modeling and analyzing the impact of location ...helmy/papers/loc-err-mc2r-04-published.pdfmaximum...

Modeling and Analyzing the Impact of LocationInconsistencies on Geographic Routing in Wireless

Networks ∗

Yongjin Kima Jae-Joon Leea Ahmed [email protected] [email protected] [email protected]

aEE Department, University of Southern California, Los Angeles, CA, U.S.A

Recently, geographic routing in wireless networks has gained attention due to several ad-vantages of location information. Location information eliminates the necessity to set upand maintain explicit routes, which reduces communication overhead and routing table size.These advantages allow scalability especially in dynamic and unstable wireless networks.However, no matter which technologies or techniques a location system uses, its measure-ments will have some amount of quantifiable inaccuracy depending on environment andsystem. These inaccuracies may affect the performance and even correctness of geographicrouting. However, thus far, these impacts have not been studied in-depth. In this paper, weanalyze the impact of location inaccuracy on geographic routing. First, we model locationinaccuracy metrics - absolute location inaccuracy, relative distance inaccuracy, absolutelocation inconsistency and relative distance inconsistency. Then, we analyze how locationinaccuracy metrics affect the building blocks of geographic routing including greedy for-warding and local maximum resolution. We use extensive NS-2 simulations to evaluate theperformance of geographic routing using a wide array of parameter settings including in-accuracy level, node degree and network diameter. In our simulation results, reasonablelocation inaccuracy (of 20 percent or less of radio range) caused packet drop reaching upto 54 percent, non-optimal path up to 53 percent and packet looping. These observationsindicate the importance of re-visiting geographic routing protocols, and the significance ofconsidering location inaccuracy in their design and evaluation.

I. Introduction

In recent years, wireless communication devices aregaining tremendous popularity due to the advanceof hardware technology and various commercial ap-plications such as conferencing, home networking,emergency services, embedded computing, and sensordust, among others. This stimulated the introductionof ad hoc networks that do not have a preestablishednetwork infrastructure. In such ad hoc networks, es-pecially in mobile ad hoc networks where nodes maymove arbitrarily, routing is a challenging task becauseof frequent topology change without prior notice.Many routing protocols for ad hoc networks have beenproposed and existing ad hoc routing protocols can bedivided into two approaches: topology-based[4] andposition-based routing[12]. Topology-based routing,such as table-driven or on-demand routing, uses linkinformation of the networks to perform packet for-warding. Position-based routing uses physical loca-tion information to perform packet forwarding.

∗Ahmed Helmy was supported by Grants from NSF CA-REER, Intel and Pratt&Whitney ICT

Geogrphic routing[1,6,10,12,20,21] has been givenattention recently due to its several merits againsttraditional topology-based routing. That is, geo-graphic routing eliminates some of the limitations oftopology-based routing by using location information.In geographic routing, the routing decision at eachnode is based on the destination’s location containedin the packet header and the location of the forwardingnode’s neighbors. Geographic routing thus does notrequire the establishment or maintenance of routes.The nodes have neither to store routing tables nor totransmit messages to keep routing tables up to date.Hence, geographic routing allows routers to be nearlystateless, and requires propagation of topology infor-mation for only a single hop. This nature of locationis the key to location’s usefulness providing followingmerits.

• Lower communication overhead to set up andmaintain routes

• Small routing table

• Natural extention to geocasting using location in-formation

48 Mobile Computing and Communications Review, Volume 8, Number 1

Location information has important meaning espe-cially in sensor networks. In sensor networks, locationinformation may be used for various purposes, such asevent reporting at specific location, data centric stor-age[18], or naming schemes[5]. Accurate location in-formation is considered important especially in rout-ing and naming schemes. Because of dense nature ofsensor networks, small location inaccuracy can affectthe performance and correctness of sensor networks.

Because of the many merits of location informationin wireless networks mentioned above, geographicrouting protocol is under active research. However,no matter which technology is used, location informa-tion has some inaccuracy depending on the localiza-tion system and the environment. Nodes could acquirethe estimates of their locations from outside sourcessuch as GPS[15,19]. GPS has inheretent inaccuracy.In addition, it is not justifiable from economic and en-ergy preservation points of view to equip each nodewith a GPS receiver in the entire wireless networks.The alternative solution proposed is to design a loca-tion discovery algorithm that uses measurements ofthe distances between nodes and estimates of the loca-tions of a small percentage of nodes( acquired throughGPS, for example ) to determine the locations of all ormajority of nodes in a network. Distance measure-ment is also inherently noisy. Mobility during beaconinterval may also induce location inaccuacy. Transientstate of location information may induce inconsistentview of location information, which may result in se-rious problems in geographic routing.Furthermore, even if every node knows its own loca-tion, there is still a need for an underlying locationdiscovery and dissemination service. Location errorsand inconsistencies may also be introduced by loca-tion dissemination services, such as the grid locationservice (GLS)[6].Existing geographic routing protocols assume accu-rate location information, which is impractical in realworld. As a result, there has been little analysis howlocation inaccuracy affects geographic routing proto-col. It is essential to know the impact of location inac-curacy on geographic routing. In this paper, we inves-tigate the impact of location inaccuracy on geographicrouting. We first model location inaccuracy metrics:absolute location inaccuracy, relative distance inaccu-racy, absolute location inconsistency and relative dis-tance inconsistency. Each inaccuracy metric may af-fect geogrphic routing in terms of performance - suchas packet drop, non-optimal path and routing loop -and even correctness of protocol. Then we study howlocation inaccuracy metrics affect the building blocks

of geographic routing including greedy forwardingand local maximum resolution using simulation andanalysis. Different node degree and network diameterwith varying inaccuracy level have different impact ongeographic routing. In our simulation, location inac-curacy caused packet drop rate up to 54 percent andnon-optimal path up to 53 percent in dense network.Routing loop occurred (up to 1.9 percent) in sparsenetwork. We used local maximum resolution schemeof GPSR[1] as a case study and found that locationinaccuracy affects even correctness of location maxi-mum resolution using analysis and simulation.It is interesting to note that we observed some benefitsfor location inaccuracy in some scenarios of greedyforwarding, but those benefits were overshadowed bysignificant performance degradation.In all, our observations strongly suggest that the de-sign of geographic routing protocols should be re-visited and that it is essential to consider location inac-curacy and inconsistency in the design and evaluationof geographic routing.

The rest of paper is organized as follows. In sectionII, we briefly review geographic routing. Section IIIdefines four inaccuracy metrics - absolute location in-accuracy, relative distance inaccuracy, absolute loca-tion inconsistency and relative distance inconsistency- and discusses possible problems of the inaccuracymetrics. Section IV investigates whether location in-accuracy always has a bad impact on geographic rout-ing. Section V shows the impact of location inaccu-racy on greedy packet forwarding. In section VI, weshow how location inaccuracy affects local maximumresolution using perimeter mode of GPSR as a casestudy. Section VII concludes our paper.

II. Background and Related Work

Geographic routing can be categorized into threeapproaches depending on the forwarding strate-gies[12]: Greedy packet forwarding, restricted direc-tional flooding and hierarchical routing.

Using greedy packet forwarding, a sender of apacket includes the position of the destination in thepacket header. This information is gathered by ap-propriate location service system. A sender sends apacket to the closest node to the destination withintransmission range as shown in figure 1.

Ideally, this process can be repeated through inter-mediate nodes until the destination has been reached.One function of greedy forwarding is to compare dis-tance between all neighboring nodes - within trans-mission range - and the destination node and decides

Mobile Computing and Communications Review, Volume 8, Number 1 49

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Figure 1: Greedy Packet Forwarding

Figure 2: Local Maximum

the closest neighbor to the destination. The otherfunction of greedy forwarding is to decide whetherit is in local maximum situation or not, where thereexists no neighbor node closer to the destination thanthe forwarding node. The proper operation of thesetwo functions can be affected by location inaccuracy.That is, location inaccuracy increases the probabil-ity of making incorrect neighbor selection and wronggreedy decision and consequent packet drop, non-optimal path and packet looping. We define wronggreedy decision as selecting a node which is not theclosest to the destination within transmission range.We will verify those phenomena in simulation. Eventhough greedy forwarding can find optimal path to thedestination, there are some situations in which greedyforwarding fails, which is called local maximum. Lo-cal maximum occurs when greedy forwarding facesa dead end and fails to find a path between a senderand a destination, even though one does exist. A localmaximum scenario is illustrated in figure 2.

As seen in the figure, a local maximum scenario oc-

curs when there are no nodes in the void region. Thissituation happens because of the greedy nature. Cur-rently, the most advanced scheme to address this localmaximum is graph-based scheme such as face-2 [14]and perimeter mode [1] with planar graph. To preventloops, constructing a plannar graph that does not havea cross edge among the nodes is the most importantfunction. However, as we shall show, constructing aplannar graph poses great challenges in the presenceof location inaccuracy. We will study how absolutelocation inaccuracy affects plannar graph constructionusing analysis and simulation.

Restricted directional flooding approach is devel-oped to relieve high overhead problem of global flood-ing in geographic routing. LAR[20] is one example.LAR proposes the use of position information to en-hance the route discovery phase of reactive ad hocrouting approaches. Reactive ad hoc routing protocolsfrequently use flooding as a means of route discovery.Under the assumption that nodes have informationabout other nodes’ location, this location informationcan be used by LAR to restrict the flooding to a certainarea. LAR defines an expected zone of the destinationbased on available location information. That is, theexpected zone is the place where the destination maybe located during a certain time interval. The expectedzone can be calculated depending on initial location,time interval and mobility speed. LAR also defines arequest zone as the set of nodes that should forwardthe routing discovery packet. The request zone typi-cally includes the expected zone. Node will forwardthe route discovery packet only if they are within thatspecific region.

In traditional networks, the complexity each nodehas to handle can be reduced tremendously by estab-lishing some form of hierarchy. Hierarchical routingallows those networks to scale to a very large num-ber of nodes. One approach that combines hierarchi-cal and geographic routing is part of the Terminodesproject[10]. In Terminodes routing, a two-level hier-archy is proposed. Packets are routed according toa proactive distance vector scheme if the destinationis close to the sending node. For long distance rout-ing a greedy approach is used. Once a packet reachesthe area close to the recipient, it continues to be for-warded by means of the local routing protocol. It isshown that the introduction of a hierarchy can sig-nificantly improve the ratio of successfully deliveredpackets and the routing overhead compared to reactivead hoc routing algorithms.

In this paper, we focus on the greedy packet for-warding scheme which is considered a basic mecha-


nistic building block in geographic routing. In addi-tion, we analyze how location inaccuracy affects thelocal maximum resolution mechanism - a mechanismsufficient to provide correct geographic routing in theabsence of location inaccuracy. To our knowledge,this is one of the first in-depth studies of the effectsof location inconsistencies on geographic routing pro-tocols. Preliminary work in [7][8][9] provides algo-rithmic analysis of face routing in presence of loca-tion inaccuracy (vs. inconsistency) in static sensornetworks, and reaches similar conclusions that the ef-fects are significant. Preliminary work in [3] investi-gates effects of mobility on geographic routing. Ourwork is unique in providing the analysis for locationinconsistencies, mainly in ad hoc networks. There isa large body of work on localization in wireless net-works. We only cite a limited number of studies forillustration. A more detailed treatment of this topic isbeyond the scope of this paper. GPS[15] is a commontechnique for localization. GPS introduces localiza-tion errors that depend on the receiver used (amongother factors). Those errors range from 1cm-100m(with 10m-100m mean error incurred for reasonablecost receivers). GPS may not work in-doors, and itmay not be feasible or economic for classes of ad hocnetworks (including sensor networks) to have GPS ca-pability. GPS-less techniques have been recently de-signed for wireless networks. Reference beacons areused for localization in[13], where the accuracy in-creases with increased number of reference beaconsand is a function of the radio range. The study cites 20percent or more (of the radio range) as the mean local-ization error with high beacon deployment. We bor-row from these studies to develop our location errormodel. All studies of localization include a section onlocalization error, and all studies cite non-negligiblelocalization errors.

III. Inconsistency Metrics

In this section, we classify location inconsistency intofour metrics - Absoulte location inaccuracy, relativedistance inaccuracy, absolute location inconsistencyand relative distance inconsistency. Each of these met-rics potentially has a negative impact on geographicrouting in terms of packet drop, non-optimal path androuting loop.We do not assume any specific location system such as[6,16,22]. Instead, we focus on the general locationinconsistency that is incurred in any location systemdepending on its implementation and the deploymentenvironment.

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Figure 3: Packet Drop by Absolute Location Inaccu-racy

III.A. Absolute Location Inaccuracy

We define the absolute location inaccuracy for eachnode i as follows.

Absolute Location Inaccuracy =√(xi − (xi +4xi))2 + (yi − (yi +4yi))2 (1)

where, Ai(xi, yi) is the true location and Ai(xi +4xi, yi +4yi) is the faulty location. Absolute loca-tion inaccuracy is the distance between the true lo-cation and the faulty location. A possible problemof absolute location inaccuracy is packet drop. Asillustrated in figure 3, node i perceives the destina-tion node d is at (xd +4xi, yi +4yi). However, thetrue location of node d is (xd, yd) which is outside thetransmission range. Absolute location inaccuracy mayinduce relative distance inaccuracy, absolute locationinconsistency and relative location inconsistency. Wedefine each of terms in the following sections.

III.B. Relative Distance Inaccuracy

In figure 4, the distance between node i, node j andthe destination node d can be calculated as follows.

Disti =√

(xd − xi)2 + (yd − yi)2 (2)

Distj =√

(xd − xj)2 + (yd − yj)2 (3)

Then the true relative distance is defined as follows.

4Dist = Disti −Distj (4)

Greedy forwarding algorithm selects node i if

4Dist < 0 (5)


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Figure 4: Distance between Node and Destination

which means i is closer to the destination than j.However, greedy forwarding may fail if high relativedistance inaccuracy exists. That is, a forwarding nodemay select a node which is not the closest to the desti-nation within its radio range. Before we define relativedistance inaccuracy, we define distance inaccuracy be-tween node i and the destination d and between nodej and the destination d as follows.

Disti−error =√

((xd +4xd)− (xi +4xi))2 + ((yd +4yd)− (yi +4yi))2

(6)

Distj−error =

√((xd +4xd)− (xj +4xj))2 + ((yd +4yd)− (yj +4yj))2

(7)Then, the relative distance inaccuracy is defined as

follows.4Dist · 4Disterror ≤ 0 (8)

where,

4Disterror = Disti−error −Distj−error (9)

Possible problems of relative distance inaccuracyinclude higher packet drop rate due to false local max-imum and non-optimal routing path due to wronggreedy decision. We define false local maximum asthe situation where forwarding node could not find thecloser neighbor node to the destination due to locationinaccuracy even though there is in fact closer neigh-bor node to the destination. This is shown in figure5, where the forwarding node s selects node j eventhough node i is the closest node to the destinationwithin the transmission range. We shall validate andquantify the impact of those possible problems in thesimulation section.

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Figure 5: Wrong Greedy Decision. The black circlerepresents the true location of a node and the whitecircle represents the faulty location.

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Figure 6: False Local Maximum by Absolute Loca-tion Inconsistency

III.C. Absolute Location Inconsistency

We define absolute location inconsistency as follows.

Absolute Location Inconsistency =√

(xik − xjk)

2 + (yik − yjk)

2 (10)

where (xik, yik) is the location of node k as perceived

by node i and (xjk, yjk) is the location of node k as

perceived by node j.Absolute location inconsistency represents the

difference of the same target locations perceived bytwo nodes. One potential problem of high absolutelocation inconsistency is the false local maximumwithin the range reachable to the destination. An ex-ample of absolute location inconsistency is illustratedin figure 6. Sender node s includes the destinationlocation in packet header as location d. However,a forwarding node, f , at the last hop perceives thedestination at a different location d′. If distance


between the d′ and d is greater than the distancebetween f and d, false local maximum occurs.

If global addressing is not available (or not used,as in some sensor networks) and the geographic rout-ing algorithm relies only on the location information,there is a high probability that the packet cannot bedelivered to the destination when absolute location in-consistency is high with high node density in radiorange since the forwarding node cannot find a neigh-bor node closer to the destination than itself. This isespecially the case when location is used as a namingin sensor networks [5]. We call this situation as thefalse local maximum within the destination range.

Absolute location inconsistency may induce rela-tive distance inconsistency, which is explained next.

III.D. Relative Distance Inconsistency

We define distance inaccuracy perceived by node i asfollows.

4Distierror = Distii−error −Distij−error (11)

In addition, we define distance inaccuracy perceivedby node j as follows.

4Distjerror = Distji−error −Distjj−error (12)

Then relative distance inconsistency is be definedas follows.

4Distierror · 4Distjerror ≤ 0 (13)

One potential problem of relative distance incon-sistency is formation of routing loop. A routing loopcase is illustrated in figure 7. Node s thinks thatnode j is the closest node to the destination withinthe transmission range and sends the packet to j. Inturn, node j thinks that node i is the closest node tothe destination and send a packet back to the nodei and this continuation results in routing loop. Thissituation can occur at the sender’s transmission rangeor in the middle of path.

There can be multi-hop relative distance incon-sistency even though there is no relative distanceinconsistency between two direct neighbor. Figure8 illustrates multi-hop routing loop. Sender node ssends a packet to node j. Then, node j sends a packetto the node i and node i sends a packet to the node k.Node k sends the packet back to node j and so on ,forming a multi-hop loop. This situation frequentlyoccurred in our simulations.

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Figure 7: Routing Loop caused by Relative DistanceInconsistency

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Figure 8: Multi-hop Routing Loop caused by RelativeDistance Inconsistency

We summarize the four inaccuracy metrics in tableI.

IV. Is Location Inconsistency AlwaysBad?

The answer is ’not always’. For example, packet dropin greedy forwarding due to a local maximum scenario(with precise location) may be avoided if location in-accuracy leads to a valid greedy forwarding path. Lo-cal maximum occurs in following condition.

{i | d(s, d) > d(i, d)} = {} (14)

Where d(s, d) is distance between s and d and s issending node, d is destination node and i is neighbornode. However, local maximum can be addressed byinaccuracy if,

{i | d(s, d) > d(i, d)} 6= {} (15)

However, the one of the following conditionsshould be satisfied.


Table 1: Inconsistency Metrics

Inaccuracy Absolute Location Inaccuracy√

(xi − (xi +4xi))2 + (yi − (yi +4yi))2

Relative Distance Inaccuracy 4Dist · 4Disterror ≤ 0

Inconsistency Absolute Location Inconsistency√

(xik − xjk)2 + (yik − y

jk)2

Relative Distance Inconsistency 4Distierror · 4Distjerror ≤ 0

{i | Path between i and d exists} 6= {} (16)

or,

{i | d(i, d) < Transmission Range} 6= {} (17)

This provides a stringent condition in the presenseof location inaccuracy in order for location inaccuracyto have good impact on geographic routing. We shallre-examine this fact in the simulation section.

V. Simulation of Greedy Forwrding

We perform extensive simulations in NS-2[11] to in-vestigate the impact of location inaccuracy on geo-graphic routing. The original NS-2 code for GPSRwas used. We use 250m of radio range and Gaussiandistributions (with zero mean) with different standarddeviation to generate location inaccuracy. Locationinaccuracy is added to true location. In our simula-tion, location error is represented as the fraction ofstandard deviation over radio range since a differentradio range may be used in different networks. Forexample, if the percentage location inaccuracy is 10percent this means that the standard deviation for theGaussian distribution is 10 percent of the radio range.Different parameters for node degree and network di-ameter are used. Node degree is defined as the averagenumber of neighbor nodes. We use different networkdiameter, 750x750, 1050x1050, 1350x1350, with thesame node degree of 17. We averaged 5 simulationsfor each node degree, network diameter and locationerror. Simulation parameters are in table II. We didnot add mobility to investigate only the impact of lo-cation inaccuracy. Mobility is expected to exacerbatethe inaccuracy problem leading to even higher per-centage inaccuracy. We perform simulation on greedymode without local maximum resolution first, then weperform analysis and simulation of perimeter mode ofGPSR as a case study of local maximum resolution.

V.A. Impact on Drop Rate

Figure 9 shows the effect of node degree and locationerror on packet drop rate. It is clear that as locationerror gets larger then the drop rate becomes higher.By carefully analyzing the simulation traces, we ob-serve that the main reason for packet drop is false lo-cal maximum. A packet is forced to drop if there isno local maximum resolution mechanism under falselocal maximum situation. Even if there is local maxi-mum resolution, longer detour is inevitable and localmaximum resolution itself may also have some prob-lem with location inaccuracy, which will be discussedin the section VI. Figure 10 shows false local max-imum rate. We can see that as location inaccuracygets higher, false local maximum rate significantly in-creases up to 30 percent. Figure 11 shows that in caseof dense network, almost over 90 percent of the packetdrop happens within the destination radio range. Thereason for higher packet drop in dense network is thehigher probability of false local maximum within thedestination range due to absolute location inconsis-tency that was explained earlier in section III-C. Fig-ure 12 shows the effect of network diameter on thepacket drop rate. Network diameter is changed from750x750 to 1350x1350 with the same node degree of17. We generated location inaccuracy with standarddeviation of 25m. Drop rate gets higher as the net-work gets larger reaching up to 43 percent. This isbecause of the fact that as the network gets larger,there is higher probability of relative distance inaccu-racy and absolute location inconsistency.

V.B. Impact on Optimal Path

We investigate the impact of location inaccuracy onoptimal path rate. Figure 13 shows the non-optimalpath rate. There is higher non-optimal path rate indense network reaching up to 53 percent. The rea-son can be explained with figure 14, which shows thewrong greedy neighbor decision rate. Since the dis-tance between nodes is closer in denser network, therate of wrong greedy neighbor decision is higher. Thisshows the more frequent relative distance inaccuracy


Table 2: Simulation Parameters

Parameter Value

Radio Range 250mStandard Deviation of Inaccuracy 0,10,20,30,40,50mLocation error 0,5,10,15,20 %Node Degree 5,8,19( area : 1350mx1350m )Network Diameter 750mx750m,1050mx1050m,1350mx1350mTraffic Sources 30

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Location Error (%)

Node Degree 19Node Degree 8Node Degree 5

Figure 9: Impact of Location Inaccuracy and NodeDegree on Packet Drop Rate

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Figure 10: False Local Maximum Rate

in denser network. This results in more non-optimalpaths. The number of non-optimal path gets higheras network diameter gets large (Figure 15), due to in-creased probability of wrong neighbor selection.

V.C. Impact on Looping

In our simulation routing loops occurred because ofrelative distance inconsistency. Figure 16 shows thatpacket drop due to loop. There is more probability ofrelative distance inconsistency in sparse network, re-sulting in more routing loops. Single hop loop andmulti-hop loop occurred in our simulation. Packets

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Figure 11: False Local Maximum Rate within theDestination Range

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Figure 12: Impact of Network Diameter on PacketDrop Rate

traversing the loop consume node power and causecontention in network path, which significantly con-sumes the network resources and affects the overallnetwork performance.

V.D. Good Impact vs. Bad Impact

As we mentioned in section IV, theoretically there canbe good effects of location inaccuracy on routing per-formance in terms of packet drop rate. However, inour extensive simulation, the percentage of good im-pact was very low (Figure 17). The bad effects far


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Figure 13: Impact of Location Inaccuracy and NodeDegree on Non-optimal Path

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Figure 14: Wrong-greedy Neighbor Decision Rate

outweighed the good effects as can be clearly seen inthe figure. This is because of stringent condition thatshould be satisfied for good impact to occur as wasmentioned in section III.

VI. Analysis and Simulation ofPerimeter Mode in GPSR:CaseStudy

Local maximum resolution is one of the most im-portant issues in geographic routing. Currently,graph-based solution such as face-2[14] or perime-ter mode[1] using planar graph are considered themost advanced schemes. We investigate how pla-nar graph construction is affected by location inaccu-racy. As a case study, we selected Gabriel Graph (GG)in the well-known greedy perimeter stateless routing(GPSR)[1] protocol.

VI.A. Correctness Analysis of PerimeterMode in GPSR

Perimeter mode routing resolves the local maximumproblem by routing around the void using the right-hand traversal rule. In order to enable such feature

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Figure 15: Impact of Location Inaccuracy and Net-work diameter on Non-optimal Path

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Figure 16: Packet Drop Rate Due to Routing Loop

all cross edges must be removed from the connectiv-ity graph using planarization. One common way toestablish planarization is through the Gabriel Graph(GG). In GG graph, there should be no witness withinthe circle between nodes to keep connection. As illus-trated in figure 18, to keep connection between nodesu and v, there should be no witness node w. Other-wise a cross edge is will be created, which prevents thepacket from traversing the perimeter correctly. Thiscondition is expressed as follows.

∀w 6= u, v : d2(u, v) < [d2(u,w) + d2(v, w)] (18)

When there is inaccuracy in the location of w, suchthat the above condition is violated, then the planargraph of GG cannot be established correctly. In otherwords, if nodes u and v think that w is outside the cir-cle (due to absolute location inaccuracy) while in factw is indeed within the circle, then an edge that shouldbe removed will not be removed. We call such sce-nario a planar graph collapse. If absolute location in-accuracy of node w is a and average distance betweennodes is 2r as described in figure 19, then the proba-bility that planar graph collapses can be calculated asfollows.


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Figure 17: Good Imapct vs. Bad Impact of LocationInaccuracy

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Figure 18: Planar Graph Construction

Probability of Planar Graph Collapse =πr2 − π(r − a)2

πr2

(19)Figure 20 shows the probability of planar graph col-

lapse with different location inaccuracy of a from 1mto 25m and node distance is 50m,100m,150m. Asshown in graph, the probability is very high even withsmall location inaccuracy. The probability becomes 1with location accuracy of 25m and average distanceof 50m. We note that this is the worst case analysisbecause the probability is affected by the direction oflocation inaccuracy of w and absolute location inac-curacy of node u, v and the existence of witness nodew within the overlapped region. In the next section,we perform simulation on GG graph of GPSR.

VI.B. Simulation of Perimeter Mode inGPSR

We perform simulation to compare with our earlieranalysis and study impact of location inaccuracy on

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Figure 19: Correctness Analysis of Perimeter Modeof GPSR

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Figure 20: Probability of Planar Graph Collapse

perimeter mode in GPSR. Figure 21 shows planargraph error rate which represents the probability ofplanar graph collapse in our analysis. Simulation re-sult trends match with our analysis result in that thereis higher probability of planar graph collapse in densernetwork. Probability value is somewhat lower in sim-ulation result because our analytical result is worstcase as we mentioned in the previous section. Wealso note that not all planar graph collapse scenariosnecessarily lead to packet drop. Furthermore, packetdrop may be due to planar graph collapse or other ef-fects of location inaccuracy (e.g., removal of edgesthat should not be removed). Figure 22 shows thepacket drop rate due to location inaccuracy. As in-accuracy increase in perimeter mode, denser networkshow higher drop rate. However, in lower locationerror, sparser network shows higher drop rate, whichis somewhat different from the probability of planargraph collapse. This occurs because there is higherprobability of disconnection between nodes with evensmall location inaccuracy in sparser network. Fig-ure 23 shows non-optimal path rate. Notice that non-optimal rate is higher in sparse network even when


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Figure 21: Planar Graph Error Rate

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Figure 22: Packet Drop Rate Due to Location Inaccu-racy in Perimeter Mode

location error is small. This is because there is higherprobability of local maximum in sparse network re-sulting in detour path. In figure 24, we can see thatmore routing loops happen in sparser network becauseof high relative distance inconsistency. One interest-ing result in this graph is that routing loop occurs inperimeter mode of GG graph even without locationerror. This is due to network disconnection and is notconsidered a protocol error per se.

VII. Summary and Conclusion

In this paper, we have introduced a classification of lo-cation errors identifying four location inaccuracy met-rics: (1) absolute location inaccuracy, (2) relative dis-tance inaccuracy, (3) absolute location inconsistencyand (4) relative distance inconsistency. Each of thelocation inaccuracy metrics affects geographic rout-ing protocol and the impacts are summarized in tableIII. In the table, each inaccuracy metric is the cause ofthe problem and the impact.

Using extensive NS simulations, we conductedstudies on greedy forwarding and perimeter moderouting. We observed the following impacts of loca-tion inaccuracy on geographic routing.

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Figure 23: Non-optimal Path Rate in Perimeter Mode

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Figure 24: Impact on Loop in Perimeter Mode

(I) For greedy forwarding, packet drops occurmainly because of (a) routing loops caused by relativedistance inconsistency and (b) no-route within desti-nation caused by absolute location inconsistency. Thepacket drop rate is affected by the node degree andthe network diameter. In denser network, there wasmore absolute location inconsistency due to shorterdistance between neighbors which resulted in higherpacket drop. In sparser network, there was more rout-ing loops due to higher relative distance inconsistency,which resulted in packet drop. For the same networkdensity, larger network diameter affects geographicrouting more negatively in the presence of location inaccuracy. This is because as the average path lengthincreases, there is higher probability to incur relativedistance inconsisency and absolute location inconsis-tency.

Non-optimal path is caused by relative distance in-accuracy. Relative distance inaccuracy causes wronggreedy neighbor selection. In addition, relative dis-tance inaccuracy causes false local maximum whichinvokes local maximum resolution. Non-optimal pathis affected by node degree and network diameter dueto reasons similar to those of packet drops. Non-optimal path is especially undesirable in wireless net-works because it consumes valuable power.


Table 3: Impact of Inaccuracy Metrics

Inaccuracy Metrics Problem ImpactInaccuracy Absolute Location Inaccuracy Wrong Neighbor Information Packet Drop

Relative Distance Inaccuracy Wrong Greedy Decision Non-optimal PathInconsistency Absolute Location Inconsistency False Local Maximum Packet Drop

Relative Distance Inconsistency Wrong Greedy Decision Routing Loop

(II) Location inaccuracy affects the correctness ofgraph-based local maximum resolution scheme. Ouranalysis of the perimeter mode (GG graph) of GPSRshows that in the presence of location inaccuracy, theplanar graph is highly unlikely to be constructed cor-rectly. Even with small absolute location inaccuracy,we observed high probability of planar graph collapse.Location inaccuracy degrades performance of perime-ter mode in terms of packet drop, optimal path androuting loop rate.

The main contributions of this paper lie in (1) in-troducing classification and conditions of location re-lated inaccuracies, (2) conducting the first in-depthanalysis of the effects of location errors on geo-graphic routing, and (3) uncovering severe perfor-mance degradation and even protocol correctness vi-olations for geographic routing even in the presenceof reasonable (relatively small) location errors, andeven without considering mobility. This final re-mark is considered the major contribution of this pa-per, which points out that even with planarization andface-routing, geographic routing behaves incorrectlywith location errors.

These observations indicate a pressing need to re-visit the design of geographic routing protocols to berobust to location errors.

In the future we would like to investigate possiblefixes to the problems uncovered in this paper. We hopethat our work stimulates further research in this area,especially that several recent applications are beingdesigned on top of geographic routing assuming theircorrectness[2,17].

References

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