent by a packet during the data transmission from source to destination.(4) Standard delay. Standard delay is the standard deviation in the delay of a packet from source to destination. The following mathematical formula is used to calculate the standard delay ():where denotes delay of packet, is average delay, and is total number of packets.(5)Throughput. Throughput is the total number of packets transmitted per unit time from source to destination.

In the simulation results, we have presented average, minimum, and maximum of the above metrics.

3.2. Simulation Model and Results

Important parameters of the simulation are shown in Table 1. All experiments are conducted on ten different node scenarios. For each scenario, we have considered ten different connection patterns, by randomly choosing source nodes at one side and destination nodes at the other side. In our simulation model, we do not consider the link quality, as we assumed that the channel is error free.

tab1
Table 1: Simulation parameters.

The first experiment is on delivery ratio. The average delivery ratio of ten different scenarios with six different routing protocols is shown in Table 2. The delivery ratio of CDT is better than RNG and GG because CDT is denser graph than RNG and GG. There is not much difference among CDT, PLDel, and LDe because many edges are common among them.

tab2
Table 2: Delivery ratios.

In the second experiment, we have computed the minimum hop count between source and destination on different network topologies. The source node sends the packet to the destination greedily on five different network topologies. The average hop count at different CBR intervals are shown in Figure 8. It clearly indicates that CDT has less average hop count than the other spanners. This happens because CDT contains constraint edges which make packet transmission from source to destination with fewer hops. Since the constraint edges are not available in RNG, GG, LDe, and PLDel, they take more hops.

60453.fig.008
Figure 8: Average hop count.

In third experiment we have checked the end-to-end packet delay on different network topologies. Here, we consider only the packets that reach the destination. The source node sends the packets greedily to the destination on five different spanners at different transfer rates. Figure 9 shows the average delays. We can observe that CDT contains less average delay compared to the other spanners. This is again because of constraint edges in CDT, which make the packets to reach destination with less number of transmissions.

60453.fig.009
Figure 9: Average delay in sec.

In the fourth experiment, we have considered the standard deviation in delay, called standard delay (). Standard delay is related to one of the quality of service (QoS) parameter called jitter. One can see from Figure 10, the standard deviation of the end-to-end packet delay on the spanner CDT is lesser than the other spanners. This could happen because the total number of transmissions on the spanner CDT is less which makes the variation to the average delay will also be less.

60453.fig.010
Figure 10: Standard delay in sec.

The minimum and maximum values of the packet end-to-end delay for different network topologies are shown in Figures 11 and 12, respectively, at different transfer rates for the five network topologies. We can observe that the minimum and maximum packet's end-to-end delays of CDT are lesser than the other spanners.

60453.fig.011
Figure 11: Minimum delay in sec.
60453.fig.012
Figure 12: Maximum delay in sec.

The graphs for minimum and maximum hop counts are shown in Figures 13 and 14, respectively. Due to the constraint edges in CDT, the minimum and maximum hop counts are lesser in CDT than the other spanners.

60453.fig.013
Figure 13: Minimum hop count.
60453.fig.014
Figure 14: Maximum hop count.

Figures 15 and 16 show the minimum and maximum standard delays, respectively. From the experiments, we have seen that the minimum and maximum standard delays are lesser in CDT.

60453.fig.015
Figure 15: Minimum standard delay in sec.
60453.fig.016
Figure 16: Maximum standard delay in sec.

We have done the final experiment on throughput. Figures 17, 18 and 19 show that the constrained Delaunay triangulation has better throughput compared to other spanners. This can happen because the constraint edges in CDT make the packets to reach the destination with fewer number of transmissions than in PLDel, LDel, GG, and RNG, which reduces the collisions, and latency thereby increase the throughput.

60453.fig.017
Figure 17: Average throughput.
60453.fig.018
Figure 18: Maximum throughput.
60453.fig.019
Figure 19: Minimum throughput.

4. Conclusion

The existing geometric spanners GG, RNG, LDe, and PLDel have smaller edges due to their geometric properties, which increase the hop count and delay. We have proposed CDT by introducing larger constrained edges to reduce hop count and delay. The experimental results substantiate our claim. Moreover, CDT is most suitable geometric spanner for multimedia and real-time applications because of lower standard delays. It would be interesting to study these spanners under various mobility conditions.

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