Greedy Routing with Anti-Void Traversal
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) that exists in the greedy routing algorithms has been studied for the wireless sensor networks. Some of the current research work cannot fully resolve the void problem, while there exist other schemes that can guarantee the delivery of packets with the excessive consumption of control overheads. In this paper, a greedy anti-void routing (GAR) protocol is proposed to solve the anti-void problem with increased routing efficiency 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 under the UDG network. The boundary map (BM) and the indirect map searching (IMS) scheme are proposed as efficient algorithms for the realization of the RUT technique. Moreover, the hop count reduction (HCR) scheme is utilized as a short-cutting technique to reduce the routing hops by listening to the neighbor’s traffic, while the intersection navigation (IN) mechanism is proposed to obtain the best rolling direction for boundary traversal with the adoption of shortest path criterion. In order to maintain the network requirement of the proposed RUT scheme under the non-UDG networks, the partial UDG construction (PUC) mechanism is proposed to transform the non-UDG into UDG setting for a portion of nodes that facilitate boundary traversal. These three schemes are incorporated within the GAR protocol to further enhance the routing performance with reduced communication overhead. The proofs of correctness for the GAR scheme are also given in this paper. Comparing with the existing localized routing algorithms, the simulation results show that the proposed GAR-based protocols can provide better routing efficiency.
Index Terms—Greedy routing, void problem, unit disk graph, localized algorithm, wireless sensor network.
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1
I
NTRODUCTIONA
wireless sensor network (WSN) consists of sensornodes (SNs) with wireless communication capabilities for specific sensing tasks. Due to the limited available resources, efficient design of localized multihop routing protocols [1] becomes a crucial subject within the WSNs. How to guarantee delivery of packets is considered an important issue for the localized routing algorithms. The well-known greedy forwarding (GF) algorithm [2] is considered a superior scheme with its low routing over-heads. However, the void problem [3], which makes the GF technique unable to find its next closer hop to the destination, will cause the GF algorithm failing to guarantee the delivery of data packets.
Several routing algorithms are proposed to either resolve or reduce the void problem, which can be classified into graph-based and graph-based schemes. In the non-graph-based algorithms [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], the intuitive schemes as proposed in [4] construct a two-hop neighbor table for implementing the GF algorithm. The network flooding mechanism is adopted within the GRA [5] and PSR [6] schemes while the void problem occurs. There also exist routing protocols that adopt the backtracking method at the occurrence of the network holes (such as GEDIR, [4], DFS [7], and SPEED [8]). The routing schemes as proposed by ARP [9] and LFR [10]
memorize the routing path after the void problem takes place. Moreover, other routing protocols (such as PAGER [11], NEAR [12], DUA [13], INF [14], and YAGR [15]) propagate and update the information of the observed void node in order to reduce the probability of encountering the void problem. By exploiting these routing algorithms, however, the void problem can only be either 1) partially alleviated or 2) resolved with considerable routing over-heads and significant converging time.
On the other hand, there are research works on the design of graph-based routing algorithms [3], [16], [17], [18], [19], [20], [21], [22], [23] to deal with the void problem. Several routing schemes as surveyed in [16] adopt the planar graph [24] derived from the unit disk graph (UDG) as their network topologies, such as GPSR [3], GFG [17], Compass Routing II [18], AFR [19], GOAFR [20] GOAFR+ [21], GOAFR++ [16], and GPVFR [22]. For conducting the above planar graph-based algorithms, the planarization technique is required to transform the underlying network graph into the planar graph. The Gabriel graph (GG) [25] and the relative neighborhood graph (RNG) [26] are the two commonly used localized planarization techniques that abandon some communication links from the UDG for achieving the planar graph. Nevertheless, the usage of the GG and RNG graphs has significant pitfalls due to the removal of critical communication links, leading to longer routing paths to the destination. As shown in Fig. 1, the nodes ðNS; NDÞ are considered the transmission pair, while
NV represents the node that the void problem occurs. The
representative planar graph-based GPSR scheme can not forward the packets from NV to NA directly since both the
GG and the RNG planarization rules abandon the commu-nication link from NV to NA. Considering the GG
planar-ization rule for example, the communication link from NV
. The authors are with the Department of Communication Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan, ROC.
E-mail: [email protected], [email protected].
Manuscript received 17 Nov. 2007; revised 1 July 2008; accepted 21 Oct. 2008; published online 6 Nov. 2008.
For information on obtaining reprints of this article, please send e-mail to: [email protected], and reference IEEECS Log Number TMC-2007-11-0346. Digital Object Identifier no. 10.1109/TMC.2008.162.
to NAis discarded since both NJ and NKare located within
the forbidden region, which is defined as the smallest disk passing through both NV and NA. Therefore, based on the
right-hand rule, the resulting path by adopting the GPSR protocol can be obtained as fNS; NV; NJ; NK; NA;
NB; NX; NY; NZ; NDg. The two unnecessary forwarding
nodes NJ and NK are observed, as in Fig. 1.
Furthermore, the planar graph-based schemes, e.g., the GPSR and GOAFR++ algorithms, will in general lose their properties of guaranteed packet delivery due to the un-expected network partition within the non-UDG networks. The reason is also attributed to the situations that critical communication links are removed by adopting the GG and RNG planarization techniques. In order to resolve the network partition problem, a cross-link detection protocol (CLDP) is therefore suggested in [27] for planarization of the underlying non-UDG networks. However, for the purposes of both detecting the cross links and planarizing the under-lying network, the CLDP planarization will introduce excessive control overhead since all communication links are required to be probed and frequently traversed. More-over, the problems of multiple cross links and concurrent probing can further enlarge the total number of communica-tion overhead within the CLDP technique.
Due to the drawbacks of link removal from the planar graph-based algorithms, the adoption of UDG without planarization for the modeling of underlying network is suggested. A representative UDG-based routing scheme, i.e., the BOUNDHOLE algorithm [23], forwards the packets around the network holes by identifying the locations of the holes. However, due to the occurrence of routing loop, the delivery of packets can not be guaranteed in the BOUNDHOLE scheme even if a route exists from the
source to the destination node. For example, as shown in Fig. 1, it is assumed that the node NX is located within the
transmission range of NB, while it is considered out of the
transmission ranges of nodes NA and NE. Based on the
minimal sweeping angle criterion within the BOUND-HOLE algorithm, NA will choose NE as its next hop node
since the counterclockwise sweeping from NV to NE
(hinged at NA) is smaller comparing with that from NV to
NB. Therefore, the missing communication link from NBto
NXcan be observed, and the resulting path by adopting the
BOUNDHOLE scheme becomes fNS; NV; NA; NE; NF;
NG; NH; NVg. It is observed that the undeliverable routing
path from the source node NS is constructed even with
unpartitioned network topology. Moreover, two cases of edge intersections within the BOUNDHOLE algorithm [23] result in high routing overhead in order to identify the network holes.
In this paper, a greedy anti-void routing (GAR) protocol is proposed to guarantee packet delivery with increased routing efficiency by completely resolving the void problem based on the UDG setting. The GAR protocol is designed to be a combination of both the conventional GF algorithm and the proposed rolling-ball UDG boundary traversal (RUT) scheme. The GF scheme is executed by the GAR algorithm without the occurrence of the void problem, while the RUT scheme is served as the remedy for resolving the void problem, leading to the assurance for packet delivery. Moreover, the correctness of the proposed GAR protocol is validated via the given proofs. The implementation of the GAR protocol is also explained, including that for the proposed boundary map (BM) and the indirect map searching (IMS) algorithm for the BM construction.
Furthermore, the associated three additional enhanced mechanisms are also exploited, including the hop count reduction (HCR), the intersection navigation (IN), and the partial UDG construction (PUC) schemes. The HCR scheme is a short-cutting technique that acquires information by listening to one-hop neighbor’s packet forwarding, while the other short-cutting method, as proposed in [28], requires information from two-hop neighbors that can result in excessive control packet exchanges. With the occurrence of the void node, the IN mechanism determines its rolling direction based on the criterion of smallest hop counts (HCs) for boundary traversal. Similar to the CLDP method [27], the IN scheme acquires information over multiple hops in order to process its algorithm. However, it is required for the CLDP technique to traverse all the communication links in the networks, while the IN scheme only exploits a small portion of network links for conducting the boundary traversal. Moreover, in order to meet the network requirement for the RUT scheme under non-UDG networks, the PUC mechan-ism is utilized to transform the non-UDG into the UDG setting for the nodes that are adopted for boundary traversal. By adopting these three enhanced schemes, both the routing efficiency and the communication overhead of the original GAR algorithm can further be improved. The performance of the proposed GAR protocol and the version with the enhanced mechanisms (denoted as the GAR-E algorithm) is evaluated via simulations under both the UDG network for the ideal case and the non-UDG setting for realistic scenario. The simulation results show that the GAR-based schemes can both guarantee the delivery of data packets and pertain better routing performance under the UDG network. On the other hand, comparing with the other Fig. 1. Example routing paths constructed by using the GAR, the GPSR,
and the BOUNDHOLE algorithms under the existence of the void problem.
existing schemes, feasible routing performance with reduced communication overhead can be provided by the GAR-based algorithms within the non-UDG network environment.
The remainder of this paper is organized as follows: Section 2 formulates the problem of interest by the under-lying network model. The proposed GAR protocol is explained in Section 3, while Section 4 provides the practical realization of the GAR algorithm. Section 5 exploits the three enhanced mechanisms, including the HCR, the IN, and the PUC mechanisms. The performance of the GAR-based protocols is evaluated and compared in Section 6. Section 7 draws the conclusions.
2
P
ROBLEMF
ORMULATIONConsidering a set of SNs N ¼ fNij 8 ig within a
two-dimensional (2D) euclidean plane, the locations of the set N, which can be acquired by their own positioning
systems, are represented by the set P ¼ fPNi j PNi ¼
ðxNi; yNiÞ; 8 ig. It is assumed that all the SNs are
homo-geneous and equipped with omnidirectional antennas. The set of closed disks defining the transmission ranges of N is denoted as D ¼ fDðPNi; RÞ j 8 ig, where DðPNi; RÞ ¼
fx j kx PNik R; 8 x 2 IR
2g. It is noted that P
Ni 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 GðP; EÞ with the edge set E ¼ fEijj Eij¼ ðPNi; PNjÞ; PNi2
DðPNj; RÞ; 8 i 6¼ jg. The edge Eijindicates the unidirectional
link from PNi to PNjwhenever the position PNi is within the
closed disk region DðPNj; RÞ. Moreover, the one-hop
neighbor table for each Niis defined as
TNi ¼ ½IDNk; PNk j PNk2 DðPNi; RÞ; 8 k 6¼ i
; ð1Þ
where IDNk represents the designated identification
num-ber for Nk. In the GF algorithm, it is assumed that the source
node NSis aware of the location of the destination node ND.
If NS wants to transmit packets to ND, it will choose the
next hop node from its TNS, which 1) has the shortest
euclidean distance to NDamong all the SNs in TNS and 2) is
located closer to ND compared to the distance between NS
and ND (e.g., 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 algorithm 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 GF algorithm is exploited for packet delivery from NSto ND. The void problem occurs while
there exists a void node ðNVÞ in the network such that no
neighbor of NV is closer to the destination as
PNkj dðPNk; PNDÞ < dðPNV; PNDÞ; 8 PNk 2 TNV
f g ¼ ;; ð2Þ
where dðx; yÞ represents the euclidean distance between x and y. TNV is the one-hop neighbor table of NV.
3
P
ROPOSEDG
REEDYA
NTI-V
OIDR
OUTING(GAR)
P
ROTOCOLThe objective of the GAR protocol is to resolve the void problem such that the packet delivery from NSto NDcan be
guaranteed. Before diving into the detail formulation of the proposed GAR algorithm, an introductory example is described in order to facilitate the understanding of the GAR protocol. As shown in Fig. 1, the data packets initiated from the source node NS to the destination node ND will
arrive in NV based on the GF algorithm. The void problem
occurs as NVreceives the packets, which leads to the adoption
of the RUT scheme as the forwarding strategy of the GAR protocol. A circle is formed by centering at sVwith its radius
being equal to half of the transmission range R=2. The circle is hinged at NV and starts to conduct counterclockwise rolling
until an SN has been encountered by the boundary of the circle, i.e., NA, as in Fig. 1. Consequently, the data packets in
NV will be forwarded to the encountered node NA.
Subsequently, a new equal-sized circle will be formed,
which is centered at sA and hinged at node NA. The
counterclockwise rolling procedure will be proceeded in order to select the next hop node, i.e., NB in this case.
Similarly, same process will be performed by other inter-mediate nodes (such as NB and NX) until the node NY is
reached, which is considered to have a smaller distance to NDthan that of NV to ND. The conventional GF scheme will
be resumed at NY for delivering data packets to the
destination node ND. As a consequence, the resulting path
by adopting the GAR protocol becomes fNS; NV; NA;
NB; NX; NY; NZ; NDg. In the following sections, the formal
description of the RUT scheme will be described in Section 3.1, while the detail of the GAR algorithm is explained in Section 3.2. The proofs of correctness of the GAR protocol are given in Section 3.3.
3.1 Proposed Rolling-Ball UDG Boundary Traversal
(RUT) Scheme
The RUT scheme is adopted to solve the boundary finding problem, and the combination of the GF and the RUT scheme (i.e., the GAR protocol) can resolve the void problem, leading to the guaranteed packet delivery. The definition of boundary and the problem statement are described as follows:
Definition 1 (boundary).If there exists a set B N such that 1) the nodes in B form a simple unidirectional ring and 2) 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
GðP; EÞ and the one-hop neighbor tables T ¼ fTNij 8 Ni2 Ng,
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 termina-tion phases.
3.1.1 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 scheme can be initialized from any SP, which is defined as follows: Definition 2 (rolling ball). Given Ni2 N, a rolling ball
RBNiðsi; R=2Þ is defined by 1) a rolling circle hinged at PNi
with its center point at si2 IR2 and the radius equal to R=2,
ball as fRB
Niðsi; R=2Þ \ Ng ¼ ;, where RB
Niðsi; R=2Þ
de-notes the open disk within the rolling ball.
Definition 3 (starting point). The SP of Ni within the RUT
scheme is defined as the center point si2 IR2of RBNiðsi; R=2Þ.
As shown in Fig. 2, each node Nican verify if there exists
an SP since the rolling ball RBNiðsi; R=2Þ is bounded by the
transmission range of Ni. According to Definition 3, the SPs
should be located on the circle centered at PNiwith a radius
of R=2. As will be proven 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 an SP, while the void problem occurs within the network, which will be explained in Section 3.2. At this initial phase, the location si
can be selected as the SP for the RUT scheme. 3.1.2 Boundary Traversal Phase
Given si as the SP associated with its RBNiðsi; R=2Þ hinged
at Ni, either the counterclockwise or clockwise rolling
direction can be utilized. As shown in Fig. 2, RBNiðsi; R=2Þ
is rolled counterclockwise until the next SN is reached (i.e., Nj in Fig. 2). The unidirectional edge Eij¼ ðPNi; PNjÞ can
therefore be constructed. A new SP and the corresponding rolling ball hinged at Nj(i.e., sjand RBNjðsj; R=2Þ) will be
assigned, and consequently, the same procedure can be conducted continuously.
3.1.3 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 Emiare
traversed. The boundary set initiated from Nican therefore
be obtained as B ¼ fNi; Nj; Nk; Nl; Nmg.
3.2 Detail Description of 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 hop node by
adopting the GF algorithm. However, the void problem prohibits NV to continue utilizing the same GF algorithm
for packet forwarding. The RUT scheme is therefore employed by assigning an SP (i.e., sV) associated with the
rolling ball RBNVðsV; R=2Þ hinged at NV. As illustrated in
Fig. 1, sV can be chosen to locate on the connecting line
between NV and ND with R=2 away from NV. It is noticed
that there always exists an SP for the void node ðNVÞ since
there is not supposed to have any SN located within the blue-shaded region (as in 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 dðPNY; PNDÞ < dðPNV; PNDÞ, the GF
algorithm is resumed at NY, and the next hop node will
be selected as NZ. The route from NSto NDcan therefore be
constructed for packet delivery. Moreover, if there does not exist a node NY such that dðPNY; PNDÞ < dðPNV; PNDÞ 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 between NS and ND.
3.3 Proof of Correctness
In this section, the correctness of the RUT scheme is proven in order to solve Problem 2, while the GAR protocol is also proven for resolving the void problem (i.e., Problem 1) in order to guarantee packet delivery.
Fact 1.A simple closed curve is formed by traversing a point on the border of a closed filled 2D geometry with fixed orientation. Lemma 1.All the SPs within the RUT scheme form the border of a shape that results from overlapping the closed disks DðPNi; R=2Þ for all Ni2 N, and vice versa.
Proof. Based on Definitions 2 and 3, the set of SPs can be obtained as S ¼ R1\ R2¼ fsij ksiPNik ¼ R=2; 9Ni2 N;
si2 IR2g \ fsjj ksj PNjk R=2; 8 Nj2 N; sj2 IR
2g by
adopting the 1) and 2) rules within Definition 2. On the other hand, the border of the resulting shape from the overlapped closed disks DðPNi; R=2Þ for all Ni2 N
can be denoted as ¼ Q1 Q2¼SNi2NCðPNi; R=2Þ
S
Ni2NDðPNi; R=2Þ, where CðPNi; R=2Þ and DðPNi; R=2Þ
represent the circle and the open disk centered at PNi
with a radius of R=2, respectively. It is obvious to notice that R1¼ Q1 and R2¼ Q02, which result in S ¼ . It
completes the proof. tu
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 disks DðPNi; R=2Þ
for all Ni2 N. Moreover, the border of a closed filled
2D 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. tu
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 RBNiðsi; R=2Þ
hinged at PNi for all Ni2 N. The closed curve can be
divided into arc segments Sðsi; sjÞ, where si is the
starting SP associated with Ni, and sjis the anchor point
while rotating the RBNiðsi; R=2Þ hinged at PNi. The arc
segments Sðsi; sjÞ can be mapped into the unidirectional
edges Eij¼ ðPNi; PNjÞ for all Ni, Nj2 U, where U N.
Due to the one-to-one mapping between Sðsi; sjÞ and
Eij, a simple unidirectional ring is constructed by Eijfor
all Ni, Nj2 U.
According to the RUT scheme, there does not exist
any Ni2 N within the area traversed by the rolling
balls, i.e., inside the light blue region, as in Fig. 2. For all Np2 N located inside the simple unidirectional ring, the
smallest distance from Np to Nq, which is located
outside of the ring, is greater than the SN’s transmission
range R. Therefore, there does not exist any Np2 N
inside the simple unidirectional ring that can commu-nicate with Nq 2 N located outside of the ring. Based on
Definition 1, the set U is identical to the boundary set,
i.e., U ¼ B. It completes the proof. tu
Theorem 2.The void problem (Problem 1) in UDGs is solved by the GAR protocol with guaranteed packet delivery.
Proof.With the existence of the void problem occurred at the void node NV, the RUT scheme is utilized by initiating an
SP ðsVÞ with the rolling ball RBNVðsV; R=2Þ hinged at NV.
The RUT scheme within the GAR protocol will conduct boundary (i.e., the set B) traversal under the condition that dðPNi; PNDÞ dðPNV; PNDÞ for all Ni 2 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) are not capable of
communicating with those located outside of the bound-ary (e.g., ND). The result shows that there does not exist a
route from the void node ðNVÞ to the destination node
ðNDÞ, i.e., the existence of network partition. On the other
hand, if there exists a node NY such that dðPNY; PNDÞ <
dðPNV; PNDÞ (as shown in Fig. 1), the GF algorithm will be
adopted within the GAR protocol to conduct data delivery
toward the destination node ND. Therefore, the GAR
protocol solves the void problem with guaranteed packet
delivery, which completes the proof. tu
4
P
ROPOSEDGAR P
ROTOCOLR
EALIZATIONThe implementation of the proposed GAR protocol is explained in this section. The format of the one-hop neighbor table TNi, as defined in (1), is realized for the
implementation purpose. TNi is considered a major
information source in the localized routing protocols, which can be obtained via the neighbor information acquisition [29]. It is noticed that the one-hop neighbor table TNi is
considered stable, while Niis making its next-hop decision,
i.e., TNi remains unchanged while Ni is determining the
next-hop SN for packet transmission. Sections 4.1 and 4.2 describe the implementation aspect of the GAR algorithm, which consists of the GF and the RUT schemes. The proofs of correctness are illustrated in Section 4.3.
4.1 Implementation of GF Scheme
As described in Section 2, the GF scheme is considered a straightforward algorithm that only requires the implemen-tation of the one-hop neighbor table TNi. The next hop node
can be found by the linear search of TNiif the void problem
does not occur; otherwise, the RUT scheme will be adopted based on the proposed GAR protocol.
4.2 Implementation of RUT Scheme
4.2.1 Problem Statement for Implementation
As mentioned in Section 3.2, the GAR protocol changes its routing mode into the RUT scheme while the void problem occurs at NV. The boundary traversal phase is conducted by
assigning an SP (i.e., sVas shown in Fig. 1) associated with the
rolling ball RBNVðsV; R=2Þ hinged at NV. While there is no
doubt regarding the description of boundary traversal, there can be considerable efforts required in order to realize the continuous rolling ball mechanism. A brute-force method can be adopted as a potential solution by rotating the rolling ball incrementally and verifying if a new SN has been encountered at each computing step. However, an infinite number of computational runs are required by adopting the brute-force method, which is considered impractical for realistic computing machines. Therefore, a feasible and efficient mechanism for the boundary traversal should be obtained in order to overcome the computational limitation. 4.2.2 Concept of Boundary Map
In order to resolve the implementation issue of the boundary traversal as mentioned above, a new parameter called BM (denoted as MNifor each Ni) is introduced in this
section. Moreover, the BM MNi is mainly derived from the
one-hop neighbor table TNi via the IMS method, as shown
in Algorithm 1. Instead of diving into the IMS algorithm, the functionality of MNi is first explained.
The purpose of the BM MNi is to provide a set of direct
output SNs with respect to Ni. Based on Theorem 1, the two
adjacent communication links formed by the input node, the node Ni, and the corresponding output node within the
RUT scheme consist part of the network boundary. There-fore, the direct mappings between the input SNs and their corresponding output SNs with respect to Nilead to the
so-called BM. An example is shown in Fig. 3 to illustrate the functionality of MNi. Based on Definition 2, the rolling balls
hinged at Ni can be constructed by rotating the dashed
circle counterclockwise from N1 to N2. The SPs associated
with the rolling balls (from Definition 3) result in the arc segment SSP
NiðP
L
1; P2RÞ between the endpoints P1L and P2R,
i.e., the dashed arc segment, as in Fig. 3. Similarly, the arc
segment SSP
NiðP
L
2; P3RÞ can be constructed by rotating the
rolling balls (hinged at Ni) counterclockwise from N2to N3.
Based on the description as above, the following definitions are introduced.
Definition 4 (SP and non-SP arc segments). Given an SN
Ni2 N and a pair of points (PA, PB) on the circle (centered at
PNi with a radius of R=2), an SP arc segment S
SP
NiðPA; PBÞ of
Niis defined by the arc from PAto PBcounterclockwise where
all points on this arc segment are SPs. Likewise, a non-SP arc segment SSP
NiðPA; PBÞ of Ni is defined by the
endpoint-excluding arc from PA to PB counterclockwise, where all
points on this arc segment are not SPs.
Definition 5 (converged SP and non-SP arc segments). Given an SP arc segment SSP
Ni ðPA; PBÞ, it is regarded as a
converged SP arc segment if there does not exist any SP arc segment SSP Ni ðPJ; PKÞ such that S SP Ni ðPA; PBÞ S SP NiðPJ; PKÞ.
Similarly, a non-SP arc segment SSP
NiðPA; PBÞ is considered as a
converged non-SP arc segment if there exists no other non-SP arc segment SSP Ni ðPJ; PKÞ such that S SP NiðPA; PBÞ S SP NiðPJ; PKÞ.
It is noticed that the converged arc segments are defined to represent the combined arc segments, e.g., the converged non-SP arc segment SSP
NiðP
R
3; P1LÞ is formed by overlapping
the non-SP segments SSP
NiðP
R
3; P3LÞ and SSPNi ðP
R
1; P1LÞ, as
shown in Fig. 3. As will be proven in Theorem 3, all incoming packets to Nithat are acquired from its neighbor
node N1 (which induces the rightmost endpoint P1L of the
converged SP arc segment SSP Ni ðP
L 1; P
R
2Þ) will be forwarded to
its neighbor node N2(which results in the leftmost endpoint
PR
2 of the same converged SP arc segment) under the
counterclockwise rolling direction. Consequently, if all the converged SP arc segments of Nican be obtained, the direct
mappings between the input SNs and their corresponding output SNs with respect to Ni can also be constructed. As
shown in Fig. 3, there exist two converged SP arc segments SNSPi ðPL 1; P2RÞ and SNSPi ðP L 2; P3RÞ, where SNSPiðP L 1; P2RÞ is
con-structed by the input SN N1 and the corresponding output
SN N2and SSPNiðP
L
2; P3RÞ is established by the input N2and
the output N3. As a result, the BM with respect to Nican be
obtained as MNi ¼ fðN1! N2Þ; ðN2! N3Þg. Therefore, all
packets from N1will be forwarded to N2, while those from
N2will be relayed to N3according to the BM.
4.2.3 Construction of Boundary Map
As mentioned above, the BM MNi can be constructed via
the converged SP arc segments with respect to Ni. However,
it is observed to be a difficult task for obtaining the converged SP arc segments directly in realization. An IMS algorithm is proposed in this section in order to acquire the BM for implementation. The definition of the neighbor-related non-SP arc segment and two associated properties are first introduced.
Definition 6 (neighbor-related non-SP arc segment). A
non-SP arc segment SSP
Ni ðPA; PBÞ of Niis given. If there exists
Nj2 N as a neighbor node of Ni such that an arc segment of
CðPNi; R=2Þ that lies inside the closed disk DðPNj; R=2Þ is
identical to SSP
NiðPA; PBÞ, this segment S
SP
NiðPA; PBÞ is called
a neighbor-related non-SP arc segment SSP
Ni NjðPA; PBÞ,
distinguished by Nj.
Two properties that are related to the SP and non-SP arc segments are described as follows:
Property 1.The circle CðPNi; R=2Þ centered at PNiwith a radius
of R=2 is entirely composed by all the converged SP and non-SP arc segments of Ni.
Proof.Based on Definitions 2 and 3, it can be observed that each point on the circle CðPNi; R=2Þ must either be an SP
or a non-SP. A number of adjacent SPs on CðPNi; R=2Þ
will establish an SP arc segment with respect to Ni; while
there must exist the largest number of adjacent SPs such that the underlying SP arc segment is a converged SP arc segments with respect to Ni. Therefore, all the adjacent
SPs on CðPNi; R=2Þ will result in converged SP arc
segments with respect to Ni. Similarly, all the adjacent
non-SPs on CðPNi; R=2Þ must be aggregated into
con-verged non-SP arc segments with respect to Ni. On the
other hand, the circle CðPNi; R=2Þ is entirely composed
by the SPs and non-SPs corresponding to Ni.
Conse-quently, all the converged SP and non-SP arc segments of Ni will construct the entire circle CðPNi; R=2Þ. It
completes the proof. tu
Fig. 3. The converged SP and non-SP arc segments with respect to Ni
Property 2. The union of all the neighbor-related non-SP arc segments with respect to Niis equivalent to the union of all the
converged non-SP arc segments with respect to Ni.
Proof. This property will be proven by contradiction as
follows: It is assumed that the union of all the neighbor-related non-SP arc segments corresponding to Ni is not
equivalent to the union of all the converged non-SP arc segments with respect to Ni. Based on Definitions 4 and 5,
and Property 1, it is stated that all the converged non-SP arc segments with respect to Niresult in the union of all
the non-SPs on CðPNi; R=2Þ. Therefore, there must exist a
non-SP PJ located on CðPNi; R=2Þ such that it does not
relate to any neighbor-related non-SP arc segments with respect to Ni, i.e., there does not exist any Nk2 TNi that
lies inside the rolling ball RBNiðPJ; R=2Þ. However, based
on Definitions 2 and 3, there should exist at least a node Nkwithin the rolling ball RBNiðPJ; R=2Þ since PJis a
non-SP on CðPNi; R=2Þ, which contradicts with the previous
statement. It completes the proof. tu
The concept of the proposed IMS algorithm is described as follows: Based on Property 1, the converged SP arc
segments for each Ni can be obtained by acquiring its
corresponding converged non-SP arc segments, i.e., the complement arc segments on the circle CðNi; R=2Þ.
More-over, according to Property 2, the converged non-SP arc
segments of Ni can be acquired via the neighbor-related
non-SP arc segments. Consequently, the problem of finding
the converged SP arc segments with respect to Ni is
transformed into the problem of obtaining the converged
non-SP arc segments with respect to Ni, which can be
acquired via merging the corresponding neighbor-related
non-SP arc segments. The BM MNi can therefore be
indirectly established by adopting the IMS method. The process flow of the IMS algorithm is summarized, as shown in Fig. 4.
In order to acquire and construct the BM MNi, as shown
in Algorithm 1, the IMS scheme is proposed. It is considered a localized algorithm where only three parameters are required within Algorithm 1, i.e., the maximum commu-nication distance R, the position of Ni (PNi), and the
one-hop neighbor table TNi.
Table 1 summarizes the notations in the IMS algorithm, and the pseudocode of the IMS method, as shown in Algorithm 1, is explained as follows: Based on that in Fig. 4, the first task within the IMS algorithm is to identify each
neighbor-related non-SP arc segment SSP
Ni NjðPA; PBÞ with
respect to Ni that is distinguished by its neighbor Nj.
Intuitively, it is feasible to utilize the two endpoints PAand
PB to represent SNSPi NjðPA; PBÞ, where each endpoint P (for
2 fA; Bg) can be characterized by an endpoint entry
defined as
NiðPÞ ¼ ½Id; F lag; Angle; Color; Counterpart: ð3Þ
The parameter Id is utilized as the identification number of the corresponding neighbor SN for this entry. F lag represents the endpoint type of this entry, which is denoted as either RIGHT or LEF T (e.g., the F lag field of the right endpoint PA is denoted as RIGHT , while that of the left
endpoint PB is indicated as LEF T ). The Angle field is
adopted to represent the polar angle with respect to Ni by
rotating counterclockwise from the x-axis. The Color field is employed to indicate whether the endpoint P is a non-SP
or not (i.e., Color ¼ T RUE denotes that P is a non-SP). The
Counterpart field provides the linkage to the counterpart endpoint entry that possesses the opposite F lag value (e.g., the counterpart of NiðPAÞ is NiðPBÞ, and vice versa).
Therefore, the neighbor-related non-SP arc segment SSP
Ni NjðPA; PBÞ can be denoted by a pair of the endpoint
entries as ½NiðPAÞ; NiðPBÞ.
In order to store and to maintain the relative locations among the entire set of endpoints IP for all the neighbor-related non-SP arc segments with respect to Ni, a circular
doubly linked list [30] sorted by the polar angle is employed as LNi ¼ ‘NiðPÞ j ‘NiðPÞ ¼ NiðPÞ Next P rev 2 4 3 5; 8P2 IP 8 < : 9 = ;; ð4Þ
where the list item ‘NiðPÞ in LNiis composed of an endpoint
entry NiðPÞ associated with two fields, Next and P rev. The
fields Next and P rev provide the addresses of the next and the previous entries of ‘NiðPÞ within LNi. Considering the
example, as shown in Fig. 3, there exist three
neighbor-related non-SP arc segments with respect to Ni as
SSP Ni N1ðP R 1; P1LÞ, SSPNi N2ðP R 2; P2LÞ, and SSPNi N3ðP R 3; P3LÞ, which
result in three pairs of endpoint entries as ½NiðP
R 1Þ; NiðP L 1Þ, ½NiðP R 2Þ; NiðP L 2Þ, and ½NiðP R 3Þ; NiðP L 3Þ,
re-spectively. Consequently, the linked list LNi will contain
these endpoint entries as LNi ¼ ½‘NiðP
R 1Þ; ‘NiðP L 3Þ; ‘NiðP L 1Þ; ‘NiðP R 2Þ; ‘NiðP L 2Þ; ‘NiðP R
3Þ. It is noticed that the endpoint
entries are connected by their Next and P rev fields, while these entries are sorted by the corresponding Angle field in the ascending order. By taking the entry ‘NiðP
L
3Þ as an
example (as in Fig. 3), the parameter Next refers to the Fig. 4. The process flow of the IMS algorithm.
TABLE 1
address of ‘NiðP
L
1Þ, while the P rev field is denoted as the
address of ‘NiðP
R
1Þ. Since LNi is a circular doubly linked list,
the parameter P rev for ‘NiðP
R
1Þ points to the address of
‘NiðP
R
3Þ, while the Next field of ‘NiðP
R
3Þ refers to ‘NiðP
R 1Þ. In
the case that two endpoint entries share the same Angle value, the F lag field will be utilized to provide the order of the entries, i.e., by taking the entry with F lag ¼ LEF T first and F lag ¼ RIGHT as the next entry in LNi. It is noticed that
the order of the endpoint entries within LNiis crucial for the
construction of MNi. The construction and sorting
mech-anisms of LNi, as described above, are summarized at
Lines 1-11 in Algorithm 1, which completes the first task in Fig. 4.
The next task within the IMS algorithm, as in Fig. 4, is to merge all the neighbor-related non-SP arc segments into the converged non-SP arc segments with respect to Ni. Given
a neighbor-related non-SP arc segment SSP
Ni NjðPA; PBÞ
represented by ½NiðPAÞ; NiðPBÞ, the combining process
is to assign the Color field of each endpoint entry NiðPCÞ,
which is located within ½NiðPAÞ; NiðPBÞ to become
colored, i.e., with the T RUE value. This indicates that the endpoint PCis merged into the neighbor-related non-SP arc
segment SSP
Ni NjðPA; PBÞ. Based on Property 2, all the
remaining uncolored endpoint entries consequently become the endpoints of the converged non-SP arc segments with respect to Ni. As shown in Fig. 3, NiðP
R
1Þ and NiðP
L 3Þ are
the colored endpoint entries, while the uncolored ones are NiðP L 1Þ, NiðP R 2Þ, NiðP L 2Þ, and NiðP R 3Þ. It is noticed
that the uncolored endpoint entries will establish the converged non-SP arc segments with respect to Ni, which
are denoted as ½NiðP R 3Þ; NiðP L 1Þ and ½NiðP R 2Þ; NiðP L 2Þ.
The combining process for the neighbor-related non-SP arc segments is summarized at Lines 12-17 in Algorithm 1, completing the second task in Fig. 4.
Within the last task listed in Fig. 4, based on Property 1, all
the converged SP arc segments with respect to Ni can be
obtained by excluding all the converged non-SP arc segments with respect to Ni on the circle CðPNi; R=2Þ. For each two
sequential converged non-SP arc segments SSP
NiðPA; PBÞ
and SSP
Ni ðPC; PDÞ (denoted as ½NiðPAÞ; NiðPBÞ and
½NiðPCÞ; NiðPDÞ), the resulting converged SP arc segment
SSP
NiðPB; PCÞ can be obtained as ½NiðPBÞ; NiðPCÞ in view of
endpoint entries. Consequently, the converged SP arc segment SSP
NiðPB; PCÞ with respect to Nican be acquired by
taking each uncolored endpoint entry NiðPBÞ with F lag ¼
LEF Tassociated with the next endpoint entry NiðPCÞ with
F lag¼ RIGHT . As shown in Fig. 3, the converged SP arc
segments with respect to Niare obtained as SNSPi ðP
L
1; P2RÞ and
SSP NiðP
L
2; P3RÞ, which are denoted as ½NiðP
L 1Þ; NiðP R 2Þ and ½NiðP L 2Þ; NiðP R
3Þ. For each converged SP arc segment
SSP
NiðPB; PCÞ denoted as ½NiðPBÞ; NiðPCÞ, the direct
map-ping (i.e., NB! NC) between input/output SNs (as
men-tioned in Section 4.2.2) can therefore be obtained by mapping NB specified in NiðPBÞ to NC denoted in NiðPCÞ. The
acquisition of the converged SP arc segments and the construction of the BM MNi are summarized at Lines 18-30
in Algorithm 1.
4.3 Proof of Correctness
Theorem 3. Given a converged SP arc segment SSP
NiðPS; PTÞ
with respect to Ni, where 1) the rightmost endpoint PS is an
SP for both Ni and its neighbor NS, and 2) the leftmost
endpoint PT is an SP for both Ni and its neighbor NT,
respectively. All incoming packets to Nithat are acquired from
its neighbor node NS will be forwarded to NT.
Proof. Based on Definitions 4 and 5, a converged SP arc
segment SSP
NiðPS; PTÞ is an arc segment composed by
some of the SPs with respect to Ni. According to
Lemma 2, a simple closed curve is constructed by the trajectory of the SPs. In order to form the closed curve, there must exist other converged SP arc segments contributed by other SNs that are connected to the endpoints PS and PT. In other words, the endpoints PS
and PT must also be owned by one of Ni’s neighbor,
respectively. On the contrary, the other points on this
converged SP arc segment SSP
NiðPS; PTÞ should only be
contributed by Ni based on Definitions 2 and 3.
More-over, it is intuitive to observe (from Definition 3) that the distances between the SNs related to the same endpoint SP (i.e., either PS or PT) must be located in their
transmission ranges. By adopting the RUT scheme (as stated in Theorems 1 and 2) starting from NS, the rolling
ball will be traversed counterclockwise via Ni to NT.
This corresponds to the situation that all the packets
coming from NS to Ni will be forwarded to NT. It
completes the proof. tu
5
E
NHANCEDM
ECHANISMS FORP
ROPOSEDGAR P
ROTOCOLIn order to enhance the routing efficiency of the proposed GAR protocol, three mechanisms are proposed in this section, i.e., the HCR, the IN, and the PUC schemes. These three mechanisms are described as follows:
5.1 Hop Count Reduction (HCR) Mechanism
Based on the rolling-ball traversal within the RUT scheme, the selected next-hop nodes may not be optimal by considering the minimal HC criterion. Excessive routing delay associated with power consumption can occur if additional hop nodes are traversed by adopting the RUT
scheme. As shown in Fig. 5, the void node NV starts the
RUT scheme by selecting N1 as its next hop node with the
counterclockwise rolling direction, while N2 and N3 are
continuously chosen as the next hop nodes. Considering the case that N3 is located within the same transmission range
of N1, it is apparently to observe that the packets can
directly be transmitted from N1 to N3. Excessive
commu-nication waste can be preserved without conducting the
rerouting process to N2. Moreover, the boundary set B
forms a simple unidirectional ring based on Theorem 1, which indicates that a node’s next-hop SN can be uniquely determined if its previous hop SN is already specified. For instance (as in Fig. 5), if NV is the previous node of N1,
N1’s next hop node N2 is uniquely determined, i.e., the
transmission sequences of every three nodes (e.g., fNV !
N1! N2g or fN1! N2! N3g) can be uniquely defined.
According to the concept as stated above, the HCR mechanism is to acquire the information of the next few hops of neighbors under the RUT scheme by listening to the same forwarded packet. It is also worthwhile to notice that the listening process does not incur additional transmission of control packets. As shown in Fig. 5, N1chooses N2as its
next-hop node for packet forwarding. while N2selects N3as
the next hop node in the same manner. Under the broadcast nature, N1will listen to the same packets in the forwarding
process from N2 to N3. By adopting the HCR mechanism,
N1 will therefore select N3as its next hop node instead of
choosing N2 while adopting the original RUT scheme.
Consequently, N1will initiate its packet forwarding process
to N3 directly by informing the RUT scheme that the
rerouting via N2can be skipped.
5.2 Intersection Navigation (IN) Mechanism
The IN mechanism is utilized to determine the rolling direction in the RUT scheme while the void problem occurs. It is noticed that the selection of rolling direction (i.e., either counterclockwise or clockwise) does not influence the correctness of the proposed RUT scheme to solve Problem 2, as in Theorem 1. However, the routing efficiency may be severely degraded if a comparably longer routing path is selected at the occurrence of a void node. The primary benefit of the IN scheme is to choose a feasible rolling direction while a void node is encountered. Consequently, smaller rerouting HCs and packet transmission delay can be achieved.
Based on the transmission pair ðNS; NDÞ, as shown in
Fig. 5, NV and NCbecome the void nodes within the network
topology. There exist three potential paths from NSto NDby
adopting the RUT scheme, i.e., PATH-R, PATH-LR, and PATH-LL. The suffixes R, LR, and LL represent the sequences of the adopted rolling direction at each encountered void node, where the symbol R is denoted as counterclockwise rolling direction, and L represents clockwise direction. It is noted that the suffix with two symbols indicates that two void nodes are encountered within the path. The entire node traversal for each path is as follows: PATH-R ¼ fNS; NV; N1; N3; N4; N5; N6; NDg, PATH-LR ¼ fNS; NV; NA;
NB; NC; NE; NF; NG; N6; NDg, and PATH-LL ¼ fNS; NV; NA;
NB; NC; NX; NY; NZ; NDg. Different HCs are observed with
each path as HCðPATH-RÞ ¼ 7, HCðPATH-LRÞ ¼ 9, and
HCðPATH-LLÞ ¼ 8.
The main objective of the IN scheme is to monitor the number of HC such that the path with the shortest HC can
be selected, i.e., PATH-R in this case. A navigation map control packet (NAV_MAP) defined in the IN scheme is utilized to indicate the rolling direction while the void node is encountered. For example, two NAV_MAP packets are initiated after NV is encountered, where NAV MAP ¼ fRg is
delivered via the counterclockwise direction to ND,
and NAV MAP ¼ fLg is carried with the clockwise direc-tion. It is noticed that the HC associated with each navigation path is also recorded within the NAV_MAP
packets. As the second void node NC is observed, the
control message NAV MAP ¼ fLg is transformed into two different navigation packets (i.e., NAV MAP ¼ fLRg and NAV MAP¼ fLLg), which traverse the two different rolling directions toward ND. As a result, the destination node ND
will receive several NAV_MAP packets at different time instants associated with the on-going transmission of the data packets. The NAV_MAP packet with the shortest HC value (i.e., NAV MAP ¼ fRg in this case) will be selected as the targeting path. Therefore, the control packet with
NAV MAP¼ fRg will be traversed from NDback to NS in
order to notify the source node NSwith the shortest path for
packet transmission. After acquiring the NAV_MAP in-formation, NS will conduct its remaining packet delivery
based on the corresponding rolling direction. Considerable routing efficiency can be preserved as a shorter routing path is selected by adopting the IN mechanism.
5.3 Partial UDG Construction (PUC) Mechanism
The PUC mechanism is targeted to recover the UDG linkage
of the boundary node Ni within a non-UDG network. The
boundary nodes within the proposed GAR protocol are defined as the SNs that are utilized to handle the packet delivery after encountering the void problem. As shown in
Fig. 6, node Ni is considered a boundary node since the
converged SP arc segment SSP
NiðPS; PTÞ exists after Ni
conducts the proposed IMS algorithm by the input of the current one-hop neighbors fN1; N2; N3; N4; Njg. It is noted
that the boundary nodes consist of a portion of the network SNs. Therefore, conducting the PUC mechanism only by the boundary nodes can conserve network resources than most Fig. 6. The PUC mechanism.
of the existing flooding-based schemes that require infor-mation from all the network nodes.
The physical links of an exemplified topology are identified by the black solid lines, as shown in Fig. 6. It is considered that the boundary node Nidoes not possess full
UDG linkages since a node Nk within Ni’s transmission
range can not directly communicate with Ni. The proposed
PUC mechanism will be initiated at the boundary node Ni
under the non-UDG networks as follows: Initially, Ni
broadcasts the PUC_REQ control packet containing its neighbor list for requesting the recovery of UDG linkages.
After the neighbor Nj receives the PUC_REQ packet,
Nj’s neighbor table will be examined to verify if there
exists any neighbor node Nk that is not in the neighbor list
of the PUC_REQ packet but is actually located within the transmission range of Ni. In the case that such node Nk is
observed, Nj will initiate a feedback message, i.e., the
PUC_REP control packet, in order to inform Ni that a
pseudo link from Ni to Nk should be constructed via the
alternative paths of the two physical links from Nito Njand
from Nj to Nk. Therefore, the UDG linkage of Ni can be
recovered, which results in the current one-hop neighbors of Nias fN1; N2; N3; N4; Nj; Nkg, while the converged SP arc
segment SSP
NiðPS; PTÞ will be changed into S
SP
NiðPS; PXÞ.
6
P
ERFORMANCEE
VALUATIONThe performance of the proposed GAR algorithm is evaluated and compared with other existing localized schemes via simulations, including the reference GF algorithm, the planar graph-based GPSR and GOAFR++ schemes, and the UDG-based BOUNDHOLE algorithm. It is noted that the GPSR and GOAFR++ schemes that adopt the GG planarization technique to planarize the network graph are represented as the GPSR(GG) and GOAFR++(GG) algorithms, while the variants of these two schemes with the CLDP planarization algorithm are denoted as the GPSR(CLDP) and GOAFR++(CLDP) protocols. The random topology is considered in both two different types of network simulations as follows: 1) the pure UDG network as the ideal case, and 2) the non-UDG network for realistic network environment. Furthermore, the GAR protocol with the enhanced mechanisms (i.e., the HCR, the IN, and the PUC schemes) is also implemented, which is denoted as the GAR-E algorithm. The simulations are conducted in the network simulator (NS-2, [31]) with wireless extension, using the IEEE 802.11 DCF as the MAC protocol. The parameters utilized in the simulations are listed, as shown in Table 2, and the following five performance metrics are utilized in the simulations for performance comparison:
1. Packet arrival rate. The ratio of the number of
received data packets to the number of total data packets sent by the source.
2. Average end-to-end delay. The average time elapsed
for delivering a data packet within a successful transmission.
3. Path efficiency. The ratio of the number of total HCs within the entire routing path over the number of HCs for the shortest path.
4. Communication overhead. The average number of
transmitted control bytes per second, including both the data packet header and the control packets.
5. Energy consumption. The energy consumption for the
entire network, including transmission energy con-sumption for both the data and control packets under the bit rate of 11 megabits per second (Mbps) and the transmitting power of 15 dBm for each SN. The simulation scenario is explained as follows: The SNs are randomly deployed with the node degree of 17.5 in the network, where the node degree is defined as the average number of nodes within a transmission range. Three pairs of source and destination nodes are respectively located around the left and the right boundaries of the network area. There also exist three equal-height void blocks of width 300 meters that are randomly placed in the network in order to simulate the occurrence of void problems. In other words, there are SNs around the peripheral of the void blocks, while none of the nodes is situated inside the void blocks. The simulations of the performance metrics versus the void height, i.e., the height of each void block, are conducted and compared with other baseline protocols under the UDG and the non-UDG networks. The non-UDG network is obtained by randomly removing some of the communication links within the original UDG network for violating the properties of the UDG setting.
6.1 Simulation Results for UDG Networks
Figs. 7a, 7b, 7c, 7d, and 7e present the performance comparison between these six algorithms with different void heights under the UDG network. As shown in Fig. 7a, both the GAR-based algorithms and the planar graph-based GPSR(GG) and GOAFR++(GG) protocols can achieve 100 percent delivery rate owing to their design nature with guaranteed packet delivery. The BOUNDHOLE and GF algorithms result in lowered delivery ratio due to the occurrence of routing loop and the ignorance of the void problem, respectively. Furthermore, with the augmentation of void height, decreased packet delivery rate can be observed from both the BOUNDHOLE and GF schemes since the probability of encountering the void problem is enlarged.
The performance of the average end-to-end delay versus the void height is shown in Fig. 7b. The smallest end-to-end delay can be found in the GF algorithm,
TABLE 2 Simulation Parameters
owing to the negligence of the void problem, while the GOAFR++(GG) scheme results in the largest delay value due to its bounding techniques [16], [21] that in general cause the back-and-forth forwarding attempts around the large void block. The planar graph-based GPSR(GG) and GOAFR++(GG) schemes possess additional delay in comparison with the proposed UDG-based GAR and GAR-E protocols, owing to the required unnecessary forwarding nodes, as illustrated in Fig. 1. With the adoption of both the HCR and IN mechanisms, the most feasible end-to-end delay performance can be observed from the proposed GAR-E protocol in comparison with the other schemes. It is also noted that the end-to-end delays from all the algorithms will be increased with the augmentation of the void height, which can be attributed to the enlarged number of forwarding hops for boundary traversal. Owing to the closely related characteristics with the end-to-end delay performance, the path efficiency obtained from these schemes follows similar trends, as can be observed in Fig. 7c.
Figs. 7d and 7e illustrate the performance comparisons for communication overhead and energy consumption versus the void height. Except for the BOUNDHOLE scheme, the performance trends from all the other protocols can be observed to be similar with those from the path efficiency in Fig. 7c due to the elongated routing path. Excessive communication overhead associated with more energy consumption will be produced from the BOUND-HOLE scheme comparing with the other algorithm, which can be attributed to its usage of excessive header bytes for preventing the routing loops. It is noted that the decreasing trend within the GF method is primarily due to its relatively low packet delivery ratio, which results in less communication overhead and energy consumption. It is
also noticed that even though the GAR-E scheme requires additional NAV_MAP control packets for achieving the IN mechanism, the total required communication overhead and energy consumption are smaller than those from the GAR method due to its comparably smaller rerouting number of HCs. Therefore, except for the reference GF scheme, it can be expected that the GAR-E algorithm possesses the lowest communication overhead and the energy consumption, which support the merits of the protocol design.
6.2 Simulation Results for Non-UDG Networks
Fig. 8a shows the performance comparison for packet arrival rate versus the void height under the non-UDG network. With the adoption of the PUC mechanism, 100 percent of packet arrival rate can be achieved by exploiting the proposed GAR-E protocol. Moreover, both the GPSR(CLDP) and GOAFR++(CLDP) schemes can also attain the same delivery rate. Nevertheless, these CLDP-enabled schemes will introduce extremely high commu-nication overhead, as illustrated in Fig. 8d. With the augmentation of the void height, it is intuitive to observe that the packet arrival rate obtained from the remaining algorithms will be decreased owing to the increasing severity of the void problem.
The performance comparisons for the average end-to-end delay and the path efficiency are shown in Figs. 8b and 8c. Owing to the guaranteed packet delivery rate, the GPSR(CLDP) and GOAFR++(CLDP) schemes will result in larger delay and worse path efficiency compared to their counterparts, i.e., the GPSR(GG) and GOAFR++(GG) pro-tocols. Other similar results can be found in Figs. 7b and 7c, respectively. It is observed that the proposed GAR-E Fig. 7. Evaluation of performance metrics versus the void height for random UDG networks. (a) Packet arrival rate (percent). (b) Average end-to-end delay (ms). (c) Path efficiency. (d) Communication overhead (byte/sec). (e) Energy consumption (uJ).
scheme can still provide better routing efficiency comparing with other algorithms under the non-UDG networks.
Figs. 8d and 8e present the performance comparisons for the communication overhead and the energy consumption versus the void height. It is especially noticed that extremely high-communication overheads are observed within the GPSR(CLDP) and GOAFR++(CLDP) schemes in comparison with the other six protocols. The main reason is that according to the CLDP algorithm, all communication links will be probed and traversed via additional control packets in order to fulfill the required tasks for planariza-tion. Other similar results can be found in Figs. 7d and 7e, respectively. As can be expected, except for the GF scheme, lowered communication overhead and energy consumption are acquired by the GAR-based algorithms in comparison with the other methods. The merits of the proposed GAR-E scheme can therefore be observed under the non-UDG networks.
7
C
ONCLUSIONIn this paper, a UDG-based GAR protocol is proposed to resolve the void problem incurred by the conventional GF algorithm. The RUT scheme is adopted within the GAR protocol to solve the boundary finding problem, which results in guaranteed delivery of data packets under the UDG networks. The BM and the IMS are also proposed to conquer the computational problem of the rolling mechan-ism in the RUT scheme, forming the direct mappings between the input/output nodes. The correctness of the RUT scheme and the GAR algorithm is properly proven. The HCR and the IN mechanisms are proposed as the delay-reducing schemes for the GAR algorithm, while the PUC mechanism is utilized to generate the required
topology for the RUT scheme under the non-UDG networks. All these enhanced mechanisms associated with the GAR protocol are proposed as the enhanced GAR (GAR-E) algorithm that inherits the merit of guaranteed delivery. The performance of both the GAR and GAR-E protocols is evaluated and compared with existing localized routing algorithms via simulations. The simula-tion study shows that the proposed GAR and GAR-E algorithms can guarantee the delivery of data packets under the UDG network, while the GAR-E scheme further improves the routing performance with reduced commu-nication overhead under different network scenarios.
A
CKNOWLEDGMENTSThis work was in part funded by the MOE ATU Program 95W803C, NSC 96-2221-E-009-016, MOEA 96-EC-17-A-01-S1-048, the MediaTek research center at National Chiao Tung University, and the Universal Scientific Industrial (USI), Taiwan.
R
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Fig. 8. Evaluation of performance metrics versus the void height for random non-UDG networks. (a) Packet arrival rate (percent). (b) Average end-to-end delay (ms). (c) Path efficiency. (d) Communication overhead (byte/sec). (e) Energy consumption (uJ).
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Wen-Jiunn Liu received the BS degree from National Chiao Tung University, Hsinchu, Tai-wan, ROC, in June 2005. Since September 2005, he has been a PhD candidate in the Department of Communication Engineering, National Chiao Tung University. His current research interests include the MAC and network protocol design for mobile ad hoc networks, wireless sensor net-works, and broadband wireless networks. He is a student member of the IEEE.
Kai-Ten Feng received the BS degree from National Taiwan University, Taipei, in 1992, the MS degree from the University of Michigan, Ann Arbor, in 1996, and the PhD degree from the University of California, Berkeley, in 2000. Since August 2007, he has been with the Department of Communication Engineering, National Chiao Tung University, Hsinchu, Taiwan, as an associ-ate professor. He was an assistant professor with the same department between February 2003 and July 2007. He was with the OnStar Corp., a subsidiary of General Motors Corporation, as an in-vehicle development manager/ senior technologist between 2000 and 2003, working on the design of future Telematics platforms and the in-vehicle networks. His current research interests include cooperative and cognitive networks, mobile ad hoc and sensor networks, embedded system design, wireless location technologies, and Intelligent Transportation Systems (ITSs). He received the Best Paper Award from the IEEE Vehicular Technology Conference Spring 2006, which ranked his paper first among the 615 accepted papers. He is also the recipient of the Outstanding Young Electrical Engineer Award in 2007 from the Chinese Institute of Electrical Engineering (CIEE). He has served on the technical program committees of VTC, ICC, and APWCS. He is a member of the IEEE.
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