Scalable Routing Modeling for Wireless Ad Hoc Networks by Using ...

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IEEE SYSTEMS JOURNAL, VOL. 7, NO. 1, MARCH 2013

Scalable Routing Modeling for Wireless Ad Hoc Networks by Using Polychromatic Sets Xinheng Wang, Member, IEEE, and Shancang Li, Member, IEEE

Abstract—Graph theory is a traditional mathematical method to analyze computer networks, including intelligent wireless networks, such as ad hoc networks, sensor networks, and wireless mesh networks. Because of the variety of the network nodes and wireless links, conventional graph theory and set theory used to describe the element of the network (node and link) are not suitable to model the modern complex wireless networks. Recent research in weighted graph and random graph theories that place a weight on the links and a probability to decide the existence of the link or not is a further step to model the wireless network in a near real-life scenario. However, it lacks the ability to describe the properties of the network node. In this paper, a new mathematical tool, polychromatic sets (PS-sets), is introduced in modeling the complex wireless networks. PS-sets have the ability to describe the property of each element, which we believe will be a perfect tool to describe the network nodes and links of wireless networks. This paper demonstrates a scalable network modeling using PS-sets theory and a routing scheme based on this model. Evaluation results show that it is simple and scalable, and its performance is superior to other conventional routing schemes. The PS-sets theory could become a new tool in studying modern complex wireless networks. Index Terms—Polychromatic sets, routing, wireless ad hoc networks.

I. Introduction RAPH THEORY has been a traditional mathematical method to analyze computer networks. In graph theory, a graph can be represented by G = (V, E), where V represents a set of vertices and E represents a set of edges. Each entity in a set V or E is called an element. Most applications of using graph theory to construct network infrastructure (physical or virtual) and solve network problems are based on two categories of graphs: unit disk graph and weighted graph [1]–[3]. Applications based on unit disk graph assume that the range of communication of a node is modeled as a disk centered at the node with radius equal to the range of communication [4]. The communication range is normally fixed. An edge exists only if the distance between two nodes is less than or equal to one unit of the communication range. Then algorithms are presented to solve

G

Manuscript received December 1, 2011; revised July 11, 2012; accepted August 17, 2012. Date of publication November 12, 2012; date of current version February 20, 2013. This work was supported in part by the National Natural Science Foundation of China under Grants 60972038 and 81101118. The authors are with the College of Engineering, Swansea University, Swansea SA2 8PP, U.K. (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JSYST.2012.2214671

the problems based on the relationships between vertex and vertex, edge and edge, or vertices and edges. Owing to the recent advances in wireless networking in wireless sensor networks (WSNs), mobile ad hoc networks, and wireless mesh networks, unit disk graph is widely used to construct these kinds of networks and develop algorithms for underlying architecture, such as using connected dominating set [5], [7]. A good survey can be found in [5]. However, this assumption is far away from real-life applications. In wired networks, the bandwidth of each path could be different, e.g., one path constructed by telephone line and another by high-speed optical fiber. Data packets are, preferably, to be delivered by the fiber cable. In wireless networks, even if the wireless nodes have the same capacity and ability to transmit the data at the same speed and cover the same range, the signal-to-noise ratio (SNR) will be different because of the attenuation, fading, and interference from its own network and other resources. This will lead to the different transmission speeds. In this case, a weighted graph is more appropriate. A weighted graph includes a numerical weight that is put on each edge to reflect the different properties of the edge, such as distance, connection cost, or the affinity. A very famous example is the development of Dijkstra’s shortest path algorithm based on weighted graph [6]. Modeling computer networks based on both unit disk graph and weighted graph assume that the edges exist between vertices. Situations where the edges exist with a probability lead the development of random graph [7]. Random graph is applied in studying the reliability of communication networks [8] and large complex networks like social networks [9]. Weighted graph and random graph consider the different properties of the network edges in terms of weight and connection probability. This, to some extent, covers the versatile properties of communication links in computer networks. However, it is still far away to model the real-time computer networks. These models lack some of key knowledge of network nodes and links, such as capacity and location. There needs to be a better tool to model the real-time computer networks [10]– [12]. About 20 years ago, a totally new graph theory was invented to model the complex manufacturing systems. This is the polychromatic graph (PS-graph) using a mathematical method called polychromatic sets (PS-sets), invented by Prof. V. V. Pavlov, Moscow State University, and published in [13], in modeling complex aviation system design [14]–[16]. PS-sets define not only the basic element in a set but also the

c 2012 IEEE 1932-8184/$31.00

WANG AND LI: SCALABLE ROUTING MODELING FOR WIRELESS AD HOC NETWORKS

properties of each element and the whole set. The previous research demonstrated that PS-sets were very successful in modeling complex systems [16]–[18]. After careful investigation, we believe that PS-sets theory and its formed graph theory have the potential to describe the various properties of the network nodes and links so that they can be used in modeling modern complex wireless networks. Preliminary research results demonstrate that it could be applied in signal recognition in cognitive radio networks [16]. The research was extended to investigate the application of the PS-sets and PS-graph theory in routing protocols, which is a fundamental problem in wireless networks. This paper reports the research results in this area, in which a method of modeling a scalable routing protocol in wireless networks is proposed. The proposed method has the advantages of simplicity and fast convergence due to application of graph theory, when compared with other high-performance scalable routing protocols based on advanced mathematical calculations. The remainder of this paper is organized as follows. Following a basic introduction of the PS-sets theory in Section II, a modeling method of applying PS-sets in wireless networks is presented in Section III, by using an example to illustrate the applications. Then a scalable routing protocol is proposed in Section IV using the information of node’s location and its relation to other nodes. Simulation results and the performance comparison between the proposed method and existing routing schemes are analyzed in Section V. Section VI concludes this paper.

Fig. 1. set.

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Relationship between elements, set, and coloring of elements and

PS-sets theory defines a color set by F , where for any element ai ∈ A, the coloring of element ai is denoted as F (ai ) = (F1 (ai ), . . . , Fj (ai ), . . . , Fm (ai ))

(2)

where m is the total number of properties of element ai . Similarly the coloring of the set as a whole is denoted as F (A) = (F1 (A), . . . , Fk (A), . . . , Fp (A))

(3)

where p is the number of properties (colorings) of set A. In both (2) and (3), F1 , F2 , . . . , Fn /Fp represent different colors. The relationship between set, elements, and coloring of elements and set can be illustrated by Fig. 1. Let color set F (a) represent coloring of all the elements, then F (a) = ∪ni=1 F (ai ).

(4)

Therefore, color set F includes color set F (A) of coloring set A as a whole and color set F (a) of coloring each element of set A. This relationship is denoted as

II. PS-Sets Theory A set is defined as a collection of elements in modern set theory that was initiated by George Cantor in 1874 [19] as

F ⊇ F (A), F (a).

A = (a1 , . . . , ai , . . . , an )

In most of the situations, color set F is an ordered set. In addition, in PS-sets theory, another set is defined to contain all the elements with the same coloring. It is denoted as

(1)

where A is the set, ai is an element of set A, and n is the number of elements. If an object ai is an element of set A, we say ai belongs to A, ai ∈ A; otherwise, ai ∈ A. In conventional set theory, objects have some properties in common, e.g., a set of all the capital cities in the world, a set of all alphabet letters, a set of all the students in one class, and so on. However, the conventional set theory does not describe the properties of each element. In the example of a set of all the capital cities, London and Beijing are two elementary cities. They are the capital cities of the U.K. and China, but they are different. The difference of these two cities is not notified in the set. In contrast to the conventional set theory, PS-sets theory defines not only the elements but also the properties of each element and the set as a whole. The property is called a color. Therefore, all the elements in a PS-set and the set itself can be painted with different colors, with each element and the set, maybe, having multiple different colors [16]–[18]. This allows any object to have different properties, which represents the real system.

(5)

A(Fl ) = {ai1 , ai2 , . . . , aiq }.

(6)

For any particular color Fl , there are q elements who share this color, where q ≤ n. The relationships between the elements of A and coloring of individual element F (a) can be represented by a matrix, which is the Cartesian product of A and F (a), having the form ci(j) = [A × F (a)] F ⎛ 1 a1 c1(1) ··· ⎜ ⎜ ··· = ai ⎜ ⎜ ci(1) ··· ⎝ ··· an cn(1)

··· ··· ··· ··· ··· ···

Fj c1(j) ··· ci(j) ··· cn(j)

··· ··· ··· ··· ··· ···

Fm ⎞ c1(m) ··· ⎟ ⎟ ci(m) ⎟ ⎟ ··· ⎠ cn(m)

(7)

where ci(j) is a logical variable whose value is defined as 1, if Fj ∈ F (ai ) ci(j) = (8) 0, if Fj ∈ F (ai ).

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which indicates that it is the ith node on level d and its father node is the jth node on level (d − 1). F (a(d, id , jd−1 )) denotes the corresponding color set of node a(d, id , jd−1 ), which describes the properties and likes parameters of the node, such as the importance index, its father node, child nodes, and so on. Then, the network can be modeled as < a(d, id , jd−1 ), F (a(d, id , jd−1 )) >, as illustrated in Fig. 3. The hierarchical model can be described by using < a(0, 0, 0), F (a(0, 0, 0)) >= in−1 ∧i=1 < a(1, i1 , 0), F (a(1, i1 , 0)) >

(13)

where < a(0, 0, 0), F (a(0, 0, 0)) > denotes the root node of the model. Since D is the depth of the hierarchical model tree, here we define that the father node of the root node is null. With (14), all child nodes of a(d, id , jd−1 ) can be modeled as < a(d, id , jd−1 ), F (a(d, id , jd−1 ) >= ∧imk+1n (d+1) =m0 (d+1) < a(d + 1, id+1 , id ), F (a(d + 1, id+1 , id )) > (14) Fig. 2.

Illustration of the PSHM.

Similarly, the relationships between the elements of set A and set A’s unitary coloring F (A) are defined as ci(k) = [A × F (A)] ci(l) = [A × A(F )].

(9)

m0 (d) = (10)

Therefore, a PS-set is represented by six components in the form of PS = (A, F (a), F (A), [A × F (a)], [A × F (A)], [A × A(F )]). (11) It is not necessary for all the components to be included in a PS-set in practical applications.

III. Modeling of a Scalable Wireless Network In this section, we present a hierarchical model defined by PS-sets, named as polychromatic sets hierarchical model (PSHM), as illustrated in Fig. 2, where the wireless network is split into several clusters with one node being selected as a cluster head (CH) based on its importance index, which will be discussed later. The CHs will become the member of a cluster at a higher level and a node from the new cluster will be selected as a CH at this level. This process continues until a sole CH exists on one particular level. In this hierarchical model, the CHs manage the network and communications. A. Polychromatic Sets Hierarchical Model (PSHM) Applying PS-sets theory in this model, we define a as the set element for one node and the properties of the node as individual color set F (a). Then a network with n nodes can be described, generally, as < ai , F (ai ) >, i = 1, 2, . . . , n.

where mn (d + 1) − m0 (d + 1) denotes the number of child nodes of a(d, id , jd−1 ) on level (d + 1), and m0 (d + 1) is the node number. Actually, there are m0 (d + 1) nodes before a(d, id , jd−1 ) on level (d + 1). They can be obtained by

(12)

Specifically, in this hierarchical model, a node is defined as a(d, id , jd−1 ) (d = 0, . . . , D, D is the total number of levels),

id −1

n(d + 1, i)

(15a)

i=1

mn (d) = m0 (d) + n(d + 1, ik )

(15b)

in which n(d +1, id ) is used to represent the number of child of id th on level d. The nd

denotes the number of nodes on level d. nd−1 In fact, n0 = 1, nd = id−1 =1 n(d, id−1 ). Similarly, a node can easily find its leaf nodes according to (15). Equations (13)– (15) represent the basic formulation of PSHM. For a more general formulation, it can be rewritten as < a(d, i, d − 1), F (a(d, i, d − 1)) >= ∪in11=1 < a(d, id , ), F (a(d, id , )) >

(16)

∀d, (1 ≤ d ≤ D), ∀id (1 ≤ id ≤ n), ∀k(1 ≤ k ≤ n), and ∀ik (1 ≤ ik ≤ nk ), where a(d, id , ) is the id th node on level d. If the network topology is changing, such as adding or deleting a node, this model can update its hierarchy easily by < a(d, id , jd−1 ), F (a(d, id , jd−1 )) >= {∧imd+1n (d+1)+1

< a(d + 1, id+1 , id ), F (a(d + 1, id+1 , id )) >}. (17)

Regarding the color set F (a(d, id , jd−1 )) of the node a(d, id , jd−1 ), it may include the important index of the node, NI , ID of its prior node, NP , ID of its child nodes, NP (for a node i, the prior node is the nearest node prior to i over a path), and the CH, NCH , that it belongs to. F (a(d, id , jd−1 )) is then defined as F (a) = {a, NI (a), NP (a), NC (a), NCH (a)}. However, it must be realized that the number of individual color in each color set is different, depending on the position of node a(d, id , jd−1 ) in PSHM. NP and NC can only be obtained when data traffic flow exists. Then the unified color set for nodes set A = {a1 , a2 , . . . , an } is defined as F (A) = {A, NI , NP , NC , NCH }.


Fig. 3.

Polychromatic sets hierarchical model.

Fig. 4.

PS-sets matrices. (a) PSM for A. (b) PSM for NI . (c) PSM for NP . (d) PSM for NC . (e) PSM for NCH .

B. Hierarchical Architecture of Wireless Networks Until now, the basic concept of modeling a hierarchical wireless network has been accomplished by applying PS-sets theory. However, it is still in a rather abstract form. In this section, a case study will be given to illustrate how the value of each color set is obtained and the whole network is modeled. Further details about the operations of PS-sets can be found in [18]. As shown in Fig. 2, an ad hoc network that contains 19 nodes is deployed in an irregular area. For simplicity, we split this network into three clusters. In cluster C1 , there are six nodes C1 = {n1 , . . . , n6 }. A polychromatic set matrix (PSM) is constructed according to its relative positions to each other. As shown in Fig. 4(a), if a node has direct links to its neighbors, the relationship is denoted as “·,” which means the Boolean value in PS-set is 1. Otherwise it is null. For example, for node n1 , it has direct links to n5 and n6 , then the adjacent vector is . Importance index is selected as the second unified color of cluster C1 , which can be calculated according to an algorithm proposed in Section IV-B and in (20). In cluster C1 , the importance index for nodes n1 , n1 , n3 , n4 , n5 , and n6 is 2.87, 2.12, 3.23, 3.56, 3.20, and 5.11, respectively. The third unified color represents the prior nodes of a node. In this case, n1 and n3 are defined as the source and destination of node cluster C1 , respectively; then it can be found that n1 is the prior node of n6 . Since the next hop of n6 is n3 , the prior node of n3 is node n6 . In this way, the PSM of NP can be derived as shown in Fig. 4(c). Similarly, the PSM of child nodes can be obtained as shown in Fig. 4(d), in which node n3 is the child node of n6 . Therefore, the child node color of n3 is NC6 = 1 and the child node color of n6 is NC1 = 1. Since

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n6 is the CH of cluster C1 , so here the unified color NCH of C1 can be easily obtained, as in Fig. 4(e). In cluster C1 , the unified color sets can be easily derived as F (C1 ) = {C1 , NI , NP , NC , NCH }. Actually, C1 can be seen as a child leaf of a PSHM < C1 , F (C1 ) >, as shown in Fig. 2. Similarly, the PSHM for C2 and C3 can also be derived as < C2 , F (C2 ) > and < C3 , F (C3 ) >. Iteratively, a PSHM for the whole network can be modeled. IV. Routing Design Based on PS-Sets Modeling In WSNs, the hierarchical routing schemes are considered superior to the flat schemes [24]. In flat routing schemes, all nodes are on the same level and each node performs the routing function. Each node maintains the global routing information (such as distance to destination, link status, and sequence number) that is flooded to all the other nodes. This will reduce the fairness and make it a simple protocol. However, in a flat routing scheme, large amount of control packet overhead is generated and flooded over the network that makes the flat routing scheme suffer from low scalability. Typically, flat routing protocols include destinationsequenced distance vector and wireless routing protocol, and the hierarchical routing schemes are further classified into two categories: dynamical hierarchical routing scheme, such as fisheye state routing (FSR) and inter-zone routing protocol, and static hierarchical routing protocols, such as cluterhead gateway switch routing. In the hierarchical routing scheme, there exists a temporary framework only for transmission. The CHs are able to organize a hierarchical structure, in which only local routing information is used. Each node only maintains local routing information and the exchange of routing information is bounded to zone/fisheye radios. The hierarchical

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routing can provide users a good fairness, small control packet overhead, and scalability, which is crucial in largescale WSNs. In general, the flat routing scheme is suitable for simple and small networks with moderate mobility, whereas the hierarchical scheme is suitable for large-scale WSNs. The constraint of hierarchical routing schemes is the complexity of the protocols. This paper aims at developing a low-complexity hierarchical routing scheme based on the PSHM, which can significantly simplify the hierarchical routing scheme. Until now, the relations between nodes have been defined and modeled with PSHM. Now this model is used to design a hierarchical routing scheme for an ad hoc network. Two key steps for this routing protocol are: 1) network partitioning and clustering, and 2) routing within the cluster and/or cross clusters. A. Network Partitioning and Clustering The parameter importance index, which was developed in the previous research work [20], is used to select a node as the cluster head. The network is then partitioned into multiple clusters around these CHs. For two 1-hop neighboring nodes ni and nj , one can calculate its link weight as I(ni , nj ) W1 (ni , nj ) = |N | (18) i I(ni , nj ) j=1

where I(ni , nj ) is the intersection area of their sensing ranges and Ni is the number of neighbors. The n-hop link weight can be calculated according to n Wn (ni , nj ) = W1 (ni , ni+1 ). (19) i=1

According to the analysis in [19], the importance index for node xi (ni ) can be calculated as 2

NI(ni ) =

Wh (ni , nk )λni ,nk (nj )

h=1

i=j=k

λni ,nk (nj )

(20)

in which only the case of 2-hop weights is considered, NI represents the importance index of node ni , and λni ,nk (nj ) denotes the hop count of the shortest path from ni to nk that passes the node nj . Different from other developed clustering methods in [21]–[23], here the importance index is used to dynamically build clusters. In the case of 2-hop, the node with the largest NI is selected as CH and all nodes within 2-hop from it are selected into this cluster. With this clustering algorithm, a large-scale wireless network can be easily partitioned into multiple clusters as small as possible when a request for clustering arrives. B. Routing Based on PSHM In this paper, we designed an on-demand routing scheme based on PSM to keep the routing scalable due to the hierarchical nature of the PS-sets. In PSHM, each node maintains

one or more PSM. If a node is assigned as the CH, it will maintain one more PSM for the upper level because it will become a member node of the upper level. If this CH is selected as the CH of an upper level, it will maintain one more PSM. In the example shown in Fig. 2, we only consider a three-level hierarchical model, in which CH2 maintains three PSMs for level 0, level 1, and level 2, respectively; CH1 and CH3 maintain two PSMs for level 2 and level 1, respectively. Iteratively, this process continues until only one CH is selected to maintain the whole network. Within a cluster, each node monitors the links to its neighbors within a period of time. If new nodes join in the network or some nodes are leaving or dead, it will update the PSM and broadcast it within the cluster. The CH summarizes the PSM within its cluster. When a node receives a request of data transmission, it will check if the destination node is within the same cluster as source node or not. This can be easily done from the definition of the node using PS-sets theory. As defined in PSHM, each node is labeled as a(d, id , jd−1 ). This actually defines the location of each node in the network that was modeled with PSHM. It can be easily found that the node is the ith node over level d and belongs to a cluster whose CH is the jth node over level d − 1. For example, as shown in Fig. 2, if the source node a(2, 1, 1) (n1 ), it is the first node on level 2 and its father node is the first one on level 1. It belongs to cluster C1 . If the destination node is a(2, 13, 2) (n13 ), it is the 13th node on level 2 and its father node is the third node on level 1 (2 − 1 = 1). It belongs to cluster C3 . When the source node and the destination node lie in a common cluster, then intra-routing algorithm will be invoked. The source node will deliver the data to the destination node directly if they are neighbors or via intermediate nodes if there are some common neighbors. In the case of non-neighbor nodes, if the distance between these two nodes is within 2hop to the CH, then the CH will act as the intermediate node. Another solution is CH assigns an intermediate node. It depends on the load on the CH. However, if the destination and the source nodes are not in the same cluster, the inter-routing algorithm will be invoked. The source node will send a transmission request to its CH, which is referred to source cluster head (S-CH). S-CH will check if the cluster head where the destination node is located, destination cluster head (D-CH), is in the same cluster or not at an upper level. If so, the route from S-CH to D-CH will be established according to the intra-routing described above. DCH will then deliver the data to the destination node. If S-CH and D-CH are not in the same cluster, S-CH will send a request to its CH on one more upper level. This process continues until S-CH and D-CH are found in the same cluster. As usual, several routes could be found in the process. The shortest one from the source to the destination will be selected. This is similar to ad hoc on-demand distance vector (AODV) routing protocol. C. Computation Complexity In this section, both the complexity of flat and hierarchical routing schemes are analyzed, in which the computation for


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graph partitioning is ignored [24]. For flat routing schemes, Dijkstra’s algorithm is known as the most commonly used algorithm [24], [25]. In a large-scale WSN with n nodes, Dijkstra’s algorithm yields a complexity of O(n2 log n). Thus, it is clear that for flat routing scheme, a total complexity of O(n2 log n) can be generated. For PSHM-based routing scheme (PSR), in case both the source node and the destination node are in a common cluster, the routing problem can be solved as a flat scheme. In case the source node and the destination node lie in two distinct clusters, the routing turns to a constrained routing problem. In this case, the PSR algorithm using PS matrices is at least as fast as O(d · n2 log n), where d denotes the average length of the path. Thus, the inter-cluster routing from source node to the CH yields a complexity of O(d · n2 log n log l), where l denotes the number of levels. In the worst case, for a WSN with C clusters, the PSR has a complexity of O(d · n2 log n log l)O(C log C). We have l n and C n, then O(d · n2 log n log l)O(C log C) ≈ O(d · C · n2 log n log l log C) for large-scale WSNs. Thus, it can be seen that the PS-based hierarchical approach yields a complexity of O(d · C · n2 log n log l log C). By refining the partition size C and level number l, an optimal bound can be obtained. Because C and l is much less than n, the computation complexity of PSR is much smaller than that of flat routing as O(n4 log n). Actually, it is also smaller than some existing hierarchical approaches such as FSR, LAR, ZRP [27], [28], and hierarchical max-flow routing [29]. V. Performance Evaluation This section analyzes the performance of the proposed routing scheme based on PS-sets theory with qualitative arguments. This proposed scheme is modified from AODV routing protocol using the control messages from AODV. A. Performance Evaluation of PSHM

Fig. 5. Packet delivery ratio versus network size. (a) 1-hop clustering. (b) 2-hop clustering.

A multihop wireless ad hoc network with 25–250 nodes is simulated with NS2 (ns−2.31). The nodes are deployed in five squares with different size. In this simulation, we use a freespace propagation channel model, the IEEE 802.11b as the access scheme, and the data rate is set at 2 Mb/s. Packet size is 1024 bytes. Transmission range is 250 m and carrier sensing range is 550 m. The buffer size at each node is of 15 packets. In this paper, we focus on three performance measures: 1) packet delivery ratio; 2) average packet delay; and 3) route discovery time. The variables are network size, number of pairs communicating with each other, and number of nodes in each cluster. The proposed scheme is compared with AODV and FSR using NS2 [21], [22]. For simplicity, we use a two-level and two-hop fisheye scoping in our simulation for FSR, and a three-level network for our proposed PSR (PSHM-based routing). In all simulations, we use scenarios with randomly placed nodes in a network, which is partitioned into multiple clusters according to the proposed clustering algorithm. In the first experiment, we focus on how the packet delivery ratio is affected when the network size is increased for all three routing schemes. In Fig. 5(a), where the network is partitioned by 1-hop, we note that for AODV, the packets

delivery ratio decreased drastically when the network size is greater than 100 nodes. The packets delivery ratio of FSR drops slightly because it is also a kind of hierarchical routing scheme; however, it is lower than that of PSR. PSR performs the best in terms of packet delivery ratio. In Fig. 5(b), the packet delivery ratio curves of a network that is partitioned by 2-hop are given. It can be seen that the results in Fig. 5(a) are better than those in Fig. 5(b). This is due to the fact that the 2-hop clustering causes larger hop count. Fig. 6 reports the average packet delay as a function of network size, in which as the network size increases, the average delays increase gradually for all three routing algorithms, but PSR remains the lowest, particularly when the network is larger than 100, which is suitable for large-scale network routing. The packet delay for AODV increased significantly due to the fact that when the network size increases, the route discovery runs with low efficiency than that of the hierarchical routing schemes. Comparing the results in Fig. 6(a) and (b), we can see that when we use the 1-hop clustering, the average delays caused by using all three protocols are all longer than that of 2-hop

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Fig. 6. Average delay versus network size. (a) 1-hop clustering. (b) 2-hop clustering.

clustering scheme. It is reasonable because the inter-routing may cause longer packet delay than intra-routing. In the next experiment, we investigate the total route discovery time for all three schemes. For the 1-hop clustering network in Fig. 7(a), the total routing discovery time for PSR is comparable to FSR and approximately two times more than AODV when the network size is smaller than 150 nodes. For the 2-hop clustering network in Fig. 7(b), the results are similar to 1-hop clustering. As the network size increases, the increase for PSR is gentler than that of AODV, so PSR is more suitable for large-scale wireless ad hoc networks. It can be seen from Fig. 7 that total path discovery time for PSR is longer at first than AODV and FSR because more paths are discovered but is not prohibitively longer even for the small-scale networks; however, for a networks with more than 150 nodes, the total path discovery time for PSR is much shorter than AODV and FSR. Fig. 7(b) shows a shorter route discovery time for PSR and FSR than that in Fig. 7(a). It is due to the fact that when using 2-hop clustering, we can obtain a suitable tradeoff between the cluster numbers and hop count.


Fig. 7. Total path-discovery time versus network size. (a) 1-hop clustering. (b) 2-hop clustering.

Last, Fig. 8 compares the packet delivery ratio and average delay with the node degree to see how the protocols are affected. Fig. 8(a) reports the packet delivery ratio with randomly placed nodes on an area of 1000 m×1000 m. For node degree smaller than 5, the network is not always in a connected state and the packet delivery ratio may be affected. When the average node degree is at 5, the network is connected completely and almost all packets are delivered successfully. However, when the node degree is too large for AODV, it will cause the collision and the delivery ratio will decrease. For PSR and FSR, the performance remains mainly unaffected. In Fig. 8(b), we report the average delay as a function of node density. For PSR and FSR, the average delay is almost constant when the density is varied. However, for AODV, the average delay is increased significantly when the nodes degree is too large. The reason in this case is that AODV produces a large amount of network overhead from the flooded route request messages. B. Complexity Analysis Compared with the existing multihop routing protocols in ad hoc networks, the PSR contains the following two features.


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Simulations show that at 1000 nodes, AODV performs poorly and only 25% packets are delivered [23]. The routing control message grows fast exponentially with number of nodes, which significantly decreases the delivery of the data packets. VI. Conclusion In this paper, a low computational complexity PS-based hierarchical routing scheme was proposed to perform routing functions as well as or better than flat routing scheme, which ignored some global information that was used in the flat routing schemes. PS-sets theory demonstrated a new mathematical tool to model scalable complex systems, such as large-scale WSNs. The reason for this was that typical topologies can be partitioned into clusters, for which the nodes within each cluster are well connected. Simulation results showed that the PS-based hierarchical routing protocol was more effective and scalable than existing hierarchical routing protocol. The uniqueness of its capability of describing the properties of the object when there are more hierarchical levels could make it useful in designing new routing schemes considering the varieties of the wireless links, which is being investigated and will be reported in future. References

Fig. 8. Packet delivery ratio versus node degree. (a) 1-hop clustering. (b) 2-hop clustering.

1) Lower computation complexity: For the proposed PSR, we can see that the number of intrinsic cycle to calculate the importance index is Nni for node ni , which is very small when the number of neighbors is small. In routing discovery process, PSR uses Boolean operations, so the complexity is O(d · C · n2 log n log l log C), in which C is the number of clusters and d is the depth of PSHM (d is a small integer constant). Normally, a 1000-node network can be effectively modeled by a PSHM with d ≤ 3. Compared with PSR, the existing multihop routing schemes, such as AODV, FSR, and others, have much higher complexity for large-scale networks. For AODV and FSR, the complexity increases exponentially. 2) Scalability: PSR is able to easily maintain large-scale networks by using the Boolean operations of PS-sets matrices, which can be adaptively updated when the topology changes. For AODV or FSR, with growth of the network scale, the average path length increases and so does the probability that a path becomes invalid. Therefore, AODV and FSR are not suited for large-scale networks. The scalability limit is about 1000 nodes.

[1] M. Gadouleau and S. Riis, “Graph-theoretical constructions for graph entropy and network coding based communications,” IEEE Trans. Inform. Theory, vol. 57, no. 10, pp. 6703–6717, Oct. 2011. [2] L. Levitin, M. Karpovsky, and M. Mustafa, “Minimal sets of turns for breaking cycles in graphs modeling networks,” IEEE Trans. Parallel Distrib. Syst., vol. 21, no. 9, pp. 1342–1352, Sep. 2010. [3] M. M. Zavlanos, M. B. Egerstedt, and G. J. Pappas, “Graph-theoretic connectivity control of mobile robot networks,” Proc. IEEE, vol. 99, no. 9, pp. 1525–1540, Sep. 2011. [4] I. Norros and H. Reittu, “Network models with a ‘soft hierarchy’: A random graph construction with loglog scalability,” IEEE Netw., vol. 22, no. 2, pp. 40–46, Mar.–Apr. 2008. [5] J. Blum, M. Ding, A. Thaeler, and X. Cheng, “Connected dominating set in sensor networks and MANETs,” in Handbook of Combinatorial Optimization. Norwell, MA: Kluwer, 2004, pp. 329–369. [6] E. W. Dijkstra, “A note on two problems in connexion with graphs,” Numerische Mathematik, vol. 1, no. 1, pp. 269–271, Dec. 1959. [7] X. Li, X. Xu, F. Zou, H. Du, P. Wan, Y. Wang, and W. Wu, “A PTAS for node-weighted Steiner tree in unit disk graphs,” in Proc. 3rd Int. Conf. Combinat. Optimiz. Applicat., vol. 5573. 2009, pp. 36–48. [8] M. Newman, “The structure and function of complex networks,” SIAM Rev., vol. 45, no. 2, pp. 167–256, 2003. [9] M. Gjoka, C. T. Butts, M. Kurant, and A. Markopoulou, “Multigraph sampling of online social networks,” IEEE J. Sel. Areas Commun., vol. 29, no. 9, pp. 1893–1905, Sep. 2011. [10] J. Eriksson, M. Faloutsos, and S. V. Krishnamurthy, “DART: Dynamic address routing for scalable ad hoc and mesh networks,” IEEE/ACM Trans. Netw., vol. 15, no. 1, pp. 119–132, Jan. 2009. [11] Q. Chen, Q. Zhang, and Z. Niu, “A graph theory based opportunistic link scheduling for wireless ad hoc networks,” IEEE Trans. Wireless Commun., vol. 8, no. 10, pp. 5075–5085, Oct. 2009. [12] P. K. Biswas, “A formal framework for multicast communication,” IEEE Syst. J., vol. 4, no. 3, pp. 353–362, Sep. 2010. [13] V. V. Pavlov, “Polychromatic graph of mathematics simulation for technical system,” in Proc. Sci. Tech. Conf., 1988, pp. 8–10. [14] V. V. Pavlov, “Mathematics simulation of discrete production system,” Inform. Technol., pp. 15–19, 1995. [15] V. V. Pavlov, Polychromatic Sets and Graphs for CALS Technology. Moscow, Russia: STANKIN Press, 2000. [16] S. Li and X. Wang, “Polychromatic set theory-based spectrum access in cognitive radios,” IET Commun., vol. 6, no. 8, pp. 909–916, 2012.

58

[17] S. Li, “The application of polychromatic sets in computer aided design,” M.S. thesis, Dept. Comput. Sci., Xi’an Jiaotong Univ., Suzhou, China, 2004. [18] Z. Li and L. Xu, “Polychromatic sets and its application in simulating complex objects and systems,” Comput. Oper. Res., vol. 30, no. 6, pp. 851–860, 2003. [19] G. Cantor, “Uber eine Eigenschaft des Inbegriffes aller reellen algebraischen zahlen,” Crelles J. Mathematik, vol. 77, pp. 258–262, 1874. [20] S. Li, X. Wang, and S. Zhao, “Multipath routing for video streaming in wireless mesh networks,” Ad Hoc Sensor Wireless Netw., vol. 11, nos. 1–2, pp. 73–92, 2011. [21] C. K. Toh, A.-N. Le, and Y.-Z. Cho, “Load balanced routing protocols for ad hoc mobile wireless networks,” IEEE Commun. Mag., vol. 47, no. 8, pp. 78–84, Aug. 2009. [22] G. Jayakumar and G. Gopinath, “Ad hoc mobile wireless networks routing protocols: A review,” J. Comput. Sci., vol. 3, no. 8, pp. 574–582, 2007. [23] S. Lee, E. M. Belding-Royer, and C. E. Perkins, “Scalability study of the ad-hoc on-demand distance vector routing protocol,” ACM/Wiley Int. J. Netw. Manag., vol. 13, no. 2, pp. 97–114, 2003. [24] C. Lim, S. Bohacek, J. P. Hespanha, and K. Obraczka, “Hierarchical max-flow routing,” in Proc. IEEE Global Telecommun. Conf., vol. 1. Dec. 2005, pp. 545–550. [25] D. Babic, J. Bingham, and A. J. Hu, “B-Cubing: New possibilities for efficient SAT-solving,” IEEE Trans. Comput., vol. 55, no. 11, pp. 1315– 1324, Nov. 2006. [26] T.-H. Chiu and S.-I. Hwang, “Efficient fisheye state routing protocol using virtual grid in high-density ad-hoc networks,” in Proc. 8th Int. Conf. Adv. Commun. Technol., vol. 3. 2006, pp. 1475–1478. [27] I. S. Thaseen and K. Santhi, “Performance analysis of FSR, LAR and ZRP routing protocols in MANET,” Int. J. Comput. Applicat., vol. 41, no. 4, pp. 20–24, Mar. 2012. [28] K. Sahadevaiah and O. B. V. Ramanaiah, “An empirical examination of routing protocols in mobile ad hoc networks,” Int. J. Commun., Netw. Syst. Sci., vol. 3, no. 6, pp. 511–522, 2010.


[29] C. Lim, S. Bohacek, J. P. Hespanha, and K. Obraczka, “Hierarchical max-flow routing,” in Proc. IEEE Global Telecommun. Conf., vol. 1. Dec. 2005, pp. 545–550.

Xinheng Wang (M’04) received the B.Eng. and M.Sc. degrees in electrical engineering from Xi’an Jiaotong University, Suzhou, China, in 1991 and 1994, respectively, and the Ph.D. degree in computer engineering and electronics from Brunel University, Uxbridge, U.K., in 2001. He was with Brunel University and Kingston University, Kingston, U.K., before he joined Swansea University, Swansea, U.K., in 2007, where he is currently a Senior Lecturer in wireless communications. His current research interests include wireless mesh and sensor networks, cognitive radio networks, mobile computing, Internet of things, and applications of wireless technologies for healthcare. He is currently investigating multiple research projects supported by the U.K. EPSRC, TSB, Welsh Government, China NNSF, China 863 Scheme, EU, and the industry. Dr. Wang is a Committee Member of BSI on ICT, Computing, and Healthcare. Shancang Li (M’08) received the B.Eng. and M.Sc. degrees in mechanical engineering and the Ph.D. degree in computer science from Xi’an Jiaotong University, Suzhou, China, in 2001, 2004, and 2008, respectively. He joined the College of Engineering, Swansea University, Swansea, U.K., in December 2008, as a Research Fellow. His current research interests include wireless mesh and sensor networks, internet of things, signal processing, and applications of wireless technologies.