planar graph routing on geographical clustersunikorn/... · candidates. consequently, greedy...

Ad Hoc Networks 3 (2005) 560–574

www.elsevier.com/locate/adhoc

Planar graph routing on geographical clusters

Hannes Frey *, Daniel Gorgen

System Software and Distributed Systems, University of Trier, 54296 Trier, Germany

Available online 16 September 2004

Abstract

Geographic routing protocols base their forwarding decisions on the location of the current device, its neighbors,and the packets destination. Early proposed heuristic greedy routing algorithms might fail even if there is a path fromsource to destination. In recent years several recovery strategies have been proposed in order to overcome such greedyrouting failures. Planar graph traversal was the first of those strategies that does not require packet duplication andmemorizing past routing tasks. This article introduces a novel recovery strategy based on the idea of planar graph tra-versal but performing routing tasks along geographical clusters instead of individual nodes. The planar graphconstruction method discovered so far needs one-hop neighbor information only, but may produce disconnection evenif there is a path from source to destination. However, simulation results show that the proposed algorithm is a goodchoice from a practical point of view, since disconnection does only concern sparse networks, while in dense networkthe proposed algorithm competes with existing solutions and even outperforms planar graph routing methods based onone-hop neighbor information. This paper finally gives an outline of further research directions which show that geo-graphical clusters may be the key to solve some problems that come along with planar graph routing in wirelessnetworks.� 2004 Elsevier B.V. All rights reserved.

Keywords: Ad-hoc networks; Sensor networks; Overlay networks; Geographic routing; Topology control

1. Introduction

Multihop ad-hoc networks are defined by mo-bile or stationary devices communicating overwireless links without using a fixed network infra-

1570-8705/$ - see front matter � 2004 Elsevier B.V. All rights reservdoi:10.1016/j.adhoc.2004.08.013

* Corresponding author.E-mail addresses: [email protected] (H. Frey), goer-

[email protected] (D. Gorgen).

structure. Due to limited transmission range com-munication between any two devices requirescollaborating intermediate network nodes in orderto forward packets from source to destinationnode. A trivial form of such collaboration maybe achieved by flooding, however, in general moreelaborate resource saving routing strategies areneeded in order to enable the practical applicabi-lity of ad-hoc networks which are comprised ofsmall low powered devices.

ed.

mailto:[email protected]



H. Frey, D. Gorgen / Ad Hoc Networks 3 (2005) 560–574 561

The first routing protocols especially designedfor ad-hoc networks follow the approach of classi-cal topology-based routing schemes, i.e. routingdecisions depend on information about the currentavailable network links [1]. Since routing pathshave to be maintained (either proactively or reac-tively) for ongoing routing tasks, topology-basedrouting might produce a significant amount oftraffic if topology changes are frequent due to de-vice mobility or alternating energy conservingsleep cycles. If physical device location can bedetermined by GPS or another kind of relativepositioning technique [2,3], geographical packetforwarding may serve as an alternative resourcesaving routing strategy, provided that the sourcenode is able to acquire the position of the packetdestination by means of a location service [4].

This article presents a novel geographic routingalgorithm based on geographical clusters, whichare used in order to construct an overlay networkon the network defined by the wireless mobile de-vices. The next section will discuss geographicalrouting methods related to the presented algo-rithm. The concept of geographic clusters is intro-duced in Section 3, and the algorithm itself ispresented in Section 4. The correctness of the pro-posed algorithm is proved in Section 5. A quanti-tative performance comparison in with existingrelated routing algorithms is given in Section 6.Finally, Sections 7 and 8 conclude the paper andgive an outline of future research topics regardingrouting along geographic clusters.

2. Related work

The first geographical routing algorithms ap-plied a greedy forwarding strategy, where routingdecisions are based on the position of the forward-ing device, its current neighbors and the final des-tination [5,6]. Each node applies this greedyrouting principle until the destination, if possible,is eventually reached. The next hop node selecteddepends on the optimization criterion applied ineach forwarding step. In order to avoid forwardingloops, only nodes reducing the distance to thedestination are allowed as possible next hopcandidates. Consequently, greedy routing cannot

guarantee delivery even if there exists a path fromsource to destination, since packets might getstuck at concave nodes having no neighbor in for-ward direction.

In order to provide guaranteed delivery in con-nected networks, a concave node might invokepartial flooding [7] or depth-first-search routing[8,9], which are recovery strategies based on mem-orizing past traffic along each node involved in arouting task. However, partial flooding inherentlyleads to an increased message complexity whichcan also be observed for depth-first-search basedapproaches when topology changes are frequent.

Planar graph routing which enables memorylessrecovery with guaranteed delivery was investigatedin [10] for the first time. The face routing algorithmproposed there is an improvement of the planargraph routing algorithm introduced in [11]. Thealgorithm is based on traversal of the faces definedby planar geometric graphs, while a geometricgraph is termed planar if there is no intersectionbetween any two edges. A packet successively tra-verses all faces F1, . . .,Fn intersected by the straightline sd connecting source s and destination node dby applying the right/left hand rule (i.e. a packet isforwarded along the next edge clockwise/counter-clockwise from the edge where it arrived). Whenthe packet reaches an edge of Fi intersected bythe straight line sd the next face Fi+1 is traversedin the same way. The algorithm terminates whena face is traversed completely (i.e. there exists nopath from source to destination) or the destinationnode is eventually reached. When used as a recov-ery strategy only, face routing will switch back togreedy mode when the forwarding node has aneighbor lying closer to the destination than theconcave node where face routing was started.

In general the graph G defined by all edges of awireless network is not planar. Thus, before per-forming face routing, a planar subgraph has tobe extracted from G. The algorithm described in[10] uses a 1-localized (i.e. only information aboutall one-hop neighbors is needed) planar graph con-struction method based on Gabriel graphs. Let thedisk with diameter juvj passing nodes u and v bedefined as U(u,v). A Gabriel graph on a finite setS of nodes is constructed by including all edgesuv where the disk U(u,v) contains no other nodes

Fig. 1. The graph resulting from the concept of nodeaggregation. Each network node is assigned to the clusterwhere it is located in. Two clusters C1 and C2 are connected ifthere is at least one connected node pair with one assigned to C1

and the other assigned to C2.

562 H. Frey, D. Gorgen / Ad Hoc Networks 3 (2005) 560–574

from S than u and v. However, the localized algo-rithm described in [10] will only produce a planarconnected graph when the network is describedby a unit disk graph. Unit graphs are modelingwireless ad-hoc networks, where each node hasthe same transmission radius 1 (radius 1 can be as-sumed without loss of generality), i.e. two networknodes are connected if and only if their Euclideandistance is less or equal to 1.

The principle of face routing is adopted in sev-eral protocols [12–14] and several improvementshave been investigated [15–17]. We will revise theconcepts of internal nodes and shortcut basedrouting [16] which are related to the algorithmdescribed in this article and will be used in oursimulation for performance comparisons.

A subset S of all network nodes G is termed adominating set, if each node of G is either ele-ment of S or has at least one neighbor in S.Nodes that belong to a dominating set are calledinternal nodes. Face routing restricted on a con-nected dominating set (resulting from a unit diskgraph) will always find the destination node,since each node is at least connected to one inter-nal node. The dominating set constructionmethod used in [16] is a 1-localized algorithm.The method preserves shortest path edges be-tween any two nodes, thus face routing appliedon that edge subset will produce paths closer tothe shortest path.

Shortcut based face routing further reduces thenumber of hops produced by face traversal. Byproviding 2-hop neighbor information, each nodeis able to construct the part of the planar graphseen by all its neighbors. Thus, a packet need notto be sent to the next node lying on the face borderbeing traversed, but the packet can be sent to thelast node on the traversed face seen by the nodecurrently holding the packet.

3. The concept of node aggregation

The algorithm described in this article is basedon the concept of face routing and assumes thenetwork to be modeled as a unit disk graph andnodes to be placed in the two-dimensional Eucli-dean space R2. In contrast to existing face routing

algorithms, routing is not performed on a per nodebasis but instead packets are forwarded along anextracted planar graph resulting from the edgesof adjacent geographical clusters. In order to definegeographical clusters, the plane is partitioned byan infinite mesh of regular hexagons, while eachhexagon has a diameter of the sending radius 1(see Fig. 1). Each hexagon defines one geographi-cal cluster, which is identified by its center p. Acluster is termed C( p) or just C if center positionhas no relevance. To obtain a unique and well de-fined definition of geographic clusters, we addi-tionally assume one cluster with center (0,0) andone with center ð3

2; 0Þ (cluster C1 and C3 in Fig. 1

for instance).Each segment of the boundary surrounding a

cluster is always shared by two clusters C( p) andC(q) and will be identified as E( p,q). The commonintersection point shared by three clusters C( p),C(q), and C(r) will be termed as V( p,q, r).

Each node is assigned to exactly one geograph-ical cluster by using its current geographicalposition. For instance, node v1 in Fig. 1 is as-signed to cluster C1 since its position is lying in-side that cluster. In order to enable a uniquemapping of points lying on cluster borders, weuse a total ordering defined on all point tuples inR2. Let (x1,y1) and (x2,y2) be two point tuples,then (x1,y1) < (x2,y2) if and only if x1 < x2 ory1 < y2 when x1 = x2. Based on this ordering, setfunctions min and max can obviously be definedfor finite nonempty point sets, and the concept


of node aggregation can formally defined asfollows.

Definition 1. Let V � R2 be a finite point set. Themapping H : V ! R2 is defined as node aggrega-

tion, while

v 7!p if v 2 CðpÞ;minfp; qg if v 2 Eðp; qÞ;minfp; q; rg if v ¼ V ðp; q; rÞ.

8><>:

Two clusters C1 and C2 are termed to be adjacentif there is at least one connected node pair with onenode lying in C1 and the other lying in C2 (e.g.there exists an edge between cluster C1 and C2

from Fig. 1, since nodes v1 and v2 are connected).Thus, we have that node aggregation defines a geo-metric graph with cluster centers being the verticesand edges between two adjacent clusters C(p) andC(q) identified by the straight line connecting thecluster centers p and q. This graph will be termedas the aggregated unit disk graph and can formallybe defined as follows.

Definition 2. Let V � R2 be a finite point set andG = (V,E) be the unit disk graph defined on V.The aggregated unit disk graph H(V ) is definedas ðeV ; eEÞ, whereeV ¼ fs 2 R2 j 9v 2 V : s ¼ HðvÞg;eE ¼ fst 2 R2 � R2 j 9uv 2 E: s ¼ HðuÞ; t ¼ HðvÞg:

Fig. 2. The principle of planar graph routing can also beapplied on the faces resulting from a planar subgraphconstructed from the edges of adjacent clusters. A packetaddressed from s to d will successively visit the faces F1, F2, andF3 until reaching the destination cluster D.

Since the sending radius of a node is equal the clus-ter diameter 1, each cluster C has a maximumnumber of adjacent clusters which is limited to18 (all clusters covered/intersected by the circlearound C with radius 3

2). By enumerating all those

18 possible edges, one can easily observe that thereexist three types of edges connecting the clusters ofan aggregated unit disk graph, namely edges hav-ing length

ffiffi3

p

2(short edges), 3

2(medium edges) andffiffiffi

3p

(long edges), respectively (note, due to regularstructure of each hexagon, it holds that clusterwidth = 1, and cluster height =

ffiffi3

p

2). For instance,

in Fig. 1 the edge between cluster C1 and C2 is ashort, between C5 and C6 is a medium, and be-tween C2 and C4 is a long one.

4. Routing along geographical clusters

The geographic cluster routing (GCR) algorithmdescribed in this section adopts the principle oflocalized routing in planar graphs. A packet ad-dressed from a source node located in cluster S

to a destination node located in cluster D visits asequence of faces which are intersected by thestraight line connecting source and destinationcluster S and D. In contrast to the original facerouting algorithm, the faces are defined by a pla-nar subgraph of the aggregated unit disk graph in-stead. For example, in Fig. 2 a packet sent from sto d, will visit faces F1, F2, F3 until reaching desti-nation cluster D. When right hand rule is applied,the sequence of subsequently visited clusters willbe SABFJD. Similar to original face routing thealgorithm proceeds until the destination cluster iseventually reached or if the first edge of the currentface traversal is traversed twice in the same direc-tion. This describes the GCR algorithm in princi-ple. We will now describe localized methods inorder to maintain the aggregated unit disk graphG, to extract a planar subgraph from G, and toperform packet forwarding along the adjacentclusters during face traversal.

Since each node is able to query its physicallocation, a node can determine its assigned clusterlocally without any additional message exchange.Additionally, the methods described subsequentlyneed information about 1-hop neighbors, thuseach node periodically broadcasts a constantamount of information (its node address, physical


location, and some additional bits as describedbelow) to all neighbors within its transmissionrange. Disappearing 1-hop neighbors are noticed,when the beacon from that node was not receivedwithin a certain timeout interval.

4.1. Constructing the aggregated graph

From a global point of view it is straightfor-ward to construct an aggregated graph G of theunit disk graph defined by a finite set of points.In order to construct G locally all nodes locatedin a cluster C have to agree on the same viewregarding the clusters C is connected to. For in-stance, in Fig. 1 the nodes v2, v3, and v4 locatedin cluster C2 have all different views about the clus-ters C2 is connected to. Since only node v2 is con-nected to v1, this node will see cluster C1 only. Inthe same way, v3 will see C4, and v4 will see C3

only.To provide a unique view of all connected clus-

ters, each node extends its beacon message withinformation about the clusters it is currently ableto see. Note this additional information will in-crease the size of each beacon message by a con-stant amount of additional information only.More precisely, due to the limited transmissionrange the number of clusters a node is able to seeis limited to 18. With one bit representing one ofthese 18 possible clusters, the beacon message isthus increased by 18 bits only.

Since the diameter of each cluster is exactly thesending radius 1, a node located in a cluster C willalways receive the beacons from all nodes whichare also located in C and is thus able to constructthe global view of all clusters C is connected to.When a node disappears, the current informationfrom the remaining nodes is used to update theglobal view of connected clusters. For instance,suppose in Fig. 1 the node v4 is removed from clus-ter C2 (due to mobility, sleep cycle or shutdown),then the edge connecting C2 and C3 disappears,too. Due to the beaconing mechanism nodes v2and v3 will notice the removal of node v4 and willalso remove the edge C2C3, since there is currentlyno other node known to v2 and v3 which sees clus-ter C3.

4.2. Extracting a planar subgraph

In general the graph defined by adjacent clus-ters is not planar. For instance, the edges C7C9

and C8C10 depicted in Fig. 1 are intersecting. Wewill use Gabriel graph construction in order to ex-tract a planar subgraph of the aggregated unit diskgraph locally. However, simply applying Gabrielgraph construction without an additional consist-ency check might lead to an inconsistent view onthe extracted planar graph. Suppose Gabriel graphconstruction is applied on edge C2C4 in Fig. 1. Allnodes within cluster C2 agree on the local view C2

being connected to C1, C3, and C4. Since the circleU(C2,C4) does not contain C1 and C3, the edgeC2C4 is preserved by those nodes. On the otherhand, the nodes located in C4 will remove the edgeC2C4, since C5 seen by those nodes is located inU(C2,C4).

Thus, each node located in a cluster C may onlyinclude a Gabriel graph edge CD to the planargraph if all nodes within cluster D preserve thisedge, too. In order to enable this additional check,the beacon message is extended by additional 36bits as follows. Additional 18 bits are used to pro-vide information about all outgoing edges whichwould be preserved by Gabriel graph constructionin a cluster C. For instance, in Fig. 1, all nodes lo-cated in cluster C2 include the information, thatoutgoing edges C2 ! C1, C2 ! C3, and C2 ! C4

are preserved. The remaining 18 bits are finallyused to disseminate information about preservedincoming edges D ! C within the receiving clusterC, since not all nodes within C will receive thatinformation from a node within the sending clusterD. For instance, in Fig. 1 the nodes v2, and v3 donot receive the information that the incoming edgeC3 ! C2 is preserved in C3 since they are notreachable from v5. In our example, node v2 wouldannounce preserved incoming edge C1 ! C2

(information received from node v1), and node v4would announce edge C3 ! C2 (information re-ceived from node v5). Node v3 will announce noincoming edge, since edge C4 ! C2 is notpreserved in cluster C4. Since all nodes within C2

receive these beacon messages, they are able toconstruct the set of incoming edges {C1 ! C2,C3 ! C2}.

(a) (b)

Fig. 3. Intersection of an edge uv with a regular hexagonresulting (a) in a triangle p1x2p2 and (b) in a square p1x2x3p2.


4.3. Forwarding along cluster links

Once the planar graph has been extracted fromthe aggregated unit disk graph, planar graph rout-ing can be applied in order to traverse the facesintersected by the straight line connecting sourceand destination cluster. In order to perform facetraversal, packets need to be forwarded along thelinks of adjacent clusters.

A device x located inside cluster C and holdinga packet which is addressed to cluster D, will per-form the following action. If there exist at least oneneighbor of x located inside cluster D, the device xwill choose the neighbor y closest to the center ofD and will forward the packet to y. If no suchneighbor exists, then there exists at least one neigh-bor y of x which is connected to a node z insidecluster D. This is due to the fact that there existsan edge between clusters C and D. Thus, thereare at least two connected nodes y and z with y lo-cated in C and z located in D. Since the diameter ofeach cluster is exactly the sending radius of eachdevice, node x will see node y located in the samecluster. Since each node announces in its beaconmessage all clusters reachable from that node,node x may select from its neighbor list the neigh-bor closest to the center of D which is also con-nected to cluster D.

Upon reaching a cluster C the packet will visitat most one intermediate hop inside C before leav-ing C and traveling to D. Thus, forwarding along acluster link is a loop-free procedure, since the num-ber of hops is limited to 2. Additional, face routingis proved to be loop-free for static planar graphs.Altogether, we have that the GCR algorithm isloop free, since the extracted subgraph is planarwhich is proved in the next section.

Note, forwarding along cluster links is not lim-ited to the nodes minimizing the distance to thecenter as described above. For instance, we mayhave also defined a rule which tries to minimize en-ergy consumption regarding a specific energy met-ric. For e.g. a node x located in cluster C currentlyholding a packet addressed to the neighbor clusterD may calculate an energy-optimal path to a nodewithin cluster C being connected to D and forwardthe packet along that path. Node x is always ableto calculate such a path, since node x knows the

positions and the clusters seen by all other nodeswithin cluster C. The last node from that energyoptimal path receiving the packet will finally per-form the hop into the next cluster D. Again, the‘‘best’’ (regarding energy metric) neighbor locatedinside D may be selected. Finally, the same proce-dure will be performed when the packet arrives atcluster D. At the moment we are, however, inter-ested in reducing the hop count from source todestination and we implemented the forwardingalgorithm ‘‘maximizing progress’’ as describedabove.

5. Discussion of planar graph construction

We will now show that the subgraph of theaggregated unit disk graph produced by theCGR algorithm is always planar. Additionally,we show that there are connected unit disk graphs,for which no connected planar subgraph of theaggregated unit disk graph can be constructed.Thus, also the localized method used by GCRmight produce a planar graph which is discon-nected even if the unit disk graph is connected.

Lemma 1. Let V be a finite set of points in R2. If

two edges uv and wx from unit disk graph UDG(V )

are crossing, then one of the four nodes is connected

to all other three nodes.

Proof. See [18]. h

Lemma 2. Let V be a finite set of points. Supposethere exists an edge uv of the unit-disk graph

UDG(V ) intersecting a regular hexagon H. If p1and p2 are the intersection points and the intersec-

tion results in a triangle (see Fig. 3(a)) then u and

(a) (b)

(c) (d)

Fig. 4. Possible intersections between edges of an aggregatedunit disk graph. If the intersection point is different from theedge end points, the different cases are intersection between (a)two long edges, (b) long and medium edges, (c) two mediumedges, and (d) medium and short edges.


v are connected to all nodes lying inside triangle

p1x2p2. If the intersection forms a square (see Fig.

3(b)) then u and v are also connected to all nodes

lying inside square p1x2x3 p2.

Proof. Let \uvw define the angle 6p betweenthe two lines vu and vw originating from v

and the unit disk with center u and radius 1 bedefined as U(u). (a) Since the hexagon is regular,the angle \p1x2p2 ¼ \x1x2x3 ¼ 2p=3 P p=2. Itfollows that the triangle p1x2p2 is contained inU(p1,p2) � U(u,v) � U(u) \ U(v). Thus, each nodelying in triangle p1x2p2 is within transmissionrange of node u and v. (b) The angle \p1x2p2 P\x1x2x4 ¼ p

2, thus, triangle p1x2p2 is contained in

U(p1,p2). Analogous, triangle p1x3p2 is containedin U(p1,p2) and we have that square p1x2x3p2 iscontained in U(p1,p2), since the square edges p1,p2, x2, and x3 are all inside U(p1,p2). It follows sim-ilar to (a), that each node lying within squarep1x2x3p2 is connected to node u and v. h

Theorem 1. Let H(V ) be the aggregated graph of

the unit disk graph defined on a finite set of points

V. For each edge uv and wx in H(V ) the following

holds. If uv intersects wx in one point different from

u, v, w and x, then at least one of the four nodes is

connected to all other ones.

Proof. By investigating the finite number of possi-ble intersections between short, medium and longedges of H(G), we get despite all symmetric casesthe following four cases depicted in Fig. 4. Theedges AB and CD depicted in Fig. 4(a)–(d) areconnecting geographical clusters A, B and C, D,respectively. Each edge AB and CD results fromat least one pair of connected devices a, b and c,d which are lying in cluster A, B, C, and D, respec-tively. Due to Lemma 1 an intersection betweenedges ab and cd results in one cluster node A, B,C or D being connected to all other cluster nodes.Thus, we have to investigate the depicted cases (a)–(d) only for the case that node edges ab and cd donot intersect. (a) Suppose ab and cd do not inter-sect, and without loss of generality both c and dare lying left from edge ab. Thus, edge ab intersectscluster D forming a triangle and node d is lyinginside that triangle. It follows by Lemma 2 that d

is connected to a and b. Thus, cluster D is con-nected to all other clusters A, B and C. (b) Supposec and d are lying left of edge ab (c lying right of abis impossible). In that case, ab intersects cluster Dby forming a square and d is lying inside thatsquare. By Lemma 2 we have again, that node d

is connected to node a and b and it follows, thatcluster D is connected to all other clusters A, Band C. The case that ab is lying left/right of cd isproved the same way as done in (a). Finally, cases(c) and (d) are handled the same way as done for(a) and (b). h

There are five (despite symmetric cases)additional possible intersections with long edges,if the intersection point is allowed to be oneof the endpoints of the intersecting edges (seeFig. 5).

Theorem 2. Let H(V ) be the aggregated graph ofthe unit disk graph defined on a finite set of points V.

For case (a) from Fig. 5 there exists an additional

edge between cluster B and C. For the cases (b) and(c) there exists always at least one additional edge

between C and A or C and B. Finally, there exists a

node placement, which results in case (d).

(a) (b)

(c) (d)

Fig. 5. Possible intersections with a long edge where theintersection point is one edge endpoint. (a) Infinite intersectionpoints of two collinear long edges. (b) Two intersecting longedges or an intersection between a long and a medium edge. (c)An intersection between a long and a short edge. (d) Infiniteintersection points between collinear long and short edges.


Proof. The edges AB and CD depicted in Fig.5(a)–(d) are connecting geographical clusters A,B and C, D, respectively. Each edge AB and CD

results from at least one pair of connected devicesa, b and c, d which are lying in cluster A, B, C, andD, respectively. (a) In order to reach node a lyinginside cluster A, node b must be located in the greyarea of cluster B, while the width of that area is

(sending radius––height of a cluster) = 1�ffiffi34

q.

The same applies to node c in order to reach noded, i.e. node c is located in the grey area of cluster C.Thus, node b and c are connected, since all nodeslying inside the grey area of B and C are con-nected, and it follows that there exists an edgebetween cluster B and C. (b) and (c) Suppose clus-ter C is neither connected to cluster A nor to clus-ter B. Since cluster A and B are connected, therehas to be with the same argument applied in (a)at least one node a inside the grey area of A andone node b inside the grey area of B. Cluster A

and C are not connected, thus, the grey area ofC must be empty. Since C and D are connected,there exists at least one node located in C which

is consequently located in the white area of C.It follows, that cluster B and C are connectedwhich contradicts the assumption. (d) Finally,the configuration depicted in (d) is possible sincea node located in B might see a node in A butnot all nodes in C (e.g. a node located at the crossinside B and with the depicted dashed sendingradius). h

Obviously, the planar graph construction algo-rithm used by GCR includes an edge CD if andonly if the local Gabriel graph condition holds inboth clusters C and D, i.e. despite C and D all clus-ters seen by C and all clusters seen by D are out-side the disk U(C,D). Due to this symmetricproperty of the algorithm, each node has the samelocal view on the extracted subgraph, since eachnode within cluster C will add an edge CD onlyif the nodes within cluster D would add this edge,too. However, it remains to show that the resultingsubgraph is also planar.

Theorem 3. Let H(S) = (V,E) be the aggregated

graph of the unit disk graph defined on a finite set of

points S. The subgraph G of H(S) produced by the

planar graph construction method used by GCR is

planar.

Proof. Fig. 4 depicts (despite symmetric cases) allpossible intersections where the intersection pointof two crossing edges AB and CD is no edge endpoint, and due to Theorem 1 we know that thereis at least one cluster being connected to all others.For case (a) suppose that without loss of generalitycluster A is connected to all others. It follows, thatcluster A will remove edge AB since U(A,B) alsocontains neighbors C and D. Due to the symmetryof the graph construction described above clusterB will thus also remove edge AB and it follows thatthe extracted subgraph contains no crossing of twolong edges where intersection point is differentfrom A, B, C, and D. For case (b) three differentsubcases have to be considered. If A is connectedto all others (B connected to all others is a sym-metric case), edge AB will at least be removed byA and B, since A sees cluster D lying in U(A,B).Similar, edge CD will be removed when C or D

is connected to all other clusters, since in bothcases the neighbor clusters A and B are contained


in U(C,D). For case (c) we have to investigate sub-case A connected to all and subcase B connected toall (the other two cases are symmetric). If A is con-nected to all others, at least edge AB is beingremoved, since A also sees cluster D withinU(A,B). If B is connected to all others, AB isremoved due to the same argument. Finally, case(d) has despite symmetry two subcases, A con-nected to all others and C connected to all others.For the first subcase at least edge AB is beingremoved, since C and D are contained inU(A,B). In the second subcase C connected to allothers, thus, cluster B will see cluster C and willthus remove edge AB, since C is contained inU(A,B).

It remains to show that all possible intersectionswhere the intersection point is one of the edge endpoints will also be removed by the graph con-struction method described above. These remain-ing possible intersections are depicted in Fig. 5(despite symmetric cases). For case (a) we knowdue to Theorem 2 that B and C are connected, thusedge AB will be removed due to Gabriel graphconstruction in B and CD will be removed due toGabriel graph construction in C. By Theorem 2 weknow that for cases (b) and (c) there exists at leastan edge between A and C or B and C. In both casesthe edge AB will be removed (either by cluster A orby cluster B). Finally, for case (d) we have thatcluster A will remove edge AB, since C is containedin U(A,B).

Altogether, we have that none of the possibleintersections between edges from the originalgraph remains in the extracted graph producedby the algorithm described above. Thus, theextracted graph has no intersecting edges and isthus planar. h

Localized Gabriel graph construction will al-ways produce a connected subgraph if it is appliedon a connected unit disk graph. However, theaggregated graph is no unit disk graph and it turnsout that the described method will of course al-ways produce a planar subgraph, but the resultingplanar graph might be disconnected, even whenthe original unit disk graph was connected. For in-stance, by Theorem 2 we have that there exists anode configuration resulting in an aggregated

graph like depicted in Fig. 5(d). The method ap-plied by GCR will remove edge AB since A is alsoconnected to C and C is within the circle U(A,B).The resulting subgraph ({A,B,C},{AC}) is obvi-ously planar but not connected. In particular, thisexample applies not only to the localized planargraph construction method used by GCR, butillustrates in general a conflict when extracting aplanar graph from the aggregated unit disk graphsince both edges AB and AC may not be removedif connectivity has to be maintained.

In order to provide connectivity within a clus-ter, the cluster diameter has to be at most the send-ing radius of each device. The proof of correctnessof the planar graph construction method used byGCR needs the unit disk graph assumption andthe fact that there exist only three possible edgelengths between connected clusters, short, med-ium, and long edges. However, one can easily ob-serve that any cluster diameter greater than 2ffiffi

7p r

will always lead to edges between connected clus-ters having a length of

ffiffi3

p

2r, 3

2r, or

ffiffiffi3

pr only. Thus,

in general we have that the cluster diameter is notrestricted to be exactly the sending radius r, butmay be one fixed value within the interval ð 2ffiffi

7p r; r�

in order to enable GCR to extract the planargraph.

6. Quantitative analysis

We performed a quantitative analysis of GCRusing the simulation environment introduced in[19]. The simulations were carried out in staticnetwork scenarios constructed according to themethod described in [10]. Let n nodes be placedrandomly on a unit square. In order to controlthe average degree between those network nodes,all possible nðn� 1Þ=2 edges between node pairsare sorted by their length in increasing order.The common sending radius r producing an aver-age degree of d corresponds to the length of thend2th edge in the sorted order. Any such random

graph being disconnected is rejected.At the moment we are interested in reducing the

number of hops a packet will travel from source todestination, thus, the performance is measured inaverage dilation [10] which is the average ratio of


hop count produced by the investigated routingalgorithm and the hop count produced by theDijkstra single source shortest path algorithm.We determined the average dilation of GCR fornetworks consisting of 100, 200, and 300 nodesand an average degree varying from 4 to 20 neigh-bor nodes. Each measuring point results from 1000independent simulated (network, source, destina-tion) triples. We observed almost the same curveprogressions for the simulations conducted forthe different node counts and decided to presentthe simulation results for 200 nodes only.

Fig. 6 depicts the average dilation of GCR as afunction of network degree. In order to judge theperformance of GCR, we compared the averagedilation to the original face routing algorithmFACE [10] and its improvements, the concept ofinternal nodes FACE-I, shortcut based routingFACE-S, and both improvements applied on facerouting FACE-I-S [16]. One can observe thatGCR tends to be the best algorithm when networkdegree is increasing. In our simulations GCR per-formed always better than FACE, outperformedFACE-I, when average degree was at least 6 neigh-bor nodes, and finally outperformed FACE-I-S

1

2

3

4

5

6

7

8

9

10

11

12

4 6 8 10 12

Fig. 6. If planar graph traversal is used on its own, GCR outperformincreased for uniformly distributed network nodes.

when average degree was greater than 10. Finally,one can observe that performance gain in sparsenetworks results mainly from the internal nodesconcept, while shortcut procedure has its main im-pact in dense networks (e.g. in Fig. 6 performancequality of FACE-I and FACE-S is interchangedbetween average degree 6 and 7). This also ex-plains the decreasing performance of GCR whenthe network tends to a sparse average node degree,since performance gain of GCR is rather related toshortcut procedure than internal nodes concept.

The main application of planar graph routing isits combination with distance-based greedy rout-ing. A packet arriving at a concave node isswitched into recovery mode and routed alongfaces until reaching a node closer to the destina-tion than the position of the concave node wheregreedy routing failed. At this node routing is per-formed in greedy mode again.

From our perspective greedy routing failuresmay appear for two possible reasons. Networkdensity has an impact on the success of greedyrouting. In sparse networks concave nodes aremore likely, thus, success of greedy routing will de-crease when average degree is reduced. The other

14 16 18 20

GCRFACE

FACE-IFACE-S

FACE-I-S

s all existing face routing methods when average node degree is

1

2

3

4

5

6

7

8

9

10

11

4 6 8 10 12 14 16 18 20

GCRFACE

FACE-IFACE-S

FACE-I-S

Fig. 7. Any recovery procedure has a decreasing impact on path lengths produced by greedy routing when average degree of uniformlyplaced network nodes is increased.


reason for greedy routing failures which may occurindependently of the network degree are concavenetwork boundaries and obstacles in the surfacewhere network nodes cannot be placed. For in-stance, even when average degree is high, a packetarriving at an obstacle may need a recovery proce-dure in order to circumnavigate that obstaclebefore it can be handled in greedy mode again.The same holds for packets arriving at a concavenetwork boundary.

We examined the impact of GCR, FACE,FACE-I, FACE-S, and FACE-I-S when they areused to recover from both classes of greedy routingfailures, i.e. failures due to a sparse network degreeand interrupted greedy routing at an obstacle. Thelatter scenario was constructed by randomly plac-ing network nodes on a unit square but rejectingeach node position located in the middle third ofthat square. The average degree is set accordingto the algorithm described above. The source desti-nation pair of each routing task is selected that way,that greedy routing will always be interrupted bythe network hole in the middle of the unit square.

Fig. 7 depicts the simulation result for greedyrouting failures due to low average degrees. GCR

performs better than the original FACE algo-rithm. However, its performance is close toFACE-S only, and FACE-I and FACE-I-Sachieve better performance in that scenario. Thiscan be explained by the fact, that the greedy rout-ing failure results at concave nodes due to lownode degree and from the simulations conductedfor Fig. 6 we know that the main performance gainfor sparse network degrees can be achieved by theinternal nodes concept (this observation was alsoobtained in [16] where face routing on internalnodes was introduced). Conforming to the obser-vation in [10] one can finally observe that in densenetworks with uniformly distributed networknodes, routing is performed mainly in greedy modeand the recovery strategy has almost no impact.

It holds that internal node concept is a goodstrategy to recover from greedy routing failuresin sparse networks, however it turns out that per-formance degrades when network degree is notminimal and greedy routing is interrupted due toobstacles or a concave structure of the networkboundary. Fig. 8 presents the simulation resultswe obtained by running greedy and recovery strat-egy in a network scenario with the non-uniform


spatial node distribution as described above. Onecan observe that shortcut based routing achieves

1

1.5

2

2.5

3

3.5

4 6 8 10 12

Fig. 8. The recovery strategy applied has a significant impact on patinterrupted at a concave network boundary or at network holes due

82

84

86

88

90

92

94

96

98

100

4 6 8 10 12

Fig. 9. The success rate of GCR in all simulated scenarios, face roucombination and applied in a network with a uniform and a nonunif

a performance gain when network degree is in-creased, while average dilation yet increases for

14 16 18 20

GCRFACE

FACE-IFACE-S

FACE-I-S

h length produced even in dense networks if greedy routing isto obstacles.

14 16 18 20

GCRGCR+Greedy uniform

GCR+Greedy nonuniform

ting applied on its own, and greedy and face routing used inorm spatial node distribution.


internal nodes concept. This holds in particular forFACE-I but can marginally also be observed forFACE-I-S. Since the GCR recovery strategy is re-lated to the shortcut procedure, GCR achieves al-most the same performance values as FACE-S,and FACE-I-S.

From our theoretical results we know that thereexist node placements of connected networks,where only disconnected planar graphs can be ex-tracted from the aggregated unit disk graph. Thus,we were additionally interested in investigating thesuccess rate of GCR, which was measured as thefraction of successful routing tasks over the totalnumber of simulated routing tasks. Fig. 9 depictsthe success rate of GCR when applied on its ownand in combination with greedy routing. Forsparse connected networks (average node degree 4)and a uniform spatial node distribution we ob-served a success rate over 90% for both GCRand the combination of GCR with greedy routing.Delivery rate degrades to 75% for sparse con-nected networks when GCR is simulated in a net-work scenario with obstacles. However, in all casesthe success rate of GCR quickly tends to almost100% when network degree is increased to 8 orabove.

7. Conclusion

This article described a geographic routingalgorithm which adopts the idea of face traversalin planar graphs. In contrast to existing solutionsrouting is not performed on a per node basis butinstead packets are forwarded along the edges ofadjacent geographical clusters. Two geographicalclusters are termed to be adjacent if there is at leastone connected node pair with each node physicallylocated in one cluster. Constructing the graph ofadjacent clusters and extracting a planar graph isachieved locally by using one-hop neighbor infor-mation only. Our theoretical results show correct-ness of the described algorithm but also that thereexist node placements where any extracted planargraph is disconnected, even when the physical net-work is connected. Simulation results show thatsuch node placements will degrade the success rateof GCR in sparse networks, while in networks

with increased average degree of 8 or above suchdisconnections are very unlikely and the successrate of GCR tends to 100%.

We compared the performance of GCR to exist-ing planar graph routing methods which try tominimize the number of hops per routing task.When used on its own, GCR outperforms all exist-ing solutions when network degree is increased. Inorder to compare the quality of GCR as a greedyrecovery strategy, we figured out that greedy rout-ing failures are possible for two reasons. From oursimulation results we conducted that network den-sity has almost no impact on greedy routing fail-ures when average node degree is greater than 9.However, when greedy routing occurs at networkholes due to obstacles or a concave networkboundary, the average node degree has an impacton the recovery strategy being used, and it turnsout that GCR applied in dense networks out-performs the existing solutions based on 1-hopneighbor information, while its performance iscomparable to those algorithms which achieverecovery by 2-hop information, i.e. in contrast toGCR for these methods the size of each beaconmessage increases linearly with the average nodedegree.

8. Future research directions

We believe that planar graph routing alongadjacent geographic clusters may be the key tosolve some problems that arise from planar graphrouting in wireless networks. It is a known factthat mobility has an impact on the success rateof face routing algorithms and in particular mobil-ity caused loops which are possible during face tra-versal need to be concerned in existing solutions.We investigated for mobile network scenarios thatthe graph structure obtained by geographical clus-tering remains more stable than extracting a pla-nar graph from the network graph directly. Inour ongoing research we will figure out if GCRis a good choice when network nodes get mobile.

An additional interesting question arises fromrobustness. The known solutions which make facerouting also possible beyond the unit disk graphassumption introduce virtual edges which might


produce arbitrary long paths between the twonodes connected by a virtual edge if network nodescan be placed arbitrarily. The cluster centers usedby GCR have fixed positions, and it is an interest-ing question if the regular structure obtained bygeographical clustering might be used to make facerouting robust without that additional harmfulproperty.

At the moment we focused on reducing the num-ber of hops produced by GCR. However, our algo-rithm is the first recovery strategy based on planargraph traversal where next hop selection can be anynode in forward direction (even shortcut procedureis restricted to the nodes lying on the path obtainedby face traversal). Thus, it remains an interestingquestion to investigate performance of GCR whenother metrics are applied to forward messagesalong cluster links (e.g. energy-aware metrics).

The number of different edge lengths betweenadjacent clusters depends on the sending radiusand cluster diameter. At the moment we use a clus-ter diameter which provides three different edgelengths only. This fact is used in order to provecorrectness of planar graph construction. How-ever, we suppose that correctness can even beproved for any cluster diameter below the sendingradius.

From our theoretical results we know that thereare node placements where any extracted planargraph resulting from the edges of node aggregationwill be disconnected. Although we investigated bysimulation that such node placements occur infre-quent in dense networks we are investigating ifintroducing additional virtual edges may resolvethose conflicts.

Finally, we observed a potential performancegain of GCR in sparse networks. The concept ofinternal nodes may also be applied on the clusterlevel. This may reduce the number of unnecessarytraversed edges we investigated in our simulationsfor sparse networks.

Acknowledgment

This work is funded in part by DFG SPP1140and Microsoft Research Embedded Systems IFP(Contract 2003-210).

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Hannes Frey is a PhD candidate withinthe System Software and DistributedSystems research group at the Uni-versity of Trier in Germany. He isworking as a full time researcher forthe German DFG research projectSPP1140. His research interestsinclude wireless networks and mobilecomputing, ad-hoc network routingprotocols, topology control, and mid-dleware support.

Daniel Gorgen is working as PhDStudent for the System Software andDistributed Systems workgroup at theUniversity of Trier in Germany. He isa full time researcher for the DFGSPP1140 research project. His mainresearch topics are middleware andapplications for mobile multihop ad-hoc networks.

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