Trace-Routing in 3D Wireless Sensor Networks: A Deterministic Approach with Constant Overhead

Transcription

1 Trace-Routing in 3D Wireless ensor Networks: A Deterministic Approach with Constant Overhea u Xia Cisco ystem Inc. 170 W Tasman Dr an Jose, CA suxia.ull@gmail.com Hongyi Wu The Center for Avance Computer tuies University of Louisiana at Lafayette Lafayette, LA wu@cacs.louisiana.eu Miao Jin The Center for Avance Computer tuies University of Louisiana at Lafayette Lafayette, LA mjin@cacs.louisiana.eu ABTRACT We propose a istribute an eterministic routing algorithm with constant storage, communication an computation overhea, ubbe trace-routing, for strong-connecte 3D wireless sensor networks. Its basic iea is to construct a virtual cutting plane that intersects bounary surface to yiel a trace, along which a routing path with guarantee elivery can be establishe. We prove the correctness of trace-routing uner both continuous an iscrete settings. We implement the trace-routing algorithm on Crossbow sensors an carry out extensive simulations to evaluate its routing efficiency. Categories an ubject Descriptors C.2.1 [Computer ystems Organization]: Computer-Communication Networks Network Architecture an Design Keywors Wireless ensor Networks; 3D; Routing 1. INTRODUCTION This work aims to achieve DeterminItic routing with Constant Overhea (or DICO routing) in the emerging three-imensional (3D) wireless sensor networks [1 12]. DICO is highly esire ue to the stringent resource constraints on iniviual noes an the often-neee large-scale eployment of 3D wireless sensor networks. Constant overhea signifies the storage, communication an computation require for routing are boune by a constant at each sensor noe, while eterministic routing means a routing path can be etermine without ranom search. The cost of employing DICO routing is the potentially sub-optimal routes, which are, however, tolerable in sensor networks where ata traffic is light. The conventional table-riven routing is obviously non-dico, because the size of a routing table grows with the size of the network. Most on-eman routing algorithms evelope for mobile a hoc networks result in non-constant communication overhea for route iscovery an thus are not DICO either. On the other Permission to make igital or har copies of all or part of this work for personal or classroom use is grante without fee provie that copies are not mae or istribute for profit or commercial avantage an that copies bear this notice an the full citation on the first page. Copyrights for components of this work owne by others than ACM must be honore. Abstracting with creit is permitte. To copy otherwise, or republish, to post on servers or to reistribute to lists, requires prior specific permission an/or a fee. Request permissions from permissions@acm.org. MobiHoc 14, August 11 14, 2014, Philaelphia, PA, UA. Copyright 2014 ACM /14/08...$ han, while the sensor network protocols that aggregate ata from sensors to sink(s) are preominantly DICO, they o not support generic communication between any peers in the network. The earliest eneavor to achieve DICO in large-scale peep-to-peer wireless sensor networks is geometric routing [13 20]. 1.1 Overview of Geometric Routing We begin our iscussion on geometric routing in 2D wireless sensor networks. Given the practical limit on raio range, the communication graph of a large-scale sensor network often well preserves its geometric characteristics. In geometric greey routing, a noe always forwars a packet to one of its neighbors closest to the estination of the packet. Geometric greey routing is istribute with constant storage, communication an computation overhea. However, geometric greey routing itself oes not guarantee elivery of ata packets, ue to the presence of local minima in the network. A noe is calle a local minimum if it is not the estination but closer to the estination than all of its neighbors. Clearly, geometric greey routing fails at local minima. As to be iscusse in ec. 3, local minima may appear at either internal or bounary noes. The former can be resolve by using local information within two hops. Therefore, most geometric routing algorithms focus on the latter, largely following two strategies: face routing [13 20] an greey embeing [21 27]. Face routing an its alternatives an enhancements are base on the fact that a voi in a 2D planar network is a face with a simple line bounary. Thus a local eterministic algorithm can be employe to search the bounary in either clockwise or counter-clockwise irection to guie the packet out of the local minimum as illustrate in Fig. 1(a). Greey embeing aims to embe a sensor network into a omain without local minima such that geometric routing uner such embeing is always successful. 1.2 Challenges in 3D Geometric Routing While increasing space imension appears irrelevant to network communication protocols at the first glance, surprising challenges are observe in efforts to exten geometric routing techniques from 2D to 3D. First, none of the greey embeing algorithms in the literature [21 27] can be generalize to 3D sensor networks. At the same time, the bounary of a voi in a 3D network is no longer a close curve or line segment, but a surface, renering face routing infeasible. More specifically, in contrast to the simple line bounary in 2D that can be searche eterministically, there exist an arbitrarily large number of possible paths on the bounary surface to be explore in orer to escape from a local minimum in 3D (as shown in Fig. 1(b)). As a matter of fact, a recent fining shows that there

2 D trace voi? D D? voi voi P (a) Face routing succees in 2D. (b) Face routing fails in 3D. (c) Propose trace-routing. Figure 1: Exposition of challenges to extening face routing from 2D to 3D an illustration of the propose trace-routing algorithm. (a) Face routing succees in a 2D planar network where a local eterministic algorithm can be employe to search the bounary in clockwise or counter-clockwise irection to guie the packet out of the local minimum. (b) Face routing fails in a 3D network because there exist an arbitrarily large number of possible paths on the bounary surface to be explore in orer to escape from a local minimum. (c) The basic iea of trace-routing is to establish a cutting plan that intersects the bounary surface to yiel a trace, along which the packet can move out of the local minimum. oes not exist a eterministic algorithm to guarantee ata elivery base on local information only in general 3D networks [28]. Given the challenges in extening 2D geometric routing algorithms to 3D, several approaches have been evelope specifically for 3D settings, which fall into four categories as summarize below. The first category focuses on topology control [29], where a critical communication istance is suggeste to eliminate local minima in 3D networks. However, such critical communication istance is often too large for practical applications, given the short sensor raio range, e.g., about 10 meters for MICAz motes. The secon category is base on imension reuction by projecting noes in 3D space onto 2D plane, where face routing is applie [6, 7]. However face routing on the projecte plane oes not guarantee a packet to move out of local minimum in the original 3D network. eparately, a 4D stereographic projection scheme is aopte in [9] for loa balancing, but it oes not aress possible local minima. The thir category aopts the ivie-an-conquer strategy to compress geometric information require to escape from local minima in a 3D network. To this en, tree structure or space partition is often employe. For example, a convex hull-base tree is introuce in GDTR-3D [10]. If a local minimum is reache, the packet is forware accoring to the tree. As another example, a 3D partial unit Delaunay triangulation algorithm is propose in [5] to ivie the network space into close subspaces such that a local minimum can be recovere within a few subspaces only. In [30], a istribute multi-imensional tree structure is applie to support guarantee packet elivery. eparately, the Ranom-Walk algorithm [8] employs a spherical ual graph structure to jump out of local minima. In all of these schemes, certain structures must be maintaine by iniviual sensors, which are often non-locally-eterministic or requires a non-constant storage space that increases with the network size. The fourth category is base on mapping schemes. For example, volumetric harmonic mapping is introuce in [31] to achieve eterministic an guarantee elivery in 3D sensor networks. However, a routing table must be maintaine for a network that contains more than one vois, with its size proportional to the number of vois in the network. Moreover, all algorithms iscusse in the above four categories suffer from network ynamics. For example, when sensor noes move, they must frequently upate their ata structures or virtual coorinates. uch upates are usually expensive or even unattainable. 1.3 Contributions of This Work As iscusse above, none of the existing schemes support DICO routing in 3D wireless sensor networks. In this research, we consier a sensor network eploye in a 3D space with one or multiple internal holes, where sensors cannot be eploye. We show that local minima are always on the bounaries of holes if noal ensity is high, an there exists a DICO algorithm to support routing between any pair of points in a strong-connecte 3D network. We propose a istribute an eterministic algorithm with constant storage, communication an computation overhea, ubbe tracerouting, to escape from local minima. Trace-routing is triggere when a packet reaches a local minimum. It constructs a virtual cutting plane that contains the local minimum an the estination an intersects the corresponing bounary surface to yiel a trace. The trace is a close loop that can be compute locally with constant overhea. The packet is route along such a trace, thus eterministically moving out of the local minimum as illustrate in Fig. 1(c). The propose trace-routing algorithm oes not eman preprocessing of the global network information, neither oes it require establishing or maintaining a global ata structure, which often consumes a storage space proportional to the network size an nees frequent upate ue to network ynamics. We further prove the trace-routing algorithm guarantees ata elivery in a strong-connecte 3D network uner both continuous an iscrete settings. In a nutshell, trace-routing supports DICO, thus highly efficient in largescale 3D sensor networks. It is worth mentioning that the propose trace-routing algorithm an relate iscussions an proofs o not rely on any particular communication moel (such as the unit ball graph moel or quasi-unit ball graph moel). Only a maximum raio range is assume, which is generally known in practical sensor networks. The rest of this paper is organize as follows: ec. 2 introuces the propose trace-routing algorithm in continuous 3D omain. ec. 3 presents a practical esign of trace-routing an proves its correctness uner iscrete 3D sensor network settings. ecs. 4 an 5 iscuss simulation an experimental results. Finally, ec. 6 conclues the paper.

3 B 0 B 0 1 a B 2 s 1 2 B 2 p min s 3 3 ^ p p min trace1 ^ p trace2 p 0 p 1 p 2 P b s 2 voi Figure 2: Bounaries an δ- vicinities. Figure 3: routing. Geometric greey Figure 4: Illustration of Figure 5: Illustration of Lemma 2. Lemma TRACE-ROUTING IN CONTINUOU 3D PACE As shown in [28], there oes not exist a universal eterministic algorithm to guarantee ata elivery base on local information only in general 3D networks. Following this result, a natural question is whether there exists a DICO routing algorithm for a class of practical 3D networks an how to evelop a istribute algorithm to achieve such DICO routing. To eliver a luci conceptual presentation an gain useful theoretic insights, we first investigate the failures in geometric greey routing an present our propose algorithm in a continuous 3D space. We will iscuss the algorithm uner iscrete 3D sensor network settings in the next section. 2.1 Greey Routing in Continuous 3D Volume Let s start with several efinitions that are necessary to our exposition an iscuss the essence of geometric greey routing in a continuous 3D Eucliean space. Let U enote an Eucliean volume in 3D space R 3, which is enclose by a set of bounaries B = {B k 0 k K}, where B 0 is the outer bounary an B k (k > 0) is an inner bounary of U. LetL(p,q) enote a straight line segment from Point p to Point q, an L(p,q) enote its Eucliean istance. Let Λ k enote the area enclose by B k (k 0). Obviously, U = Λ 0 \ Λ k (1 k K}). DEFINITION 1. Let C(s, ) be a sequence of line segments, C(s,) = L(p 0,p 1),L(p 1,p 2),..,L(p m 1,p m), where p 0 = s an p m =. We call C(s,) a routing path connecting s an, if an only if L(p i,p i+1) oes not intersect with L(p j,p j+1), i j, i+1 j an i j +1; an L(p i,p i+1) oes not penetrate any bounary of U, L(p i,p i+1) C(s,). The first conition of Definition 1 requires the routing path free of self-intersection, while the secon conition inicates that the path must be completely containe insie U. Next we introuce the geometric greey routing algorithm via the following two efinitions. DEFINITION 2. We call V(p,δ) the δ-vicinity of Point p, if ˆp in V(p, δ), L(p, ˆp) is completely containe in U an L(p, ˆp) δ where δ is a given positive real number. Fig. 2 illustrates two examples of δ-vicinity of points at ifferent locations. The δ-vicinity of Point a, an internal noe, is a ball centere at a with a raius of δ, while Point b has a efecte δ- vicinity ue to the nearby bounary. DEFINITION 3. A routing path C(s, ) is a geometric greey path (or greey path for conciseness), enote as C(s,,δ), if p i onc(s,,δ),p i+1 is the closest point inv(p i,δ) to Destination. The above efinition suggests a simple geometric greey routing algorithm. It starts from the source as its current point, an chooses the point in its current δ-vicinity that is the closest to the estination as the next point. The process repeats until it reaches the estination. An example of routing from ource s 1 to Destination 1 is shown in Fig. 3. Note that the geometric greey routing algorithm may fail to fin the next point, when the current point is closer to the estination than all other points in its δ-vicinity. Formally we efine the point where geometric greey routing fails as a local minimum. DEFINITION 4. Pointp min is a local minimum for Destination uner a given δ, if p min an L(p min,) L(p,) p V(p min,δ). An example of geometric greey routing failure is illustrate in Fig. 3 from ource s 2 to Destination 2. Greey routing often fails when the routing path meets the bounary of an internal hole. Particularly, we have the following observation regaring geometric greey routing failures. LEMMA 1. If there oes not exist a geometric greey routing path C(s,,δ) between s an uner a given δ, L(s,) must intersect at least one of the bounaries of U. PROOF. We prove it by contraiction. Assume L(s, ) oes not intersect any bounary. ince s an are points in U, L(s, ) must be completely containe in U (otherwise it woul intersect a bounary). Then a greey path C(s,,δ) along the straight line L(s,) can be reaily constructe accoring to Definition 3, contraicting the given conition that there is no greey path between s an. Thus, the lemma is proven. Equivalent to Lemma 1, ifl(s,) oes not intersect any bounary of U, we can always fin a greey routing path from s to. It gives the sufficient conition for the existence of a greey path. But it is not the necessary conition, i.e., even if L(s, ) intersects a bounary, a greey routing path may still be establishe from s to. Fig. 3 illustrates such an example where L(s 3, 3) intersects a spherical bounary, but there exists a greey routing path along the spherical bounary to Destination 3. Base on Lemma 1 an Definition 4, we arrive at the following conclusion. LEMMA 2. Greey routing only fails at local minimums an local minimums are always on the bounaries of holes. PROOF. The first part of the lemma is obvious from Definitions 3 an 4. The secon part of the lemma can be proven by contraiction. Assume there is a local minimum point ˆp that is an internal point in U (i.e., not on any bounary). Let s raw a line L(ˆp,). ince ˆp is an internal point, there must exist a δ such that the intersection ofl(ˆp,) anv(ˆp,δ) (highlighte in re in Fig. 4) is a line segment, on which all points are closer tothan ˆp is. This contraicts the assumption that ˆp is a local minimum. The lemma is proven.

4 B 0 B 0 B 0 B 0 B 2 (a) Volumes with convex bounaries are always s-con. (b) A volume with a C-shape inner voi is s-con. (c) A volume with complex shapes of inner vois is s-con. () A volume with a bowlshape inner voi is not s-con. Figure 6: Examples of s-con an non-s-con volumes. Each volume has a cubic outer bounary (i.e., B 0) an one or two inner vois elineate by the colore bounaries. An example of cutting planes is shown in each figure, where the shae area(s) illustrate its intersection with the volume. Lemma 2 reveals an important fact: greey routing only fails at bounaries, inspiring us to evelop a eterministic bounary navigation algorithm to escape from local minimums. 2.2 Trace-Routing in Continuous 3D Volume In a 3D space for practical sensor network eployment, the bounary surfaces are close (a close surface is a compact one with empty bounary) an the 3D volume is often trong-connecte (s-con). Here close means no breaches on the surface. The s-con property is efine below. DEFINITION 5. A 3D volume U is s-con (trong-con-necte), if an only if the intersection of any plane an U is a connecte graph on the plane. Figs. 6(a)-6(c) epict several examples of s-con 3D volumes. Each volume has a cubic outer bounary (i.e., B 0) an one or two inner vois elineate by the colore bounaries. An example of cutting planes is shown in each figure, where the shae area(s) illustrate its intersection with the volume. It is obvious that a 3D volume with convex bounaries is always s-con as shown in Fig. 6(a). But s-con volumes may have non-convex bounaries too. For instance, volumes in Fig. 6(b) an Fig. 6(c) are s-con with nonconvex inner bounaries. Note that, although the plane intersects the volume in Fig. 6(c) to yiel multiple vois on the intersection plane, the intersection itself (i.e., the shae area) is still connecte. Therefore it is s-con. Fig. 6() shows an example of non-s-con volume, with a bowl shape inner voi, resulting in isconnecte components on the intersection plane (see the isconnecte shae areas). Although DICO routing is not achievable in general 3D networks as iscusse earlier, we now introuce the propose tracerouting algorithm, an prove that it supports DICO routing in any 3D wireless sensor networks that are trong-connecte (s-con). There exist other types of bounary surfaces in theory, but they are rare uner practical sensor network settings an thus exclue from our iscussions below. As shown by Lemma 2, if greey routing fails, it must stuck at a local minimum, an such local minimum must be on a bounary of U. Letp min enote the local minimum where geometric greey routing fails, an assume p min is on Bounary B i. Now we construct an arbitrary plane that contains L(p min,). It intersects B i, resulting in one or multiple traces. By examining the traces, we have the following observation. LEMMA 3. A trace on a bounary surface is a close loop with no self-intersection. PROOF. If a trace is not a close loop, the space between two enpoints on the plane forms a breach on the surface, contraicting p min b Figure 7: Lemma 5. a c h p min Illustration of Figure 8: Prove of Theorem 2. the fact of close bounary surface. ince the trace is the intersection of the surface an the plane, it cannot self-intersect, given that the surface is homeomorphic to sphere with no self-intersecting component. If we raw a line segment from p min to, L(p min,) must intersect B i at two or more points incluing p min, as shown in Fig. 5. Let p 0 enote one of such intersection points, which is the closest top min. Clearly, p 0 must be closer tothan p min is, since it is on the line segment betweenp min an. LEMMA 4. In an s-con volume, the trace containingp min must also containp 0 an there is a loop-free path fromp min top 0 along this trace. PROOF. Obviously p min an p 0 must belong to some trace(s). Accoring to the way of constructing the plane, L(p min,) must intersect one or multiple traces, yieling a sequence of intersection points on L(p min,). Now let us look at the trace that contains p min, enote by Γ pmin. ince L(p min,) starts from p min that is on BounaryB i an goes towar the insie ofb i, it cannot reach another point before it meets Γ pmin again. Let ˆp enote this intersection point as shown in Fig. 5. ince ˆp is the first such intersection point, it is the closest to p min. Clearly, ˆp must be co-locate withp 0 introuce earlier. Therefore, we have provenp min anp 0 must be on the same trace. As shown by Lemma 3, a trace is a close loop with no selfintersection. Thus it is straightforwar to choose a loop-free route along the trace in either clockwise or counterclockwise irection to route from p min top 0. Till now we are reay to formally introuce the trace-routing algorithm an prove its correctness as follows. THEOREM 1. There exists a eterministic algorithm with constant storage, communication an computation overhea, which can always successfully navigate the routing path out of local minimums in an s-con volume. a b h c

5 PROOF. First we construct a simple algorithm, ubbe tracerouting as follows. When geometric greey routing reaches a local minimum p min on Bounary B i, it chooses a cutting plane that is etermine by p min, Destination, an another ranom point ˆp. The plane intersects B i, yieling a trace that contains p min. The routing path avances along the trace in clockwise or counterclockwise irection until it reaches a point that is closer to Destination than p min is. Then geometric greey routing follows. Now we prove the correctness of the algorithm. Accoring to Lemma 4, there must exist at least one point (i.e., p 0) on the trace containing p min, which is closer to Destination than p min is. ince the trace is a close loop with no self-intersection, if one starts from p min an navigates along the trace in clockwise or counterclockwise irection, it will visit all points on the trace exactly once before it returns back to p min. Therefore the routing path must reach p 0, from which it can escape from the local minimum. Trace-routing is clearly eterministic as evient by the escription of the algorithm itself. Next, we show its storage, communication an computation overhea are all constant. The algorithm requires each noe to maintain the coorinates of itself an its neighbors only, an to pass merely a constant amount of information (i.e., the cutting plane efine by three points p min, ˆp, an ) along the routing path. Assume the routing path reaches Point p. The trace segment aroun p can be foun by computing the intersection of the cutting plane an the bounary surface in V(p,δ). Thus, the routing ecision is mae locally with a constant computation complexity. In summary we have prove the existence of a local eterministic algorithm with constant storage, communication an computation overhea that can always successfully move out of local minimums in an s-con volume. Theorem 1 reveals salient properties of trace-routing, which supports efficient an istribute implementation uner practical sensor network settings to be iscusse next. 3. TRACE-ROUTING IN DICRETE 3D NET- WORK While trace-routing has been introuce above to escape from local minimums, it remains challenging to aapt the concepts an ieas to a practical sensor fiel. In contrast to the continuous 3D Eucliean volume consiere in ec. 2, a sensor network is uner a iscrete setting, which presents an approximation of the 3D volume only, renering part of the earlier iscusse methos an results invali. For example, sensor noes rarely resie perfectly on the trace compute in a continuous space, calling for approximate solutions. Furthermore, the routing algorithm must be istribute in sensor networks. However, the eployment of sensor noes exhibits ranomness an their raio transmissions are often irregular, which together lea to new challenges to realize istribute tracerouting an to achieve constant storage, communication an computation overhea. In this section, we examine our conclusions presente in the previous section uner a iscrete setting, an introuce a practical esign of the trace-routing algorithm for 3D sensor networks with internal holes. The following iscussions o not rely on any particular communication moel (such as the unit ball graph moel or quasi-unit ball graph moel). Only a maximum transmission range is assume, which is generally known in practical wireless sensor networks. 3.1 Challenges of Trace-Routing in 3D ensor Networks Let us first revisit the concepts an conclusions given in ec. 2 for continuous 3D Eucliean volume, an evaluate their valiity in 3D sensor networks Tetraheron tructure To facilitate our exposition, we first establish a tetraheral structure base on iscrete sensors as iscusse in [31]. A triangular face in a tetraheral structure is a bounary face if it is associate with one an only one tetraheron. A set of connecte bounary faces form a close triangular bounary surface. In the following iscussions, a bounary refers to a triangular bounary surface Local Minimums Lemma 2 has shown that local minimums are always on the bounaries of holes in a continuous 3D Eucliean volume. Unfortunately, this result no long hols in iscrete settings, where local minimums may appear at not only bounary but also internal sensors. However, the internal local minimums can be resolve by using local information only as reveale by the following lemma. LEMMA 5. Given an internal local minimump min, the routing path can move out the local minimum by using local information in2δ-vicinity of p min, where δ is the maximum raio transmission range. PROOF. Base on the tetraheron structure, a greey routing path can be establishe accoring to faces [31]. More specifically, when the routing path reaches an internal local minimum p min, a line segment from p min to Destination can be compute by p min. The line segment passes through a sequence of faces in the tetraheron structure. Let s consier the first such face, enote by Face(a,b,c), where a, b, an c are vertex noes (as illustrate in Fig. 7). Obviously, Face(a,b,c) must be closer to the estination compare to the current face, i.e., Face(p min,a,b). Clearly, Face(a,b,c) is within two-hop neighborhoo ofp min. As a result, the routing path can move out of the local minimum by using local information in2δ-vicinity of p min. Accoring to Lemma 5, we slightly moify the greey routing algorithm to let each noe check its two-hop neighborhoo to ientify the closest noe to the estination, thus eliminating internal local minimums. Note that, the moifie algorithm oes not help to escape from bounary local minimums, which are yet to be aresse by the propose trace-routing Traces As iscusse in ec. 2.2, a trace on an inner bounary is a close loop without self-intersection. It can be compute by using local information. But in a iscrete setting, sensors rarely resie perfectly on the compute trace. To this en, we reefine traces uner iscrete settings as follows. DEFINITION 6. In a iscrete 3D space, a trace forme by a given plane an a bounary consists of a sequence of line segments on the bounary surface, which intersect the plane an are connecte en-to-en to form a loop. LEMMA 6. Given a tetraheral structure of the 3D sensor network an a plane that intersects a triangular bounary surface of the tetraheral structure, a close-loop trace can be constructe eterministically.

6 PROOF. For a given tetraheral structure of the 3D sensor network, the faces on the bounary of an internal hole form a close an connecte triangular surface. A plane intersects the triangular bounary surface at a set of triangular faces, enote by Φ. For each face f Φ, it must have at least two eges intersecting the plane. ince each ege is share by two faces on the triangular bounary surface, Face f must have at least two neighboring faces in Φ. Therefore, the faces in Φ are connecte to form a close strap. It is thus straightforwar to select a sequence of eges on such faces, which intersect the plane, to buil a close loop, i.e. the trace. Base on Lemma 6, we have the following result that shows the correctness of trace-routing. THEOREM 2. A eterministic routing path can be ientifie by following the trace constructe accoring to Lemma 6 to escape from local minimums. PROOF. To prove this theorem, we show that there exists at least one noe on the trace, which is closer to the estination than the local minimum (i.e., p min) is. imilar to our iscussion uner the continuous setting, if we raw a line segment from p min to Destination, L(p min,) must intersect the bounary surface at one or more points besies p min. Let h enote one of such intersection points that is the closest to p min, an assume h is on an arbitrary face, e.g., Face(a,b,c) as illustrate in Fig. 8. Clearly, L(a,b), L(b,c) an L(a,c) are no greater than the maximum raio range r because they are connecte in the communication graph. ince h is on Face(a,b,c), L(a,h), L(b,h) an L(c,h) must be no greater than r either. At the same time, L(p min,a), L(p min,b), L(p min,c) an L(p min,h) are all greater than r, since there is a hole between p min an Face(a,b,c). Thus we have L(a,) < L(a,h) + L(h,) r+l(h,) < L(p min,h) + L(h,) = L(p min,). Therefore a is closer to than p min is. imilarly, b an c are closer to than p min is. Note that at least one ege of Face(a,b,c) must be on the trace constructe accoring to Lemma 6. Therefore one can always fin a noe on the trace closer to than p min is. The theorem is thus proven. Theorem 2 shows trace-routing is eterministic an always succees in iscrete sensor networks. Moreover, similar to the proof of Theorem 1, its storage, communication an computation overhea are all boune by a constant. 3.2 Propose Trace-Routing Algorithm in 3D ensor Networks The basic iea of trace-routing is to move clockwise or counterclockwise aroun the trace constructe by Lemma 6, which is a close loop an thus can navigate out of local minimums. The iscussions so far have assume known bounary noes an triangular bounary surfaces. Next we show that such information can be acquire locally with constant overhea or is unnecessary at all in practice Bounary Awareness We aopt the bounary etection algorithm propose in [32], which is localize, requiring information within one-hop neighborhoo only for each noe to etermine if it is on a bounary. Therefore, when greey routing fails at a local minimum p min, the bounary etection algorithm [32] is employe to ientify bounary noes in the δ-vicinity of p min as input for the trace-routing algorithm Triangular Bounary urface The tetraheron structure an triangular bounary surfaces are assume to facilitate our iscussions in ec. 3.1, especially the proof of Lemmas 5 an 6. However, it is unnecessary to explicitly construct them for the implementation of trace-routing. The algorithm can simply choose one of the bounary noes in its neighborhoo as the next hop along the trace. More specifically, the tracerouting algorithm is outline in Algorithm 1. It is initiate when a greey routing path reaches a local minimum p min. It efines a plane base on p min, an an arbitrary noe p in the δ-vicinity of p min. Let enote the plane. The algorithm then enters an iterative process. In each iteration, a bounary noe in the δ-vicinity of the current noe (i.e.,p cur) is ientifie to be the next noe (i.e., ˆp) along the trace. To keep the next noe as close to the ieal trace (compute uner continuous setting) as possible an to avance along the trace as fast as possible, ˆp is chosen such that L(p cur, ˆp) intersects Plane an forms the largest angle with L(p pre, p cur), wherep pre is the previous hop noe on the routing path. The iterative process repeats until it reaches a noe closer to the estination thanp min is, i.e., L(p cur,) < L(p min,). It is clear that Algorithm 1 only results in constant storage, communication an computation overhea. It oes not eman preprocessing of the global network information, neither oes it require establishing or maintaining a global ata structure, which often consumes a storage space proportional to the network size an nees frequent upate ue to network ynamics. Thus it is highly efficient for large-scale 3D sensor networks. Algorithm 1: Trace-Routing Algorithm Input: Local minimum p min, Destination; 1 Define Plane base onp min,an an arbitrary noe p V(p min,δ); 2 p cur p,p pre p min; 3 while L(p cur,) L(p min,) o 4 Ientify a bounary noe ˆp V(p cur,δ) such that L(ˆp,p cur) intersects Plane an forms the largest angle withl(p pre, p cur); 5 p pre p cur;p cur ˆp; 3.3 Peer-to-Peer Routing in 3D Wireless ensor Networks In summary, when a packet is route from ource s to Destination, it first follows geometric greey routing until a local minimum is reache. Then it employs trace-routing to move out of the local minimum. When the packet arrives at a non-local-minimum noe, it switches to greey routing again. The process repeats until the packet arrives at Destination. Examples of the routing paths are illustrate in Fig. 1(c) an Figs IMULATION REULT We have implemente the propose trace-routing algorithm (enote by TR ) an two other state-of-the-art 3D wireless sensor network routing algorithms, namely GDTR-3D [10] an HVE (harmonic volumetric embeing) [31], for performance evaluation an comparison. ince HVE works in networks with one hole only, we focus on the comparison between TR an GDTR-3D in most simulation scenarios. Various 3D sensor networks with ifferent sizes (ranging from 1,800 to 11,000 noes) an shapes are simulate. Fig. 9 illustrates three examples, where the inner bounary

7 0.35 (a) Moel 1 (eabe Network). (b) A sample path in Moel 1. Percent of Total Routing Paths (x100%) TR GDTR 3D [ ) [ ) [ ) [ ) [ ) [ ) [ ) > 2.50 Routing Path tretch Figure 10: tretch uner UDG. (c) Moel 2 (pace Network). () A sample path in Moel 2. local minimum recovery proceure that follows one of the two trees often results in significantly stretche routing paths in comparison with corresponing shortest paths. Moreover, uner complex an multiple hole settings, there are more overlappe convex hull partitions, which cause longer paths uring the tree routing phase. In a contrast, TR follows the geometric bounary of the holes in local minimum recovery, resulting in a stable stretch factor in average. (e) Moel 3 (Unerwater Network). (f) A sample path in Moel 3. Figure 9: The left column shows 3D sensor network moels an the right column illustrates sample routing paths in each moel. The sensors on the bounary of holes are highlighte in ifferent colors. The blue segments inicate greey forwaring, while the re segments follow trace-routing. noes are highlighte to elineate the holes in the networks. ensors are ranomly istribute. While our propose scheme oes not rely on a particularly communication moel, we carry out simulations base on unit ball graph an quasi-unit ball graph moels, with an average noal egree between 10 to 18. Our simulations show that the propose scheme works well for networks with a noal egree greater than 7 (which represents a sparse network in three imensional space). 4.1 tretch Factor As iscusse in previous sections, TR, GDTR-3D an HVE all ensure successful elivery. Therefore we focus on stretch factor in performance evaluation. The stretch factor of a route is the ratio of the actual routing path length to the shortest path length. 20,000 pairs of noes are ranomly selecte to calculate the average stretch factor for each network moel. The following table compares the average stretch factors of GDTR-3D an TR in ifferent networks. The results of HVE are not inclue because it can be applie for networks with one hole only, where its stretch factor is aroun Both algorithms achieve goo an comparable stretch factor (less than 1.3) in all network moels. GDTR-3D is more sensitive to the hole size an the number of holes, exhibiting higher stretch factors uner Moels 2-4. This is mainly because that the GDTR-3D uses two trees roote at two furthest away noes. When there are multiple holes or complex shape holes, the Moels TR GDTR-3D Ave The istribution of stretch factor is illustrate in Fig. 10. As can be seen, more than 50% routing paths have their stretch factor lower than 1.15 uner both schemes. Compare with TR, the stretch factor istribution of GDTR-3D has a noticeable shift to the right sie with more routes experiencing a stretch factor of 1.2 or higher. As iscusse in the previous sections, TR an GDTR-3D o not rely on any particular transmission moel. Here we test them uner a general QUAI-UBG" moel, where two noes are connecte if their istance is less than ; or isconnecte if their istance is beyon the maximum transmission range r; or have a probability of p to be connecte if their istance is between an r. We vary an p, an compare the performance of TR an GDTR-3D. Both schemes support perfect ata elivery rate. Their stretch factors are shown in Fig. 11. We observe that the stretch factor of GDTR3D increases fast with the ecrease of, because the connections between noes become more ranom, leaing to more overlap between convex hulls. Consequently, a non-optimal branch is more frequently chosen to establish a routing path, resulting in a higher stretch factor. On the other han, TR only increases stretch factor moerately ue to the lower noal ensity. 4.2 Overhea an tability TR oes not require any global information to be store by iniviual noes. Each noe only collects its 1-hop neighbor information, which is proportional to noal egree, i.e., the ensity of the network, which is usually regare as constant for a given network setting. The information necessary for greey forwaring is simply the estination ID an location. The information nees for trace-routing is the plane efine by the estination, local minimum, an another arbitrary noe. The computation overhea inclues the ientification of bounary neighbors an the next hop on the trace, which is proportional to the noal egree as well. o

8 1 TR GDTR 3D 1.5 Average tretch of Routing Paths TR p=0.5 GDTR 3D p=0.5 TR p=0.7 GDTR 3D p=0.7 TR p=0.9 GDTR 3D p=0.9 Delivery Ratio Probability P of QUAI UBG Moel Parameter of QUAI UBG Moel 0.5 0% 5% 10% 15% 20% 25% 30% 40% 50% Location Errors (x r) Figure 11: tretch uner Quasi-UDG. Figure 12: Delivery ratio against location errors. both communication an computation overhea are boune by a small constant (i.e., noal egree). GDTR-3D constructs two convex hull-base trees, where each noe aggregates the locations of its chilren by using 2D convex hulls. Its time complexity an communication cost are ominate by the computation of 2D convex hulls (i.e.,o(nlogn)), while the per noe storage cost can be close to a constant after compression. Location information is require in both TR an GDTR-3D, which is, however, often subject to errors in practical sensor network settings. Fig. 12 epicts the results of TR an GDTR-3D uner ifferent localization errors. We consier a path fails if its stretch factor excees 6.0. The elivery ratio of GDTR-3D rops much faster than TR, because the location errors on iniviual noes are aggregate in establishing the convex hull tree, increasing the chance to search into an incorrect tree branch. ince TR uses only local information only, the localization errors o not sprea out. Therefore TR exhibits esire stability against localization errors. For example, uner 30% errors, it still keeps its elivery ratio over 94%, while GDTR-3D can eliver merely81% of ata. 4.3 Network Dynamics ince TR oes not rely on any global network structure, it is aaptive to network ynamics (e.g., sensors movement). Here we evaluate the performance of TR, GDTR-3D an HVE in mobile 3D sensor networks. We assume the holes are obstacles in the fiel where sensor noes cannot access, an assume a simple ranom waypoint mobility moel. More specifically, each noe chooses a ranom position within a istance of ρ as its new position, where ρ is the maximum moving range uring a time interval t. We let ρ = 0.2r in our simulations. In each time interval, we measure the elivery ratios of three algorithms base on10,000 pair of ranomly chosen noes. imilar to the above iscussion, a path is eeme unsuccessful if its stretch factor excees 6.0. The simulation lasts for 20 time intervals. As can be seen in Fig. 13, TR achieves a stable elivery ratio which is very close to 1 as expecte, since the geometric shape of the network remains the same, even though the noes change their positions. The faile routes, which are rare, are cause by the irregularity istribution of bounary noes that results in very low noe ensity in a local area. In a sharp contrast, the elivery ratios of GDTR-3D an HVE ecreases ramatically after 9 intervals, because the convex hull tree for GDTR-3D an tetraheron mesh for HVE become outate, proviing wrong routing information. Delivery Ratio TR GDTR 3D HVE Time Intervals Figure 13: Delivery ratio uner network ynamics. 5. EXPERIMENT We have also implemente the TR algorithm on Crossbow MI- CAZ motes (MPR2400CA). We set up a small testbe with 98 motes, where the maximum transmission range is set to aroun 4.5 inches for easy configuration of network topology. We place the motes uniformly aroun the bounary of a cubic hole of inches an manually measure the relative coorinates of each mote. The istance between neighboring motes is 4 inches an the noal egree is aroun 4, as shown in Fig 14. ince all motes are bounary noes, we skip bounary ientification in our implementation. Each mote keeps a table to store its neighboring motes coorinates, which is an very small array (4 4 in size), which contains noe ID an 3D-coorinates. The exchange of neighboring information is one uring the bootstrap phase, which takes less than 20 secons. The TR algorithm involves mostly vectors operations, such as cutting plane ientification, intersection between line segment an plane, an calculation of angle between two vectors, which are very convenient to implement on MICAZ motes. A ata packet contains a ummy ata segment (of 512 bytes), the IDs an coorinates of the source an estination, an a flag to inicate current routing metho (i.e., greey or trace-routing). If trace-routing is employe, the ata packet also inclues the ID an coorinates of the local minimum noe, an the cutting plane (represente as Ax + By + Cz + D = 0 where A, B, C an D are compute only once at the local minimum noe base on the coorinates of

9 Figure 14: Experiment setup base on MicaZ motes. the local minimum, the estination an an arbitrary neighboring noe). The total overhea per packet is 52 bytes. We select 40 pairs of motes on ifferent sies of the hole an transmit 50 packets for each route. All ata packets are successfully elivere from source to estination in our experiment as expecte, with an average stretch factor of CONCLUION AND FUTURE WORK We have propose trace-routing, a istribute an eterministic routing algorithm with constant storage, communication an computation overhea for 3D wireless sensor networks. Its basic iea is to construct a virtual cutting plane that intersects bounary surface to yiel a trace, along which a routing path with guarantee elivery can be establishe. We have proven the correctness of trace-routing uner both continuous an iscrete settings. We have implemente trace-routing on Crossbow sensors an carrie out extensive simulations to evaluate its routing efficiency. Acknowlegements. Hongyi Wu is partially supporte by NF CN an CN Miao Jin is partially supporte by NF CCF , CN an CN REFERENCE [1] J. Allre, A. B. Hasan,. Panichsakul, W. Pisano, P. Gray, J. Huang, R. Han, D. Lawrence, an K. Mohseni, ensorflock: An Airborne Wireless ensor Network of Micro-Air Vehicles, in Proc. of enys, pp , [2] J.-H. Cui, J. Kong, M. Gerla, an. Zhou, Challenges: Builing calable Mobile Unerwater Wireless ensor Networks for Aquatic Applications, IEEE Network, pecial Issue on Wireless ensor Networking, vol. 20, no. 3, pp , [3] X. Bai, C. Zhang, D. Xuan, J. Teng, an W. Jia, Low-Connectivity an Full-Coverage Three Dimensional Networks, in Proc. of MobiHOC, pp , [4] X. Bai, C. Zhang, D. Xuan, an W. Jia, Full-Coverage an K-Connectivity (K=14, 6) Three Dimensional Networks, in Proc. of INFOCOM, pp , [5] C. Liu an J. Wu, Efficient Geometric Routing in Three Dimensional A Hoc Networks, in Proc. of INFOCOM, pp , [6] T. F. G. Kao an J. Opatmy, Position-Base Routing on 3D Geometric Graphs in Mobile A Hoc Networks, in Proc. of The 17th Canaian Conference on Computational Geometry, pp , [7] J. Opatrny, A. Aballah, an T. Fevens, Ranomize 3D Position-base Routing Algorithms for A-hoc Networks, in Proc. of Thir Annual International Conference on Mobile an Ubiquitous ystems: Networking & ervices, pp. 1 8, [8] R. Flury an R. Wattenhofer, Ranomize 3D Geographic Routing, in Proc. of INFOCOM, pp , [9] F. Li,. Chen, Y. Wang, an J. Chen, Loa Balancing Routing in Three Dimensional Wireless Networks, in Proc. of ICC, pp , [10] J. Zhou, Y. Chen, B. Leong, an P. unaramoorthy, Practical 3D Geographic Routing for Wireless ensor Networks, in Proc. of enys, pp , [11] D. Pompili, T. Meloia, an I. F. Akyiliz, Routing Algorithms for Delay-insensitive an Delay-sensitive Applications in Unerwater ensor Networks, in Proc. of MobiCom, pp , [12] W. Cheng, A. Y. Teymorian, L. Ma, X. Cheng, X. Lu, an Z. Lu, Unerwater localization in sparse 3 acoustic sensor networks, in Proc. of INFOCOM, pp , [13] P. Bose, P. Morin, I. tojmenovic, an J. Urrutia, Routing with Guarantee Delivery in A Hoc Wireless Networks, in Proc. of Thir Workshop Discrete Algorithms an Methos for Mobile Computing an Communications, pp , [14] B. Karp an H. Kung, GPR: Greey Perimeter tateless Routing for Wireless Networks, in Proc. of MobiCom, pp. 1 12, [15] E. Kranakis, H. ingh, an J. Urrutia, Compass Routing on Geometric Networks, in Proc. of CCCG, pp , [16] F. Kuhn, R. Wattenhofer, Y. Zhang, an A. Zollinger, Geometric A-hoc Routing: Theory an Practice, in Proc. of The 22n ACM ymposium on the Principles of Distribute Computing, pp , [17] F. Kuhn, R. Wattenhofer, an A. Zollinger, Worst-case Optimal an Average-case Efficient Geometric A-hoc Routing, in Proc. of MobiHOC, pp , [18] B. Leong,. Mitra, an B. Liskov, Path Vector Face Routing: Geographic Routing with Local Face Information, in Proc. of ICNP, pp , [19] H. Frey an I. tojmenovic, On Delivery Guarantees of Face an Combine Greey-face Routing in A Hoc an ensor Networks, in Proc. of MobiCom, pp , [20] G. Tan, M. Bertier, an A.-M. Kermarrec, Visibility-Graph-base hortest-path Geographic Routing in ensor Networks, in Proc. of INFOCOM, pp , [21] C. Papaimitriou an D. Ratajczak, On A Conjecture Relate to Geometric Routing, Theoretical Computer cience, vol. 344, no. 1, pp. 3 14, [22] P. Angelini, F. Frati, an L. Grilli, An Algorithm to Construct Greey Drawings of Triangulations, in Proc. of The 16th International ymposium on Graph Drawing, pp , [23] T. Leighton an A. Moitra, ome Results on Greey Embeings in Metric paces, in Proc. of The 49th IEEE

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