A New Localized Geometric Routing withGuaranteed Delivery on 3-D Wireless Networks

Jun Duan∗, Donghyun Kim‡, Wenping Chen∗, and Deying Li∗†∗ School of Information, Renmin University of China, Beijing 100872, China.

† Corresponding author. E-mail: deyingli@ruc.edu.cn‡ Department of Mathematics and Computer Science, North Carolina Central University, Durham, NC 27707

E-mail: donghyun.kim@nccu.edu

Abstract—Recently, geometric routing has emerged as anefficient routing strategy on wireless networks. An ideal geometricrouting is memoryless and does not suffer from the drawbacksof traditional proactive/reactive routings. All existing geometricroutings on 3-D wireless networks either work deterministicallyonly on the networks with special properties or do not guaranteedelivery. In this paper, we divide the memoryless requirementinto two sub-requirements, node-memoryless-ness and message-memoryless-ness. Then, we propose a new node-memorylessgeometric routing, which is still free from the drawbacks oftraditional routings. Our algorithm partitions the 3-D space withregular cubes and converts the routing problem over nodes into arouting problem over cubes. With minimal information attachedto the header of a message, our algorithm deterministicallydelivers a message to its destination in any connected 3-D wirelessnetworks. The forwarding decision on the message is made in acompletely localized manner. The simulation results indicate thatour algorithm outperforms its competitors on average.

Index Terms—Geometric routing, mobile ad-hoc networks,wireless sensor networks, mobile computing, computational geo-metric, graph theory.

I. INTRODUCTION

Recently, wireless networks such as mobile ad-hoc wirelessnetworks and wireless sensor networks have received hugeattention due to their important applications. Routing algo-rithm is one of the most important primitives to build thewireless networks. Since the topology of a wireless networktends to be highly dynamic, designing an efficient routingstrategy on wireless networks is a challenging task. Largely,most existing routing algorithms on wireless networks canbe classified into reactive routing algorithms and proactiverouting algorithms [1]. Reactive ones require on-demand-basisflooding for route discovery and proactive ones need to updatetheir routing table each time topology changes. Consequently,both of them are quite demanding on resource restrictedwireless networks.

Geometric routing1 is a routing strategy on wireless net-works in which the forwarding decision of a node on amessage is made solely based on the geographical positions(which can be obtained through a location service [2], [3] ora hardware such as global positioning system (GPS)) of thenode, its neighbors, and the source and destination of the mes-sage. Ideally, a geometric routing should be memoryless, which

1Geometric routing is also known as “geographic routing” [5]. This paperwill refer it as “geometric routing” for unification.

means a wireless network do not remember the messages it hasseen [1]. Due to the minimal operation and maintenance costs,it is widely believed that geometric routing is a promisingrouting strategy on wireless networks.

The most popular geometric routing strategy is greedy for-warding such as distance-greedy [4] and direction-greedy [5],where a message is forwarded to the neighboring node geo-graphically closest to the destination, until it reaches at thedestination. While greedy forwarding is simple and efficient,it suffers from the local-minimum (i.e. dead end) problem, inwhich a message may reach at a node all whose neighboringnodes are no closer to the destination than the node, and thusthe routing fails [6]. On a 2-D wireless network, face routing,which utilizes the planar graph (e.g. a Gabriel Graph) inducedfrom the network, and its variations are frequently used to dealwith the local-minimum problem [5], [6], [7], [8], [9], [10],[11], [12].

Nowadays, wireless networks are also deployed on variousenvironment such as underwater, space, and air, and thus 3-D wireless networks attract lots of attentions. In these cases,the topology of the wireless networks can be modeled usinga 3-D graph such as unit ball graph (UBG) rather than a 2-Dgraph model such as unit disk graph (UDG). Unfortunately,3-D wireless networks do not have the concept of planartopology unlike its 2-D counterparts. As a result, the facerouting strategy does not apply to 3-D wireless networks.In [13], the authors proved that with a fixed k, there isno deterministic memoryless k-local2 geometric routing algo-rithm that guarantees delivery on 3-D wireless networks. Sofar, all existing geometric routing algorithms on 3-D wirelessnetworks guarantee delivery only in some graphs with specialsubstructures or conditions [13], [14], [15], or simply do notguarantee delivery [1], [16].

The main idea of this paper is that we divide the mem-oryless requirement of ideal geographic routing [13] intotwo sub-requirements, node-memoryless-ness and message-memoryless-ness. In detail, we define a geographic routing isnode-memoryless only if no node in the system records theinformation of the messages it has seen so far, and message-

2A geometric routing algorithm is “k-local” if a forwarding decision of anode on a message is made solely based on the geographical information ofat most k-hop neighbors for some constant k, itself, and the destination ofthe message.

978-1-4673-1544-9/12/$31.00 ©2012 IEEE

memoryless only if a message does not keep the information ofthe nodes it has visited so far. Based on this idea, we proposea new localized node-memoryless geometric routing algorithmwith guaranteed delivery, which has following two outstandingcharacteristics.(a) Our algorithm is node-memoryless and localized, which

means that the only information each node needs to keepis the geographical information of its direct neighbors.

(b) Each message is guaranteed to reach at its destination inany connected 3-D UBG. A message encountered a local-minimum will be rescued and will continue moving towardthe destination.

In our algorithm, a forwarding decision of a node fora message is based on the positions of the neighbors ofthe node, the source and destination of the message, andsome extra information attached to the message. There isno need for on-demand flooding (like reactive routing) orfrequent knowledge update (like proactive routing), except onebroadcasting to initialize our algorithm. Meanwhile, the sizeof extra information attached to the message is no greaterthan 2 × n + c bits, where n is the size of a given networkand c = �log2 n�. That is, if n = 8192, then the size ofthis extra information is only 2013 bytes. In conclusion, ournew geometric routing algorithm on 3-D wireless networksis very efficient and practical. Our simulation result indicatesthat our algorithm outperforms the competitors on average (seeSection IV for details).

The rest of this paper is organized as follow. Section IIoverviews several related works. We introduce our new de-terministic geometric routing algorithm on 3-D wireless net-works in Section III. The simulation results are presented inSection IV. Finally, Section V concludes this paper.

II. RELATED WORK

Most of the existing routing algorithms on wireless net-works such as mobile ad-hoc networks and wireless sensornetworks are either reactive or proactive [1]. A reactive routingalgorithm is an on-demand routing scheme which relies onflooding to determine the path between the source and thedestination of a message, and thus the routing path discov-ery phase causes huge overhead [17]. On the other hand,a proactive routing scheme predetermines the routing pathand each node maintains corresponding information within itslocal routing table. While this strategy drastically reduces theburden of flooding in reactive routing, the information in therouting table has to be updated each time the topology of anetwork changes. Therefore, proactive routing tends to be veryinefficient in wireless networks with highly dynamic topology.

An ideal (memoryless) Geometric routing enables each nodeto make a forwarding decision on a message solely based onthe geographical positions of the node, its neighbors, and thesource and destination of the message. It is widely knownthat such memoryless geometric routing is highly efficientin the resource-restricted wireless networks with dynamictopology. The most popular geometric routing strategy isgreedy forwarding, in which a message is forwarded to the

neighboring node geographically closest to the destination,until it reaches at the destination. The geometric routing al-gorithms based on distance-greedy [4] or direction-greedy [5](i.e. Compass routing I) are very simple yet efficient. Theyare also applicable on both 2-D and 3-D wireless networks.However, greedy forwarding suffers from the local-minimumproblem (i.e. dead end), in which a message may reach at somenode all whose neighborhoods are no closer to the destination,and thus the routing fails [6]. In case of Compass RoutingI [5] and the greedy path vector face routing (GPVFR) [10],the geometric routing algorithms may fail to deliver messagesdue to routing loops [9].

On a 2-D wireless network, face routing, which uses theplanar graph (e.g. a Gabriel Graph) induced from the network,and its variations are frequently employed to deal with thelocal-minimum problem [5], [7], [8], [6], [9], [10], [11], [12].For example, the authors in [5] proposed a face routing strat-egy known as Compass Routing II. In the greedy-face-greedyapproach [7], [18], which is a hybrid of greedy forwarding andface routing, a message is routed in a greedy manner until itconfronts a local-minimum. Once stuck in the local-minimum,the message is forwarded using face routing to find anothernode closer to the destination. In [11], [19], the authors haveproposed another hybrid geometric routing algorithms whoseaverage performances are improved. Other than those, thereare several hybrid routing approaches in the literature [6], [9],[10], [12]. As identified in [9], however, some of the hybridroutings, such as [6] and [11] do not guarantee data delivery asdone by Compass Routing II under some specific conditions.

Unlike its 2-D counterparts, 3-D wireless networks do nothave the concept of planar topology. Therefore, the facerouting-like algorithms do not work on 3-D wireless networks.Due to the reason, several geometric routing algorithms specif-ically designed for 3-D wireless networks are appeared inthe literature [1], [13], [14], [15], [16]. Most of them usea greedy forwarding strategy on 2-D wireless networks as abasis to design an efficient geometric routing algorithm on3-D wireless networks. Unfortunately, all of them guaranteedelivery only on 3-D networks with some specific topology orconditions.

In [13], the authors proved that with a fixed k, there is no k-local geometric routing algorithm that guarantees delivery on3-D wireless networks, and proposed a deterministic k-localgeometric routing on some graphs with special conditions.In [1], the authors have proposed a randomized approachwhich is the combination of the greedy forwarding with arandom walk strategy to recover from a local-minimum. Thisrouting algorithm is memoryless and makes its forwardingdecision only based on O(1) neighbors information. How-ever, this algorithm does not guarantee delivery due to therandomness. In [16], the authors proposed an improvementof greedy forwarding on 3-D wireless networks. Naturally,their algorithm still suffers from the local-minimum problem.Each of the geometric routings on 3-D wireless networks byLiu and Wu [14], and Duan et al. [15] assume 3-D wirelessnetworks with specific topology (polygon structure, Delaunay

Triangulation, etc.) and thus does not guarantee delivery onrandom 3-D wireless networks.

In [20], the authors introduced a geometric routing forunderwater wireless sensor networks, which is also 3-D wire-less networks. They simplified the local-minimum problemby assuming that the routing in such network will be froma sensor node at the bottom to the sink on the surface ofthe water. Then, they proposed an algorithm which alwayspicks the cluster of nodes with the highest expected packetadvance(EPA). Since this work considers a special networkin which communication is from bottom to up, this algorithmdoes not apply to random 3-D wireless networks.

III. MAIN CONTRIBUTIONS

In this paper, we assume each node has a uniform maximumtransmission range, rmax. We use rmax as the unit distanceand construct an UBG of a given 3-D wireless network toabstract its topology as done in the other related literatures [1],[13], [14]. In the rest of paper, G = (V,E) is an UBG, whereV = V (G) is the set of nodes and E = E(G) is the setof edges. We also assume that a given network is connected,otherwise there might be no routing path between some pairof nodes independent from the routing algorithm in use.

In this section, we introduce a new localized geometricrouting with guaranteed delivery on random 3-D wirelessnetworks, which are connected and homogeneous (i.e. samemaximum transmission range). It is proven that a memorylessk-local geometric routing with guaranteed delivery does notexist on 3-D wireless networks for any integer k ≥ 1 [13].Therefore, so far, all of the existing geometric algorithmseither 1) are memoryless but non-deterministic or 2) guaranteedelivery only on wireless networks with special properties.Different from those, our approach is deterministic, localized,and node-memoryless, but not message-memoryless. That is,by embedding some additional information on a message beingforwarded, each node can locally make a forwarding decision,and eventually the message will be delivered to the destination.

Given an UBG of a wireless network, our geometric routingbegins by partitioning the UBG with imaginary regular cubes.The size of the cubes is carefully determined such that eachnode can learn 1) which cube it is in by itself and 2) whichnodes are in the neighboring cubes. During this process, thenodes will form a 2-layered communication hierarchy whichpreserves the topology of the original UBG. Once a messageis arrived, each node will be able to make a local forwardingdecision. In the following subsections, we will discuss about3-D space partition with regular cubes and the constructionof the communication hierarchy, the structure of additionalinformation on each message, and our localized geometricalgorithm with guaranteed delivery on random connectedUBGs.

A. 3-D Space Partition and 2-Layering of UBG

We first divide the 3-D space containing a give UBG G intocubes with equal size. It is clear that by properly selecting theedge length l of the regular cube, the set of nodes within the

U

D

B

F

L R0

12

34

5

NW

ES

+-

(a) (b)

iv�

v

Fig. 1. (a) The six unit directions of a cube. (b) Forwarding priorities of thesix directions from a cube.

same cube forms a complete subgraph in the original UBG.For example, by selecting l such that

l ≤ rmax√3

, (1)

the distance between any two nodes within such regular cube isat most rmax, and therefore all of nodes within the same cubeare connected in G. Let GL be the union of such completesubgraphs. Now, we consider GU be the set of edges whichare in E(G), but not in GL. Clearly, the edges in GU areconnecting two nodes, each of which is in a different cube.Now, consider a 2-layered communication hierarchy with GH

as upper layer and GL as lower layer. Clearly, the union ofthe two layers still keeps the entire topological structure of theoriginal UGB G of a given network N . Now, we introduce thedetail of the 3-D space partitioning and 2-layering of an UBG.

1) Details: The partition of the space containing a givenUBG G can be done throughout the following two steps.First, one of the nodes becomes an anchor node and randomlydetermines its coordinate in the virtual network space such as(1, 1, 1). Then, it broadcasts a message with this coordinateinformation and its geographical information, which can beobtained by a hardware (e.g. global positioning system) or aprotocol (e.g. location service). Once the message is received,every other node can calculate its own unique coordinate inthe virtual network space. Second, the virtual space will becut into regular cube cells (hexahedrons) with the same edgelength l. Note that each node is within some cube and anedge in G may cross the border between cubes (an edge inthe upper layer). Hereinafter, we will use “cube”, “cube cell”,and “hexahedron” interchangeably.

Now, suppose l, the edge length of the cubes, satisfiesEquation (1). Remind that in a regular cube whose edge lengthsatisfies this inequality, all of nodes within the cube forma complete graph (by the edges in the lower layer). This isbecause the maximum distance between two nodes within thesame cube, body diagonal, is no greater than rmax. We wouldlike to emphasize that

• Property 1-(a): each node does not lose their knowledgeon the topology of G after the 3-D space partition,

• Property 1-(b): any two nodes within the same cube cellcan communicate with each other, and

• Property 1-(c): each node knows which cube it is in.From now on, we notate the unit vectors (with length l)

from the center of the cube to the center of each surface by−→d

112D

113D

111D

121D

211D 311D

123D131D

122D

132D

133D

212D

213D221D

222D

223D231D

232D

233D

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313D321D

322D

323D331D

332D

333D

1a 1b

2a

1c

2c3c

1d

1e

1f

1g2g

1h

1k

1p

1q 2q

1u2u

3u

1w

1z

1m

Fig. 2. The 3-D space is divided into regular hexahedrons, and a 2-layeredcommunication hierarchy of a given UBG is constructed.

(down), −→r (right), −→u (up),−→l (left),

−→f (front), and

−→b (back)

(toward D, R, U , L, F , and B, respectively in Fig. 1(a)).Once the space partition is done, each node v in a cube

cell H can easily learn in which hexahedron each of its 1-hop neighbors stays (e.g. by exchanging a “hello” message).Without loss of generality, we do not consider the case thatnodes are on the boundaries of cubes (this can be preventedby properly adjusting l). Now, suppose l satisfies

l =rmax√

12(2)

instead of Equation (1). Sincermax√

12<

rmax√3

,

all of Properties 1-(a) ∼ 1-(c) are still good with such l. Fur-thermore, this change will bring following another importantfeature: while a node v is still connected to all of nodes withinthe cube H where it stays, v is also connected to all of thenodes within the six cubes adjacent to H .

For example, in Fig. 2, node g2 has four 1-hop neighbors:g1, f1, p1 and m1. Because of the 3-D space partition, g2 andg1 are in the same cube D331, while f1 and p1 are in the D331’sneighbor cell, D231 and D321, respectively. The neighbor m1

stays in a farther cell labeled D333. All the nodes can connectother nodes in the adjacent cubes. Some can even connect thenodes in further cubes, like the edge g2m1.

B. Structure of Extra Information on Message Header

As we mentioned earlier, in our routing strategy, someadditional information is attached to a message being for-warded. In this section, we describe the detailed structure ofthis information.

Suppose s and t are the source and destination nodes of amessage. Based on the geographical locations of s and t, anynode can easily (and completely locally) identify C(s) andC(t) which are the cubes containing each of them (s and t),

L R

B

F

s

t

1D

2D 3D

4D5D 6D

7D 8D 9D 10D

v

Fig. 3. A routing from s to t and the vector records on the status of themessage m in v, being sent from s to t, with Qpst : [�b, �r,�b], Qups : [�r,�b, �r]and Ccur = D5

respectively. Let −→vst be the vector from the center of Cs tothe center of Ct. Note that it is always possible that −→vst canbe expressed using a subset X of the six unit vectors (withlength l),

−→d ,−→r ,−→u ,

−→l ,

−→f , and

−→b .

In our algorithm, a message carries a collection X of thesubsets of X in each of following two queues, passed-vectorqueue Qpst and unused-vector queue Qups. In detail, thetravel from the center of C(s) to the center of C(t) canbe decomposed with a series of movements between twoconsecutive cubes. Each time the message moves from onecube C(u) to anther consecutive cube C(w), −−→vuw is equivalentto one of the vectors in X . Then, we push (add) this vectorto the queue Qpst and the rest in X to Qups (see Fig. 3).In addition to the two queues, we also embed Ccur, the cubecurrently holding the message, into each message m. Note that

• Property 2-(a): any two unit vectors in Qpst are notopposite,

• Property 2-(b): the sum of the unit vectors in Qpst isthe source-destination vector −→vst, and

• Property 2-(c): the unit vectors in Qpst belong to at mostthree 3 directions of the 6 basic directions.

C. A New Localized Geometric Routing with GuaranteedDelivery

The core idea of our routing algorithm is that we maintainsome extra information in two vector queues embedded oneach message. Each node uses this information to make aproper forwarding decision on a message in a completelydistributed and localized manner. In this way, our algorithmis node-memoryless, but is still able rescue a message froma local-minimum, never fall into a loop, and deterministicallydelivers the message to the destination.

Our routing algorithm operates in two forwarding modes: 1)normal forwarding mode and 2) recover forwarding mode. Thefirst mode is for casual greedy routing, and the second mode isfor recovering a message from a local-minimum. Essentially,the second mode is a variation of back-tracking technique.However, by restricting each node to receive the same message

1D

2D

3D

1D 2D

3D

1D 2D

3D 4D

5D

v

vv

ur

ur

urur

rr rr br

(a) (b) (c)

t

tt

w

w

w

Fig. 4. The vector from the center of a node v to the center of another cubecontaining w, a 1-hop neighbor of v, may consist of at most 3 unit vectors.

at most one time, the worst case time and message complexityof our algorithm is O(n2), where n is the size of the node,and thus even in the worst case, our algorithm works veryefficiently. Now, we introduce the detail of each mode andprove the correctness of our approach.

1) Normal Forwarding Mode: Consider a network N towhich the procedures in Section III-A have been applied.When a node v in a cube C(v) receives a message m whosedestination is t, v first checks if v = t or any of its neighborsis t. If so, then we are done. Otherwise, it computes a vector−→p from the center of C(v) to the center of another cubeC−→p , which is closest to C(t) among those whose surface isno further than rmax from v (a signal from v can reach atthis cube). Due to our choice of l (Equation (2)), −→p consistsof at least 1 unit vector. For example, in Fig. 4(a), (b), (c),−→p = −→u + −→u , −→p = −→r + −→u , and −→p = −→r +

−→b + −→u ,

respectively.

Suppose Cvnbr be the set of cubes which contain some 1-hop

neighbors of v except C(v). Once −→p is determined, v needsto find if C−→p contains w, a 1-hop neighbor of v (i.e. check ifC−→p ∈ Cv

nbr.)

• Case 1: C−→p ∈ Cvnbr. In this case, C−→p includes w, a

1-hop neighbor of v. We add one or more unit vectorsin −→p into the passed-vector queue Qpst of the messagem we try to deliver. At the same time, we discard thesame unit vectors from the unused-vector queue Qups ofm. We also change the current cube record Ccur fromC(v) to C−→p (or equivalently C(w).) Then, v forwardsm to w. Observe that our forwarding strategy for thiscase is similar to a regular greedy forwarding. That is,our approach finds the cube nearest to the destinationand sends messages to a node in this cube.

• Case 2: C−→p /∈ Cvnbr. In this case, no such w exists in C−→p ,

which means that v cannot forward any node in C−→p . Dueto our choice of the edge length of each cube l, there isno node in C(v) which has a 1-hop neighbor w′ in C−→p(see Lemma 1). As a result, v is in a local-minimum, andthus the routing algorithm changes its forwarding modefrom normal forwarding mode to the recover forwardingmode below.

2) Recover Forwarding Mode: Once a node v in a cubeC(v) confronts a local-minimum, the recover forwarding modebegins. The node first identifies the four directions which areperpendicular to the newest unit vector −→vi in Qpst. Note that−→vi is a unit vector from the center of some cube C(w) to thecenter of C(v). Let us call these four by candidate directions.For example, in Fig. 1(b), N,W,S,E are four candidatedirections of −→vi . From now on, we assign 0 to 5 (0 is thehighest) as the priorities of the six direction, −→vi , N,W,S,Eand -−→vi . Once the algorithm fell into recover forwarding mode,a node v with the message m should search another next-hopnode to deliver the message to the destination.

(a) Traverse Over Local Cubes: To recover from a local-minimum, we first search the four cubes neighboring toC(w) in the decreasing order (1 to 4) of their directionalpriority. Meanwhile, we skip searching a cube if it isempty (has no node in it) or some node in the cube hasreceived the message m before. There are two possiblecases.

• Case 1: During the search procedure, we may find a1-hop neighbor vpri of w in a cube C(vpri). Then,we stop searching and record unit vector −→vj , whichis from the center of C(w) to the center of C(vpri),to Qpst. We also check if Qups has −→vj . If so, removeone −→vj from Qups. Otherwise, add an opposite unitvector of −→vj to Qups. In addition, set Ccur to the newcube C(vpri), and w forwards m to vpri.Now, m moved from a local-minimum to anothernode, we go back to the normal forwarding mode.However, it is possible that the normal forwardingfails again (since vpri is also a local-minimum). Insuch case, we continue forwarding in recover forward-ing mode. Essentially, recover forwarding mode andnormal forwarding mode will be alternated until weare completely out of a series of local-minima.

• Case 2: It might be possible that we have tried all ofthe four neighboring cubes before and failed. Then,the algorithm will step-back the message m like aback tracking algorithm (see below).

(b) Step-back: The algorithm is stepping-back since itsearched the four candidate directions and failed to escapefrom one or more consecutive local-minima. In such case,the newest vector(s) −−→vnew in Qpst is opposite to the vector−→vi . Then, the message m should be sent back to a node inthe cube whose center is the origin of −−→vnew. At the sametime, the newest vector(s) −−→vnew should be removed fromQpst.

3) Analysis of Our Approach: The face routing algorithmswhich works well on planar graphs do not apply to UBGs, andthus a new strategy is needed to have a deterministic geometricrouting on UBGs. By Lemma 1 and Corollary 1, our algorithmpartitions the 3-D space with regular cubes and converts therouting problem on an UBG into a routing problem over thecubes such that a routing path on the cubes can be convertedto a routing path on the UBG.

Essentially, our algorithm is a hybrid algorithm. That is,normally, it uses a greedy strategy to find a path over the cubes.Once a local minimum is encountered, our approach can visitall the cubes of a network with a deterministic order until it canget out from the local minimum (Theorem 2). With the helpof the extra information in the message header, the algorithmdoes not visit a failed cube twice (a cube tried to use to get outof a local minimum, but did not work), and thus the messagewill be eventually delivered to the destination. Note that ouralgorithm is not completely memoryless but node-memoryless,and thus we did not disprove the results of [13] and [1],which showed that with a fixed k, there is no deterministicmemoryless localized geometric routing algorithm.

Lemma 1. Suppose {v1, v2, · · · , vn} are the set of nodes inthe same cube C. For each vi, 1 ≤ i ≤ n, let Wi be the set of1-hop neighbors of vi in the six cubes adjacent to C. Then,for any 1 ≤ i, j ≤ n, Wi = Wj (i.e. equivalent).

Proof: Remind that the edge-length of a cube C is l ≤rmax√

12. Therefore, the distance between any two nodes inside

C is at most √3 · l2 =

rmax

2,

and the distance between one node in C and another nodeinside a cube adjacent to C is at most rmax. As a result,Wi = Wj for any 1 ≤ i, j ≤ n and this lemma is true.

From Lemma 1, following corollary naturally follows.

Corollary 1. Suppose a node u in a cube C has a messagem whose destination v in another cube adjacent to C. Then,u and v are neighbors.

Based on Lemma 1 and Corollary 1, each cube can beconsidered as a node with at most six neighboring nodes. Asa result, a routing path from the cube with a source of node tothe cube with the destination of a message on the cubes canbe easily converted to the routing path from the source to thedestination over the nodes.

Theorem 2. Once a message encounters a local minimum,our algorithm makes sure that the message is saved from thelocal minimum and never fall into the same local minimumagain.

Proof: Once a message fell into a local minimum at acube C, it searches four possible directions, N,W,S,E inFig. 1(b), perpendicular to the direction it came from theprevious cube C ′ to the current cube. When it checked the fourdirections, if there is another cube Cnext closer than the localminimum or there is another cube whose four neighborhoodsCnext are not checked yet, the algorithm forwards the messageto Cnext. If all of C’s four neighborhoods have been checkedalready and it turned out that non of them are useful, thealgorithm sends the message back to the cube C ′. Now, C ′

replaces C at the beginning of this paragraph and repeatsthe whole paragraph. This whole procedure is essentiallybacktracking. That is, as long as there is a path from asource to the destination, a message will find a node closer

Fig. 5. Our algorithm (RVG) is deterministic and thus success rate ofdata delivery of our proposed algorithm is almost 1.

than the local minimum (or the destination). By not allowingthe message to be forwarded over the same link twice (thisis accomplished with the help of vector queues), we canguarantee that a message is saved from a local minimum andnever fall into the same local minimum again. Therefore, thistheorem is true.

Clearly, our algorithm is a distributed algorithm. The infor-mation to be used to determine which direction the messageshould be sent is attached to the message and moves together.Any node which needs to make a forwarding decision can dothe job based on this massage-attached information and itslocal information.

IV. SIMULATION

We perform two simulations to validate the performance ouralgorithm. We used Matlab and Eclipse for the simulations.We randomly generated connected UBGs in 1000 × 1000 ×1000 unit distance space. We set the transmission range to

√6

unit distance. Each node stores the information of its 1-hopneighbors and the coordinate of itself and randomly generatesmessages.

We first compare the performance of four routing algo-rithms, a greedy routing (GR) (distance-greedy [4]), greedyrouting combined with random walks (GRG) [1], CompassRouting I (CRI) (direction-greedy [5]), and our approach (sayRVG). We assume that the lifetime of each message is 5 sec-ond, and thus if a routing algorithm does not deliver a messagewithin this time period, we assume that the routing algorithmfailed. In Fig. 5, we pick arbitrary sender-receiver pairs and leteach sender to transmit a message to corresponding receiver.To evaluate the scalability of the algorithms, we consider thenetworks with four different scales in which the number ofnodes is 500, 1000, 1500, and 2000. Based on our simulationresult, the successful delivery rate of our approach is higherthan other three approaches. Moreover, from result, we canlearn that unlike hybrid routing approaches, stateless greedyrouting algorithms like GR and CRI does not ensure messagedelivery due to the lack of any recovery mechanism from local-minimum and loop.

Fig. 6. Time cost (average time to deliver a message) of our algorithmis still as low as GRG.

In Fig. 6, we compare the efficiency of the four algorithms.To evaluate the scalability of the algorithms, we vary thenumber of nodes from 300 to 600. In this simulation, 3000sender-receiver pairs are randomly selected and a message isexchanged between each pair. From our simulation, we canlearn that time cost (average time to deliver a message) ofCRI and GR are greater than those of GRG and RVG. Thisis because that there is no recovery mechanism in CRI andGR when a local-minimum encountered. Therefore, routingmethods without any recovery mechanism can hardly beefficient. In conclusion, the results from the two simulationsshow our method is more efficient than GRG.

V. CONCLUSIONS

In this paper, we proposed a new localized geometricrouting algorithm with guaranteed delivery on 3-D wirelessnetworks. We observe the memoryless requirement of idealgeometric routing is very demanding and split the requirementinto two sub-requirements, namely node-memoryless-ness andmessage-memoryless-ness. In fact, node-memoryless-ness issufficient for a geographic routing to be free from the draw-backs of traditional routing algorithms on wireless networks.Based on this key idea, we attach a minimal meta data on themessage header. The size of this extra data is very small andis manageable in practice. With the extra data on the messageheader and the kind of information only allowed to use inan ideal geometric routing algorithm, each node can make aforwarding decision on a message in a completely localizedmanner, and the message is guaranteed to be delivered in anyconnected 3-D wireless networks. In simulation, we show ouralgorithm outperforms its competitors on average.

ACKNOWLEDGMENT

This research was jointly supported in part by NationalNatural Science Foundation of China under grants 61070191and 91124001, the Fundamental Research Funds of RenminUniversity of China under grant 10XNJ032, and ResearchFund for the Doctoral Program of Higher Education of ChinaNo. 20100004110001. This work was also supported in partby US National Science Foundation (NSF) CREST No. HRD-0833184 and by US Army Research Office (ARO) No.W911NF-0810510.

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