Three-dimensional greedy routing in large-scale random wireless sensor networks

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Three-dimensional greedy routing in large-scale random wireless

sensor networks

Yu Wang

a,⇑

, Chih-Wei Yi

b

, Minsu Huang

a

, Fan Li

c

a

Department of Computer Science, University of North Carolina at Charlotte, Charlotte, NC 28223, United States b

Department of Computer Science, National Chiao Tung University, Hsinchu City 30010, Taiwan, ROC c

School of Computer Science, Beijing Institute of Technology, Beijing 100081, China

a r t i c l e

i n f o

Article history:

Available online 12 October 2010 Keywords:

Greedy routing Localized routing Delivery guarantee Energy-efﬁciency

3D wireless sensor networks

a b s t r a c t

In this paper, we investigate how to design greedy routing to achieve sustainable and scal-able in a large-scale three-dimensional (3D) sensor network. Several 3D position-based routing protocols were proposed to seek either delivery guarantee or energy-efficiency in 3D wireless networks. However, recent results[1,2]showed that there is no deterministic localized routing algorithm that guarantees either delivery of packets or energy-efficiency of its routes in 3D networks. In this paper, we focus on design of 3D greedy routing protocols which can guarantee delivery of packets and/or energy-efficiency of their paths with high probability in a randomly deployed 3D sensor network. In particular, we first study the asymptotic critical transmission radius for 3D greedy routing to ensure the packet delivery in large-scale random 3D sensor networks, then propose a refined 3D greedy routing protocol to achieve energy-efficiency of its paths with high probability. We also conduct extensive simulations to confirm our theoretical results.

1. Introduction

Most existing wireless sensor systems and protocols are based on two-dimensional (2D) design, where all wireless sensor nodes are distributed in a two-dimensional plane. This assumption is somewhat justiﬁed for applications where sensor nodes are deployed on earth surface and where the height of the network is smaller than transmis-sion radius of a node. However, 2D assumption may no longer be valid if a wireless sensor network is deployed in space, atmosphere, or ocean, where nodes of a network are distributed over a 3D space and the difference in the third dimension is too large to be ignored. In fact, recent

interest in under-water sensor networks[3]or space

sen-sor networks [4] hints at the strong need to design 3D

wireless networks. However, the design of networking pro-tocols for 3D wireless networks is surprising more difﬁcult

than that for 3D networks. In this paper, we focus on one particular problem in 3D networks: 3D localized posi-tion-based routing.

Localized position-based routing makes the forwarding decision based solely on the position information of the destination and local neighbors. It does not need the dis-semination of route discovery information and the mainte-nance of routing tables. Thus, it enjoys the advantages of lower overhead and higher scalability than other tradi-tional routing protocols. This makes localized routing pro-tocols much suitable for large-scale sensor networks. The most common and efﬁcient localized routing is greedy routing, in which a packet is greedily forwarded to the clos-est node to the dclos-estination in order to minimize the aver-age hop-count. Greedy routing can be easily extended to 3D case. Actually, several under-water routing protocols

[5,6]are just variations of 3D greedy routing.Fig. 1 illus-trates the basic idea of 3D greedy routing. Let t be the

des-tination node. As shown inFig. 1a, current node u ﬁnds the

next relay node

v

who is the closest to t among all

neigh-bors of u. But, it is easy to construct an example (see

doi:10.1016/j.adhoc.2010.10.003

⇑Corresponding author. Tel.: +1 7046878443; fax: +1 7046873516. E-mail addresses:[email protected] (Y. Wang), [email protected] (C.-W. Yi),[email protected](M. Huang),ﬂ[email protected](F. Li).

Contents lists available atScienceDirect

Ad Hoc Networks

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Fig. 1b) to show that greedy routing will not succeed to reach the destination but fall into a local minimum (at a node without any ‘‘better” or ‘‘closer” neighbors). This is true for both 2D and 3D networks.

However, to guarantee packet delivery of 3D greedy routing is not straightforward and very challenging. Face routing can be used on planar topology to recovery from the local minimum of greedy routing and guarantee the delivery in 2D networks, as did in many 2D localized

rout-ing protocols[7–9]. However, there is no planar topology

concept any more in 3D networks and simple projection from 3D to 2D may break the network connectivity. In fact,

Durocher et al.[1]recently proved that there is no

deter-ministic localized routing algorithm for 3D networks that guarantees the delivery of packets. On the other hand, even a localized routing method can ﬁnd the route to deliver the packet, it may not guarantee the energy-efﬁciency of the path, i.e., the total power consumed compared with the optimal could be very large in the worst case. Several

energy-aware localized 2D routing protocols[10–12]

al-ready took the energy concern into consideration, but none of them can theoretically guarantee the energy-efﬁciency of their routes. This is true for all existing 3D localized

routing methods too. Recently, Flury and Wattenhofer[2]

showed an example 3D network where the path found by any deterministic localized routing protocol to connect two nodes s and t has energy-consumption asymptotically at leastH(d3_{) in the worst case. Here d is the optimal}

en-ergy-consumption to connect s and t.

Therefore, in this paper, we are interested in (1) how to achieve delivery guarantee of 3D greedy routing in large-scale random networks; and (2) how to achieve energy-efﬁciency of paths in large-scale 3D networks so that the networks can be sustainable. In particular, we make the following contributions on 3D greedy routing:

We prove that 3D greedy routing can guarantee the delivery of packets between any source–destination pairs if the underlying topology is Delaunay translation. We study on the critical transmission radius (CTR) of 3D greedy routing that guarantees the delivery of packets between any source–destination pairs. We prove that for a 3D random network, formed by nodes that are generated by a Poisson point process of density n over a convex compact region of unit-volume, the CTR for 3D greedy routing is asymptotic almost surely (a.a.s). at

most ffiffiffiffiffiffiffiffiffiffi3b ln n 4pn

3

q

for any b > b0and at least

ffiffiffiffiffiffiffiffiffiffi 3b ln n 4pn 3 q for any b< b0. Here, b0= 3.2.

We extend our previous 2D energy-aware routing

method[13]to an energy-efﬁcient restricted 3D greedy

routing, which is a simple variation of 3D greedy rout-ing. The proposed routing method can guarantee energy-efﬁciency of its path with high probability if it ﬁnds one in 3D networks. We also study its CTR in ran-dom 3D networks and show it is in the same formation of that of 3D greedy routing, except for b0¼1cos2 awhere

a

is an parameter used by the proposed method.

We conduct extensive simulations on 3D random net-works to study the distributions of CTRs of both 3D greedy routing and restricted 3D greedy routing and evaluate their routing performances.

The rest of the paper is organized as follows. In Section

2, we ﬁrst review related work on 3D position-based

rout-ing and critical transmission radius of greedy routrout-ing. Then we present our network model and several preliminaries in

Section3. In Section4, we study how to achieve delivery

guarantee of 3D greedy routing by deriving the asymptotic almost sure bounds on the critical transmission radius of

3D greedy routing. In Section5, we further extend the 3D

greedy routing to an energy-efﬁcient localized routing and derive its CTR bounds. We present simulation results

in Section6and summarize this paper in Section7.

2. Related work

Due to its wide-range potential applications, 3D wiless sensor network has recently emerged as a premier re-search topic. Most current rere-search in 3D sensor networks

primarily focuses on coverage [14–17], connectivity

[15,18–20], and routing issues[5,6,21–24]. Since we focus on design of 3D position-based localized routing in this pa-per, we will ﬁrst review the status on 3D position-based localized routing.

2.1. 3D localized routing: delivery guarantee and energy-efﬁciency

As the most widely used position-based routing, greedy routing has been used by Pompili and Melodia and Xie

et al.[5,6]for 3D under-water sensor networks. However,

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all of these greedy-based routings cannot guarantee the delivery, since they may fail at the local minimum. In 2D

networks [7–9], delivery guarantee can be achieved by

applying face routing as a backup method to get out of the local minimum after simple greedy heuristic fails. The idea of face routing is to walk along the faces which are intersected by the line segment st between the source s and the destination t. To guarantee the packet delivery, face routing requires the underlying 2D routing topology to be a planar graph (i.e., no link/edge intersection). How-ever, 3D networks cannot be planarized any more. Fevens

et al.[21,22]proposed several 3D position-based routing

protocols and tried to ﬁnd a way to still use face routing to get out of the local minimum. Their basic idea is

project-ing the 3D network to a 2D plane (as shown in Fig. 2a),

then applying the face routing in the plane. However, as

shown inFig. 2b[21], a planar graph cannot be extracted

from the projected graph. It is clear that removing either

v

0

3

v

04 or

v

01

v

02 will break the connectivity. Furthermore,

Durocher et al. [1] have recently proven that there is no

deterministic localized routing algorithm for 3D networks that guarantees the delivery of packets. Flury and Wattenhofer

[2]then proposed a randomized 3D routing which adopts

a randomized recovery technique when 3D greedy fails. Beside the delivery guarantee of packets, the energy-efﬁciency of paths is also very important for large-scale

sensor networks. Given a routing method A, let PAðs; tÞ

be the path found by A to connect the source node s and the destination node t. A routing method A is called energy-efﬁcient if for every pair of nodes s and t, the

en-ergy-consumption of path PAðs; tÞ is within a constant

fac-tor of the least energy-consumption path connecting s and t in the network. Even a 3D localized routing method can ﬁnd the route to deliver the packet, it may not guarantee the energy-efﬁciency of the path, i.e., the total power con-sumed compared with the optimal could be very large in the worst case. Several energy-aware localized 2D routing

protocols [10–12] already took the energy concern into

consideration, but none of them can theoretically guaran-tee the energy-efﬁciency of their routes. This is true for all existing 3D localized routing methods too. For path

en-ergy-efﬁciency, recently, Flury and Wattenhofer[2]proved

that no deterministic localized routing method is energy-efﬁcient in 3D networks. They proved the claim by

constructing an example of a 3D network (Fig. 1 of[2])

where the path found by any localized routing protocol to connect two nodes s and t has energy-consumption (or

hop-count or distance) asymptotically at least H(d3) in

the worst case, where d is the optimum cost. Therefore, we are also interested in reﬁning 3D greedy routing into a 3D

energy-efﬁcient routing. In particular, we extend our previous 2D

energy-aware routing method[13] to an energy-efﬁcient

restricted 3D greedy routing.

2.2. Critical transmission radius for greedy routing

One way to guarantee the packet delivery for greedy routing in 2D/3D networks is letting all nodes have sufﬁ-ciently large transmission radii to avoid the existence of lo-cal minimum. It is clear that this can be achieved when the transmission radius is inﬁnite. Assume that V is the set of all wireless nodes in the network and each wireless node has a transmission radius r. Let B (x, r) denote the open disk of radius r centered at x. Let

qðVÞ ¼ max

ðu;vÞ2V2

u–v

min

w2Bðv;kuvkÞkw uk: ð1Þ

In the equation, (u,

v

) is a source–destination pair. Since

w 2 B(

v

, ku

v

k), we have kw

v

k < ku

v

k. It means w

is closer to

v

than u. If the transmission radius is not less

than kw uk,w might be the one to relay packets from u to

v

. Therefore, for each (u,

v

), the minimum of kw uk over all nodes on B(

v

,ku

v

k) is the transmission radius that en-sures there is at least one node that can relay packets from

u to

v

, and the maximum of the minimum over all

(u,

v

)pairs guarantees the existence of relay nodes between

any source–destination pair. Clearly, if the transmission

ra-dius is at least

q

(V), packets can be delivered between any

source–destination pairs. On the other hand, if the

trans-mission radius is less than

q

(V), there must exist some

source–destination pair, e.g., the (u,

v

)that yields the value

q

(V), such that packets cannot be delivered. Therefore,

q

(V) is called the critical transmission radius (CTR) for gree-dy routing that guarantees the delivery of packets between any source–destination pair of nodes among V.

Previously, several studies (e.g.[25–28]) focused on the critical transmission radius for certain network

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ties such as connectivity, k-connectivity, and coverage. Surprisingly, there is not much study for the critical trans-mission radius for certain routing methods, except for the recent results[29,30]for 2D greedy routing. Traditionally it is assumed that the network nodes are represented by a

Poisson point process of density n, denoted as Pn, over a

unit area disk or square. Wan et al.[29] proved that for

any constant

e

> 0, it is a.a.s. that ð1

e

Þ ffiffiffiffiffiffiffiffiffiffi b0ln n pn q 6

q

_ðP_n_{Þ 6} ð1 þ

e

Þ ffiffiffiffiffiffiffiffiffiffiffi b0ln n pn q , where b0¼ 1= 23 ffiffi 3 p 2p

. The same authors

further improved asymptotic bounds on

q

ðPnÞ in[30].

Spe-ciﬁcally, they proved that for any constant c, the

asymp-totic probability of

q

ðPnÞ 6 ffiffiffiffiffiffiffiffiffiffiffiffiffi b0ln nþc pn q is at least 1 1 1=b21=3 b0 2

ec _{and at most e}b02ec. In this paper, we

will apply similar techniques used by Wan et al.[29]to

derive the CTR for 3D greedy routing and the proposed restricted 3D greedy routing.

3. Preliminaries

In this section, we present our models and several use-ful results which are used by our analysis on critical trans-mission radius of 3D greedy routing.

3.1. Assumptions and notations

We consider a set V of n wireless sensor devices (called nodes hereafter) uniformly distributed in a compact and

convex 3D region D with unit-volume in R3_{. By proper}

scaling, we assume the nodes are represented by a Poisson

point process Pnof density n over a unit-volume cube D.

Each node knows its position information and has a

uni-form transmission radius r (orrn). Then the communication

network is modeled by a unit disk graph G(V, r), where two

nodes u and

_v

are connected if and only if their Euclidean

distance is at most r. Hereafter, we use ku

v

k to denote

For a set A R3_{, we use jAj to denote the volume of A}

and use @A to denote the topological boundary of A. Let B(x, r) denote the open sphere of radius r centered at x.

For any two points u;

v

2 R3, the intersection of two

spheres of radii ku

v

kcentered respectively at u and

v

, de-noted by Luv, is called the biconvex of u and

v

, i.e. Luv = B(u,k

u

v

k = 1, a straightforward calcula-tion yields that j Luvj¼512p. The volume of such a biconvex

with respect to the volume of a unit-volume ball is

5p=12 4p=3¼

5

16. Let b0¼165¼ 3:2. Then, the volume of a

bicon-vex with depth r is 1

b0

4 3

p

r3

. The following lemma gives a lower bound of the volume of two intersecting biconvexes.

Lemma 1. Assume R > 0 and a1;b1;a2;b22 R3 . Let z1¼ 1

2ða1þ b1Þ; r1¼ ka1 b1k; z2¼12ða2þ b2Þ, and r2= ka2

b2k. If r1;r22 12R; R ;kz1 z2k 6 ffiffiffi 3 p R; a1;b1RLa2b2, and

a2;b2RLa1b1, there exist a constant

c

such that

jLa1b1[ La2b2j jLa1b1j P

cR

2

kz1 z2k:

For any convex compact set C R3_{, we use C}

rto denote

the set of points in C that are away from @C by at least r. The

next lemma gives a lower bound of the volume of Cr.

Lemma 2. Given a convex compact set C R3_{with diameter}

at most d,

jCrj P jCj

pd

2r:

An

e

-tessellation is a technique that divides the 3D

space by vertical planes perpendicular to either x-axis or y-axis and horizontal planes perpendicular to z-axis into equal-size cubes, called cells, in which cells are with width

e

. Without loss of generality, we assume the origin is a cor-ner of cells. In a tessellation, a polycube is a collection of cells intersecting with a convex compact set. The x-span (and y-span, z-span, respectively) of a polycube is the dis-tance measured in the number of cells in the x-direction (and y-direction, z -direction, respectively). If the span of a convex compact set is s and the width of cells is l, the span of the corresponding polycube is at most ds/le + 1. We have the following lemma.

Lemma 3. If a convex compact set S consists of m cubes and

s

is a positive integer constant, the number of polycubes with

span at most

s

and intersecting with S isH(m).

3.3. Probabilistic preliminaries

The following lemma from[29]gives a lower bound for

the minimum of a collection of Poisson RVs.

Lemma 4 [29]. Assume that limn!1ln nkn ¼ b for some b > 1.

Let Y1;Y2; ; YInbe InPoisson RVs with means at least kn. If

In¼ oðn

ffiffiffiffiffiffiffiffi ln n p

Þ, then for any 1 < b0_<_b;_minIn

i¼1Yi>Lðb0Þ ln n

a.a.s.

Here, LðxÞ is deﬁned as a function L over (0,1) by

LðxÞ ¼ x/1_{ð1=xÞ when x P 1 and =0 otherwise. / is the}

function over (0, 1) deﬁned by /(x) = 1 x + x ln x and

/1is the inverse of the restriction of / to (0, 1]. It can

be veriﬁed that L is a monotonic increasing function of b.

At last, we state the Palm theory[31]on the Poisson

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Theorem 5 [31]. Let n > 0. Suppose k 2 N, and hðY; X Þ is a bounded measurable function deﬁned on all pairs of the form ðY; X Þ with X R3_{being a ﬁnite subset and Y being a subset}

of X , satisfying hðY; X Þ ¼ 0 except when Y has k elements. Then E X Y # Pn hðY; PnÞ " # ¼n k k!E½hðXk;Xk[ PnÞ

where the sum on the left side is over all subsets Y of the ran-dom Poisson point set Pn, and on the right side the set Xkis a

binomial process with k nodes, independent of Pn.

We need to estimate the number of subsets with some speciﬁed topology, e.g., two nodes are local minima w.r.t. each other. But it is not so easy to estimate this among Poisson point processes. The Palm theory allows us to place a set of random points ﬁrst and then estimate the expecta-tion over the Poisson point process. This technique will be

used in proof ofTheorem 7.

4. Delivery guarantee of 3D greedy routing

In this section, we study how to guarantee the packet delivery of 3D greedy routing. We ﬁrst prove that 3D gree-dy routing can guarantee the delivery on Delaunay trian-gulation. Then, we investigate the critical transmission radius of 3D greedy routing in random networks.

4.1. 3D greedy routing on delaunay trianglation

In a d-dimensional Euclidean space, a Delaunay

triangu-lation[32]is a triangulation Del(V) such that there is no

point in V inside the circum-hypersphere of any d-simplex in Del(V). For example, in 3D space the 3-simplex is a tet-rahedron, while in 2D scarce the 2-simplex is a triangle.

In[33], Morin proved that 2D greedy routing can guarantee

the packet delivery on Delaunay triangulation. Here, we extend his proof to 3D space.

Theorem 6. The 3-dimensional greedy routing can guarantee the packet delivery on any Delaunay triangulation Del(V). Proof. Assume that t is the destination. We ﬁrst prove that

every node

v

in Del(V) has a neighbor that is strictly closer

to t than

v

is. In other words, there is no local minimum for

Delaunay triangulation has been used as routing

topol-ogy for wireless ad hoc networks[34,24]. Since building

the Delaunay triangulation needs global information and the length of a Delaunay edge could be longer than the

max-imum transmission radius, both methods[34,24]use some

local structures to approximate the Delaunay triangulation. This can break the delivery guarantee of 3D greedy routing. 4.2. Critical transmission radius of 3D greedy

Next we will prove the following theorem on critical

transmission radius

q

ðPnÞ of 3D greedy routing in random

sensor networks.

Theorem 7. Let b0= 3.2 and n43

p

r3n

¼ ðb þ oð1ÞÞ ln n for some b > 0. Then, for 3D greedy routing,

1. If b > b0, then

q

ðPnÞ 6 rnis a.a.s.

2. If b < b0, then

q

ðPnÞ > rnis a.a.s.

To simplify the argument, we ignore boundary effects by assuming that there are nodes outside D with the same distribution. So, if necessary, packets can be routed through those nodes outside D.

4.2.1. Upper bound of Theorem 7

The upper bound in Theorem 7is going to be proved

through a technique called minimal scan statistics. For a ﬁ-nite point set V and a real number r > 0, we deﬁne

SðV; rÞ ¼ min

u;v2D;kuvk¼r#ðV \ LuvÞ:

S(V, r) is the minimal number of nodes of V that can be cov-ered by a biconvex with depth r. In other words, S(V, r) is the minimal number of ‘‘better” neighboring nodes that any intermediate node u can choose for any possible desti-nation

v

. As proved in[29], SðPn;rnÞ > 0 implies the event

q

ðPnÞ 6 rn. Therefore, it sufﬁces to prove that SðPn;rnÞ > 0

is a.a.s. Instead, we now prove a stronger result shown in the following lemma.

Lemma 8. Suppose that n4

3

p

r3n

¼ ðb þ oð1ÞÞ ln n for some b> b0. Then for any constant b12 (b0,b), it is a.a.s. that

SðPn;rnÞ > L b1 b0 ln n:

t

v

u

f

Fig. 3. Any nodevcan ﬁnd a neighbor u which is strictly closer to t thanv is.

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Proof. To have the lower bound of minimal scan statistics, we apply the tessellation technique to discretize the scan-ning process. We tessellate the deployment region by properly choosing cell size such that: (1) each copy of the

biconvex contains a polycube with volume at least

g

ln n

n

for some

g

> 1, and (2) the number of polycubes is O n

ln n

.

Let d ¼pffiffiffi3rn which is the largest distance between any

two points in a biconvex. For a given b1, choose a constant

b22 (b1,b), and let

e

¼27b40 1

b2

b

. Consider an

e

d-tessel-lation. (Note that

e

is chosen such that each copy of the

biconvex contains a polycube with volume at least

g

n

ln n

for some

g

> 1.) To prove this inequality, it is sufﬁcient to show that any biconvex of two points in D that are

sepa-rated by a distance of rncontains a polycube with span at

most1

eand volume at least

b2 b0 4 3

p

r3n 1 b.

For a biconvex L, let P denote the polycube induced by Lpﬃﬃ₃

ed. Then, P # L, and the span of P is at most d2

ﬃﬃ 3 p ed ed l m þ 1 <1

e. By Lemma 2 and the fact that jLj ¼43

p

r 3 nb10¼ 4 9pﬃﬃ3

p

d 3 1 b0, we have jPj P jLpﬃﬃ3edj P jLj

pd

2 ðpffiffiffi3

edÞ ¼ jLj

pffiffiffi3

epd

3 ¼ jLj 27b0 4

ejLj ¼ jLj 1

27b0 4

e

¼b2 bjLj ¼b2 b0 4 3

pr

3 n ₁ b:

Let Indenote the number of polycubes in D with span at

most 1

e and volume at least

b2 b0 4 3

p

r 3 n 1 b¼ b2 b0þ oð1Þ ln n n,

and Yibe the number of nodes on the ith polycubes. Then

Yiis a Poisson RV with rate at least bb20þ oð1Þ

ln n. Since

the number of cells in D is O 1

ed 3 ¼ O n ln n , byLemma 3, In¼ O ln nn

. ByLemma 4, it is a.a.s. that

minIn i¼1Yi ln n PL b2 b0 >L b1 b0 : Thus, SðPn;rnÞ P min In i¼1 Yi>L b1 b0 ln n:

4.2.2. Lower bound of Theorem 7

The second half ofTheorem 7can be proved by showing

that if rn¼ ffiffiffiffiffiffiffiffiffiffi 3b ln n 4pn 3 q

for any b < b0, there a.a.s. exists local

minima. The space is going to be tessellated into equal-size cube cells. For each cell, an event that implies the existence of local minima in the cell is introduced, and a lower bound for the probability of the event is derived. Since these events are identical and independent over cells, we can estimate a probability lower of existence of local minima. By showing the lower bound is a.a.s. equal to 1, we prove

the second part ofTheorem 7. The detail is given below.

Let b1 and b2 be two positive constants such that

max 1

8b0;b

<b1<b2<b0. In addition, let R1 and R2be

given by n 4 3

p

R 3 1 ¼ b1ln n and n 43

p

R 3 2 ¼ b2ln n, respec-tively. Since1 8b0<b1<b2<b0, we have12R26R16R2. Di-vide D by 4 ffiffiffiffiffiffiln n np 3 q

-tessellation. Let Indenote the number

of cells fully contained in D. Here we have In¼ O ln nn

. For

each cell fully contained in D, we draw a ball of radius

1 2 ffiffiffiffiffiffi ln n np 3 q

at the center of the cell. For 1 6 i 6 In, let Eibe the

event that there exists two nodes X; Y 2 Pnsuch that their

midpoint is in the ith ball, their distance is between R1and

h1(T = {x1, x2},V) = 1 only if 21ðx1þ x2Þ 2 A; R16kx1

x2k 6 R2, and there is no other node of V in Lx1x2; otherwise,

h1(T, V) = 0. In addition, under Boolean addition, for any

{x1, x2, x3} # V, let h2({x1, x2, x3}, V) = h1({x1, x2}, V) h1({x1,

x3}, V) + h1({x2, x1}, V) h1({x2, x3}, V) + h1({x3, x1}, V)h1({x3,

x2}, V); for any {x1, x2,x3, x4} # V, let h3({x1, x2, x3, x4},V) =

h1({x1, x2}, V) h1({x3, x4}, V) + h1({x1, x3}, V) h1({x2, x4},V) +

h1({x1, x4}, V) h1({x2, x3}, V). E1is the event that there

ex-ists two nodes X; Y 2 Pn such that h1ðfX; Yg; PnÞ ¼ 1. In

the remaining of this subsection, we use X0

1;X 0 2;X 0 3and X 0 4

to denote elements of Pn. Let F01 X 0 1;X 0 2

be the event that h1 X01;X02 ;Pn ¼ 1; F0 2 X01;X02;X03

be the event that h2 X01;X 0 2; X0 3g; PnÞ ¼ 1; and F03 X 0 1;X 0 2;X 0 3;X 0 4 be the event that h3 X01;X 0 2;X 0 3;X 0 4 ;Pn ¼ 1. Applying Boole’s inequalities, we have Pr½E1 P X fX0 1;X02g # Pn Pr F0 1 X 0 1;X 0 2 X fX0 1;X 0 2;X 0 3g # Pn Pr F02 X 0 1;X 0 2;X 0 3 X fX0 1;X02;X03;X04g # Pn Pr F0 3 X 0 1;X 0 2;X 0 3;X 0 4 : ð2Þ

For the sake of clarity, we use X1,X2,X3 and X4to denote

independent random points with uniform distribution over

D and independent of Pn. Let F1 be the event that

h1ðfX1;X2g; fX1;X2g [ PnÞ ¼ 1; F2 be the event that

h2ðfX1;X2;X3g; fX1;X2;X3g [ PnÞ ¼ 1, and F3 be the event

that h3ðfX1;X2;X3;X4g; fX1;X2;X3;X4g [ PnÞ ¼ 1. According

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X fX0 1;X02g #Pn Pr F0 1 X 0 1;X 0 2 ¼ E X fX0 1;X02g#Pn h1 X01;X 0 2 ;Pn 2 4 3 5 ¼n 2 2!E½h1ðfX1;X2g; fX1;X2g [ PnÞ ¼n 2 2Pr½F1; ð3Þ X fX0 1;X02;X03g # Pn Pr F0 2 X 0 1;X 0 2;X 0 3 ¼ E X fX0 1;X02;X03g # Pn h2 X01;X 0 2;X 0 3 ;Pn 2 4 3 5 ¼n 3 3!E½h2ðfX1;X2;X3g; fX1;X2;X3g [ PnÞ ¼n 3 2Pr½F2; ð4Þ and X fX01;X 0 2;X 0 3;X 0 4g # Pn Pr F0 3 X 0 1;X 0 2;X 0 3;X 0 4 ¼ E X fX0 1;X 0 2;X 0 3;X 0 4g # Pn h3 X01;X 0 2;X 0 3;X 0 4 ;Pn 2 4 3 5 ¼n 4 4!E½h3ðfX1;X2;X3;X4g; fX1;X2;X3;X4g [ PnÞ ¼n 4 8Pr½F3: ð5Þ

From Eqs.(2)–(5), we have

Pr½E1 P n2 2Pr½F1 n3 2Pr½F2 n4 8Pr½F3: ð6Þ

In the next, we will derive the probabilities of F1, F2, and

F3. Let S1 denote the set fðx1;x2Þj12ðx1þ x2Þ 2 A; R16

kx1 x2k 6 R2:g. We have Pr½F1 ¼ ZZ S1 Pr½F1jX1¼ x1;X2¼ x2 dx1dx2 ¼ ZZ S1 enjL_x1x2j_dx 1dx2¼ ZZ S1 enb01ð 4 3pkx1x2k3Þ_dx 1dx2: Let z ¼x1þx2 2 and r ¼12kx1 x2k. Then, Pr½F1 ¼ Z z2A Z R2 2 r¼R12 eb0n323pr3_32pr2_drdz ¼ Z z2A Z R2 2 r¼R12 eb0n323pr3_d 32 3

pr

3 dz ¼ b0 ne n b0 32 3pr3 R2 2 r¼R12 0 @ 1 AjAj ¼ b0 6n2 n b1 b0_nb2b0 ln n: ð7Þ

Let S2 denote the set

ðx1;x2;x3Þ x1þx2 2 ; x1þx3 2 2 A; R16kx1 x2k 6 R2;x1;x2RLx1x3; R16kx1 x3k 6 R2;x1;x3RLx1x2 8 < : 9 = ;: ApplyingLemma 1, if (x1, x2, x3) 2 S2, we have

Pr½F2 ¼ ZZZ S2 Pr F2 Xi¼ xi

8

i ¼ 1; 2; 3 2 4 3 5dx1dx2dx3 6₃ ZZZ S2 enjL_x1x2[L_x1x3j_dx 1dx2dx3 6₃ ZZZ S2 en 1 b0 4 3pkx1x2k3þcR22kx1þx22 x1þx32 k dx1dx2dx3:

Let z1¼x1þx₂2;z2¼x1þx₂3;r ¼kx1x₂2k, and

q

= kz1 z2k. Then,

Pr½F2 6 3 Z z12A Z R2 2 r¼R12 Z z22A en 1 b0 32 3pr3þcR 2 2kz1z2k 256pr2_drdz 1dz2 624 Z z12A ZR2 2 r¼R12 eb0n 32 3pr 3 ð _{Þd 32} 3

pr

3 dz1 Z z22A ecnR2 2kz1z2k_dz 2 6₂₄ Z z12A ZR2 2 r1¼R12 eb0nð323pr3Þd 32 3

pr

3 dz1 Z 1 q¼0 ecnR2 2q4pq2dq ¼ 24 b0 6n2 n b1_b0 nb2b0 ln n _8p

cnR

2 2 3 0 B @ 1 C A ¼ 32pb0

c

3 _nR3 2 2 n3 nb1b0_n b2 b0 ln n: ð8Þ

Let S3 denote the set

ðx1;x2;x3;x4Þ x1þx2 2 ; x3þx4 2 2 A; R16kx1 x2k 6 R2;x1;x2RLx3x4; R16kx3 x4k 6 R2;x3;x4RLx1x2 8 < : 9 = ;. ApplyingLemma 1, if (x1, x2, x3, x4) 2 S3, we have

Pr½F3 ¼ ZZ ZZ S3 Pr F3 Xi¼ xi; 8i ¼ 1; 2; 3; 4 2 4 3 5dx1dx2dx3dx4 6₃ ZZ ZZ S3 enjLx1x2[Lx3x4j_dx 1dx2dx3dx4 6₃ ZZ ZZ S3 en 1 b043pkx1x2k3þcR22kx1þx22 x3þx42 k dx1dx2dx3dx4: Let z1¼x1þx₂2;r1¼kx1x₂2k;z2¼x3þx₂4;r2¼kx3x₂4k, and

q

= kz1 z2k. Then, Pr [F3]

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n1 b1 b0 ! 1:

This complete the proof of the second half ofTheorem 7.

5. Energy-efﬁciency of 3D greedy routing

Since Flury and Wattenhofer[2]showed no

determinis-tic localized routing protocol is energy-efficient in 3D networks, the simple 3D greedy routing may lead to energy-inefficient paths in the worst case. Therefore, we are interested in designing a localized routing method that is energy-efficient with high probability for random

3D networks. Here a routing method is

energy-efficient with high probability if (1) with high probability, the routing method will find a path successfully; and (2) with high probability, the found path is energy-efficient.

5.1. Energy-efﬁcient restricted 3D greedy routing (ERGrd) Our energy-efﬁcient localized 3D routing method is a variation of classical 3D greedy routing and an extension

of a localized routing method[13]we designed for 2D

net-works. In 3D greedy routing, current node u selects its next hop neighbor based purely on its distance to the destina-tion, i.e., it sends the packet to its neighbor who is closest to the destination. However, such choice might not be the most energy-efﬁcient link locally, and the overall route might not be globally energy-efﬁcient too. Therefore, our routing method use two concepts energy mileage and

re-stricted region to reﬁne the choices of forwarding nodes in 3D greedy routing.

Energy mileage. Given a energy model e(x), energy mileage is the ratio between the transmission distance and the

en-ergy-consumption of such transmission, i.e., x

eðxÞ. Let r0be

the value such that r0

eðr0Þ¼ maxx

x

eðxÞ. We call r0as the

maxi-mum energy mileage distance1_{under energy model e(x). We}

assume that the energy mileage x

eðxÞis an increasing function

when x < r0 and a decreasing function when x > r0. This

assumption is true for most of commonly used energy mod-els. For example, if e(ku

v

k) = ku

v

k2_{+ c is the energy used by}

sending message from u to

v

, the maximum energy mileage

distance r0¼ ffiffiffic

p

. Our 3D localized routing greedily selects the neighbor who can maximize the energy mileage as the forwarding node.

Restricted region. Instead of selecting the forwarding node from all neighbors of current node u (a unit ball in 3D

as shown in Fig. 4a), our 3D routing method prefers the

forwarding node

v

inside a smaller restricted region. The

region is deﬁned inside a 3D cone with an angle parame-ter

a

<

p

/3, such that angle \vut 6

a

, as shown inFig. 4b.

The use of

a

(restricting the forwarding direction) is to

bound the total distance of the routing path. Then the re-stricted region is a region inside this 3D cone and near the

maximum energy mileage distance r0, such that every

node

v

inside this area satisﬁes

g

1r06ku

v

k 6

g

2r0, as

shown inFig. 4b. Here,

g

1and

g

2are two constant

param-eters. This can help us to prove the energy-efﬁciency of the route.

Notice that both these ideas are not completely new. Restricted region with an angle has been used in some localized routing methods, such as

nearest/far-thest neighbor routing [34], while concepts similar to

energy mileage have been used in some energy-aware

localized routing methods [11,12,35]. However,

combin-ing both of these techniques to guarantee

energy-efﬁ-ciency is ﬁrst done in our previous work [13] for 2D

networks. In this paper, we further adapt them into 3D routing.

Our energy-efﬁcient localized 3D routing protocol is

gi-ven inAlgorithm 1. There are four parameters used by our

method. Three adjustable parameters 0 <

a

<p₃ and

g

1< 1 <

g

2deﬁne the restricted region, while r0is the best

energy mileage distance based on the energy model. For example, the following setting of these parameters can

be used for energy model eðxÞ ¼ x2_{þ c :}

_a

_¼p

4;r0¼ ffiffiffic

p ;

g

1¼ 1=2 and

g

2= 2. Hereafter, we denote the routing

algo-rithm, energy-efﬁcient restricted greedy, as ERGrd if no

gree-dy routing (Grd) is used when no node

v

satisﬁes that

\_{vut 6}

a

_{. If Grd is applied afterward, then the routing} pro-tocol is denoted ERGrd+Grd. Notice that if Grd fails to ﬁnd a

forwarding node, randomized scheme [2] could also be

applied.

The path efﬁciency of 3D ERGrd is given by the follow-ing two theorems. The detail proofs of these two theorems

1

Here, we assume that d eðxÞ x

=dx is monotone increasing, thus, r0is unique.

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are exactly the same with the proofs of Theorems 1–3 in

[13]for 2D network, thus are ignored here.

Theorem 9. When 3D ERGrd routing indeed ﬁnds a path PERGrd(s,t) from the source s to the target t, the total Euclidean

length of the found path is at most dkt sk where d ¼ 1

12 sina

2,

thus, a constant factor of the optimum.

Theorem 10. When 3D ERGrd routing indeed ﬁnds a path PERGrd(s, t) from the source s to the target t, the total

energy-consumption of the found path is within a constant factor

r

of the optimum. When r0Pr,

r

depends on

a

; otherwise,

depends on

g

1,

g

2and

a

.

Algorithm 1. Energy-efﬁcient restricted 3D greedy routing (3D ERGrd)

1: while node u receives a packet with destination t

do

2: if t is a neighbor of u then

3: Node u forwards the packet to t directly.

4: else if there are neighbors inside the restricted

region and r0< r then

5: Node u forwards the packet to the neighbor

v

such that its energy mileage kuvk

eðkuvkÞis maximum

among all neighbors w inside the restricted region, as shown inFig. 4b.

6: else if there are neighbors inside the 3D cone

then

7: Node u ﬁnds the node

v

inside the 3D cone

(Fig. 4c) with the minimum kt

v

k.

8: else

9: Greedy routing (Fig. 4d) is applied, or the

packet is simply dropped.

10: end if

11: end while

5.2. Critical transmission radius of 3D ERGrd

Notice that 3D ERGrd routing may fail, as all other gree-dy-based methods do, when an intermediate node cannot ﬁnd a better neighbor to forward the packet. We now study the critical transmission radius for ERGrd routing in ran-dom 3D wireless networks. Given a set of nodes V distrib-uted in a region D, the critical transmission radius

q

(V) for successful routing by 3D ERGrd is

max

u;v w:\wuminv6akw uk: ð10Þ

By setting the r =

q

(V), ERGrd can always ﬁnd a forwarding

node inside the 3D cone region, thus can guarantee its packet delivery. Now, we can prove a similar result for 3D ERGrd as we did for 3D greedy routing.

Theorem 11. Let b0¼1cos2 aand n 43

p

ffiffiffi 3 p

rn.

Again the same tessellation technique can be used. The

only difference is that jLj ¼4

r3n

₁

b. The remaining parts are the same with

the proof ofLemma 8.

5.2.2. Lower bound of Theorem 11

We now show that, if rn¼

ffiffiffiffiffiffiffiffiffiffi

3b ln n 4pn

3

q

for any b < b0, a.a.s.,

there are two nodes u and

v

such that we cannot ﬁnd a

node w for forwarding by node u, i.e., there does not exist node w inside the 3D cone. Again we partition the space

using equal-size cubes (called cells) with side-length

g

rn

for a constant 0 <

g

to be speciﬁed later. Thus the number

of cells, denoted by Inhere, that are fully contained inside

the compact and convex region D with unit-volume, is

H 1 g3_r3 n ¼H n ln n

. Let Eu,v denote the event that no forward-ing node w (in the 3D cone) exists for node u to reach node

v

. Then to prove our claim, it is equivalent to prove that the probability of at least one of the event Eu,v happens a.a.s.,

v

will also be inside this cell, and the shaded

3D cone where the possible forwarding node could locate is also inside this cell. Thus, events Eu1;v1and Eu2;v2are

inde-pendent if u1and u2are selected as above from different

cells.

For each cell i, we compute the probability that event Eui;vi happens, where uiis selected from the shaded cube

of cell i and

_v

i is selected such that rn< k

v

i uik 6

(1 + d)rn. Recall that for any region A, the probability that

it is empty of any nodes is enjAj_{. Clearly, the probability}

that node uiexists is 1 enðg22dÞ

3_r3

nsince the shared cube

has volume ð

g

2 2dÞ3r3

n; the probability that node

v

i

ex-ists is 1 en4 3pðð1þdÞ

3

1Þr3

n _{since the torus has volume}

4

3

p

ðð1 þ dÞ 3

1Þr3

n. Given node ui and

v

i, the probability

that event Eui;vi happens is e

n2

3pð1cosaÞr3n_¼

eb=b0ln n_{¼ n}b=b0. Consequently, event E_u,_{v happens for}

some node pairs uiand

v

iis PrðEui;viÞ P 1 e

nðg22dÞ3_r3 n 1en4 3pðð1þdÞ31Þr3n nb=b0_{¼ ð1n}bðg22dÞ33=4p_Þð1nbðð1þdÞ31Þ_Þ

nb=b0_{. Thus, the probability that ERGrd routing fails to ﬁnd a}

path for some source/destination pairs is Pr (at least one of events Eu,v happens) PPr (at least one of Eu_i;vi happens)=

1-Pr (none of Eui;vi happens)= 1ð1PrðEui;viÞÞ

In_{¼ 1}

eInlnð1PrðEui;viÞÞP1eInPrðEui;viÞ. Notice that I_nPrðE_u i;viÞ PH n lnn ð1nbðg22dÞ33=4p_Þ _ð1nbðð1þdÞ31Þ_Þnb=b0 _’n1b=b0 lnn ,

which goes to 1 as n ? 1 when b < b0,

g

2

2d > 0, and d > 0. This can be easily satisﬁed, e.g., d = 1,

g

= 5. Thus, limn!11eInPrðEui;viÞ_{¼ 1. This completes the proof.}

6. Simulation

6.1. Critical transmission radius for random networks We have analyzed the theoretical bounds of the critical transmission radius for 3D greedy routing and 3D ERGrd routing. To conﬁrm our theoretical analysis, we conduct several simulations to see what is the practical value of

transmission radius rn such that greedy can guarantee

the packet delivery with high probability in random net-works. We randomly generate 1000 networks with n nodes in a 100 100 100 cubic region, where n is from 50 to 500. For each network V, we compute the critical

transmis-sion radius

q

(V) of 3D greedy and 3D ERGrd by their

def-initions (Eqs.(1) and (10)). For 3D ERGrd routing, we let

a

=

p

/6 or

a

=

p

/4.Fig. 6gives the histograms of the

distri-bution of

q

(V) of these 3D greedy routing methods for 1000

random networks.Fig. 7show the probability distribution

function of

q

(V) for these methods. It is clear that the CTRs of all methods satisfy a transition phenomena, i.e., there is

a radius r0such that 3D Grd/ERGrd can successfully deliver

all packets when rn> r0and cannot deliver some packets

when rn< r0. Notice that the transition becomes faster

Fig. 5. Illustrations of the proof of lower bound: (a) a cubic cell and the region where we select a node u; (b) the event that node u cannot ﬁnd a forwarding node w to reach a nodev.

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when the number of nodes increases. This confirms our theoretical analysis on the existence of CTR. In addition, from these figures, we can find that larger node density al-ways leads to smaller value of CTR. The practical value of

q

(V) is larger than the theoretical bound in our analysis,

since the theoretical bound is standing for n ? 1. How-ever, the practical value will approach the theoretical bound with the increasing of n . For example, when

n = 500,_{ffiffiffiffiffiffiffiffiffiffiffi} the theoretical bound of 3D greedy is

3b0ln n

4pn

3

q

100 ¼ 0:212 100 ¼ 21:2 for a 100 100 100

cubic region. FromFig. 7a, the CTR of 3D greedy is around

25, which already becomes very near the theoretical bound. . Compared the two cases of ERGrd method with

a

=

1and

g

2are 1/2 and 2. By

set-ting various transmission radii, we generate random net-works with 100 wireless nodes again in a 100 100

100 cubic region.Fig. 8shows a set of random networks

generated on the same set of nodes. We select 100

Fig. 6. The distributions ofq(V) for random networks with 100–500 nodes. (a–e) For 3D greedy, (f–j) for 3D ERGrd witha=p/6, and (k–o) for 3D ERGrd with

a=p/4. 20 25 30 35 40 45 50 55 60 0 0.2 0.4 0.6 0.8 1 Transmission Radius, r Probability n=50 n=100 n=200 n=300 n=400 n=500 30 40 50 60 70 80 90 100 0 0.2 0.4 0.6 0.8 1 Transmission Radius, r Probability n=50 n=100 n=200 n=300 n=400 n=500 20 30 40 50 60 70 80 0 0.2 0.4 0.6 0.8 1 Transmission Radius, r Probability n=50 n=100 n=200 n=300 n=400 n=500

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connected random networks for each setting, then for each network we randomly select 100 source–destination pairs and test ﬁve greedy-based 3D routing. All results presented hereafter are average values over all routes and networks. In all ﬁgures, ERGrd+G denotes ERGrd+Grd, which is the re-stricted greedy routing with classical greedy routing as the back up.

Fig. 9illustrates the average delivery ratios of the ﬁve routing methods. Clearly, the delivery ratio increases when rnincreases. After rnis larger than a certain value, it always

guarantees the delivery. This also conﬁrms our theoretical analysis of CTRs. In addition, we can conclude that the CTR for 3D greedy routing (approaching 100% delivery ratio when rnis around 35 inFig. 9) is just a little bit larger than

the CTR for connectivity (network becomes connected

when rnis around 30 inFig. 8). Notice that ERGrd methods

without greedy backup have lower delivery ratio under the same circumstance, since they have smaller region to se-lect the next hop node. With greedy backup, the delivery 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 x y z 0 20 40 60 80 100 0 20 40 60 80 1000 20 40 60 80 100 x y z 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 x y z 0 20 40 60 80 100 0 20 40 60 80 100 y z 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 x y z 0 20 40 60 80 100 0 20 40 60 80 100 y z 0 20 40 60 80 100 0 20 40 60 80 100 y z 0 20 40 60 80 100 0 20 40 60 80 100 y z 0 20 40 60 80 100 x 0 20 40 60 80 100 x 0 20 40 60 80 100 x 0 20 40 60 80 100 x

Fig. 8. Network topologies with 100 nodes when rnis from 10 to 80.

25 30 35 40 45 50 55 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Transmission Radius

Average Delivery Ratio

Greedy ERGrdα=π/6 ERGrdα=π/4 ERGrd+G α=π/6 ERGrd+G α=π/4

Fig. 9. Average delivery ratios of 3D greedy and 3D ERGrd in random networks. 25 30 35 40 45 50 55 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.1 Transmission Radius

Average Length Stretch Factor

Greedy ERGrdα=π/6 ERGrdα=π/4 ERGrd+Gα=π/6 ERGrd+Gα=π/4 25 30 35 40 45 50 55 1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2 Transmission Radius

Average Energy Stretch Factor

Greedy ERGrdα=π/6 ERGrdα=π/4 ERGrd+Gα=π/6 ERGrd+Gα=π/4

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ratios of ERGrd+Grd methods are almost the same with those of Grd (simple 3D greedy).

Fig. 10a and b illustrate the average length stretch fac-tors and energy stretch facfac-tors of all routing methods, respectively. Here, the length/energy stretch factor of a path from node s to node t is the ratio between the total length/ energy of this path and the total length/energy of the opti-mal path connecting s and t. Sopti-maller stretch factor of a routing method shows better path efﬁciency. For the

length stretch factor, the ERGrd with

a

=

p

/6 has the best

length efﬁciency. It is surprising that with

a

=

p

/4 the

length of ERGrd path could be longer than simple greedy. However, when considering the energy-efficiency, all ERGrd methods can achieve better path efficiency than simple greedy method. Notice that smaller restricted re-gion leads to better path efficiency, however it also has lower delivery ratio. Therefore, it is a trade-off between path efficiency and packet delivery. It is also clear that when the network is dense (with large transmission ra-dius), ERGrd and ERGrd+Grd are almost the same, since ERGrd can always find nodes inside the 3D cone. Notice that all the stretch factors in our simulations are near to 1.0, this is due to the uniform distribution of nodes. In pra-tice, the stretch factors of simple greedy routing could be very large in the worst case.

Besides deploying random networks in a cubic region, we also performed simulations for networks deployed in a spherical region. The conclusions from these simulations are consistent with the simulations for random network deployed in cubic region.

7. Conclusion

In this paper, we study the design of 3D greedy routing for large-scale sensor networks. We ﬁrst provide a theoret-ical analysis on the crittheoret-ical transmission radius for 3D gree-dy routing which leads to a delivery-guaranteed 3D localized routing. We theoretically prove that for a random 3D network, formed by nodes that are generated by a Poisson point process of density n over a convex compact region of unit volume, the critical transmission radius for 3D greedy routing is a.a.s.

ffiffiffiffiffiffiffiffiffiffiffi

3b0ln n

4pn

3

q

, where b0= 3.2. This

the-oretical result answers a fundamental question about how large the transmission radius should be set in a 3D net-works, such that the greedy routing guarantees the deliv-ery of packets between any two nodes. We then refine the 3D greedy routing to a new localized routing protocol 3D ERGrd, which achieves the energy-efficiency by limiting its choice inside a restricted region and picking the node with best energy mileage. We also derive its critical trans-mission radius in random networks. Finally, we conduct extensive simulations to confirm our theoretical results. We believe that the proposed energy-efficient localized routing protocol is crucial for achieving sustainable and scalable in large-scale sensor networks.

Acknowledgment

The work of Y. Wang and M. Huang was supported in part by the US National Science Foundation under Grant

No. CNS-0721666, CNS-0915331, and CNS-1050398. This work of C.-W. Yi was partially supported by NSC under Grant No. NSC97-2221-E-009-052-MY3 and NSC98-2218-E-009-023, and by the MoE ATU plan. His research is also supported by the Information and Communications Research Laboratories (ICL), Industrial Technology Re-search Institute (ITRI), Taiwan, Republic of China (ITRI Grant Project Code 9365C52200). The work of F. Li was partially supported by the National Natural Science Foun-dation of China (NSFC) under Grant 60903151.

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Yu Wang is an Associate Professor of com-puter science at the University of North Carolina at Charlotte. He received his Ph.D. degree in Computer Science from Illinois Institute of Technology in 2004, his B.Eng. degree and M.Eng. degree in computer science from Tsinghua University, China, in 1998 and 2000. His research interest includes wireless networks, ad hoc and sensor networks, mobile computing, complex networks, and algorithm design. He has published more than 80 papers in peer-reviewed journals and conferences. He has served as program chair, publicity chair, and program committee member for several international conferences (such as IEEE INFOCOM,

IEEE IPCCC, IEEE GLOBECOM, IEEE ICC, and IEEE MASS). He was the pro-gram co-chair of the ﬁrst/second ACM International Workshop on Foun-dations of Wireless Ad Hoc and Sensor Networking and Computing (FOWANC 2008/2009), and was the program co-chair of the 26th IEEE International Performance Computing and Communications Conference (IEEE IPCCC 2007). He is a recipient of Ralph E. Powe Junior Faculty Enhancement Awards from Oak Ridge Associated Universities in 2006 and a recipient of Outstanding Faculty Research Award from College of Computing and Informatics at UNC charlotte in 2008. He is a member of the ACM and a senior member of the IEEE, and IEEE Communications Society.

Chih-Wei Yi received his PhD degree from the Illinois Institute of Technology in 2005, and BS and MS degrees from the National Taiwan University in 1991 and 1993, respectively. He is currently an Associate Professor in Com-puter Science at the National Chiao Tung University. He is a member of the IEEE and the ACM. He had been a Senior Research Fellow of the Department of Computer Science, City University of Hong Kong. He was awarded the Outstanding Young Engineer Award by the Chinese Institute of Engineers in 2009. His research focuses on wireless ad hoc and sensor networks, vehicular ad hoc networks, network coding, and algorithm design and analysis.

Minsu Huang received his BS degree in com-puter science from Central South University in 2003 and his MS degree in computer science from Tsinghua University in 2006. He is cur-rently a PhD student in the University of North Carolina at Charlotte, majoring in computer science. His current research focuses on wire-less networks, ad hoc and sensor networks, and algorithm design.

Fan Li received the PhD degree in computer science from the University of North Carolina at Charlotte in 2008, M.Eng. degree in elec-trical engineering from the University of Delaware in 2004, the M.Eng. degree and B.Eng. degree in communications and infor-mation system from Huazhong University of Science and Technology, China. She is cur-rently an associate Professor of school of computer science at Beijing institute of Technology. Her current research focuses on wireless networks, ad hoc and sensor net-works, and mobile computing.