A geometric deployment and routing scheme for directional wireless mesh networks

A Geometric Deployment and Routing Schemefor Directional Wireless Mesh Networks

Weisheng Si, Member, IEEE, Albert Y. Zomaya, Fellow, IEEE, and Selvadurai Selvakennedy

Abstract—This paper first envisions the advent of the wireless mesh networks with multiple radios and directional antennas in future.Then, based on the observation that simplicity induces efficiency and scalability, the paper proposes a joint geometric deploymentand routing strategy for such mesh networks, and also gives a concrete approach under this strategy. The main idea of this strategy isto deploy mesh networks in certain kind of geometric graph, and then design a geometric routing protocol by exploiting the routingproperties of this graph. The proposed concrete approach comprises two parts: (1) a topology generation algorithm based on Delaunaytriangulations and (2) a geometric routing protocol based on the greedy forwarding algorithm. Both parts are characterized by simplicityand appealing properties, with formal proofs provided when possible. The simulation results validate our proposed approach.

Index Terms—Directional antennas, Delaunay triangulations, geometric routing, network deployment, wireless mesh networks

1 INTRODUCTION

WIRELESSmeshnetworks (WMNs) typically consist of a setof mesh routers that communicate with each other via

wireless links and form a mesh topology [1]-[5]. The basicfunctionalities of these mesh routers are (1) providing thebackhaul connection for Wireless LANs (WLANs) and(2) routing the traffic in the backhaul. These mesh routersmay also have additional functionalities, based onwhich theycan be classified into the following three categories:

gateways: also interface with the Internet.APs: also serve as Access Points (APs) for WLANs.pure mesh routers: only have the aforementioned basicfunctionalities.

In this paper, we solely focus on the basic functionalities ofthese mesh routers, aiming to improve their performance inthe backhaul. For brevity, we also refer to these mesh routersas “nodes” hereafter.

According to [1]-[5], the recent years saw the following twonew technologies for the WMNs:

Multiple radios andmultiple channels: each node is equippedwith multiple radios, each of which uses a distinct non-overlapping channel. By enabling multiple channels tocarry network traffic simultaneously, this trend essentiallymultiplies the available bandwidth for network nodes.Directional antennas: these antennas are used in the back-haul connections, enabling the nodes to communicate in a

point-to-point fashion. Thus, the interference among linkssharing identical channels is basically eliminated. Notethat, the directional antennas are still not perfect ineliminating interference, and the effects of interferencebetween directional antennas are discussed in [6] and [7].

Until now, not all WMN deployments have employedthe above two technologies, and only some of them did(e.g., [1], [3]). However, with the decreasing hardware cost,we envision that the WMNs in future will employ both.Accordingly, in this paper, we focus on such WMNs withmultiple radios/channels and directional antennas, and referto them asDirectionalWMNs (DWMNs). For a DWMN, if wedepict its nodes on a ( , ) coordinates system and draw astraight line for each point-to-point communication linkamong the nodes, we will get a geometric graph (see Fig. 1).

Based on the abovemodel,we propose a geometric schemefor the DWMNs, which advocates that the position informa-tion (i.e., the ( , ) coordinates) of the nodes will play a majorrole in the deployment and routing of the DWMNs. Themotivation for proposing this scheme is to perform what canbe planned in advance to simplify the operation of WMNs,thus achieving efficiency and scalability. In retrospect, animportant lesson from Internet deployment is the SimplicityPrinciple [8], which states that ‘complexity is the primaryaspect which impedes efficient scaling, and is the primarydriver of increases in both capital expenditure andoperationalexpenditure for the network carriers’.

For this geometric scheme, we propose a joint geometricdeployment and routing strategy, and also give a concreteapproach to validate this strategy. The idea of this strategy isto deploy the DWMNs with certain kind of geometric graph as thenetwork topology, and then design a geometric routing protocol byexploiting this graph’s routing properties. The justifications forthis strategy are as follows:

Feasibility: the ( , ) coordinates of mesh routers can beobtained by the GPS devices, and are accurate enough foruse in a city area. Recall that the Internet uses the prefix ofIP addresses to perform routing; however, since the

• The authors are with the Centre for Distributed & High PerformanceComputing, School of Information Technologies, University of Sydney,Sydney, New South Wales 2006, Australia. W. Si is also with the Schoolof Computing, Engineering and Mathematics, University of WesternSydney, Sydney, NSW 2751, Australia.E-mail: [email protected]; [email protected];[email protected].

Manuscript received27Aug. 2009; revised 02May2010; accepted 28May2010.Date of publication 30 June 2010; date of current version 09 June 2014.Recommended for acceptance by Y. Yang.For information on obtaining reprints of this article, please send e-mail to:[email protected], and reference the Digital Object Identifier below.Digital Object Identifier no. 10.1109/TC.2010.169

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0018-9340 © 2010 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

Internet turns out to be deployed haphazardly, the prefixof IP addresses cannot reflect the actual locations ofsubnetworks or computers. However, the ( , ) coordi-nates of mesh routers can achieve this purpose.Locality: geometric routing protocols are localized [9]protocols in that the routing decision is solely based ona constant amount of information stored in the packetsand the positions of the current forwarding node, itsneighbors and the destination. Thus, geometric routingprotocols are inherently efficient and scalable.Ease of maintenance: the localized property also signifi-cantly eases the maintenance of WMNs, since the recon-figuration to one part of the network does not need to benotified to the other parts.Fixedness: with the nodes in WMNs not mobile, the peri-odical exchange of position information among them is nolonger needed. In contrast, such exchanging overhead issignificant for the mobile wireless networks [10], [11].Directionality: with the interference among backhaullinks eliminated by the directional antennas, the metricsrelated to positions become practical for making routingdecisions. Otherwise, the interference-aware routing me-trics (e.g., WCETT in [12] and MGF in [13]) are morereasonable.Dynamics: though the nodes in WMNs are not mobile,there exist significant network dynamics such as linkfailure and congestion, making the static routing proto-cols unsuitable. However, since the geometric routingprotocols basically rely on the one-hop neighbor infor-mation that can be updated quickly, they are well suitedfor dynamic networks.

To validate this strategy, we present a concrete approachthat consists of the following two parts dealing with deploy-ment and routing respectively: the Pruned Delaunay Triangu-lation (PDT) generation algorithm and the Limited BackwardGreedy Forwarding (LBGF) protocol.

The PDT generation algorithm uses Delaunay triangula-tions (DT, see Section 3 for its definition and routing proper-ties) as the basis to generate the network topologies. Basically,it produces first refined and then pruned DTs. The PDTshave the advantages of (1) retaining the desirable routing

properties of the complete DTs and (2) being feasible for theWMN deployment and (3) having a low computation com-plexity. Note that though the PDT generation algorithm is thesame as that presented in our previouswork [14], in this paperwe prove a theorem stating its low computation complexityand conduct more evaluations on its performance.

The LBGF protocol extends the well-known greedy for-warding algorithm [10] by allowing the delivery of a packetfarther from the destination in limited number of times.More-over, it proactively detects two kinds of loops to reduce thepacket drop ratio. Compared with other geometric routingprotocols, LBGF achieves more simplicity by discarding themechanism of face routing [15], which are exploited by mostother geometric protocols to recover from local failure whenthe greedy forwarding does not work. In addition, LBGFprovably causes no loops even under the network dynamicsand non-planar network topologies,while the geometric rout-ing protocols with the face routing mechanism cannot achievethis. We note here that LBGF is a significant modification to theBackward-Enabled Greedy Forwarding protocol presented inour previouswork [14], which incurs a larger packet drop ratio.

To the best of our knowledge, we are the first to advocatesuch a combined geometric deployment and routing strategyfor WMNs. We show by extensive simulations that:

The PDT Generation algorithm can generate topologiesthat not only enable LBGF to find paths very close to theactual shortest paths, but also reduce the installation costfor deployers.The LBGF protocol can scale to a large number of nodes.Moreover, it performs well in terms of packet drop ratioand path qualities.

The rest of this paper is organized as follows. Section 2reviews relatedwork anddistinguishes ourwork fromothers.Section 3 introduces the theoretical background for our con-crete approach. Section 4 presents the PDT generation algo-rithm and its evaluation. Section 5 presents the LBGF protocoland its evaluation. Finally, Section 6 concludes this paper anddiscusses the future work.

2 RELATED WORK

As our work covers two areas, node deployment and geo-metric routing, the related work in these two areas is dis-cussed respectively in this section. Generally speaking, ourwork has its uniqueness in both areas.

2.1 Node DeploymentThe node deployment issue exists in the context of WMNs,wireless sensor networks (WSNs), and cellular mobile net-works. Generally, the node deployment solutions aim atoptimizing different criteria for these three types of networks.

Since WMNs just emerged in the recent years and thedeployment issue has not received enough attention fromthe academic community, we only discovered one study [16]that investigates the impact of several factors such as thenumber of radios and backhaul connectivity on the deploy-ment of WMNs. As a simulated study on network perfor-mance, this work is based on extensive Monte Carlo simula-tions, which reveal several practical guidelines on the WMNdeployment.

Fig. 1. Modeling a DWMN with a geometric graph.

1324 IEEE TRANSACTIONS ON COMPUTERS, VOL. 63, NO. 6, JUNE 2014

For WSNs, the study on deployment mainly considers thecriteria of coverage area, connectivity, and installation cost[17]-[19]. It is worth noting that in WSNs, coverage area andconnectivity are generally optimized instead of routing, sincethe WSNs are more concerned with gathering data or detect-ing events. A similarwork to our PDT generation algorithm isthe Adaptive Triangular Deployment algorithm (ATRI) [19],which also exploits the Delaunay triangulations to conductdeployment. The difference is that the ATRI algorithm isdesigned to maximize the coverage area and minimize thecoverage gaps, while our PDT generation algorithm is de-signed to preserve the routing property of the complete DTsand save the installation cost.

For cellularmobile networks, the study on the deploymentof base stations (BS) is called cellular BS planning, which is amajor research topic in this area. For cellular BS planning, themain criteria to consider include coverage area, traffic distri-bution, signal quality, and installation cost [20], [21]. Notethat, routing is not considered here, since routing is basicallydone at the mobile switching centers instead of BSs in cellularnetworks. The current BS planning approaches generally usethe hexagonal deployment strategy.

In brief, the uniqueness of our approach forWMNdeploy-ment lies in the following:

The main factors considered are routing and installationcost, since we believe that routing is the foremost func-tionality provided by theWMNs,while installation cost isa primary concern of deployers.A novel kind of geometric graph, PDT, is proposed as thenetwork topology.

2.2 Geometric RoutingAccording to the two surveys [15], [22], there are plenty ofapproaches that apply geometric routing (also called geo-graphic routing and position-based routing) on wireless net-works. The geometric graphs adopted by those approachesmainly include the relative neighborhood graph (RNG), Gabrielgraph (GG), Yao graph (YG), and Delaunay triangulation.Despite using different kinds of graphs, most of theseapproaches have the following common characteristics.

First, the wireless network environment is modeled by theunit disk graph (UDG), in which all network nodes use omni-directional antennaswith an identical transmission range andtwo nodes have a link between them if their distance is nomore than the transmission range. Thus, there exists signifi-cant interference among links that are near each other, whichmakes the routing protocols solely based on position infor-mation not appealing.

Second, the geometric graphs used as the network topolo-gy needs to be established and maintained by exchangingcontrol messages among the nodes, which incurs significantoverhead.

Finally, in routing a packet to the destination, if the proces-sing node does not have a neighbor closer to the destinationthan itself, face routing is used to overcome this communica-tion void phenomenon and to guarantee the packet delivery[15]. Though face routing cannot generate loops in staticnetworks, it can do so in dynamic networks.

For example, face routing will fall in a loop in the dynamicnetwork scenario depicted in Fig. 2, where the source has a

packet destined for , and the link is temporarily broken.We assume that a combined greedy-face routing algorithm isused here. In the beginning, will send to according to thegreedy forwarding algorithm.At , since no neighbor is closerto than , will start the face routing process.Without loss ofgenerality, here we assume that uses the left-hand rule toforward . Thus, will traverse nodes , , sequentially.When arrives at , suppose the link becomes available. Inthis new topology, will be trapped in the loop bcab, since in aface routing algorithm, loops are detected by rememberingthe first edge of the current face being traversed [9]. In thisexample, the face routing algorithm will remember the edgeua, so it will not be able to discover the loop bcab.

Compared with these previous approaches, the unique-ness of our routing approach lies in the following:

The UDG model is replaced with the point-to-point linkmodel due to the use of directional antennas, thus theinterference from nearby links is avoided.With the fixedness of mesh routers and the planning inadvance, the overhead of maintaining network topologyis obviated.Since no fixed faces exist in dynamic networks and nomechanism can guarantee packet delivery in dynamicnetworks (which can bedisconnected anyway),wedonotsee the need to exploit faces to assist routing. Instead, weuse a mechanism of imposing a limit to the number ofrouting backward to overcome the communication voidphenomenon. This mechanism provably causes no loopseven when the network topology is dynamic or non-planar. Moreover, we also employ two loop detectiontechniques to proactively break the loops. As to be de-scribed later, our approach is simpler than the face rout-ingapproach in twoways: (1) less source codes areused inimplementation and (2) less number of fields is added inthe packet header.

3 THEORETICAL BACKGROUND

This section describes the concepts and properties of thegreedy forwarding (GF) algorithm and the Delaunay trian-gulation (DT) graph, and also introduces some other defini-tions used in this paper.

GF is a geometric routing algorithm thatfinds apath fromasource node to a destination node in the following method[10]: at each node (say ) along the path to , chooses theneighbor that has the smallest d( , ) as the next hop, whered( , ) denotes the Euclidean distance between node and ;and ties are broken arbitrarily. GF is very simple, since it is notonly localized, but also makes the routing decision by just onesearch of the neighbor list.

Fig. 2. A dynamic network scenario where face routing generates loops.

SI ET AL.: GEOMETRIC DEPLOYMENT AND ROUTING SCHEME FOR DIRECTIONAL WIRELESS MESH NETWORKS 1325

DT is widely used as network topologies by geometricroutingprotocols. It is defined as a triangulation graph that nofourth node lies inside the circumcircle of any its triangle [23].Fig. 3 shows an example of the DT graph. A DT has thefollowing appealing properties for geometric routing:

For any triangulation graph, let denote the number ofnodes, the number of edges, the number of convex hulledges, we have [23]. This implies that in aDT, the total number of links in aDT is less than and theaverage node degree is less than 6, thus simplifying theoperation of routing.In a DT, for any node to any destination , there alwaysexists a neighbor of satisfying that < , thusGF can always find a path between any two nodes [24].In aDT, the Euclidean length of the shortest path betweenany two nodes and is guaranteed to be less than

, where is proved to be between 1.5846 and1.998 [25], [26].As an aside, determining exactly is one ofthe most challenging problems in the area of computa-tional geometry.

In this paper, we introduce the following definitions. Anode is said to forward a packet if routes this packet to aneighbor with < ; otherwise, is said to back-ward this packet. Correspondingly, we say that a packet isalways transmitted in two modes: the forward mode or thebackward mode. And we say that a graph is backward-free, ifin this graph from any node to any node , always hasa neighbor such that < . Obviously, a DTgraph is backward-free according its properties describedabove.

4 THE PDT GENERATION ALGORITHM

4.1 Problem FormulationIn formulating our topology generation problem, we makethe following assumptions on the DWMNs.

The positions of the AP nodes are decided by the sites ofhotspots (e.g., restaurants and hotels), and are taken asgiven inputs.The positions of the other two types of nodes (gatewaysandpuremesh routers) and the existence of links betweenany pair of nodes can be decided by the deployers.Due to the imperfection of directional antennas, radios onthe same node need to use distinct channels. Besides, theangle between any two links incident on a node should beno less than a threshold value , so as to fully avoid theinter-channel interference. Hereafter, an angle less than

is called a bad angle in this paper.

Due to the regulation on transmitting power, the length ofa point-to-point link cannot exceed a threshold value .Hereafter, a link longer than is called a bad link.

Since a DT has the routing properties mentioned in Sec-tion 3, it is straight-forward to use the DT obtained by simplytriangulating the given AP nodes as the network topology(denoted by hereafter). However, there usually existbad angles and links in , making it impractical fordeployment. Fig. 4 illustrates a of 39 given AP nodes,where are examples of bad angles, and , areexamples of bad links. To remove these bad angles and links,and also to keep the backward-free property, a possible wayfor deployers is to add puremesh routers into the topology tochange the triangulation. Thus, we formulate the followingtopology generation problem:

Given a set of AP nodes and the thresholds and , find agraph as the network topology by adding pure mesh routers intothe and recalculating the triangulation, such that:

1) T has no bad angles or bad links.2) T is backward-free.3) The number of pure mesh routers added is as small as possible.Note that in this formulation, objective 2 is to guarantee the

support to GF algorithm, and objective 3 is to reduce theinstallation cost for deployers.

4.2 Algorithm DescriptionTo solve the problem formulated above, we propose the PDTgeneration algorithm, which has the following three stages:

1) DT construction: construct the of the given APnodes.

2) DT refinement: add pure mesh routers to remove badangles and links in the , producing a refined DT(denoted hereafter) that has bad angles and linksonly near the boundary.

3) DT pruning: remove the remaining bad angles and linksnear the boundary in by eliminating some relevantedges, while guaranteeing that the resulting graph(called PDT) is still backward-free.

Note that, both and are DTs, but the PDTs arethe subgraphs of DTs. Since there is extensive literature incomputational geometry on DT construction and refinementby adding new nodes, stages 1 and 2 are accomplished using

Fig. 3. An example of DT.

Fig. 4. A of given 39 AP nodes.


the existing state-of-the-art algorithms, which are implemen-ted in the open-source software Triangle [27].

Concretely, for stage 1,we use thewell-knowndivide-and-conquer algorithm [23] to construct the . For stage 2, therefinement algorithm implemented in Triangle is a hybrid ofthree algorithms [28]-[30] ever proposed in the literature, andit can refine an input DT into another DT that satisfies variousspecified criteria (e.g., having no bad angles or bad links).Software Triangle calls a triangle that does not satisfy thespecified criteria a bad triangle (e.g., a trianglewith a bad angleor a bad link), and its basic refinement operation is to add anew node at either the circumcenter [28] or the off-center [30]of each bad triangle to split it, until there are no bad triangles.

In using Triangle to do the refinement, we setand respectively. The is used because Tri-angle is proved to terminate when [28], [30],otherwise it may infinitely split the triangles. Actually, it canusually terminate at a larger in practice [30]. For ,Triangle can terminate at any value.

When a bad triangle is near the boundary of a DT, itscircumcenter or off-center can possibly lie outside the bound-ary of thisDT. In this case, the software Trianglewill not add anew node at the circumcenter or off-center. Instead, severalnew nodes will be added on the boundary of this DT toremove this bad triangle. This phenomenon is illustrated inFig. 5, where the in Fig. 4 is the input, and bold pointsdepict those newnodes added on the boundary. It can be seenthat the number of such new nodes is large and these newnodes are not in the critical positions for relaying networktraffic.

Based on this observation, we use the options provided inTriangle to prohibit it from adding new nodes on the bound-ary when dealing with the bad triangles near the boundary.After this stage 2 of refinement, for those bad triangles left, wepropose aDTpruning algorithm as stage 3 to remove thembyeliminating some relevant edges. Thus, considerable cost forinstalling new nodes is saved for deployers. In addition, wewant this pruning algorithm to keep the backward-free prop-erty of DTs. To illustrate this pruning idea, Fig. 6 (using the

in Fig. 4 as the input) shows the actual obtained bystage 2 with some bad triangles left near the boundary (e.g.,

), andFig. 7 shows thePDTobtainedbyourpruningalgorithm. For both figures, the bold points depict the puremesh routers addedwithin the boundary of the duringthe refinement.

According to [28], [30], a bad triangle left in stage 2 satisfiesthe following two properties: (1) it is a right or obtuse trianglewith its circumcenter or off-center outside the boundary of the

and (2) it either has avertex on theboundaryof (e.g.,in Fig. 6), or is recursively adjacent to another bad triangle

that has a vertex on the boundary of DTR (e.g., in Fig. 6).Based on these two properties, the basic idea of our pruningalgorithm is as follows: keep traversing around the boundaryof to remove those bad triangles that have a vertex on theboundary, until no such triangles are found; to remove a badtriangle, delete the edge that opposites the right orobtuse angle of this bad triangle. The details of our pruningalgorithm are given in Fig. 8, and comments are inserted forexplanation.

To justify the backward-free property of PDTs and thesimplicity of the PDT generation algorithm, we prove thefollowing two theorems.

Fig. 5. The DTR with new nodes added on the boundary. Fig. 6. The DTR without new nodes added on the boundary.

Fig. 7. The PDT obtained by our pruning algorithm.


Theorem 1. The PDTs obtained by our proposed topologygeneration algorithm are backward-free.

Proof. In stage 1 and stage 2 of our algorithm, the graphsobtained are complete DTs, so they are backward-free.Thus, we only need to prove that our pruning algorithmpreserves the backward-free property.

Since the basic operation of this algorithm is todelete theedge that opposites the right or obtuse angle of a badtriangle, the proof can be accomplished by proving that ifthe graph before deleting such an edge is backward-free, itremains backward-free after this deletion. Consider Fig. 9,where depicts a bad triangle left in a , and isits right or obtuse angle, and is the circumcenter of ,and , , and are the perpendicular bisectors of edges

, , and respectively, and is an arbitrary destinationnode. According to [30], [31], the circumcenter oflies outside the boundary of , so an assisting line canbe drawn by extending one of the boundary edges of this

such that all nodes in this is in one side of andis in the other side.

Since the deletion of edge only affects the neighbor-hood of and , to prove the graph remains backward-freeonly requires to prove that for both and (without loss ofgenerality, we prove it for here), given any destination ,there exists a neighbor satisfying < . Asshown in Fig. 9, , , and divide the half plane whereresides into three areas: I, II, and III. Suppose is in I,

which completely lies in the left side of the bisector , sowe have < . Moreover, is the only neighborthat loses when is deleted, so must have anotherneighbor , such that < . Suppose is in area IIor III, which completely lies above the bisector , so wehave < , thus is the nearer neighbor of to .And similar proof can be done for node . ◽

Theorem 2. The PDT generation algorithm has a time complexityof , where is the number of givennodes and the total number of nodes at the end of the algorithm.

Proof. The time complexity of the PDT generation algorithmis the sum of the time complexities of its three stages. Forstage 1, according to [23], the construction of DT costs

, where is the number of the given APnodes. For stage 2, according to [30], [31], the refinementalgorithm costs , where is the total number ofnodes after the refinement. For stage 3, according to ourpruning algorithm described in Fig. 8, it at most processesall the triangles in the , so it costs , where isthe number of triangles in the . Since the number oftriangles has a linear relationshipwith the number of nodesin a triangulation [23], the stage 3 also costs .Combining the costs of the three stages, the total timecomplexity of the PDT generation algorithm equals

. ◽

As to the relationship between and , it is empiricallyshown that the softwareTriangle addsvery limitednumber ofnew nodes during the refinement [30], so will be notsignificantly larger than . Theoretically, it is proved that theminimum distance between any two nodes in is no lessthan the parameter , which denotes the minimum dis-tance between any two nodes in the original input DT [30],[31]. Thus, in , each node at least occupies a circular areaof , so is bounded by the total area of divided by

. In light of this, the parameter is critical in deter-mining the upper bound of . In our evaluations below, weuse (we believe this minimum distance can berealizedby the deployers), andour experiments show that ouralgorithm atmost adds new nodes to the network topology.Therefore, the PDT generation algorithm actually has a timecomplexity of in our settings.

4.3 Algorithm EvaluationThis subsection first describes the experiment setup, and thengives experimental results on the following four metrics: theratio of added pure mesh routers, the ratio of pruned edges,the link deviation ratio and Euclidean deviation ratio of thePDTs. The first two metrics reflect the installation cost for

Fig. 9. The proof of backward-free property.

Fig. 8. The DT pruning algorithm.


using the PDTs as the network topologies, and the last twometrics reflect the path quality of PDTs.

4.3.1 Experiment SetupThe given AP nodes are assumed to be randomly distributedin a square areawith a constant density, and this density is setto . For instance, if the number of nodes is 20, theside length of the square area is approximately 1414m. Since aDT exists for any set of points on a plane [23], our PDTgeneration algorithm works for any kind of distribution ofAP nodes, and we only use the random distribution here toconduct experiments. Our PDT generation algorithm is im-plemented by modifying the aforementioned software Trian-gle. With this implementation, experiments are conducted onnetwork sizes of 50, 100, 200, 400, 600, 800, and 1000 given APnodes respectively. For each network size, 200 random topol-ogies are generated and the average result of them is obtained.

4.3.2 The Ratio of Added Pure Mesh RoutersThe ratio of addedpuremesh routers for a topology equals thenumber of added pure mesh routers divided by the totalnumber of nodes after the topology generation. Fig. 10 plotsthe average ratio of 200 topologies for each experimentednetwork size. From this figure, we see that (1) this ratioincreases with the growth of the network size, and the in-creasing slope levels out gradually; and (2) this ratio is lessthan 0.5 for all plotted network sizes. Thus, we can infer thatthe number of added pure mesh routers is below half thenumber of all mesh routers, and tends to increase linearly inrelation to the network size.

4.3.3 The Ratio of Pruned EdgesThe ratio of pruned edges for a topology equals the number ofpruned edges divided by the total number of edges after thetopology generation. Fig. 11 plots the average ratio of prunededges of 200 topologies for each experimented network size.From this figure, we see that (1) this ratio decreases with thegrowth of the network size, since the pruned edges are onlylocated near the boundary of PDTs; and (2) this ratio is quitesmall (below 0.1) for all plotted network sizes.

According to the formula ‘ ’ mentioned inSection 3, the total number of edges in a PDT is less than 3times the total number of nodes. Since Fig. 11 shows that theedge pruning does not reduce the number of edges signifi-cantly, the installation cost of network links remains a linearrelationship with the network nodes. Though the pruning ofthe edges does not help significantly in saving the installation

cost, it does remove the bad angles and links, thusmaking thePDTs feasible for deployment.

4.3.4 The Link Deviation Ratio of PDTFor a given routing algorithm, the link deviation ratio of asource/destination pair ( , ) in a graph is defined as thenumber of links (or hops) in the path found by the givenalgorithm versus the minimum number of links from to .Further, the link deviation ratio of a graph is defined as theaverage link deviation ratio for all ( , ) pairs in the graph.Similarly, with a given routing algorithm, the Euclidean devia-tion ratio for a ( , ) pair or for a graph canbedefined in termsofEuclidean distance instead of the number of links.

The implication of the deviation ratio is that the closer theratio is to 1, the less deviation from the actual shortest paths agiven routing algorithm can achieve on a given networktopology. As to be described later, since our LBGF protocolreduces to the GF protocol if no network dynamics exist onPDTs, we can use GF here to evaluate the path qualityprovided by PDTs. Hereafter, we will not explicitly mentionGF when discussing the deviation ratios.

Fig. 12 plots the average link deviation ratio of 200 topolo-gies for each experimented network size. To see the effect of ourrefinement and pruning algorithms, both the deviation ratiosfor and PDT are plotted. From this figure, we see that(1) the link deviation ratios for both and PDT increaseslowly, reflecting that the paths found by GF tend to deviatemore from the shortest paths when the network size grows;(2) PDT improves the link deviation ratio significantly over

; and (3) for PDTs, the linkdeviation ratio remains below1.1 for all experimented network sizes. Thus, it can be seen thatthe PDTs provide very high path quality in terms of number of links.

4.3.5 The Euclidean Deviation Ratio of PDTsFor the definition of Euclidean deviation ratio, please see theprevious subsection. Fig. 13 plots the average Euclidean

Fig. 10. The ratio of added pure mesh routers.

Fig. 11. The ratio of pruned edges.

Fig. 12. The link deviation ratios for and PDT.


deviation ratio of 200 topologies for each experimented net-work size. Also for the comparison purpose, the deviationratios for both and PDT are plotted. From this figure,we see that (1) the Euclidean deviation ratio is lower than itslink counterpart, because the structure of DTs is more helpfulfor finding short paths in the Euclidean metric than the linkmetric; (2) for the Euclidean deviation ratio, PDT also im-proves significantly over ; and (3) for PDTs, this devia-tion ratio remains below 1.05 for all plotted network sizes.Thus, it can be seen that the PDTs also provide very high pathquality in terms of Euclidean distance.

5 THE LBGF PROTOCOL

5.1 Protocol OverviewGivenPDTs as network topologies, LBGF is designedwith thefollowingmain ideas. First, when no network dynamics exist,it reduces to the GF algorithm and fully exploits the back-ward-free property of PDTs. Second, it extends GF by consid-ering two kinds of network dynamics: link failure and linkcongestion. The former can be notified by the link-layer func-tionalities, and the latter can be obtained by examining theoccupancy of link packet buffers. If the packet buffer for a linkis full, the LBGF simply does not send packets to this link.Thus, both kinds of network dynamics can be monitoredlocally by nodes, without the overhead of exchanging controlpackets in the network-layer. Finally, with the presence ofnetworkdynamics, the underlyingnetwork topologymaynotbe backward-free. To address this issue, LBGF allows back-ward a packet to reduce the packet drop ratio. For loopprevention, LBGF imposes a limit to the number of timesthat a packet can be backwarded (denoted byhereafter). Later, wewill prove that this technique guaranteesno routing loops.

Besides guaranteeing no loops by imposing the, LBGF also detects the following two kinds of

loops to break them proactively.two-hop loops: the loops between two neighboring nodes.backward-then-forward loops: the loops that consist of aseries of consecutive transmissions in backward modeand then a series of consecutive transmissions in forwardmodes. Fig. 14 gives two examples of such backward-then-forward loops with node as the destination.

We believe these two kinds of loops are the most commonloops that occur in our network settings. By breaking thesetwo kinds of loops, the packet drop ratio shall be significantlyreduced, which is verified in our simulations.

5.2 Protocol DescriptionRequired by LBGF, the following three fields are added to apacket’ network-layer header:

Dst Coords: the ( , ) coordinates of the destinationnode.Backward Number (BN): the number of times that thispacket has been backwarded. Moreover, we use the mostsignificant bit (msb) of BN to indicate the transmissionmode of a packet, with ‘0’ indicating the forward modeand ‘1’ the backward mode. When creating a packet, themsb of BN is initialized to ‘0’.Backward Head (BH): for each series of consecutivebackward transmissions, this field records the first nodethat starts this backward series. For comparison purpose,when creating a packet, the BHfield is initialized to a non-existing value.

Note that, LBGF uses less number of fields in the packetheader than the combined greedy-face routing mechanism,which has several variants. All these variants typically needthe followingfivefields in their packet header: the destinationcoordinates, the forwardingmode, the nodewhere the greedyforwarding fails, the first edge of the current face beingtraversed, and the nearest node or point on an edge everreached toward the destination [9], [10].

Exploiting these three fields in the packet header as well asthe neighbor lists at nodes, the LBGF protocol is detailed inFig. 15, where the processing node is denoted , the packetbeing processed , ’ destination node . For explanation,comments are also inserted in Fig. 15.

As seen from Fig. 15, to break the two-hop loops, a node isprohibited from returning to its predecessor (i.e., ’ neigh-bor that sent to in the last step), unless is a dangling node.Here a dangling node is defined as a node that only has oneneighbor, excluding those neighbors connected by broken orcongested links. Note that, the reason for permitting a dan-gling node to return to its predecessor is to enable to comeout of the dangling node situation. And to break the backward-then-forward loops, a node is prohibited fromsending to thenode recorded inBH. Since aBHnodewill definitely appear ina backward-then-forward loop, avoiding sending to the BHnode again breaks this kind of loops.

To exemplify the LBGFprotocol,wenext give twonetworkscenarios involving two-hop loop and backward-then-forwardloop respectively. For the two-hop loop, in Fig. 16, the sourcehas a packet to transmit to the destination , and node

constitutes a dangling node. In this scenario, will be sent toand then . Since is a dangling node, it is allowed to returnto . Not a dangling node, is prohibited to send to , thushas to send to . Similarly, will send to . Finally, willfollow the path cdet to reach , thus overcoming the two-hoploop and the dangling node situation.

Fig. 14. Two examples of the backward-then-forward loops.

Fig. 13. The Euclidean deviation ratios for and PDT.


For the backward-then-forward loop scenario depicted byFig. 17, the source also has a packet to transmit to thedestination . Initially, will send to node , set thetransmission mode to backward, and record itself in BH of. Since is prohibited from sending back to , will send

to . Then, will follow thepath bcda to reach again,with thetransmission mode now changed to forward. Since is re-corded in BH of , is not allowed to send to . Thus, willsend to , set the transmission mode to backward, andrecord itself in BHof . Afterwards, the packetwill follow thepath bcd to reach . At , since is recorded in BH of ,cannot send to . Instead, will send to , thus breakingthe two backward-then-forward loops sabcdas and abcda.

We next prove the following two theorems regarding theloop-free property and the complexity of LBGF respectively.These two proofs also give some insights to this protocol.

Theorem 3. For any network topology, LBGF is loop-free in that iteither delivers a packet to the destination or drops .

Proof. This proof is done in two cases. First, suppose isnever backwarded. Then in each transmission, getscloser to the destination, thus each node visited isdifferent from all the previously-visited ones. Since thereare finite number of nodes in the network, will definitelyreach the destination in finite transmission steps or getdropped due to no available neighbors.

Second, suppose is ever backwarded. Because thenumber of backward transmissions is bounded, and alsobecause the number of consecutive forward transmissionsbetween any two backward transmissions cannot be infi-nite as revealed in the previous paragraph, after finitetransmission steps, will be backwarded for the last time.After thisfinal backward transmission, all transmissions of

will be forward transmission, thus will reach thedestination or get dropped in light of the same argumentin the previous paragraph. ◽

Note that, his proof reveals the following facts aboutthe loop-free property of LBGF:

It does not require the network topology to be backward-free.Hence, the purpose for PDTs to be backward-free is notto guarantee the loop-free property of LBGF, but to reducethe length of the routing paths and the packet drop ratio.It holds for arbitrary network topologies, not necessarilyplanar ones as in the face routing.It holds under the conditions of link failures orcongestions.

Theorem 4. For any network topology with its maximum nodedegree bounded by a constant, LBGF runs with O(1) complexityin both time and space at a node.

Proof. For the time complexity, the only non-trivial compu-tation in LBGF is one search of the neighbor list. If themaximum node degree is bounded by a constant, thissearch can be completed in O(1) time.

For the space complexity, a node needs to store itsneighbor list in the memory, which costs O(1) space dueto the boundedmaximum node degree. Moreover, a pack-et, which also costs a node’ memory, only carries threefields in its header for LBGF. Since both types of space costsare constant, LBGF consumes O(1) space at a node. ◽

Note that, this theorem certainly holds when the networktopologies are PDTs, in which the maximum node degree isbounded by 12 due to .

5.3 Protocol EvaluationWe implement the LBGF protocol in ns-2 [32], into which twoparts of codes are added: a routing module that realizes the

Fig. 16. An example scenario for the two-hop loop with a dangling node.

Fig. 17. An example scenario for the backward-then-forward loop.

Fig. 15. The LBGF protocol.


steps in Fig. 15 and a link object that simulates the point-to-point wireless links. Besides, to simulate these point-to-point links, theWirelessChannel object in ns-2 is modified,so that it only delivers the wireless signal to the twoneighboring nodes in a point-to-point link. To compare withother works, we implement another well-known geometricrouting protocol called GOAFR [33] under ns-2. GOAFRclaims itself to be worst-case optimal and average-caseefficient in terms of path qualities. Also, a theoretical work[9] shows that GOAFR guarantees delivery under arbitrarystatic network topologies, while other well-known geometricrouting protocols such as [34] and GPVFR [35]cannot.

Using the network topologies produced by our PDT gen-eration algorithm, we conduct experiments on LBGF andGOAFR with network sizes of 100, 200, 400, 600, 800, and1000 nodes. To simulate the dynamic networks, the Exponen-tial Model included in ns-2 is used to generate link failures. Inthis model, the up time and down time for a link are exponen-tially distributed, and theirmeans can be set by the command-line arguments up-interval and down-interval respectively.Using thismodel, experiments are conductedon the followingthree scenarios for each network size:

Static: no link failuresDyn1: for every link, ,

Dyn2: for every link, ,

For each scenario of a given network size, 200 experimentswith different random topologies are conducted and theaverage result is obtained. In each experiment, each node isfound a peer randomly, and a CBR flow is generated fromeach node to its peer. In the later evaluation figures, the datalines depictingLBGFandGOAFRperformances in these threescenarios are labeled LBGF0, GOAFR0, LBGF1, GOAFR1,LBGF2, and GOAFR2.

In the following, we first present evaluation results solelyrelated to LBGF: (1) the impact of to the packetdrop ratio and the linkdeviation ratio and (2) the effectivenessof the two loop detection techniques. Then, we present theevaluation results on the following three metrics for bothLBGF and GOAFR: (1) packet drop ratio, (2) link deviationratio, and (3) Euclidean deviation ratio. Note that the defini-tions of link/Euclidean deviation ratios here are slightlydifferent from those when evaluating PDTs by being definedon a packet instead of being defined on a ( , ) pair.

Specifically, we define the link deviation ratio of a packet asthe number of links traversed by this packet from a sourceto a destination versus the minimum number of links

from to . Further, we define the link deviation ratio of arouting protocol as the average link deviation ratio of allpackets routed by this protocol in an experiment. Similarly,the Euclidean deviation ratio of a routing protocol isdefined in terms of Euclidean distance instead of the numberof links.

5.3.1 Impact ofOur choice of is based on the concept of Graphdiameter,which isdefinedas the length (numberof links) of thelongest shortest path between any two nodes of a graph [36].For random DTs, the mean of the graph diameters is ,where is the number of nodes in a DT [37]. Since the numberof backward transmissions is directly related to the length ofthe path traversed by a packet, we conjecture thatshould be set to , where is a configurable coefficient.Below, we present our experimental results on the packetdrop ratio and link deviation ratio of LBGF when the valueranges from 0.5 to 5 with a step of 0.5. From these experimen-tal results, wewill select an appropriate value for ourWMNenvironment. All these experiments are conducted in theDyn2 scenario, since we believe that a sufficiently largevalue for Dyn2 will be also sufficient for Dyn1 and Staticscenarios. To make our experiments encompass differentnetwork sizes, the networks of 100, 400, 900 nodes are tested,with their data lines labeled N100, N400, and N900,respectively.

Fig. 18 shows the following impact of to thepacket drop ratio: (1) the packet drop ratio decreases with theincreasing of and levels out gradually; (2) around

, the decreasing slope becomes small, implying that alarger value does not help significantly in reduc-ing the packet drop ratio.

Fig. 19 shows the following impact of to thelink deviation ratio of LBGF: the link deviation ratio increasesslowly with the growth of , reflecting that whenthe value increases, packets can travel longerpaths, thus more network capacity will be consumed.

Based on the above observations, we use in con-ducting all the subsequent experiments. Note that for thenetwork environment other than ours, a different valuemaybe more appropriate.

Fig. 19. Impact of to the link deviation ratio.Fig. 18. Impact of on the Packet Drop Ratio.


5.3.2 Effectiveness of the Two Loop DetectionTechniques

To evaluate the effectiveness of our two loop detection tech-niques, we show the ratio of number of loops detected to thenumber of packets dropped. Since when a loop is detected, apacket drop is avoided, so this ratio reflects how successfulour loop detection techniques are in avoiding loops. Specifi-cally, we measure the ratio of the number of two-hop loopsdetected versus the number of packet dropped (denoted by

) and the ratio of the number of backward-then-forward loops detected versus the number of packet dropped(denoted by ) in each experiment. The experimentsare conducted in theDyn2 scenario with network sizes of 100,200, 600, and 1000 nodes. These two ratios are summarized inTable 1, which shows that both ratios have very large values,so our loop detection techniques are very effective.

5.3.3 Packet Drop RatioPacket drop ratio is defined as the total number of packetsdropped by all nodes versus the total number of packetsgenerated by all nodes during an experiment. Fig. 20 plotsour experimental results on it, showing that (1) bothLBGFandGOAFR exhibit very low packet drop ratios (less than 0.017evenunderDyn2) and (2) in the Static scenario,where the onlynetwork dynamics is network congestion, both protocolsachieve a drop ratio of nearly zero, and (3) in the scenariosDyn1 and Dyn2, LBGF outperforms GOAFR considerably.

Note that, the drop ratio of LBGF decreases with thegrowth of the network size. This reflects that a larger

(set to ) tends to offer a packet morechance to reach thedestination, despite that the path traversedby a packet becomes longer. On the contrary, the drop ratio ofGOAFR increases with the growth of the network size. This isbecause GOAFR drops a packet when (1) the packet encoun-ters again the first edge that it traverses on the current facesuch that a loop is discovered or (2) the packet falls into a loopthat cannot be discovered by the face routing mechanism butthe TTL (set to 128) is exceeded. Andwhen the path traversedby a packet is longer, the packet has a larger chance to fall intoa loop.

5.3.4 Average Link Deviation RatiosFig. 21 plots the average link deviation ratios of 200 experi-ments for both protocols in all the three scenarios. It showsthat: (1) for both protocols in all the three scenarios, the linkdeviation ratios increase with the growth of network size,

reflecting that the paths found by the geometric routingprotocols are more likely to deviate from the optimal whenthe network size is large; and (2) LBGF achieves low linkdeviation ratios, especially in the Static scenario, which in-dicates that LBGF performs well in finding paths with shortlengths; and (2) LBGF performs slightly better than GOAFR.

5.3.5 Average Euclidean Deviation RatiosFig. 22 plots the average Euclidean deviation ratios of 200experiments for both protocols in all the three scenarios. Inaddition to what are already reflected in Fig. 21, this figuremainly shows that for both protocols in all the three scenarios,the Euclidean deviation ratio has a smaller range than its hopcounterpart, which is because the structure of DTs is morehelpful forfinding short paths in theEuclideanmetric than thelink metric.

Fig. 20. Packet drop ratios.

TABLE 1The Two Loop Detection Ratios with Respect to the

Number of Packets Dropped

Fig. 21. Average link deviation ratios.

Fig. 22. Average Euclidean deviation ratios.


6 CONCLUSION

In this paper, we first foresee the advent of the DWMNs withmultiple radios and directional antennas in future. Then,based on the observation that simplicity brings the efficiencyand scalability, we propose a joint geometric deployment androuting strategy, and then give a concrete approach under thisstrategy for the DWMNs. The proposed concrete approachhas the following two parts: the PDT generation algorithm andthe LBGF routing protocol. We prove theoretically or showempirically the following merits of these two parts.

The PDT generation algorithm (1) has a low time com-plexity of , where is the number of APnodes in the network, and (2) guarantees that the gener-ated PDTs are backward-free, and (3) incurs low installa-tion cost for the deployers.The LBGF protocol (1) has a constant complexity in bothtime and space at nodes, (2) guarantees that the route of apacket is loop-free, (3) incurs a low packet drop ratio, and(4) finds high quality paths with both low link deviationratio and low Euclidean deviation ratios.

Being the first to propose the geometric deployment androuting strategy forWMNs,we believemanyproblems underthis strategy are worth exploring, especially the follows.

The backward-free property is newly introduced for thegeometric graphs in this paper. Since this propertyenables the geometric routing protocols to produce highquality paths, finding the sufficient and necessaryconditions for a geometric graph to be backward-freeconstitutes an interesting open problem.Our proposed PDT is only an example of the backward-free graphs, and other kinds of such graphs are worthexploring to suit different requirements of networkdeployments.Since the face routing mechanism does not functionproperly in dynamic networks asmanifested in Section 2,designing other kinds of geometric routing protocolscapable of tackling network dynamics becomes a chal-lenging problem. While our LBGF protocol attempts tosolve this problem by imposing a backward limit anddetecting loops proactively, other different mechanismscan be further explored.

Finally, we note that no current WMN deployments usesuch a geometric deployment and routing strategy yet. This isbecause the implementation of this strategy requires themodification to the TCP/IP protocol stack at themesh routers(i.e., enabling these mesh routers to insert, exploit, and thenremove the position information from the headers of networkpackets). Though this modification is not difficult, no WMNdeployers will be tempted to do this if the benefit of thisgeometric strategy is not apparent. Thus, we believe thatwhen the scale of the WMNs becomes large, the scalabilityand efficiency advantages of this geometric strategy willbecome evident, and this geometric strategy will be adopted.

REFERENCES

[1] BelAirNetworks, “BelAir300 Converged Multi-Service WirelessNode,” http://www.belairnetworks.com, Data Sheet, 2007.

[2] MeshDynamics, “Technology—Performance Analysis,” http://www.meshdynamics.com, 2006.

[3] Nortel, “Wireless Mesh Network Basics,” http://www.nortel.com,Standard Document Release 3.0, 2007.

[4] StrixSystems, “Solving the Wireless Mesh Multi-Hop Dilemma,”http://www.strixsystems.com, White Paper, 2006.

[5] TroposNetworks, “Tropos Networks MetroMesh Architecture,”http://www.tropos.com, White Paper, 2005.

[6] V. Angelakis, N. Kossifidis, S. Papadakis, V. Siris, andA. Traganitis,“The Effect of Using Directional Antennas on AdjacentChannel Interference in 802.11a: Modeling and Experience withan Outdoors Testbed,” Proc. Int’l Symp. Modeling and Optimizationin Mobile, Ad Hoc, and Wireless Networks and Workshops (WiOPT),pp. 24-29, 2008.

[7] T. Ireland, A. Nyzio, M. Zink, and J. Kurose, “The Impact ofDirectional Antenna Orientation, Spacing, and Channel Separationon Long-Distance Multi-Hop 802.11g Networks: A MeasurementStudy,” Proc. Int’l Symp. Modeling and Optimization inMobile, Ad Hocand Wireless Networks and Workshops (WiOPT), pp. 1-6, 2007.

[8] R. Bush andD.Meyer, “Some Internet Architectural Guidelines andPhilosophy,” RFC 3439 Informational, 2002.

[9] H. Frey and I. Stojmenovic, “On Delivery Guarantees of Face andCombined Greedy-Face Routing in Ad Hoc and Sensor Networks,”Proc. ACM Int’l Conf. Mobile Computing and Networking (MobiCom),pp. 390-401, 2006.

[10] B. Karp andH.T. Kung, “GPSR: Greedy Perimeter Stateless Routingfor Wireless Networks,” Proc. ACM Int’l Conf. Mobile Computing andNetworking (MobiCom), pp. 243-254, 2000.

[11] J.Gao,L.J.Guibas, J.Hershberger, L.Zhang, andA.Zhu,“GeometricSpanners for Routing in Mobile Networks,” IEEE J. Selected Areas inComm., vol. 23, no. 1, pp. 174-185, Jan. 2005.

[12] R. Draves, J. Padhye, and B. Zill, “Routing in Multi-Radio, Multi-Hop Wireless Mesh Networks,” Proc. ACM Int’l Conf. Mobile Com-puting and Networking (MobiCom), pp. 114–128, 2004.

[13] P. Kyasanur and N.H. Vaidya, “Routing and Link-Layer Protocolsfor Multi-Channel Multi-Interface Ad Hoc Wireless Networks,”Proc. ACM SIGMOBILE Mobile Computing and Comm. Rev., vol. 10,pp. 31-43, 2006.

[14] W. Si and S. Selvakennedy, “A Position-Based Deploymentand Routing Approach for Directional Wireless Mesh Networks,”Proc. Int’l Conf. Computer Comm. Networks (ICCCN), pp. 1-8,2008.

[15] D. Chen and P.K. Varshney, “A Survey Of Void Handling Techni-ques For Geographic Routing In Wireless Networks,” IEEE Comm.Surveys & Tutorials, vol. 9, no. 1, pp. 50-67, May 2007.

[16] J. Robinson and E.W. Knightly, “A Performance Study of Deploy-ment Factors in Wireless Mesh Networks,” Proc. IEEE Int’l Conf.Computer Comm. (INFOCOM), pp. 2054-2062, 2007.

[17] J. Pan, Y.T. Hou, L. Cai, Y. Shi, and S. Shen, “Topology Controlfor Wireless Sensor Networks,” Proc. ACM Int’l Conf. Mobile Com-puting and Networking (MobiCom), pp. 286-299, 2003.

[18] C.-F. Huang and Y.-C. Tseng, “The Coverage Problem in aWirelessSensor Network,” Mobile Networks and Applications, vol. 10,pp. 519-528, 2005.

[19] M. Ma and Y. Yang, “Adaptive Triangular Deployment Algorithmfor Unattended Mobile Sensor Networks,” IEEE Trans. Computers,vol. 56, no. 7, pp. 847-946, July 2007.

[20] E. Amaldi, A. Capone, and F. Malucelli, “Planning UMTS BaseStation Location: Optimization Models with Power Control andAlgorithms,” IEEE Trans. Wireless Comm., vol. 2, no. 5, pp. 939-952,Sept. 2003.

[21] R. Mathar and T. Niessen, “Optimum Positioning of Base Stationsfor Cellular Radio Networks,”Wireless Networks, vol. 6, pp. 421-428,2000.

[22] M. Mauve, J. Widmer, and H. Hartenstein, “A Survey on Position-Based Routing in Mobile AdHoc Networks,” IEEE Network, vol. 15,no. 6, pp. 30-39, Nov./Dec. 2001.

[23] M.d. Berg, M.v. Kreveld, M. Overmars, and O. Schwarzkopf,Computational Geometry: Algorithms andApplications, 2nd ed. Springer,2000.

[24] P. Bose and P.Morin, “Online Routing in Triangulations,” Proc. Int’lSymp. Algorithms and Computation, pp. 113-122, 1999.

[25] G. Xia, “Improved Upper Bound on the Stretch Factor of DelaunayTriangulations,” Proc. 27th Ann. Symp. Computational Geometry, pp.264-273, 2011.

[26] P. Bose, L. Devroye, M. Loffler, J. Snoeyink, and V. Verma, “TheSpanningRatio of theDelaunayTriangulation IsGreater thanPi/2,”Proc. Canadian Conf. Computational Geometry, pp. 1-3, 2009.


[27] J.R. Shewchuk,Triangle version1.6, http://www.cs.cmu.edu/~pquake/triangle.html, http://www.cs.cmu.edu/~pquake/triangle.html,2005.

[28] P. Chew, “Guaranteed Quality Mesh Generation for Curved Sur-faces,” Proc. Ann. Symp. Computational Geometry, pp. 274-280, 1993.

[29] J. Ruppert, “A Delaunay Refinement Algorithm for Quality2-DimensionalMeshGeneration,” J. Algorithms, vol. 18, pp. 548-585,1995.

[30] A. Ungor, “Off-Centers: A New Type of Steiner Points for Comput-ing Size-Optimal Guaranteed-Quality Delaunay Triangulations,”Proc. Latin Am. Symp. Theoretical Informatics, pp. 152-161, 2004.

[31] L.P. Chew, “Guaranteed-Quality Triangular Meshes,” Dept. ofComputer Science, Cornell Univ., Technical Report TR-89-983, 1989.

[32] NS-2, Network Simulator 2, http://www.isi.edu/nsnam/ns. 2008.[33] F. Kuhn, R. Wattenhofer, and A. Zollinger, “Worst Case Optimal

andAverageCase Efficient Geometric AdHoc Routing,” Proc. ACMInt’l Symp. Mobile Ad Hoc Networking and Computing (MobiHoc), pp.267-278, 2003.

[34] F. Kuhn, R. Wattenhofer, Y. Zhang, and A. Zollinger, “GeometricAd-Hoc Routing: Of Theory and Practice,” Proc. ACM Principles ofDistributed Computing (PODC), pp. 63–72, 2003.

[35] B. Leong, S. Mitra, and B. Liskov, “Path Vector Face Routing:Geographic Routing with Local Face Information,” Proc. IEEE Int’lConf. Network Protocols (ICNP), pp. 147–158, 2005.

[36] F. Harary, Graph Theory. Addison-Wesley, 1994.[37] P. Bose and L. Devroye, “On the Stabbing Number of a Random

Delaunay Triangulation,” Computational Geometry: Theory and Ap-plications, vol. 36, pp. 89-105, 2007.

Weisheng Si received the BS, MS, and PhDdegrees in computer science from Peking Univer-sity, China, University of Virginia, Charlottesville,and University of Sydney, Australia, respectively.He is nowa lecturer with the School of Computing,Engineering and Mathematics, University ofWesternSydney, Australia. His research interestsinclude routing inwirelessnetworks, graph theory,and green networking.

Albert Y. Zomaya is the chair professor of HighPerformance Computing & Networking and Aus-tralian Research Council Professorial Fellow withthe School of Information Technologies, SydneyUniversity, Darlington, Australia. He is also thedirector of the Centre for Distributed and HighPerformance Computing which was establishedin late 2009.Hepublishedmore than500 scientificpapers and articles and is author, co-author oreditor of more than 20 books. He is currently theeditor in chief of the IEEE Transactions on Com-

puters and Springer’s Scalable Computing and serves as an associateeditor for 22 leading journals. He is the founding editor of the Wiley BookSeries on Parallel and Distributed Computing. From 1999 to 2003, he wasthe chair of the IEEE Technical Committee on Parallel Processing, andcurrently, serves on its executive committee. He is the vice-chair of IEEETask Force on Computational Intelligence for Cloud Computing andserves on the advisory board of the IEEE Technical Committee onScalable Computing and the steering committee of the IEEE TechnicalArea in Green Computing. He has delivered more than 130 keynoteaddresses, invited seminars, and media briefings and has been activelyinvolved, in a variety of capacities, in the organization of more than 600conferences. His research interests span several areas in parallel anddistributed computing. He is a Fellow of the American Association for theAdvancement of Science and the Institution of Engineering and Technol-ogy (UK). He received the 1997 Edgeworth David Medal from the RoyalSociety of New South Wales for outstanding contributions to AustralianScience. Hewas the recipient of the IEEEComputer Society’sMeritoriousService Award and Golden Core Recognition in 2000 and 2006, respec-tively. He received the IEEE TCPP Outstanding Service Award and theIEEE TCSC Medal for Excellence in Scalable Computing, both in 2011.

Selvadurai Selvakennedy obtained the PhDdegree in computer networking in 1999 from theUniversity of Putra, Darul Ehsan, Malaysia. Hecurrently works as a consulting engineer with aprofessional services company, and is also asso-ciated with the University of Sydney, Darlington,Australia. His research interests lie in developingprotocols and algorithms for radio resource man-agement, MAC and network protocols, middle-ware, and topology control issues in wirelessmesh networks. He has served on the technical

program committees of many international conferences. He is a profes-sional member of ACM.

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A geometric deployment and routing scheme for directional wireless mesh networks