optimizing deployment of internet gateway in wireless mesh networks

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www.elsevier.com/locate/comcom

Computer Communications 31 (2008) 1259–1275

Optimizing deployment of Internet gateway in Wireless Mesh Networks

Bing He, Bin Xie, Dharma P. Agrawal *

OBR Center of Distributed and Mobile Computing, Department of Computer Science, University of Cincinnati, 833 Rhodes Hall,

Cincinnati, OH 45221-0030, USA

Available online 9 February 2008

Abstract

In a Wireless Mesh Network (WMN), Mesh Routers (MRs) are interconnected by wireless links and constitute a wireless backbone toprovide ubiquitous high-speed Internet connectivity for mobile clients (MCs). The wireless backbone is tightly integrated with the Inter-net by some selected nodes called as Internet Gateways (IGWs). An IGW has more capabilities than a simple MR and is more expensive.In this paper, we address the IGW deployment problem which is shown to be NP-hard. We first formulate it as a linear program (LP)issue, then develop two heuristic algorithms: Degree based Greedy Dominating Tree Set Partitioning (Degree based GDTSP) and Weightbased Greedy Dominating Tree Set Partitioning (Weight based GDTSP), for the purpose of cost-effective IGW deployment. We evaluatethe effectiveness of these two algorithms by extensive simulations and comparisons with two major approaches.� 2008 Elsevier B.V. All rights reserved.

Keywords: Interfaces; Internet Gateways; Mesh Networks; Mesh Routers; Multiple channels; Multiple radios; Network capacity; Throughput

1. Introduction

A Wireless Mesh Network (WMN) integrates multi-hopcommunication with the Internet to provide a cost-efficientInternet connectivity for mobile clients (MCs). In a WMN,a mobile client can access the Internet through a wirelessbackbone formed by wireless Mesh Routers (MRs) whilesome MRs provide Internet accessibility, called as theInternet Gateways (IGWs). In the wireless backbone,MRs are interconnected in a multi-hop fashion while IGWsact as the communication bridges between the wirelessbackbone and the Internet. To increase the network capac-ity, both the MR and IGW have the ability to simulta-neously use multiple non-overlapping channels for packettransmission and reception.

Designing a WMN is a fundamental issue in decidingthe network efficiency such as coverage and throughputcapacity, and thus should be addressed carefully. TheWMN design includes many issues such as the interfaceconfiguration and the placement of MRs and IGWs. In a

0140-3664/$ - see front matter � 2008 Elsevier B.V. All rights reserved.

doi:10.1016/j.comcom.2008.01.061

* Corresponding author. Tel.: +1 513 556 4756; fax: +1 513 556 7326.E-mail addresses: [email protected] (B. He), [email protected]

(B. Xie), [email protected] (D.P. Agrawal).

certain degree, the interface configuration determines theMR’s throughput capacity and the Internet throughput,while different locations of MRs and IGWs lead to differentnetwork topologies and architectures. Since in a WMN allInternet traffic has to pass through one of IGWs, a WMNmay be unexpectedly congested at an IGW, even if everyMR provides enough throughput capacity. On the otherhand, each IGW has to be configured with Internet access,which make it more expensive than setting up anMR. Therefore, it is crucial to minimize the number ofIGWs.

Design of a WMN is in its infancy and many challengesstill remain open. A number of independent research effortshave recently been made to develop WMN network archi-tectures and protocols in order to have an effective WMN.For example, distributed multi-channel assignment proto-cols [1] intend to deploy non-overlapping channels in sucha way that each interface of MRs is maximally utilized fordata transmission. Multi-hop and load-balancing routingschemes [2] target to efficiently forward traffic flows fromMRs to the destination. However, all these protocolsassume the WMN is already deployed in advance. In otherwords, these protocols assume that every node is located ata fixed location, with a given interface configuration.

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1260 B. He et al. / Computer Communications 31 (2008) 1259–1275

In this paper, we address the optimization of the IGWplacement which is one of essential issues in the WMNdesign. The IGW placement concerns where the IGWshould be located and how to minimize the number ofIGWs while satisfying the MR Internet throughputdemand. Although there are some gateway placement tech-nologies [3–6] in the literature, none of them captures theunique traffic and network characteristics over a multi-channel and multi-radio WMN. The key contributions ofthis paper are:

� We formulate the IGW placement problem in a genericWMN model, such as IEEE 802.11a/b/g to impose nec-essary constraints on the IGW placement, and show theoptimal placement problem to be NP-hard.� We address the WMN network architecture from the

aspect of IGW placement, and present some theoreticalresults.� We model the IGW placement problem as an integer lin-

ear program problem, and develop two heuristic algo-rithms. Our experimental results verify the effectivenessof our algorithms by comparing their performance withtwo major gateway placement algorithms.

The remainder of this paper is structured as follows. InSection 2, related work is addressed. In Section 3, a gen-eric WMN model and the design objectives are defined forIGW placement. Then, we discuss the network architec-ture and the IGW placement in Section 4. In Section 5,the IGW placement problem is formulated as a linearprogram and two heuristic algorithms are presented.Section 6 evaluates the effectiveness of the proposed algo-rithms. Finally, the concluding remarks are included inSection 7.

2. Related work

The placement of IGWs in WMNs is critical in deter-mining the network performance such as the Internetthroughput. Optimal selection of the IGW from a givenWMN is a complex problem, involving many issues suchas network partition, traffic effort from MRs to IGWs,node and link capacity, computational efficiency, installa-tion cost, etc. However, there are only a few approaches[3–6] on this subject and none of them considers all of theseissues simultaneously.

Chandra et al. [3] proposed a gateway placementapproach, aiming to minimize the number of gatewayswhile guaranteeing MR’s bandwidth requirement. Thegateway placement is formulated as a network flow prob-lem. A max-flow min-cut based algorithm is developedfor gateway selection. An MR may be attached to multipleIGWs through multiple paths. In this approach, the pathlength between MR and IGW is not considered as an opti-mization parameter and thus long paths may be selected.As a result, the traffic effort from MR to IGW could notbe effectively addressed.

Prasad and Wu [4] have modeled the IGW placement asan Integer Linear Programming (ILP) with the focus on theIGW and link capacity, faulty tolerance, load-balancing,and minimization of the number of IGWs. Further, theauthor proposed an OPEN/CLOSE heuristic algorithm,starting from an arbitrary solution and repeatedly selectinga random solution until a predefined cost could be satisfied.The gateways are selected from a given gateways sets, andthus the efficiency of such a random IGW strategy needs tobe improved for a large WMN where a large number ofpotential MRs can serve as IGWs.

Bejerano [5] presented an approach to connect a staticmulti-hop wireless network by selecting some nodes asinfrastructure gateway nodes such that the multi-hop net-work is attached to the wired backbone. In this approach,the multi-hop wireless network is partitioned into disjointclusters that satisfy the requirement on bandwidth anddelay. The cluster is further organized as a spanning treewhich is rooted at the cluster header. The cluster head isaccordingly considered as gateway node and each clusteris a connected graph, with an upper bound on its radius.Since this approach is designed for generalized staticmulti-hop wireless networks such as sensor networks andbased on Time Division Multiple Access (TDMA) scheme,the defined network model could not capture the uniquefeatures of WMNs, e.g., traffic constraints of the MR thathas multiple interfaces in the IEEE 802.11a/b/g-basedWMN.

Aoun et al. [6] improved Bejerano’s work [5] and pro-posed a recursive Quality of Service (QoS)-based IGWselection approach for WMNs. The goal of this approachis to minimize the number of IGWs while satisfying theQoS requirements. An one-hop dominating-set is firstformed from the original network graph and this result isused as the input to the next recursion. The greedy domi-nating-set searching operation continues until the clusterradius reaches R, which is the predefined upper bound ofcluster radius. This approach does not consider the capac-ity of the MR which operates multiple interfaces. At thesame time, the channel interference of multiple radios hasnot been addressed in the selection of the IGW and itsattached MRs.

It can be seen that all of the above approaches fail toconsider the multi-channel, multi-interface, and otherunique traffic characteristics of the WMN. Theseapproaches pursue the minimal number of IGW as theoptimization objective without considering the channelinterference in a WMN, which significantly affects thenetwork performance. In this paper, we address theIGW placement in a WMN from two novel aspects: (i)modeling the throughput capacity of MRs and IGWsin terms of multi-channel and multi-radio, and (ii) devel-oping effective heuristic algorithms to minimize the num-ber of IGWs with the consideration of variousconstraints. We further compare the performance ofour approach with [5,6] and show the effectiveness ofour approach.

B. He et al. / Computer Communications 31 (2008) 1259–1275 1261

3. Network model and problem formulation

In a WMN shown in Fig. 1, MRs are wirelessly con-nected in a multi-hop fashion and form a wireless back-bone to enable the ubiquitous Internet connectivity formobile clients. The wireless backbone is again tightly inte-grated with the Internet through the IGWs. Each MR maybe equipped with single or multiple interfaces. The multipleinterfaces have the ability to deploy multiple non-overlap-ping channels (e.g., 12 for IEEE 802.11a and 3 for IEEE802.11b/g) with which MR can simultaneously transmitand receive packets from different neighboring MRs. EachIGW is configured with wireless and wired links, and actsas the gateway between the Internet and wireless backbone.The wireless links enable the IGW to communicate with itsneighboring MRs while the wired links connects to theInternet or other wired networks. For clarification, we for-mally define the network model as follows.

3.1. Network model

A WMN can be formulated as a multi-radio, multi-channel, multi-hop, and infrastructure-based wireless net-work, which can be represented by an undirected networkgraph G = (V,E). V = {v1, . . . ,vn} is the set of n networknodes, representing MRs and IGWs. The physical locationof a node vi 2 V is static after deployment, and accordinglydonated by Xi. Additionally, each node vi 2 V is connectedto a rich power supply, and thus is not subjected to anypower constraints as in a mobile ad hoc network. Amongn MR nodes, there are m nodes configured as IGWs whichprovide the Internet connection. The other (n � m) nodesare simply MRs. Every MR vi is configured with a set ofwireless interfaces (i.e., radios) donated by q(vi) ={1, 2, . . . , jq(vi)j}. In order to efficiently and fully use thenetwork resources, any two radios at the same node aretuned to different channels for simultaneous transmissionor reception. In addition to multiple radios, each IGW isconfigured with wired Internet connection. It may be notedthat the wired connection of an IGW as well as the Internetbandwidth, is assumed to have unlimited bandwidth as

Fig. 1. WMN network model.

compared to wireless links. The IGW is expensive due tothe construction of wired link. In WMN, the number ofIGWs is usually less than the number of MRs (i.e.,m 6 (n � m)). For simplicity, we assume all backboneradios of MRs have the same transmission range Rtran.Set CH = {1, 2, . . . ,c} donates c non-overlapping channels(e.g., 12 for IEEE 802.11a in 5 GHz and 14 for IEEE802.11b/g in 2.4–2.5 GHz) in the wireless system. The datarate possible in a channel i 2 CH is wi bits/s. For example,the 802.11b channel operates in the 2.4 GHz band with thecapacity up to 11 Mbps and the 802.11a/g channel cansupport up to 54 Mbps. Two nodes (MRs or IGWs) areconnected by an edge if and only they are within the trans-mission range of each other such that their radios can betuned to the same channel to communicate. E = {e1, . . . ,ek}is the set of edges between two nodes. It is noted that thewireless connection between two neighboring nodes, whichis represented by an edge, may consist one or more wirelesslinks.

In a WMN, every MR has the functionality of aggregat-ing traffic from mobile clients. Unlike a pure ad hoc networkwhere traffic is randomly generated between peer nodes, in aWMN, the traffic of an MR is predominantly directed eithertowards the IGW or from the IGW to the MR. It means thatthe aggregated traffic in an MR is forwarded in a multi-hopfashion towards an IGW, which in turns forwards to theInternet. In the opposite direction, all traffic from the Inter-net passes through an IGW before it is routed to the destina-tion MRs and the corresponding mobile clients. Given anMR vi 2 V, its traffic may include two parts:

� Local Internet traffic (i.e., Tl(vi)), which is Internet trafficgenerated from its own mobile clients in its servicingarea, and� Relaying Internet traffic (i.e., Tr(vi)), which is relayed for

other MRs.

The local traffic of an MR vi 2 V can be regarded as thetraffic demand of the MR. The relaying traffic of an MRvi 2 V can be defined as the traffic generated from otherMRs, which is relayed through the MR vi toward theIGW. We further model the capacity of the IGW andMR in Appendix A.

3.2. Problem formulation

Definition 1. IGW placement problem: The IGW placementproblem basically is, given a network with n MRs, to selecta set of m nodes I = {I1, . . . ,Im}, and Ii 2 V, to serve asIGWs with which the WMN can satisfy the Internet trafficdemand for each MR (i.e., Tl(vi)).

From the definition, we know I � V. The IGW place-ment problem is an off-line issue and should be solved atthe stage of the network construction. If not all n MR posi-tions can be IGW candidates, the IGW placement in thiscase is a variation of that n IGW potential positions. When


the number of potential IGW positions decreases, the spacecomplexity of the IGW placement is accordingly reduced.The approach for the IGW placement with n IGW poten-tial positions, can be applied to variations that the IGWlocations are geographically restricted. Thus, we considerthe IGW placement with n nodes as the potential locations.We further ignore peer-to-peer traffic between MRs. Thisassumption is based on the observation that the selectionof IGW has a minor impact on the peer-to-peer through-put. It is because the user traffic in a WMN is dominantlydirected between IGWs and MRs, and the location deter-mination of IGW does not change the link capacity andthe network topology for peer-to-peer communication,which is decided by G = (V,E). Thus, if we neglect theeffect of peer-to-peer throughput on the Internet traffic,the above assumption can hold well.

Determination of m IGWs from n nodes ought to satisfythe traffic demand at each MR with additional constraints.We model the IGW placement problem with the followingconstraints.

� Full coverage: IGWs should be located such that allMRs are attached to IGWs. We say ‘‘attached” meansthat each MR is connected to at least one IGW so thatall MRs have the Internet accessibility via single hop ormulti-hop paths.� IGW throughput capacity: According to Lemma 3 proved

in Appendix A, each IGW has its maximum throughputcapacity Wigw. The WMN network capacity can beenhanced by increasing the number of IGWs. WhenMRs and their traffic demand increase, the number ofIGWs have to be added accordingly in a way that the net-work capacity should be at least equal to the total trafficdemand imposed by MRs. Otherwise, the IGWs couldbecome the bottleneck for the network throughput, evenwhen each MR has enough throughput capacity. Accord-ing to Lemma 3, we can derive the minimal number ofIGWs if we know the traffic demand on each MR. Forinstance, if every IGW has the same configuration, weobtain the number of IGWs with the following inequality:

m PXn

j¼1

T lðvjÞ=XjqðvigwÞj

i¼1

wi: ð1Þ

� MR throughput capacity: According to Lemma 4 provedin Appendix A, each MR has its maximum throughputcapacity Wmr. The traffic pass through an MR cannotexceed the MR throughput capacity. For an MR nodevi, the sum of its own local traffic and its relaying trafficcan not exceed its throughput capacity W vi , i.e.,

mW vi P T lðviÞ þ T rðviÞ: ð2Þ

When IGWs are located at different places, the Internettraffic of MR flows through different MRs and pathsthat are bounded by MR throughput capacity.

� Co-channel interference: In multi-hop wireless network,the co-channel interference between neighboring nodesseverely limits the network performance [7–9], especiallyfor the network configured with omnidirectional radios.In a WMN, both the throughput capacity of IGW andMR are affected by the interference. The interferencemay be due to adjacent hops in the same path and/orfrom neighboring paths. In the following subsection,we discuss three interference models.� Investment cost: Minimizing the investment cost is a fun-

damental issue in deploying a WMN. Increasing thenumber of IGWs may augment the network throughputcapacity, but it also increases the investment cost due toexpensive construction of wired links in IGWs.

3.3. Interference models

In the network design phase, it is very difficult to modelthe interference in a WMN due to changing traffic pattern,dynamic channel assignment, and the presence of multi-hop routing. In addition, the interference among linksrelies on many factors, such as transmission power, linkquality, and reception threshold in the interface. These fac-tors are also hard to evaluate at the time of network setup.Therefore, it is extremely difficult to consider a fine-grainedinterference model, which is even prohibitive in a smallWMN [3]. Due to this, it is necessary to find an approxi-mate way to model the impact of interference on theIGW selection. In this part, we discuss three interferencemodels: ideal link model, hop-based throughput degrada-tion model, and collision-based model.

1) Ideal link model: In such an ideal model, the flowthroughput on a path will not degrade if the numberof hop is within a given hop such as Rhop. If the trans-mission hop of the flow is beyond this limited hop,the throughput over the path becomes zero.

2) Hop-based throughput degradation model: This modelconsiders that the consumed bandwidth of a flowincreases in accordance with the number of hop dueto increased chance of interference and contention[10,3] when flow has to travel a larger number ofnodes. Considering a path with hop p from MR vi

to an IGW, in order to achieve a required throughputof Tl(vi), the practical required throughput denotedby T 0lðvi; pÞ, can be estimated as:

T 0lðvi; pÞ ¼ T lðviÞ � ð1þ aÞðp�1Þ; ð3Þ

where a is the additional percentage of per hopthroughput consumption.

3) Collision-based model: This model proposed in [11]considers that a node can successfully use a channelfor transmission only when other neighboring nodesdon’t use this channel for transmission or reception.Let Rint be the interference range of a channel, whichis larger than the transmission range Rtran, i.e.,Rint > Rtran. In this range, the channel can not be


reused. Let Wint be the maximal throughput capacityusing the channel in this range. When a traffic flowneeds to be forwarded within this range given by Rint,it has to validate whether existing traffic in this rangeis saturated or not. This means that it is unacceptableif the available channel bandwidth in Rint is less thanthe throughput demand of the new flow. This modelignores the interference caused by the transmissionoutside Rint and implicitly avoids the hidden-nodeeffect. Furthermore, the above process considers asingle shared channel for use and is applicable formultiple orthogonal channels since they are no inter-ference between orthogonal channels. In this case,Wint is the maximal throughput capacity using allorthogonal channels in Rint.

3.4. Objectives on the IGW placement

Based on our analysis of the problem and the aforemen-tioned constraints, we define three objectives for IGWplacement algorithm.

� Minimizing the number of IGWs: The number of IGWsm in the network should be minimized. We need todeterminate the locations of IGWs that results in min-imum number of IGWs while providing enough net-work throughput capacity. If IGWs are situated atwrong positions, more IGWs should be deployed tomeet the traffic requirements of MRs, thus the invest-ment cost is accordingly increased. However, when thenumber of IGWs is not enough, the traffic demand ofMRs may not be fulfilled due to limited networkcapacity.� Minimizing the MR-IGW Hop: The MR-IGW path indi-

cates the number of transmission hops between the MRand the IGW. As proposed by Li et al. [7], the capacityavailable to each node k, i.e., the data throughput whichis originated at an MR, is approximately bounded bythe following inequality:

k 6Cmax=n

Lpath=Rtran

; ð4Þ

where Cmax is total one-hop throughput capacity for alink, n is the number of nodes in the network, Lpath isthe Euclidean distance (i.e., path physical length) fromthe source to the destination, Rtran is the radio transmis-sion range, and thus Lpath/Rtran is the minimal numberof hops the data packet has to travel. This inequalityshows that as the number of hops increases, the band-width obtainable for an originating node linearlydecreases. The above inequality considers an ad hoc net-work with a single shared channel, and we can applythis inequality to multiple radios by replacing the Cmax

with the one-hop throughput capacity for a set of MRlinks (i.e., Clinks-max ¼

PjqðvmrÞji¼1 Ci

max), where Cimax is ith

link one-hop throughput capacity. It yields:

k 6

PjqðvmrÞj

i¼1

Cimax=n

Lpath=Rtran

; ð5Þ

The reason behind this extension is that each interfaceoperates in non-overlapping channels at the MR with-out co-channel interference. It also shows that the realnetwork throughput scales better when the average dis-tance between source and destination nodes remainsmall. Due to this, the locations of the IGWs shouldbe selected in the places that necessitates fewer hopcounts from each MR to the IGW. For this purpose,we use the average MR-IGW hop as the evaluation in-dex, which coarsely reflects the MR-IGW transmissiondistance for all MRs.

� Affordable computational complexity: Computation com-plexity is another important factor to be considered whileevaluating the IGW placement algorithm. Based onabove constraints, finding the optimum placement ofIGW is NP-hard, which is proven later. When the numberof MRs increases, the computation overhead may beextremely high due to exponential increase in the compu-tational complexity. Most of the existing approaches tar-get to find a polynomial time approximation algorithm.In this paper, we will show that our heuristic IGWplacement algorithms have polynomial time complexity.

Among these objectives, some are competing againstother. As discussed earlier, there is a trade-off betweenthe number of IGWs and the Internet throughput capacity.At the same time, decreasing the number of MR-IGW hopsrequires more IGWs which increases the number of IGWsand consequently increases the investment cost.

4. Network architecture and IGW placement

In the last few years, many WMN architectures havebeen proposed in order to efficiently deploy a WMN. Inthis section, we discuss two types of WMN network archi-tectures: IGW-directed and connected clusters and IGW-rooted trees. Both of these architectures can be generatedby placing the IGWs in the WMN.

4.1. IGW-directed and connected cluster

Definition 2. IGW-directed and connected cluster: In aWMN represented by G = (V,E), an IGW-directed andconnected cluster is formed as a connected graph Clus-teri = (Vi,Ei), where V i ¼ fI i; v10 ; . . . ; vi0 g, Ii 2 I is the headof the cluster, v10 ; . . . ; vi0 2 V � I , and Ei ¼ fe10 ; . . . ; ek0 g isthe set of edges toward the IGW Ii. We say ‘‘toward the Ii”

means all edges are directed toward the IGW Ii.

Fig. 2(a) is a given initial network represented as anundirected graph that consists of nine MR nodes, andFig. 2(b) and (c) shows IGW-directed and connected clus-ters. In Fig. 2(b), node 5 (i.e., v5) is selected as the IGW I1,


while node 2 (i.e., v2) is chosen as the IGW I1 in Fig. 2(c). Itcan be seen from the definition that an IGW-directed andconnected cluster is a directed graph G0 that has the follow-ing properties:

� Clusteri is headed at the node Ii that is selected as theIGW, and is connected without cyclic path.� Each edge is directed from an MR toward the IGW.� Every MR is connected to the IGW by single hop or

multiple hops, and an MR may have multiple pathstoward the IGW. As shown in Fig. 2(b), v9 has twopaths to the IGW (v9 ? v7 ? I1 and v9 ? v8 ? I1).

We use a directed cluster for the purpose of identifyingthe path from the MR to the IGW. From a given IGW-directed and connected cluster, we can clearly see all thepaths from each MR to the IGW. When an IGW-directedand connected cluster is formed, we can obtain a realWMN network architecture by removing the direction ofthe edge in the IGW-directed and connected cluster. Thismeans that in an undirected cluster, the Internet trafficcan be directed from the IGW to MRs as well. Accordingto the above definition, given an initial WMN with n nodes,we can partition the network into m disjoint clusters, witheach IGW as the head of the cluster. Fig. 2 has m = 1. Inaddition, each IGW has the capability to satisfy the trafficdemand of all MRs belonging to the cluster. In the follow-ing section, we can further reduce IGW-directed and con-nected clusters into IGW-rooted trees.

4.2. IGW-rooted tree

Definition 3. IGW-rooted tree: An IGW-rooted tree is adirected graph as a tree and all edges are directed towardsthe IGW node, which is the root of the tree.

Fig. 2(d) shows an IGW-rooted tree that consists ofnine nodes as in Fig. 2(b). These nodes are connected tothe IGW through the tree-based routing path. A tree-based WMN architecture, which consists of IGW-rootedtrees, has been extensively discussed in the literature

Fig. 2. IGW-directed and connected cluster, IGW-rooted tree.

[12–14]. Such a network architecture offers benefits suchas low routing overheads and efficient aggregation offlows [14]. For example, a tree-based routing mechanismis suitable for the IGW-oriented traffic because: (i) it sim-plifies the routing path from MRs to the IGW and, (ii) itcan easily control traffic and routing path so as toincrease the network channel utilization and ensure anoptimal utilization of bandwidth. The tree-based channelassignment and routing schemes have been developed in[12]. The network performance of this tree-based architec-ture has been analyzed in [13]. Therefore, in this paper,we focus on the tree-based network architecture and for-mulate the IGW placement in a way that all MRs areassociated with one IGW by using a tree structure. Inother words, we can determine the IGW positions suchthat a given WMN can be organized into multipleIGW-rooted trees.

Lemma 1. An IGW-directed and connected cluster Clusteri

can be reduced to an IGW-rooted tree, and conversely aIGW-rooted tree may be transformed to an IGW-directedand connected cluster.

Proof. According to our definition, an IGW-rooted tree isa directed graph which is a tree and all edges are directedtowards a particular root vertex, which is the IGW. Again,according to the definition of IGW-directed and connectedcluster, every node has at least one path towards the IGW.Therefore, if a Clusteri maintains only one path for eachnode towards the IGW and other redundant edges areremoved, the Clusteri is reduced to an IGW-rooted tree.On the contrary, an IGW-rooted tree can be converted toan IGW-directed and connected cluster by finding the addi-tional paths from an MR to the IGW. h

Lemma 1 indicates when a WMN is partitioned intoIGW-directed and connected clusters, we can establishIGW-rooted trees as the tree-based WMN network archi-tecture. On the contrary, if an initial WMN is divided intoIGW-rooted trees, we can also construct correspondingIGW-directed and connected cluster architecture. There-fore, it is essential to divide an initial WMN into m

IGW-rooted trees while satisfying the constraints discussedin Section 3. In other words, we can first find m IGW-direc-ted and connected clusters and then prune each cluster intoIGW-rooted tree. Upon removing the direction of the edgeon the tree, we can obtain a real WMN network, allowingbidirectional traffic flow. Due to the particular networkarchitecture of IGW-rooted trees, the design objective ofminimizing the number of IGWs m can be regarded asthe issue to find minimum number of IGW-rooted treeswhile each IGW in a tree can satisfy the tree traffic demandof nodes. We further present the constraints discussed inSection 3 in the following way.

� Bound of the tree size: The constraint of IGW through-put capacity can be translated into an upper bound ofthe tree size:


W igw ¼XjqðvigwÞj

i¼1

wi PXntree

j¼1

T lðvjÞ; ð6Þ

where ntree is the size (i.e., the number of MRs) of anIGW-rooted tree.� Bound of the MR-IGW hops: The first two interference

models considered in the Section 3 can be transformedto a maximum MR-IGW hops in an IGW-rooted tree.More hops also lead to longer packet delays [13]. Ithas been shown that in multi-hop transmission, main-taining a high packet delivery fraction requires thatthe number of multi-hop path should be less than fouror five hops [15] for IEEE 802.11a/b/g. We can use theseparameters as the upper bound on the number of hopswhile partitioning an initial WMN into IGW-rootedtrees. In this paper, MR-IGW hop denotes the hopcount in the shortest MR-IGW path.� Bound of relaying load: The interfaces and link through-

put capacity of MR can be reflected by the MR through-put capacity depicted by Eq. (2), which impose an upperbound on the relaying load. For an MR vi in an IGW-directed tree, its relaying load can be calculated by:

T rðmiÞ ¼X

mk2uðmiÞT lðmkÞ; ð7Þ

where u(vi) denote the set of MRs in an IGW-directedtree that transmit their traffic through the MR vi.

IGW placement problem is an NP-hard problem asproved in Appendix B.

5. IGW placement algorithms

In this section, we formulate the IGW placement prob-lem as a linear program issue and then provide two heuris-tic algorithms. All these approaches solve the IGWplacement problem by finding the minimal set of IGW-rooted trees from a given initial WMN. We model theproblem as a Capacitated Facility Location Problem(CFLP) issue with multiple optimization objectives andadditional constraints in the linear program. For efficiency,we develop two heuristic algorithms and obtain near-opti-mal solution.

5.1. Linear program formulation for IGW placement

As we discussed in Section 3, two objectives must beachieved: minimize the number of IGWs and minimizethe number of MR-IGW hop respectively. With thesetwo goals in mind, we formulate the IGW placement as amulti-objectives optimization problem in a linear program.

For simplicity, we exchangeably use vi and i to representa node in the graph G. We use the binary variable yi toidentify whether an MR vi 2 V is set up as an IGW ornot. If the MR vi is selected as an IGW, yi = 1; otherwiseyi = 0. When an MR vi 2 V is selected as the IGW, vi 2 V

corresponds an I i0 2 I . Again, let p(i,j) be a binary variableto represent the relationship between the MR vj and anIGW vi. p(i, j) = 1 means MR vj is attached to the IGW vi

by single or multi-hop paths. Otherwise, it has p(i, j) = 0.We use h(i, j) to represent the minimum number of hopsbetween two nodes. When an MR (vj 2 V) is attached toan IGW (vi 2 I), the h(i, j) acts as the MR-IGW hop. Inaddition, a binary variable kk

ði;jÞ is defined to indicatewhether or not the shortest path from vj to vi passesthrough node vk.

The linear program searches the entire space to find theoptimal results. Before searching, it has to specify the max-imal MR-IGW hops Rhop. At the same time, it has to knowthe interfaces configuration and the link capacity, and thus,computes the node throughput capacity. Our linear pro-gram considers the IGW placement as the CFLP, withadditional constraints as bellows:

minXn

i

yi; ð8Þ

and

minXn

i

Xn

j

hði;jÞ � pði;jÞ; ð9Þ

subject to:Xvi2I

pði;jÞ ¼ 1; 8vj 2 V ; ð10Þ

pði;jÞ 6 yi; 8vj 2 V ; vi 2 I ; ð11ÞXvi2I

pði;jÞ � hði;jÞ 6 Rhop; 8vj 2 V ; ð12ÞXvj2V

pði;jÞ � T lðvjÞ 6 W igw;8vi 2 I ; ð13ÞXvj2V

pði;jÞ � kkði;jÞ � T lðvjÞ 6 W vk � T lðvkÞ; 8vk 2 V ; vi 2 I ; ð14Þ

pði;jÞ � kkði;jÞ 6 pði;kÞ; 8vk 2 V ; vi 2 I ; vj 2 V ; ð15Þ

yi 2 f0; 1g; 8vi 2 V ; and ð16Þpði;jÞ 2 f0; 1g; 8vi 2 V ; vj 2 V : ð17Þ

The first objective (i.e., (8)) means that a minimum numberof nodes are selected as the IGW (i.e., yi = 1) among n

nodes, and thus it achieves the objective of minimizingnumber of IGWs. As the second objective (i.e., (9)), weminimize the number of MR-IGW hops. When everyMR vj is attached to one of m IGWs (i.e., p(i, j) = 1), the to-tal number of MR-IGW hops (i.e.,

Pni

Pnj hði;jÞ � pði;jÞ) is

minimized. The above objectives are subjected to the fol-lowing constraints. The first equation ensures each MRto be attached to an IGW (i.e., p(i, j) = 1), satisfying therequirement of full coverage. For every node vj, it hasP

vi2Ipði;jÞ ¼ 1, meaning that each node is only attachedto one IGW. This is a necessary condition for generatingm IGW-rooted trees. Inequality (11) guarantees that anMR vj is only attached to the node that has been selectedas an IGW. By inequality (12), the MR-IGW hop for an

Fig. 3. Steps of heuristic for IGW placement.


MR is within the maximum hop Rhop. Inequality (13) en-sures the IGW throughout capacity as the upper boundof the traffic in each cluster. When we only consider onepath to the IGW in the linear program, the final resultsare m IGW-rooted trees. Inequality (14) further ensuresthat all the traffic traveling through the MR vk is withinits throughput capacity W vk . Inequality (15) means anynode vj and its parent node vk are attached to the sameIGW vi, which is a necessary condition to guarantee theformulation of IGW-rooted tree structure. The last twoconstraints define the binary values (0 or 1) for two vari-ables yi and p(i, j). When yi = 1 for a node vi, it means thatthis node is selected as an IGW. The total number of IGWm is the sum of yi for all vi. If yj = 0 for a node vj, p(i,j) = 1indicates which IGW vi the node vj is attached to.

As discussed above, the IGW placement problem inWMN can be formulated as a multiple objective IntegerLinear Program (ILP). There are many methods for solvingthis type of problem in the literature. For example, themultiple objectives can be reformulated as a single-objec-tive by forming a weighted combination of different objec-tives. On the other hand, we can transfer one of theobjectives to a constraint. After converting two optimiza-tion objectives into a single objective, some standard LPsolver, such like Matlab or LP-solve can be utilized toaddress the ILP problem.

5.2. Heuristic IGW placement

According to Lemma 5 proved in Appendix B, thesearch space of the linear program of IGW placementincreases exponentially with the number of nodes n. There-fore, we have to develop heuristic algorithm for large-scalenetworks. In this section, we discuss two algorithms, bothbased on Greedy Dominating Tree Set Partitioning(GDTSP), including a degree-based GDTSP and aweight-based GDTSP. In both these algorithms, the graphG = (V,E) is divided into disjoint IGW-rooted trees, andthese trees again can be transformed to IGW-directedand connected clusters according to Lemma 1. Both algo-rithms aim at minimizing the number of IGWs as wellreducing MR-IGW hops, while satisfying the constraintslisted in Section 3. Our degree-based GDTSP algorithmemphasizes the connectivity degree of IGW, meaninghow many MRs are connected to the IGW within the max-imal MR-IGW hop. On the other hand, our weight-basedGDTSP have a strength in minimizing the MR-IGW hops.

Fig. 3 shows four steps of our heuristic IGW placementprocedure, including network information collection, R-hop connecting graph formation, IGW selection, and gen-eration of IGW-rooted trees.

1) Network Information Collection: In this stage, the nec-essary network information is collected. A networkgraph G = (V,E) contains the number of nodes, thenumber of interfaces on each node, the physical posi-tion of each node, and the link and neighboring rela-

tionship. Based on the link capacity and the numberof interfaces, we can calculate the node throughputcapacity. This information is used as the input tothe heuristic IGW placement algorithm. In addition,the heuristic algorithm has to specify a maximalMR-IGW hop, which is referred to as Rhop. As dis-cussed earlier, the MR-IGW hop plays one of decisiveroles on the packet delivery fraction for multi-hoptransmission and interference. The Rhop acts as themaximal hops allowable from an MR to the IGW.Note that the final maximal MR-IGW hop of eachIGW-rooted tree may be less than the specific Rhop.Once the network information has been collected, theheuristic IGW placement algorithm generates an adja-cency matrix for all nodes in the graph G = (V,E). InG = (V,E), two nodes are neighboring only when eachnode is located in the direct transmission range of theother. For instance, the adjacency matrix of graph inFig. 2(a) is a matrix as shown in Fig. 4(b). This adja-cency matrix of G is used to create R-hop connectinggraph in the following step.

2) R-hop connecting graph GR = (VR,ER) Formation: Aresidual graph Gresidual = (V0,E0) is a graph thatexcludes the nodes that have been assigned to IGW-rooted trees. Given a graph G = (V,E), or a residualgraph Gresidual = (V0,E0), this stage is to generate a R-hop connecting graph (GR = (VR,ER)), which is rep-resented by the R-hop adjacency matrix, and denotedas AdjacencyMatrixR. Contrary to the adjacencymatrix generated in the above step, in the R-hop adja-cency matrix, any two nodes are regarded as neigh-bors, if they are within the Rhop hop range in theoriginal graph G or Gresidual. For example, in a4-hop connecting graph, any two neighboring nodes

0 1 1 0 1 0 0 0 0

1 0 1 1 1 1 1 1 0

1 1 0 0 1 1 0 0 0

0 1 0 0 1 1 1 1 0

1 1 1 1 0 1 1 1 1

0 1 1 1 1 0 1 1 0

0 1 0 1 1 1 0 1 1

0 1 0 1 1 1 1 0 1

0 0 0 0 1 0 1 1 0

0 1 0 0 0 0 0 0 0

1 0 1 0 1 0 0 0 0

0 1 0 0 0 1 0 0 0

0 0 0 0 1 0 0 0 0

0 1 0 1 0 1 1 1 0

0 0 1 0 1 0 0 0 0

0 0 0 0 1 0 0 0 1

0 0 0 0 1 0 0 0 1

0 0 0 0 0 0 1 1 0

Fig. 4. Original and R � hop adjacency matrix.


can be connected by four hops or less in originalgraph. Fig. 4(c) demonstrates the 2-hop adjacencymatrix for original graph Fig. 2(a).

3) IGW selection from a GR = (VR,ER): In this stage, anIGW is selected from a GR = (VR,ER). We define twoapproaches on the IGW selection: degree-based IGWselection and weight-based IGW selection.

� Degree-based IGW selection: In the degree-basedGDTSP placement algorithm, IGW is selected in termsof the R � hop degree of the network nodes. We definethe R-hop degree of a node as the total number of othernodes connected within Rhop (i.e., from 1 hop to Rhop) ingraph G. Based on the definition of R � hop connectinggraph, the node degree in R � hop connecting graph isexactly the R � hop degree of the node in the originalgraph. For example, considering Fig. 2(a) and settingRhop = 2, nodes v1 and v2 has 2-hop degrees of 3 and7, respectively, which can be obtained directly from its2-hop adjacency matrix (i.e., Fig. 4(c)). The higher theR � hop degree for a node is, the more MR nodes areconnected within R hops. In other words, the morenodes can be covered in the range of Rhop. In theGR = (VR,ER), every node’s degree is R � hop degreefor the original graph and the degree-based IGW selec-tion algorithm selects the node that has the highest con-nectivity degree as the IGW. Therefore, the degree-based IGW selection achieves the target of minimizingthe number of IGWs. In the process of the degree-basedGDTSP algorithm, we greedily select IGW, meaningthat this algorithm computes all degrees for every nodein the GR = (VR,ER), and uses the node has the highestconnectivity degree as the IGW.� Weight-based IGW selection: In the weight-based

GDTSP IGW placement algorithm, the IGW selectionis based on the connectivity weight of R-hop, which isdefined as the following:

W ðvi;RÞ ¼X

8vj2NRðviÞ1=Hopðvi; vjÞ; ð18Þ

where NR(vi) is the set of nodes that are neighbors of vi

in GR; Hop(vi,vj) is the shortest hops between node vi

and node vj in the original graph. W(vi,R) measuresthe path length from all other nodes to a specific nodevi. In the degree-based GDTSP algorithm, all nodeswithin R � hop are treated as the same in terms of con-nectivity. For example, a two hops away node is con-sidered as same connection as a three hops awaynode. Distinguishing to degree-based algorithm, inthe weight-based GDTSP algorithm, the two hopsaway node has a better connectivity than the three hopsaway node due to higher value 1/Hop(vi,vj) for the twohops away node (i.e., 1/2 > 1/3). In the weight-basedIGW selection, each node computes its W(vi,R), andthe algorithm selects a node having the highestW(vi,R) as the IGW. Thus, the weight-based GDTSPIGW algorithm not only reflects the coverage but alsothe MR-IGW hop distance in the IGW selection.

Fig. 2 illustrates the difference in IGW selection of thesealgorithms. Node index will be used to break the tie if twonodes have the same metric. Nodes v2 and v5 have the same3-hop connectivity degree (i.e., 8) in the 3-hop connectinggraph (Rhop = 3). However, they have W(v2,3) = 5.3 andW(v5,3) = 6.5, respectively. Weight W(v5,3) > W(v2,3) saysthat v5 has more closer nodes connected to it than that ofnode v2.

4) Generation of IGW-rooted tree: After the IGW isselected, an IGW-rooted tree is created by selectinga set of MRs. A breadth first search (BFS) [16] proce-dure is used to select the nodes from 1 hop awaynodes to Rhop hop away nodes. In this way, the closerneighboring MRs are assigned first before far awaynodes, which achieves a shorter MR-IGW pathlength. Node index will be used to break the tie iftwo nodes have the same distance metric. Once anode is selected as IGW or attached to a tree, it isdeleted from the network graph and Gresidual will beupdated accordingly.

At the same time, the attachment of any MR node to theIGW-rooted tree should satisfy the constraints imposed bythe IGW throughput capacity and the MR throughput


capacity. The entire traffic due to addition of MRs in theIGW-rooted tree should be less than the maximal IGWthroughput capacity. The traffic passes through MR mustsatisfy the MR throughput capacity constraints. Based ondifferent interference models used, the capacity constraintsare validated as follows.

Ideal channel model: In the ideal link model, the interfer-ence of the MR vj is ignored if the number of hops from theMR to the IGW is within Rhop from the IGW. Therefore, ituse the traffic demand Tl(vj) for MR vj to validate the MRthroughput capacity constraints given by Eq. (2) on eachparent node.

Hop-based throughput degradation model: In the hop-based throughput degradation model, rather than directlyusing the local traffic demand Tl(vj) for each node vj, theestimated bandwidth demand to communicate with IGWvi is given by Eq. (3), i.e., T 0lði; jÞ ¼ T lðvjÞ � ð1þ aÞðhði;jÞ�1Þ,where h(i,j) is the shortest hops between node vi andnode vj.

Collision-based interference model: In the collision-basedinterference model [11], it needs to verify if it has enoughbandwidth available for the new added MR vj, having thetraffic demand Tl(vj). Rint is the interference range for theradio (e.g., the circle centered at vA in Fig. 5). To determinewhether there are enough bandwidth available for the MRto attach to the tree, two bandwidths are defined [11]:

� The available channel bandwidth Bavailable(vj): It is deter-mined by the aggregated traffic within the interferencerange of the node.� The bandwidth to be consumed at MR vi due to addi-

tion of MR vj (i.e., Tl(vj)): In order to specify the associ-ated IGW of MR vj, we use Bconsumed(vi,Tl(vj)) torepresent the bandwidth for simplicity.

In the following part, we show the calculation of Bavail-

able(vj) and Bconsumed(vi,Tl(vj)), and then show the validationof traffic caused by adding a node. In order to estimateBavailable for any node vi (e.g., vA in Fig. 5), we need to cal-culate the existing total traffic load Bagg(vi) in the interfer-ence range. Let Tvivj be the bi-directional traffic betweennode vi and vj. Given a Rint as depicted in Fig. 5, thereare three types of traffic that could contribute to Bagg(vA)for node vA [11].

Fig. 5. Three traffic contribute to traffic aggregation.

� Self-traffic Tself(vA): The total traffic between node vA

and its neighbors such as the traffic between vA and vB

in Fig. 5, denoted by T vAvB .� Neighborhood-traffic Tneighbour(vA): The total traffic of

vA’s neighbors in the range of Rint, such as the trafficbetween vC and vD in Fig. 5, denoted by T vCvD .� Boundary-traffic Tboundary(vA): The total traffic between

vA’s neighbors and nodes that are outside vA’s interfer-ence range, such as the traffic between vE and vF inFig. 5, denoted by T vEvF .

Fig. 6 shows a sample scenario of traffic aggregation inthe tree based WMN structure. For MR v1, we have

T aggðv1Þ ¼ T selfðv1Þ þ T neighbourðv1Þ þ T boundaryðv1Þ; ð19Þ

where,

T selfðv1Þ¼ T v1v2þT v1v7

; ð20ÞT neighbourðv1Þ¼ T v2v4

þT v3v5; ð21Þ

T boundaryðv1Þ¼ T v2I1þT v3I1

þT v6v4þT v9v7

þT v10v7þT v8v5

: ð22Þ

Thus,

T aggðv1Þ ¼ T v1v2þ T v1v7

þ T v2v4þ T v3v5

þ T v2I1þ T v3I1

þ T v6v4

þ T v9v7þ T v10v7

þ T v8v5

In the graph, we observe that,

T v1v2¼ T uplinkðv1Þ; T v1v7

¼ T uplinkðv7Þ; ð23Þ

and we can represent other node traffic T vavb using the samelogic, where T vavb is the existing traffic between nodes va andvb, and Tuplink(vi) is the traffic between vi and its parent inthe tree. Tuplink(vi) can be computed by summing the trafficdemand on the descendent nodes of vi. For instance, T v1v7

isthe sum of Tl(v7), Tl(v9), and Tl(v10) in Fig. 6. Thus,

T aggðviÞ ¼ T uplinkðviÞ þX

vj2NðviÞT uplinkðvjÞ þ

Xvk2Y ðviÞ

T uplinkðvkÞ;

ð24Þ

where N(vi) is the set of nodes whose links are inside vi’sinterference range, and Y(vi) denotes the set of the bound-ary nodes that are outside vi’s interference range and theirconnections across the boundary of node vi’s interferencerange. It is noted that the above computation is based on

Fig. 6. WMN network traffic aggregation.


all the traffic toward the IGW and this can be applied forthe traffic from the IGW to each destination node in thetree. In this case, we need to divide the traffic demand intwo directions so as to separately compute their aggrega-tion traffic. Then, Tagg(vi) is the sum of aggregation trafficin both directions. Given the maximum throughput Wint,the available bandwidth at node vi can be calculated as:

BavailableðviÞ ¼ W int � T aggðviÞ: ð25Þ

In the next step, we calculate the consumed bandwidthBconsumed(vi,Tl(vj)). Once a traffic flow f is added into agiven path, i.e., a new MR is attached to the path and itrequires additional bandwidth demand Tl(vj), the band-width to be reserved for the flow at all nodes vi along thatpath is given by [11]:

Bviðf Þ ¼T lðvjÞ; vi is IGW

2T lðvjÞ; vi is the MR along the path:

�ð26Þ

In all intermediate nodes, it requires to receive and transmitthe flow f in the process of traffic forwardness, and there-fore consumes double bandwidth. The consumed band-width due to the addition of a node is estimated asfollowing. Fig. 7 illustrates a path v7 ? v1 ? v2 ? I1. Oncea traffic flow f is added into this path, e.g., a new node v10

having f = Tl(v10) is considering to attach to the path vianode v7, the consumed bandwidth at v1 is:

Bconsumedðv1; f Þ ¼ T v10v7þ T v7v1

þ T v1v2þ T v2I1

¼ Bv7ðf Þ þ Bv2

ðf Þ: ð27Þ

In general, we have:

Bconsumedðvi; f Þ ¼ BUpðviÞðf Þ þ BDownðviÞðf Þ: ð28Þ

where Up(vi) and Down(vi) represent the parent nodes anddescendent nodes of node vi within Rint respectively.

Based on the above calculation of Bavailable(vi) andBconsumed(vi,f), we can validate if an MR can attach to aIGW-rooted tree or not. If Bavailable(vi) is higher thanBconsumed(vi, f) at all nodes along the path, the MR isacceptable.

5.3. Algorithm illustration

We illustrate the above IGW placement heuristics inFig. 8 and Fig. 9, respectively. We consider an example net-work topology as shown in Fig. 8(a), which consists of 20

Fig. 7. Estimation of consumed traffic.

nodes. We set the hop count bound Rhop as 2 and the treesize bound as 7 nodes. For simplicity of illustration, werelax the constraint of the MR throughput capacity.

In the degree based GDTSP, the 2-hop degree of eachnode is calculated and marked in the superscript, i.e.,vðdegreeÞ

i . For example, node v2 has the 2-hop degree of 7 andv5 has the 2-hop degree of 9. It says v2 and v5 has 7 and 9 2-hop neighboring nodes that are marked as vð7Þ2 and vð9Þ5 respec-tively. In the first iteration, we consider that the greedydegree-based IGW selection selects the node v5 as the firsttree root. Once the root v5 is chosen, the 2-hop neighbor ofv5 is the set NeighbourSet = {v1,v2,v3,v4,v6,v7,v8,v9,v16}, inwhich the set of {v2,v4,v6,v7,v8} is the level one neighbors(one hop distance to root v5) and the set of {v1,v3,v9,v16} con-tains the level two neighbors. Hereafter, by following theBFS process, the node set of {v2,v4,v6,v7,v8,v1,v3} is selectedsequentially and assigned to the v5-rooted tree which satisfiesthe tree size requirement. Nodes v9 and v16 can not beassigned to the v5-rooted tree due to constraint of the tree size(i.e., 7). Therefore, the first tree is formed with root v5 andmember set of {v1,v2,v3,v4,v6,v7,v8}. The heuristic continuesto the next loop. New degrees are recalculated by using theresidual graph Gresidual (Fig. 8(b)). In this loop, nodesv10,v12,v16,v17 have the same biggest 2-hop degree. We breakthe tie by node index so that node v10 is chosen as the newroot and node set {v9,v11,v12,v13,v14} is accordingly assignedto the tree (see Fig. 8(c)). With the same logic, in loop 3 (i.e.,Fig. 8(d)), node v16 is chosen as the root of a new tree byselecting the node set as {v15,v17,v18,v19,v20}. Until now,three tree are generated that are rooted at nodes v5,v10, andv16 respectively. At the same time, all nodes are attached toone of these three trees.

The similar operations for the weight-based GDTSPalgorithm are illustrated in Fig. 9. The difference is thatthe 2-hop degree is replaced by the 2-hop weight, whichis calculated by Eq. (18). The weight for each node i ismarked with vðweightÞ

i . In the first iteration, the greedyIGW selection operation will select node v5 as the first treeroot NewTreeRoot, which has the maximum 2-hop weight7 among all nodes, as shown in Fig. 9(a). Consequently, thefirst tree is formed at root v5 and the member node set is{v1,v2,v3,v4,v6,v7,v8}, similar to the degree based GDTSP.When the heuristic runs to the next loop, the new weight isupdated at each node in the residual graph Gresidual

(Fig. 9(b)). As shown in Fig. 9(b), nodes v10 and v17 havethe same largest 2-hop degree. By breaking the tie withnode index, node v10 is chosen as the root of the new tree.As a result, the new tree has the node set of{v9,v11,v13,v12,v14}. In loop three, v17, which has the high-est weight 4.5, is chosen as the tree root. In the newly gen-erated tree, it has the node set of {v18,v19,v20,v16,v15}sequently. Finally, three tree clusters are generated rootedat {v5,v10,v17}, respectively.

In the example, the weighted-based GDTSP algorithmachieves the better results. It is because it considers the con-nectivity as well as the MR-IGW path. The detailed algo-rithms for the heuristic are presented in following section.

Fig. 8. Steps of heuristic degree based GDTSP algorithm.

Fig. 9. Steps of heuristic weight based GDTSP algorithm.


Algorithm 1. Heuristic IGW placement algorithm

Input: G = (V,E),Rhop

Output: IGWForest

Gresidual = G;IGWForest = null;AdjacencyMatrixR = BuildAdjacencyMatrix(Gresidual,Rhop);while Gresidual 6¼ null do

NewTree = null;NewTreeRoot = IGW_Selection(AdjacencyMatrixR,Gresidual);NewTree = NewTreeRoot;NeighbourSet = FindNeighbour(NewTreeRoot,AdjacencyMatrixR, Rhop);while NeighbourSet 6¼ null do

PossibleChildNode = MR_Selection(NeighbourSet);


PossibleNewTree = BuildTree(NewTree, PossibleChildNode);if Verify_Constraints (PossibleNewTree) then

NewTree = PossibleNewTree;Gresidual = Gresidual � NewTree;

end

NeighbourSet = NeighbourSet � PossibleChildNode;end

IGWForest = IGWForest [ NewTree;AdjacencyMatrixR = BuildAdjacencyMatrix(Gresidual,Rhop);

end

5.4. Heuristic IGW placement algorithm

Algorithm 1 shows the heuristic IGW placement proce-dure as illustrated in Fig. 3. It should be noted that thedegree-based GDTSP algorithm and weight-basedGDTSP algorithm follow the same procedure, and aredistinguished by the functions in the algorithm. In thedegree-based GDTSP algorithm, the function IGW_Selec-tion() in Algorithm 1 executes a degree-based IGW selec-tion process, and in the weight-based GDTSP algorithm,it executes a weight-based IGW selection process. Theheuristic IGW placement algorithm can be explained asfollows. Given a G = (V,E) and Rhop, the algorithm gen-erates IGW-rooted trees which are saved in the IGWFor-

est. For simplicity, we use G = (V,E) to represent allnodes, links, physical position, node and link capacity.In the algorithm, Gresidual is a residual graph of G fromwhich the IGW-rooted tree nodes have been excluded.It represents the set of nodes that haven’t been assignedto any tree. In the function BuildAdjacencyMatrix

(Gresidual,Rhop), the node adjacency relations are calculatedbased on a given Gresidual, and expressed as R � hopadjacency matrix AdjacencyMatrixR. Two nodes areneighbors if they are within Rhop transmission ranges.The algorithm continues until all nodes are included inthe trees (i.e., Gresidual = null). NewTree is used to storea new generated tree in each iteration. IGW_Selectionhas the functionality of selecting one node as the IGW,which is then saved in NewTreeRoot. The function ofFindNeighbour searches all nodes within Rhop of NewTree-

Root. MR_Selection completes the MR selection to buildan IGW-rooted tree. Every time, it selects a node (i.e.,PossibleChildNode) following the BFS sequence. Mean-while, it validates the constraints of node capacity in thefunction of Verify_Constraints() until all neighbor nodeshave been visited. Only the neighboring node that satisfiesthe constraints, is assigned to NewTree. After a new treeis constructed, the nodes of the tree are deleted fromresidual graph (i.e., Gresidual = Gresidual � NewTree), andthen the algorithm continues to the next iteration untilall nodes have been assigned.

Lemma 2. The GDTSP IGW placement algorithm (degree-based GDTSP or weight-based GDTSP) has a polynomialtime complexity.

Proof. In the degree-based or weight-based GDTSP IGWalgorithm, the first while loop has maximum O(n) iterationsfor a graph G = (V,E) with n nodes. In each iteration, func-tion BuildAdjacencyMatrix generates R-hop adjacencymatrix AdjacencyMatrixR. It can be implemented by theshortest path algorithm like Floyd’s algorithm [16], andresults in the complexity of O(n3). The degree/weight canbe calculated in the same procedure. Function IGW_Selec-tion selects the node having maximum degree/weight, withcomputational complexity O(n). The FindNeighbour canalso be implemented in O(n). Therefore, the complexityof functions in each iteration of first while, i.e., IGW selec-tion (IGW_Selection), neighbour selection (FindNeighbour)and adjacency matrix updating (BuildAdjacencyMatrix), is(O(n)+O(n)+O(n3)), then is bounded by O(n3). In the sec-ond while loop, the BFS-based MR_Selection operationhas at most O(n) operations. It means that the time com-plexity of every iteration of first while loop is bounded byO(n3). Thus, the GDTSP algorithm is with the polynomialtime complexity of O(n4). h

6. Simulation and comparison

In this section, we perform a simulation-based analysison our proposed heuristic IGW placement algorithms(degree-based GDTSP and weight-based GDTSP algo-rithms). We evaluate the two objectives as discussed in Sec-tion 3.4.

� Number of IGWs, reflecting the investment cost as morenumber of IGWs for a given network is, the higher willbe the investment cost.� Average number of MR-IGW hops, reflecting the net-

work connectivity of the IGW-rooted tree as fewer theaverage MR-IGW hops are, the better will be the net-work connectivity.

Our simulations are implemented for different networkenvironments so as to evaluate the effectiveness of our algo-rithms from many different aspects. We vary the MRthroughput capacity, the IGW throughput capacity andthe network size. In the simulation, our results are com-pared with two previous works proposed by Bejerano [5]and Aoun et al. [6]. The approach proposed by Bejerano


[5] is a greedy Dominating Independent Set (DIS) algo-rithm. The approach proposed by Aoun et al. [6] includesa basic and a weighted recursive algorithm. The weightedrecursive algorithm is an enhancement of the basic recur-sive algorithm.

We have implemented our algorithms, greedy DIS algo-rithm, and two recursive algorithms (basic and weighted)with the following configuration. Each node is configureda set of links and each link has the same transmissionrange. We normalize the link capacity of one link as 1 unitwhich corresponds to a real link capacity such as 11 Mbpsfor IEEE 802.11b. Each MR has 1 unit of local trafficdemand. When a node is selected as the IGW, we assumesuch a node may be reconfigured with more interfaces.The simulations consider the maximal MR-IGW hop tobe Rhop = 6 as the implementation in Aoun et al. [6]. Thenodes are randomly and independently distributed in asquare network domain. We consider a square networkdomain as 200 � 200 m2 and the link transmission rangeis 30 m. In the experiments for every network size, it runsevery algorithm 20 times and uses the average results asthe final results. Every time, the nodes are randomly redis-tributed in the network domain.

6.1. Comparison on different MR throughput capacity

In this part of the simulation experiment, we consider afixed network size (i.e., 200 nodes in the network domain).In order to see the impact of MR throughput capacity, werelax the IGW capacity constraint. When an MR is selectedas the IGW, we configure its throughput capacity as infi-nite. This configuration allows a large size tree to accom-modate all nodes within Rhop = 6 from the viewpoint ofthe IGW capacity. Thus, the real tree size is limited onlyby the MR throughput capacity on the path from a nodeto the IGW. The residual throughput capacity of an MRdecreases when a children node is added through its path

4 6 8 10 12 14 16 18 20 225

10

15

20

25

30

35

MR Throughput Capacity

Num

ber

of IG

Ws

Greedy DIS (Bejerano)Basic Recursive (Aoun)Weighted Recursive (Aoun)Degree based GDTSPWeight based GDTSP

Fig. 10. Number of IGWs by varying MR throughput capacity.

to the IGW. We vary the MR throughput capacity from4 to 22 units. It can been seen from Fig. 10 that when theMR has a higher throughput capacity, the number ofIGW decreases. It is because a high throughput capacityMR allows more nodes to attach to itself as an intermedi-ate node towards the IGW. The experimental results inFig. 10 show that the degree-based and weight-basedGDTSP algorithms require lower number of IGWs as com-pared to the greedy DIS algorithm. In the greedy DIS algo-rithm, the IGW-rooted trees are constructed with twoseparate steps: network partition and tree formulation. Inthe network partition step, a G = (V,E) is divided into dis-joint hop-limited clusters. Then, in the stage of tree formu-lation, spanning trees are formed in each cluster. As thesetwo steps are independent, each IGW-rooted tree is sepa-rately constructed in a partitioned cluster, and thereby thisapproach causes a high number of IGWs. In our approach,the IGW is selected one by one from the entire residual net-work Gresidual, thus resulting in a better global optimiza-tion. Fig. 10 also shows that the weight-based GDTSPalgorithm achieves a better performance on the IGW selec-tion than the degree-based GDTSP algorithm. It is becausethe weight-based GDTSP algorithm selects more nodes clo-ser to the IGW for each tree, thus accommodating morenodes in an IGW-rooted tree. At the same time, Fig. 10shows that the two recursive algorithms need fewer IGWsthan the greedy DIS algorithm. However, the recursivealgorithms cause more IGWs than our proposed algo-rithms. This is because the recursive algorithm could notmerge two clusters formed in previous iteration if the mer-gence is unable to satisfy its QoS requirement in any itera-tion of forming dominating set.

When the MR has a higher throughput capacity, thenumber of MR nodes on a path to the IGW increases,which causes a larger MR-IGW hops as shown inFig. 11. Since more nodes are accommodated in the clus-ters generated in our degree-based and weight-based

4 6 8 10 12 14 16 18 20 221

1.5

2

2.5

3

3.5

MR Throughput Capacity

Ave

rage

MR

-IG

W H

ops


Fig. 11. MR-IGW hop by varying MR throughput capacity.


GDTSP algorithms, then result in a higher MR-IGW hopas compared to greedy DIS algorithm and the recursivealgorithms.

6.2. Comparison on different IGW throughput capacity

To evaluate the impact of IGW capacity, we fix the MR-IGW hop bound (i.e., Rhop = 6) as well as the network size(i.e., 200 nodes), and relax the MR throughput constraint.We vary the maximal IGW throughput capacity from 6 to28 units. Increasing IGW throughput capacity results inlower number of IGWs as shown in Fig. 12, and accord-ingly larger MR-IGW hops as shown in Fig. 13. Fig. 12and Fig. 13 also show when the MR throughput capacityis relaxed, both the degree-based and weight-based GDTSPalgorithms need fewer IGWs as compared to greedy DISalgorithm. We can also see from Fig. 12 that the recursive

5 10 15 20 25 3010

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IGW Throughput Capacity

Nu

mb

er

of

IGW

s


Fig. 12. Number of IGWs by varying IGW throughput capacity.

5 10 15 20 25 301

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

IGW Throughput Capacity

Ave

rage

MR

-IG

W H

ops


Fig. 13. MR-IGW hop by varying IGW throughput capacity.

algorithms proposed by Aoun et al. [6] needs fewer IGWsthan the greedy DIS algorithm but more than our two pro-posed algorithms. Again, the weight-based GDTSP algo-rithm needs fewer IGWs than the degree-based GDTSPalgorithm. The weight-based GDTSP algorithm considersthe coverage as well as the MR-IGW hop, and thus selectsmore MRs close to the IGW, which results in lower num-ber of IGWs compared with all other algorithms as well asa low MR-IGW hop.

6.3. Comparison on different network sizes

We assume a maximal total through capacity of an MRis 6 units in this set of simulation. When a node is selectedas the IGW, we assume such node can be reconfigured with20 units throughput capacity. The network size is variedfrom 100 to 240 nodes with a step of 20 (i.e., 100, 120,140, 180, 200, 220, 240 nodes). The results in Fig. 14 indi-cate that when the network sizes are increased, the numberof IGWs also increases. Fig. 14 also shows that the greedyDIS algorithm proposed by Bejerano [5] needs higher num-ber of IGWs than all other approaches. The separation ofcluster partition with IGW selection in the greedy DISalgorithm also causes a fast increase in the number ofIGWs when the network size increases. For example, whenthe network size is increased to 240 nodes, the weight-basedGDTSP algorithm selects 15 IGWs while the greedy DISalgorithm selects 23 IGWs, which is unacceptable fromthe perspective of the investment cost. Fig. 14 furthershows that the recursive algorithms proposed by Aounet al. [6] needs fewer IGWs than the greedy DIS algorithmbut more than our two proposed algorithms.

Fig. 15 shows that the MR-IGW hop decreases when thenetwork size increases. When the number of nodes in a net-work increases within the fixed network domain, the nodedensity increases. Thus, the MR-IGW hops decrease whenmore IGWs are selected in the network domain. The

100 150 200 2508

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Network Size

Num

ber

of IG

Ws


Fig. 14. Number of IGWs by varying network sizes.

100 150 200 2501

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Ave

rage

MR

-IG

W H

ops


Fig. 15. MR-IGW hop by varying network sizes.


throughput capacity of an IGW can be saturated withoutattaching long MR-IGW hop nodes. The greedy DIS algo-rithm results in higher MR-IGW hops than all otherapproaches. As indicated in Fig. 14, our weight-basedGDTSP selects fewer IGWs for a given WMN. Meanwhile,the MR-IGW hop does not significantly increase due tofewer IGWs as compared to the recursive algorithms.Fig. 15 shows that the MR-IGW hop of our weight-basedGDTSP is still close to that of the recursive algorithms.This indicates that our weight-based GDTSP has lowernumber of IGWs while maintaining a small MR-IGW hop.

7. Conclusion

In this paper, we address one of the fundamental prob-lems of designing a WMN: how to place IGWs for optimalperformance. We provides an analytical framework undertwo network architectures: IGW-directed and connectedcluster, and IGW-rooted tree. We further formulate theIGW placement problem as a linear program. Theoreticalanalysis on the determination of IGW placement is pre-sented with the target of partitioning of a global WMNinto IGW-rooted trees. We develop two heuristic algo-rithms and compare our results with two existingapproaches to show the effectiveness of our algorithms.

Appendix A. IGW and MR capacity

Lemma 3. In a WMN, the maximum Internet throughputof an IGW is W igw ¼

PjqðvigwÞji¼1 wi and the maximum Internet

throughput of a WMN is W net ¼Pm

igw¼1

PjqðvigwÞji¼1 wi, where

|q(vigw)| is the number of interfaces configured on the IGWvigw and m is the number of IGWs.

Proof. At any given time slot, there are at most |q(vigw)|(i.e., |q(vigw)| 6 c) interfaces in an IGW that can simulta-neously transmit/receive data packets to/from its neighbor-ing MRs. The upper bound of wireless transmission

capacity of an IGW is when all interfaces (i.e., jq(vigw)j)are involved in the transmission/reception using orthogo-nal channels. Since there is no constraint on the wired linksof the IGW, the maximal Internet throughput is when allinterfaces are fully deployed, which yields toW igw ¼

PjqðvigwÞji¼1 wi. In a WMN, its Internet throughput is

limited by m IGWs since all Internet traffic from eachMR travels through one of m IGWs. The IGWs have toforward all the traffic aggregated by each MR to the Inter-net or distribute the Internet traffic to corresponding MRs.Therefore, the maximum Internet throughput is W net ¼Pm

igw¼1

PjqðvigwÞji¼1 wi. If each IGW is assumed to have the same

configuration on the wireless interfaces, the maximal Inter-net throughput is W net ¼ m�

PjqðvigwÞji¼1 wi. h

Lemma 4. In a WMN, the maximum throughput capacityof an MR is W mr ¼

PjqðvmrÞji¼1 wi=2, where jq(vmr)j is the num-

ber of interfaces configured on the MR vmr.

Proof. At any given time slot, there are at most jq(vigw)j(i.e., jq(vigw)j 6 c) interfaces in an MR that can simulta-neously transmit/receive data packets to/from its neighbor-ing MRs. The upper bound of data throughput capacity ofan MR achieves when all interfaces (i.e., jq(vmr)j) are activein transmission/reception using orthogonal channels.Unlike the IGW, the forwardness of any bit of the dataflow in an MR involves two steps: the bit reception fromits mobile clients or a neighboring node, and the bit trans-mission toward the next hop. Therefore, the throughputcapacity of an MR is bounded by W mr ¼

PjqðvmrÞji¼1 wi=2. h

Appendix B. Complexity analysis of the IGW placement

problem

Lemma 5. The defined IGW placement problem (i.e.,Constructing m IGW-rooted trees) in a WMN network isNP hard.

Proof. To prove this, we use a reduction from the Capac-itated Facility Location Problem (CFLP) [17], which is aservice-open issue for a set of clients D, the intercity dis-tances, and a set of facilities F. Each client j 2 D, a demanddj must be served by one or more open facilities. Each facil-ity i 2 F is specified by a cost fi, which is incurred whenfacility i is open, and by a capacity ui, which is the maxi-mum demand that facility i can serve. The cost for servingone unit demand of a client j from facility i is cij. The pur-pose of CFLP problem is to open a subset of facilities suchthat the demands of all clients are met by correspondingassignment and the total cost of facilities opened and clientservice is minimized. Then, we consider the IGW place-ment issue. We first simplify our problem by consideringthat each link has the same capacity wi so that each IGWhas a service capacity (i.e., Wigw), which can be mappedto ui. We set the cost of deploying one IGW is fi = 1 so thatthe total cost of the IGW placement is equal to the number


of IGWs, which follows the same objective of minimizingthe number of the IGWs. Meanwhile, the cost for IGW i

to serve one unit demand of MR j, is set as the path dis-tance to connecting the IGW i and the MR j, which is rep-resented by hop count hij. In this way, the CFLP problemcan be reduced to the IGW placement problem. The CFLPis a well-known NP-hard problem. Therefore, the IGWplacement problem is an NP-hard problem. h

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