562 IEEE COMMUNICATIONS LETTERS, VOL. 11, NO. 7, JULY 2007
Greedy Anti-Void Routing Protocol for Wireless Sensor Networks
Wen-Jiunn Liu, Student Member, IEEE, and Kai-Ten Feng, Member, IEEE
Abstract— The unreachability problem (i.e. the so-called void
problem) which exists in the greedy routing algorithms has been studied for the wireless sensor networks. However, most of the current research work can not fully resolve the problem (i.e. to ensure the delivery of packets) within their formulation. In this letter, the Greedy Anti-void Routing (GAR) protocol is proposed, which solves the void problem by exploiting the boundary finding technique for the Unit Disk Graph (UDG). The proposed Rolling-ball UDG boundary Traversal (RUT) is employed to completely guarantee the delivery of packets from the source to the destination node. The proofs of correctness for the proposed GAR protocol are also given at the end of this letter.
Index Terms— Greedy routing, void problem, unit disk graph,
localized algorithm, wireless sensor network.
I. INTRODUCTION
A
Wireless Sensor Network (WSN) consists of Sensor Nodes (SNs) with wireless communication capabilities for specific sensing tasks. Due to the limited available re-sources, efficient design of localized routing protocols [1] becomes a crucial subject within the WSNs. How to guar-antee delivery of packets is considered an important issue for the localized routing algorithms. The well-known Greedy Forwarding (GF) protocol [2] is considered a superior scheme with its low routing overheads. However, the void problem [3] that occurs within the GF algorithm will fail to guarantee the delivery of data packets. Several localized routing algorithms as surveyed and proposed in [4] resolve the void problem by using the planar graphs. Nevertheless, the usage of the planar graphs has significant pitfalls due to the removal of critical communication links [5]; while the adoption of the Unit Disk Graph (UDG) for modeling the underlying network is suggested. A representative UDG-based greedy scheme, i.e. the BOUNDHOLE algorithm [6], forwards the packets around the network holes by identifying the locations of the holes. However, the delivery of packets can not be guaranteed in the BOUNDHOLE scheme even if a route exists from the source to the destination node. In this letter, the Greedy Anti-void Routing (GAR) protocol is proposed to completely resolve the void problem based on the UDG setting. The Rolling-ball UDG boundary Traversal (RUT) scheme is utilized within the GAR algorithm with the assurance for packet delivery.Manuscript received March 2, 2007. The associate editor coordinating the review of this letter and approving it for publication was Prof. H.-H. Chen. This work was supported in part by the National Science Council (NSC) under Grant 95-2218-E-009-014, the MOE ATU Program 95W803C, and the MediaTek Research Center at the National Chiao Tung University.
W.-J. Liu and K.-T. Feng are with the Department of Communication Engineering, National Chiao Tung University, Hsinchu, Taiwan (e-mail: [email protected]; [email protected]).
Digital Object Identifier 10.1109/LCOMM.2007.070311.
sV sA sB sY sX R GAR BOUNDHOLE 0 ND NS NV NA NB NE NF NG NH NX NY NZ NW NU N0 N1 N2 N3 N4 N5 N6 N7 N8 N9 d(PNS, PND) d(PNV, PND) d(PNY, PND)
Fig. 1. The example routing paths constructed by using the GAR and the BOUNDHOLE algorithms under the existence of the void problem: (NS, ND) is the transmission pair and NV is the Void Node. Node NX is within the transmission range ofNB; while it is out of the range ofNA and NE. The GAR protocol utilizes the RUT scheme (with red solid arcs denoted as the trajectory of the SPs); while the minimal angle criterion is employed by the BOUNDHOLE algorithm. The resulting paths obtained from these two schemes are {NS, NV, NA, NB, NX, NY, NZ, ND} using the GAR protocol and{NS, NV, NA, NE, NF, NG, NH, NV} by adopting the BOUNDHOLE algorithm, which is observed to be undeliverable. The blue-shaded region associated with each SN is utilized to determine if the SN is a Void Node or not.
II. NETWORKMODEL ANDPROBLEMSTATEMENT Consider a set of SNs N = {Ni| ∀ i} within a two-dimensional Euclidean plane, the locations of the set N, which can be acquired by their own positioning systems, are represented by the set P = {PNi| PNi = (xNi, yNi), ∀i}. It is assumed that all the SNs are homogeneous and equipped with omni-directional antennas. The set of closed disks defining the transmission ranges of N is denoted as D = {D(PNi, R) | ∀ i}, where D(PNi, R) = {x | x − PNi ≤
R, ∀ x ∈ R2}. It is noted that PNi is the center of the closed disk with R denoted as the radius of the transmission range for each Ni. Therefore, the network model for the WSNs
can be represented by a UDG as G(P, E) with the edge set E = {Eij| Eij = (PNi, PNj), PNi ∈ D(PNj, R), ∀ i = j}. The edge Eij indicates the unidirectional link from PNi to
PNj whenever the position PNi is within the closed disk region D(PNj, R). Moreover, the one-hop neighbor table for each Ni is considered available via the neighbor information 1089-7798/07$25.00 c 2007 IEEE
LIU and FENG: GREEDY ANTI-VOID ROUTING PROTOCOL FOR WIRELESS SENSOR NETWORKS 563
acquisition [7] asTNi = {PNk| PNk∈ D(PNi, R), ∀ k = i}. In the Greedy Forwarding (GF) algorithm, it is assumed that the source nodeNS is aware of the location of the destination
node ND. If NS wants to transmit packets to ND, it will choose the next hopping node from its TNS which (i) has the shortest Euclidean distance to ND among all the SNs in TNS and (ii) is located closer to ND compared to the distance between NS and ND (e.g. node NV as in Fig. 1).
The same procedure will be performed by the intermediate nodes (such as NV) until ND is reached. However, the GF
will be inclined to fail due to the occurrences of voids even though some routing paths exist from NS to ND. The void
problem is defined as follows.
Problem 1 (Void Problem). The Greedy Forwarding (GF) algorithm is exploited for packet delivery from NS to ND. The void problem occurs while there exists a Void Node (NV) in the network such that
{PNk| d(PNk, PND) < d(PNV, PND), ∀ PNk∈ TNV} = ∅, (1)
where d(x, y) represents the Euclidean distance between x andy. TNV is the neighbor table ofNV.
III. THEPROPOSEDGREEDYANTI-VOIDROUTING (GAR) PROTOCOL
A. The Rolling-ball UDG boundary Traversal (RUT) Scheme The RUT scheme is adopted to solve the boundary finding problem. The definition of boundary and the problem state-ment are described as follows.
Definition 1 (Boundary). If there exists a setB ⊆ N such that (i) the nodes in B form a simple unidirectional ring and (ii) the nodes located on and inside the ring are disconnected with those outside of the ring,B is denoted as the Boundary Set and the unidirectional ring is called a Boundary. Problem 2 (Boundary Finding Problem). Given a UDG G(P, E) and the one-hop neighbor tables T = {TNi| ∀ Ni∈
N}, how can a Boundary be obtained by exploiting the distributed computing techniques?
There are three phases within the RUT scheme, including the initialization, the boundary traversal, and the termination phases.
1) The Initialization Phase: No algorithm can be executed without the algorithm-specific trigger event. The trigger event within the RUT scheme is called the Starting Point (SP). The RUT technique can be initialized from any SP, which is defined as follows.
Definition 2 (Rolling Ball). Given Ni ∈ N, a Rolling Ball
RBNi(si, R/2) is defined by (i) a rolling circle hinged at PNi
with its center point at si∈ R2 and the radius equal toR/2; and (ii) there does not exist anyNi located inside the rolling ball as {RBN∼i(si, R/2) ∩ N} = ∅, where RBN∼i(si, R/2)
denotes the open disk within the rolling ball.
Definition 3 (Starting Point). The Starting Point of Ni
within the RUT scheme is defined as the center point si∈ R2 ofRBNi(si, R/2).
As shown in Fig. 2, each nodeNican verify if there exists a SP since the rolling ballRBNi(si, R/2) is bounded by the
si sj sk sl sm R 1/2R Nl Nm Ni Np Nj Nk Nq Eij
Fig. 2. The Rolling-ball UDG boundary Traversal (RUT) scheme: Given si
and Ni, the RUT scheme rotates the rolling ball RBNi(si, R/2) counter-clockwise and constructs the simple closed curve (i.e. the flower-like red solid curve). The Boundary SetB = {Ni, Nj, Nk, Nl, Nm} is established as a
simple unidirectional ring by using the RUT scheme.
transmission range ofNi. According to Definition 3, the SPs
should be located on the circle centered atPNi with a radius of R/2. As will be proved in Lemmas 1 and 2, all the SPs will result in the red solid flower-shaped arcs as in Fig. 2. It is noticed that there should always exist a SP while the void problem occurs within the network, which will be explained in subsection B. At this initial phase, the location si can be selected as the SP for the RUT scheme.
2) The Boundary Traversal Phase: Given sias the SP asso-ciated with itsRBNi(si, R/2) hinged at Ni, either the counter-clockwise or counter-clockwise rolling direction can be utilized. As shown in Fig. 2, RBNi(si, R/2) is rolled counter-clockwise until the next SN is reached (i.e. Nj in Fig. 2). The
unidirec-tional edgeEij = (PNi, PNj) can therefore be constructed. A new SP and the corresponding rolling ball hinged at Nj (i.e.
sj and RBNj(sj, R/2)) will be assigned, and consequently the same procedure can be conducted continuously.
3) The Termination Phase: The termination condition for the RUT scheme happens while the first unidirectional edge is revisited. As shown in Fig 2, the RUT scheme will be terminated if the edge Eij is visited again after the edges
Eij, Ejk, Ekl, Elm, and Emi are traversed. The boundary
set initiated from Ni can therefore be obtained as B =
{Ni, Nj, Nk, Nl, Nm}.
B. The Proposed GAR Protocol
As shown in Fig. 1, the packets are intended to be delivered from NS to ND. NS will select NV as the next hopping
node by adopting the GF protocol. However, the void problem prohibitsNV to continue utilizing the same GF algorithm for
packet forwarding. The RUT scheme is therefore employed by assigning a SP (i.e. sV) associated with the rolling ball RBNV(sV, R/2) hinged at NV. As illustrated in Fig. 1, sV can be chosen to locate on the connecting line between NV and
ND withR/2 away from NV. It is noticed that there always
exists a SP for the void node (NV) since there is not supposed to have any SN located within the blue-shaded region (as in
564 IEEE COMMUNICATIONS LETTERS, VOL. 11, NO. 7, JULY 2007
Fig. 1), which is large enough to satisfy the requirements as in Definitions 2 and 3. The RUT scheme is utilized until NY is reached (after traversing NA, NB, and NX). Since
d(PNY, PND) < d(PNV, PND), the GF algorithm is resumed atNY and the next hopping node will be selected asNZ. The
route fromNS toND can therefore be constructed for packet
delivery. Moreover, if there does not exist a NY such that
d(PNY, PND) < d(PNV, PND) within the boundary traversal phase, the RUT scheme will be terminated after revisiting the edge EV A. The result indicates that there does not exist a routing path betweenNS andND.
IV. PROOF OFCORRECTNESS
Fact 1. A simple closed curve is formed by traversing a point on the border of a closed filled two-dimensional geometry. Lemma 1. All the SPs within the RUT scheme form the border of the resulting shape by overlapping the closed disks D(PNi, R/2) for all Ni∈ N, and vice versa.
Proof: Based on Definitions 2 and 3, the set of SPs can be obtained asS = R1∩R2= {si| si−PNi = R/2, ∃Ni∈
N, si ∈ R2} ∩ {sj| sj− PNj ≥ R/2, ∀Nj ∈ N, sj ∈ R2} by adopting the (i) and (ii) rules within Definition 2. On the other hand, the border of the resulting shape from the overlapped closed disks D(PNi, R/2) for all Ni ∈ N can be denoted as Ω = Q1− Q2 = N
i∈NC(PNi, R/2) −
Ni∈ND(PNi, R/2), where C(PNi, R/2) and D(PNi, R/2) represent the circle and the open disk centered at PNi with a radius of R/2 respectively. It is obvious to notice that R1= Q1andR2= Q2, which result inS = Ω. It completes
the proof.
Lemma 2. A simple closed curve is formed by the trajectory of the SPs.
Proof: Based on Lemma 1, the trajectory of the SPs forms the border of the overlapped closed disksD(PNi, R/2) for allNi ∈ N. Moreover, the border of a closed filled
two-dimensional geometry is a simple closed curve according to Fact 1. Therefore, a simple closed curve is constructed by the trajectory of the SPs, e.g. the solid flower-shaped closed curve as in Fig 2. It completes the proof.
Theorem 1. The Boundary Finding Problem (Problem 2) is resolved by the RUT scheme.
Proof: Based on Lemma 2, the RUT scheme can
draw a simple closed curve by rotating the rolling balls RBNi(si, R/2) hinged at PNi for all Ni ∈ N. The closed curve can be divided into segments S(si, sj), where si is
the starting SP associated with Ni; and sj is the anchor
point while rotating the RBNi(si, R/2) hinged at PNi. The segmentsS(si, sj) can be mapped into the unidirectional edges
Eij= (PNi, PNj) for all Ni, Nj∈ U, where U ⊆ N. Due to the one-to-one mapping between S(si, sj) and Eij, a simple unidirectional ring is constructed byEij for allNi, Nj∈ U.
According to the RUT scheme, there does not exist any Ni ∈ N within the area traversed by the rolling balls, i.e.
inside the light blue region as in Fig. 2. For allNp∈ N located
inside the simple unidirectional ring, the smallest distance fromNptoNq, which is located outside of the ring, is greater
than the SN’s transmission rangeR. Therefore, there does not exist any Np ∈ N inside the simple unidirectional ring that
can communicate with Nq ∈ N located outside of the ring.
Based on Definition 1, the setU is identical to the Boundary Set, i.e. U = B. It completes the proof.
Theorem 2. The Void Problem (Problem 1) is solved by the GAR protocol with guaranteed packet delivery.
Proof: With the existence of the void problem occurred at the Void NodeNV, the RUT scheme is utilized by initiating a SP (sV) with the rolling ball RBNV(sV, R/2) hinged at
NV. The RUT scheme within the GAR protocol will conduct Boundary (i.e. the set B) traversal under the condition that d(PNi, PND) ≥ d(PNV, PND) for all Ni∈ B. If the Boundary within the underlying network is completely traveled based on Theorem 1, it indicates that the SNs inside the Boundary (e.g.NV) is not capable of communicating with those located
outside of the Boundary (e.g. ND). The result shows that
there does not exist a route from the Void Node (NV) to
the destination node (ND), i.e. the existence of network
partition. On the other hand, if there exists a NY such that
d(PNY, PND) < d(PNV, PND) (as shown in Fig. 1), the GF scheme will be adopted within the GAR protocol to conduct data delivery toward the destination nodeND. Therefore, the
GAR protocol solves the void problem with guaranteed packet delivery, which completes the proof.
V. CONCLUSION
In this letter, the Greedy Anti-Void Routing (GAR) pro-tocol is proposed to completely resolve the void problem incurred by the conventional Greedy Forwarding (GF) al-gorithm. The Rolling-ball UDG boundary Traversal (RUT) scheme is adopted within the GAR protocol to solve the boundary finding problem, which results in the guarantee of packet delivery. Finally, the correctness of the RUT and the GAR algorithms are properly proved.
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