The r-Neighborhood Graph: An Adjustable
Structure for Topology Control in
Wireless Ad Hoc Networks
Andy An-Kai Jeng and Rong-Hong Jan, Member, IEEE
Abstract—In wireless ad hoc networks, constructing and maintaining a topology with lower node degrees is usually intended to mitigate excessive traffic load on wireless nodes. However, keeping lower node degrees often prevents nodes from choosing better routes that consume less energy. Therefore, the trade-off is between the node degree and the energy efficiency. In this paper, an adjustable structure, named the r-neighborhood graph, is proposed to control the topology. This structure has the flexibility to be adjusted between the two objectives through a parameter r, 0 r 1. More explicitly, for any set of n nodes, the maximum node degree and power stretch factor can be bounded from above by some decreasing and increasing functions of r, respectively. Specifically, the bounds can be constants in some ranges of r. Even more, the r-neighborhood graph is a general structure of both RNGand GG, two well-known structures in topology control. Compared with Y Gk, another famous adjustable structure, our method always results in a connected planar with symmetric edges. To construct this structure, we investigate a localized algorithm, named PLA, which consumes less transmitting power during construction and executes efficiently in Oðn log nÞ time.
Index Terms—Wireless ad hoc networks, topology control, energy-efficient, localized algorithm.
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1
I
NTRODUCTIONW
IRELESS ad hoc networks enhance the conventionaldeployment of communicating environments for many applications, such as conferences, hospitals, battle-fields, search and rescue teams, etc. In these environments, the performance of network operations heavily depends upon the underlying topology [4]. For instance, the delivery rate would be significantly lower as the underlying topology breaks. Therefore, appropriately controlling the topology is a crucial stage in communication. The topology control problem in wireless ad hoc networks has been widely studied in recent years [3], [15], [18], [19], [20], [23], [29], [32]. Generally speaking, the core of this problem is to determine a set of wireless links such that the composed topology is able to achieve certain goals [23]. These goals would be variant depending upon the circumstances and could be either qualitative features or quantitative objectives. Since wireless nodes usually struggle with limited bandwidth and computation power, a genius way should be able to simultaneously achieve several goals. In this paper, we aim to control the topology with the following goals, which are extremely desired in wireless environments:
1. Symmetry. The existence of asymmetric links may complicate many communication primitives. For instance, the MAC layer’s ACK is hard to implement
when some links are not bidirectional [21]. Besides, asymmetric links in topology would also cause inconsistent routing qualities at two ends.
2. Connectivity. Connectivity is unquestionably the most essential prerequisite in any communicable topology [23]. Two nodes u and v are strongly connected if there is a directed path from u to v and vice versa. A directed topology is strongly connected if all pairs of nodes are strongly connected. If the links are symmetric, we should aim at the connectiv-ity of an undirected topology instead.
3. Energy efficiency. Energy is the most crucial resource in wireless nodes. Due to the severe path loss in radio carriers, transmitting with large ranges would exponentially run out of nodes’ energy. Therefore, relaying messages through multiple hops with shorter ranges could usually consume less energy [24]. How to choose the links between nodes for relaying is a critical point in this goal.
4. Sparseness. Numerous distributed and localized routing protocols are based on flooding [13]; how-ever, this may burden networks with unavoidable redundant messages. Thus, keeping a sparse topol-ogy, consisting of linear number of links [15], would be an ingenious way to shrink the expenditure from network operations.
5. Maximum node degree. For some nodes with overly large degrees, the network flows will concentrate on them and rapidly draw out their energy. Besides, a larger node degree means tighter dependency among nodes, which is not expected when wireless nodes move frequently. Therefore, the maximum node degree over a topology should be bounded from above by some constant.
. The authors are with the Department of Computer and Information Science, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu, 300, Taiwan, ROC. E-mail: {andyjeng, rhjan}@cis.nctu.edu.tw.
Manuscript received 7 July 2005; revised 24 Jan 2006; accepted 17 Mar. 2006; published online 25 Jan. 2007.
Recommended for acceptance by K. Nakano.
For information on obtaining reprints of this article, please send e-mail to: [email protected], and reference IEEECS Log Number TPDS-0323-0705. Digital Object Identifier no. 10.1109/TPDS.2007.1004.
6. Planarity. A graph is planar if it has no crossed links inside. It is helpful for many geometric problems: The shortest path (least energy unicast route) can be quickly found in linear time when the underlying topology is planar [12]. Besides, in many position-based routing algorithms, the successful delivery can be guaranteed only if the underlying topology is planar [2], [11].
Taking a further look, a topology having constant node degree must be sparse. So, we can be concerned with the fifth goal only. Unfortunately, keeping nodes with lower node degree would possibly sacrifice some potential links composing more energy efficient routes in topology. There-fore, empirically, a trade-off is between the node degree and energy efficiency [15]. For this reason, we aim to design an adjustable way so that the trade-off can be adjusted flexibly. In wireless ad hoc networks, due to the absence of a central arbitrator and the limited sensing range, centralized approaches [3], [30] are rarely attainable. Therefore, a variety of distributed approaches were proposed [17], [19], [29]. A distributed protocol passes messages hop-by-hop. However, this may cause considerable overhead through the entire network. So, a localized approach is more preferred. According to Stojmenovic and Lin [27], a node using the localized topology control method requires information within constant hop(s). However, in some localized approaches [15], [16], [18], [27], the operations should recursively depend upon the computed status or partial results from nearby nodes, which may hurt their practicability. Therefore, in the following, we define a new type of methodology for more practicability:
Definition 1. An algorithm L is purely localized if it is localized and all operations depend upon only the information inherent1in nodes, available before any execution of L. A purely localized topology control algorithm is more useful to large-scale and high mobility environments, since the operation of a node is completely isolated from any execution of other nodes. Further, we say that a structure is purely localizable if we can construct it by a purely localized algorithm. Our goal is to investigate a purely localizable structure so that all the desired goals listed above can be achieved.
The rest of this paper is organized as follows: Section 2 specifies the network model and formally describes the problem under study. In Section 3, we review and summarize the related works. The main structure, components, and their theoretical results are presented in Section 4. Some detailed derivations are given in the Appendix. In Section 5, we investigate an extended version of the main structure to comprehend our theoretical properties. In Section 6, a purely localized algorithm is investigated to construct our structure. Finally, concluding remarks and some directions for further research are given in the last section.
2
The M
ODEL ANDP
ROBLEMThe wireless ad hoc network discussed in this paper consists of a set V of n wireless nodes distributed on a
two-dimensional plane <2. Each node is equipped with an
omnidirectional antenna and can change its transmission range by adjusting the transmitting power at any level. The maximum transmission ranges are equal among all nodes. In other words, we can normalize the maximum transmis-sion ranges of all nodes to be 1 for simplicity. In addition, each node u can obtain its position P ðuÞ through a lower-power GPS or some other ways [14], and a unique idðuÞ is also available to each node u.
This network can be modeled as a unit disk graph, UDGðV Þ. In this graph, an edge uv exists if and only if the euclidean distance between u and v, denoted as kuvk, is at most 1.
The least power required to transmit immediately between u and v is modeled as kuvk, where is typically taken on a value between 2 and 4, depending on the attenuation strength of the communication environment [5]. To measure the power efficiency of a topology, Li et al. [15] defined a well-formed measure, named power stretch factor. We reintroduce it below. Let ðu; vÞ ¼ v0v1. . . vh1vh be a
unicast path connecting nodes u and v, where v0¼ u and
vh¼ v. The total transmission power consumed by path
ðu; vÞ is defined as
pððu; vÞÞ ¼X
h
i¼1
kvi1vik:
Let GðV Þðu; vÞ be the least-energy path connecting u and v in graph GðV Þ. Given a subgraph G0ðV Þ in UDGðV Þ, the power
stretch factor of G0ðV Þ with respect to UDGðV Þ is defined as
ðGðV ÞÞ ¼ max u;v2V p G0ðV Þðu; vÞ p UDGðV Þðu; vÞ :
On the other hand, the maximum node degree of graph GðV Þ is defined as
dmaxðGðV ÞÞ ¼ max
u2V dGðV ÞðuÞ;
where dGðV ÞðuÞ is the degree of node u in graph GðV Þ.
3
R
ELATEDW
ORKMany localizable structures, used to control the network topology, have been proposed in the literature [15], [16], [18], [26], while only a few of them are purely localizable. In the following, we list four well-known structures. Most of them or their extensions are purely localizable:
. The constrained Relative Neighborhood Graph [28], denoted by RNGðV Þ, has an edge uv if and only if kuvk 1 and the intersection of two open disks2
centered at u; v with radius kuvk contains no node w 2 V ; see Fig. 1a.
. The constrained Gabriel Graph [6], denoted by GGðV Þ, has an edge uv if and only if kuvk 1 and the open disk using kuvk as diameter contains no node w 2 V ; see Fig. 1b.
1. The node’s position and id are usually assumed to be inherited in nodes. See Section 2 for more explanation.
2. An open disk centered at point x with radius d is the collection of points with distance less than d from P ðxÞ.
. The constrained Yao Graph [33] with a parameter k 6, denoted by Y G!kðV Þ, is constructed as follows:
For each node u, define k equal cones by k equal-separated rays originated at u. At each cone, a directed edge uv exists if kuvk 1 and the cone contains no vertex w 2 V such that kuwk < kuvk. Ties are broken arbitrarily. Y GkðV Þ is denoted as the
underlying undirected graph of Y G!kðV Þ; see Fig. 1c.
. A Delaunay Triangulation, denoted by DelðV Þ, is a triangulation of V in which the interior of the circumcircle of each uvw contains no node w 2 V . The unit Delaunay Triangulation, denoted by UDelðV Þ, has all edges of DelðV Þ except those longer than 1 [8], [18]; see Fig. 1d.
Let us discuss the properties of these structures and their extensions. We say an objective fð:Þ of a structure SðV Þ is bounded if there is a constant C such that fðSðV ÞÞ C, for any set V of n nodes. Li et al. [15] showed that dmaxðRNGðV ÞÞ
is unbounded if there is a node u 2 V having an unbounded number of neighbors adjacent to u at exactly the same distance in the underlying UDGðV Þ. To overcome this problem, Wattenhofer and Zollinger [32] proposed an algorithm to find a structure, denoted by XT CðV Þ. They showed that XT CðV Þ is a subgraph of RNGðV Þ and the dmaxðXT CðV ÞÞ is at most 6. Especially, if there is no node
having two or more neighbors at exactly the same distance in V, XT CðV Þ is identical to RNGðV Þ [24]. Their results infer the following theorem:
Theorem 1. Given a set V of nodes on <2, if there is no node
having two or more neighbors at exactly the same distance, then dmaxðRNGðV ÞÞ 6.
We denote the condition in Theorem 1 as Assump-tion AS. That is,
Assumption AS. There is no node in V having two or more neighbors at exactly the same distance.
This theorem reveals that even RNGðV Þ has no constant bound on its node degree. It is still useful since the distances of nodes in the real world are rarely exactly the same. The constrained Gabriel Graph GGðV Þ has the least power stretch factor 1 in comparison with the unbounded power stretch factor n 1 of RNGðV Þ [15]. However, dmaxðGGðV ÞÞ could be as large as n 1. An extended
structure, Enclosure graph [16], [14], [24], denoted by EGðV Þ, is generalized from GGðV Þ. It can always result in a subgraph of GGðV Þ [16]. Even so, its maximum node degree is still unbounded [20], [24].
To overcome the trade-off between the maximum node degree and the power stretch factor, an adjustable structure, having the flexibility to be adjusted between the two objectives, becomes more attractive. Y G!kðV Þ is an adjustable
structure. It can be adjusted through a parameter k such that, for any given k, the maximum out-degree is at most k, and the power stretch factor is at most 1=ð1 ð2 sin =kÞÞ [15]. We say an objective fð:Þ of an adjustable structure SkðV Þ
with parameter k is partially bounded if there is at least one k0
such that fðSk0ðV ÞÞ is bounded. According this definition, the maximum out-degree and power stretch factor of Y G!kðV Þ
are partially bounded since, for some ranges of k, k and 1=ð1 ð2 sin =kÞÞ are constants. However, the asymmetric edges of Y G!kðV Þ may lead to large in-degrees even when k is
very small [15]. So, dmaxðY GkðV ÞÞ can be neither bounded
nor partially bounded. To improve this, an extension of Y G
!
kðV Þ, named Yao and Sink, was proposed [15], [17], [29]. It
can limit the maximum node degree in ðk þ 1Þ2 1 and result in symmetric edges. Unfortunately, in this structure, the neighbors of some node should be recursively deter-mined by one another so that it cannot be purely localizable. The unit Delaunay triangulation UDelðV Þ has a bounded power stretch factor. However, neither DelðV Þ nor UDelðV Þ can be computed locally. So, Li et al. [18] suggested a localized version of the Delaunay graph, denoted by LDelðhÞðV Þ, where h means that each node uses at most k-hopinformation. The power stretch factor of LDelðkÞðV Þ is bounded for all k 1. Even so, its maximum node degree is not bounded for any h.
The relations among these structures were studied in several papers [7], [10], [16], [22], [24], [33]. We summarize them in Fig. 2, where EMST ðV Þ is the euclidean minimum spanning tree of UDGðV Þ. With these relations, their connectivity and planarity can be easily inferred.
Regarding their connectivity, we know that EMST ðV Þ is connected if UDGðV Þ is itself a connected component of V . Therefore, when UDGðV Þ is connected, all graphs containing EMSTðV Þ are connected. That is, RNGðV Þ, GGðV Þ, EGðV Þ, UDelðV Þ, LDelðkÞðV Þ, and Y G
kðV Þ are all connected. The
connectivity of XT CðXÞ was proven in a different way [24]. Regarding their planarity, LDelðkÞðV Þ is planar for any k 2 [18]. Therefore, all subgraphs of LDelð2ÞðV Þ are planar.
That is, UDelðV Þ, GGðV Þ, EGðV Þ, RNGðV Þ, XT CðV Þ, and EMSTðV Þ are all planar. On the contrary, Y G!kðV Þ and
LDeð1ÞlðV Þ cannot avoid producing a crossed link, so they are not planar [15], [18]. Table 1 summarizes the above discussion.
From Table 1, we can see that no presented structure can bound or even partially bound the two objectives. Besides, to the best of our knowledge, no other structure can be purely localizable and achieve this goal. Therefore, we will propose the first purely localizable structure, named r-Neighborhood Graph, to fill this gap. This structure is adjustable and can always result in a connected planar with symmetric edges. In addition, we can show that our structure is a generation of both GGðV Þ and RNGðV Þ.
Apart from the purely localizable structures, several composite methods, based on combining two or more existent structures, were investigated in the last few years [17], [19], [25], [31]. Conceptually, the main idea is to use the virtue of one structure to patch up the fault in the other structures. For examples, the ordered Yao structure, denoted as OrdY aoðV Þ [1], is a variation of Y G
kðV Þ. It has the
partially bounded maximum node degree and length stretch factor. However, the planarity cannot be guaranteed. Therefore, Li and Wang [19], [31] applied OrdY aoðV Þ onto LDelð2ÞðV Þ to avoid the crossed edges produced by OrdY aoðV Þ; Song et al. [25] improved it by applying the OrdY aoðV Þ on GGðV Þ using only one-hop information. However, the construction of OrdY aoðY Þ requires exchan-ging the computed status as well as partial results between nodes. Consequently, none of them is purely localized or purely localizable.
4
T
HEr-N
EIGHBORHOODG
RAPHIn this section, we introduce a new adjustable structure. First, we define a region on <2. It will be used to compose
our structure. Let x be any point on <2and the open disk and
circle centered at P ðxÞ with radius d are denoted as Dðx; dÞ and Cðx; dÞ, respectively. The region is defined as follows: Definition 2. Given a node pair ðu; vÞ on <2, the
r-neighborhood region of ðu; vÞ, denoted as NRrðu; vÞ, is
defined as
NRrðu; vÞ ¼ Dðu; kuvkÞ \ Dðv; kuvkÞ \ Dðmuv; luvÞ;
where muvis the middle point on uv,
luv¼ ðkuvk=2Þð1 þ 2r2Þ1=2;
and 0 r 1.
When not confused, we use m and l instead of muv and
luv, respectively. In Fig. 3, the shaded region intersected
by the three open disks sketches an example of the r-neighborhoodregion. This region is obviously equivalent to the following point set:
NRrðu; vÞ ¼ fP ðxÞ 2 R2jkuxk < kuvk; kvxk < kuvk; kmxk < lg:
ð1Þ For any node w located on NRrðu; vÞ, this region limits the
power consumed by path uwv. This property is shown in Lemma 2 and derived in the Appendix.
Fig. 2. The relations of the purely localizable structures and their extensions.
TABLE 1
The Properties of the Four Main Purely Localizable Structures
Lemma 2.Given two nodes u and v on <2, for any node w such
that P ðwÞ 2 NRrðu; vÞ, pðuwvÞ < kuvkð1 þ rÞ, for all 2.
This lemma explains why we call such a plane a neighborhood region: For any node w located in the region NRrðu; vÞ, it should be an alternative neighbor for u with
respect to v in the sense that the power required for relaying from u to v through w is no greater than 1 þ r times the
immediate transmission. Based on this region, the structure is defined below:
Definition 3.Given a set V of nodes on <2, the r-neighborhood
graph of V , denoted as NGrðV Þ, has an edge uv if and only if
kuvk 1 and NRrðu; vÞ contains no node w 2 V , where
0 r 1.
By Definition 3, if edge uv is not in UDGðV Þ or a node w is inside NRrðu; vÞ, there is no direct link connecting u and v
in NGrðV Þ, which means that all transmissions between u
and v should be relayed through some other node(s) in NGrðV Þ. Now, we explore the desired properties in our
structure. Before this, we shall discussion the following relations:
Lemma 3.For any set V of nodes on <2,
RNGðV Þ NGrðV Þ GGðV Þ;
for all 0 r 1.
Proof. Consider the open disk Dðm; kuvk=2Þ defining GGðV Þ. Suppose uv 2 NGrðV Þ, the region NRrðu; vÞ,
has no node inside. Since Dðm; kuvk=2Þ is obviously a subregion of NRrðu; vÞ, for any 0 r 1, there is also no
node in Dðm; kuvk=2Þ. Therefore, according to the definition of GGðV Þ, we get uv 2 GGðV Þ. On the other hand, consider the two open disks Dðu; kuvkÞ and Dðv; kuvkÞ defining RNGðV Þ. Suppose uv 2 RNGðV Þ, no node is inside the intersection of Dðu; kuvkÞ and Dðv; kuvkÞ, which obviously covers the region NRrðu; vÞ,
for any 0 r 1. Therefore, no node can be inside NRrðu; vÞ and we get uv 2 NGrðV Þ. tu
Specifically, as r ¼ 0, NR0ðu; vÞ Dðm; kuvk=2Þ, which
is the disk defining GGðV Þ. On the contrary, as r ¼ 1, NR1ðu; vÞ Dðm; kuvkÞ, which is the disk defining
RNGðV Þ. Therefore, GGðV Þ NG0ðV Þ and RNGðV Þ
NG1ðV Þ: So, we can conclude the following theorem:
Theorem 2. The r-neighborhood graph is a generalized structure of both the restricted Gabriel graph and the restricted relative neighborhood graph.
Since a subgraph of a planar graph is always planar, and a supergraph of a connected graph is always connected, with the planarity of GGðV Þ and connectivity of RNGðV Þ, we can infer the following two theorems:
Theorem 3.For any set V of nodes on <2, NR
rðV Þ is planar, for
all 0 r 1.
Theorem 4. For any set V of nodes on <2, if the underlying
UDGðV Þ is connected, NRrðV Þ is connected, for all
0 r 1.
Now, we consider the energy efficiency and node degree of NRrðV Þ. We will show that the upper bound of
ðNGrðV ÞÞ is increased by r and, contrarily, the upper
bound of dmaxðNGrðV ÞÞ is decreased by r. In other words,
the r-neighborhood graph is adjustable to the two objectives through the parameter r. With these results, we can further show that the power stretch factor and maximum node degree are partially bounded in our structure. Before these, a property proposed by Li et al. [15] shall be mentioned. It can be used to simplify our proof.
Lemma 4 [15]. Given a subgraph G0ðV Þ UDGðV Þ and a
constant C, ðG0ðV ÞÞ C if and only if, for any edge uv in GðV Þ, there is a path ðu; vÞ in G0ðV Þ such that
pG0ðV Þðu; vÞ Ckuvk.
This lemma indicates that to derive an upper bound for ðNGrðV ÞÞ, it is sufficient to consider only those node
pairs having direct links in UDGðV Þ. So, we aim to derive a strictly decreasing function F ðrÞ, such that, for any uv in UDGðV Þ, a path ðu; vÞ is in NRrðV Þ such that
pððu; vÞÞ F ðrÞkuvk. To achieve this, we investigate an algorithm called EXPANSION with an input of any two nodes ðu; vÞ that outputs subgraph S of NRrðV Þ related to
ðu; vÞ. Let P ðSÞ be the total transmission power of edges in S, i.e., P ðSÞ ¼Pst2Spðs; tÞ. We can show that there is some path in S connecting ðu; vÞ and P ðSÞ F ðrÞkuvk. ALGORITHMEXPANSION
Input: A node pair ðu; vÞ in V .
Output: A subgraph S and a positive value P . Step 1: S ¼ fg, S0¼ fðu; vÞg, Q ¼ fu; vg, P ¼ kuvk; Step 2: When some node pair ðs; tÞ is in S such that a node w2 NRrðs; tÞ S0¼ S0 ðs; tÞ; If w =2 Q then S0¼ S0[ ðs; wÞ [ ðw; tÞ; Q¼ Q [ fwg; P¼ P þ ðkstkrÞ; Otherwise, S0¼ S0[ ðs; wÞ;
Step 3: S ¼ fxy 2 NGrðV Þjðx; yÞ 2 S0g;
Step 4: Stop and output E and P .
In this algorithm, S0 is a set of node pairs in which an
edge st in NRrðV Þ can be a part of S only if its two ends
ðs; tÞ are in S0as described at Step 3. So, to determine S, we
have to discuss the S0first. Initially, S0contains only ðu; vÞ. Then, it will be recursively expanded as follows: For each ðs; tÞ in S0, if a node w is in NR
rðs; tÞ and not considered
before, replace ðs; tÞ with ðs; wÞ and ðw; tÞ; if a node w is in NRrðs; tÞ but considered before, replace ðs; tÞ with ðs; wÞ;
otherwise, keep ðs; tÞ unchanged. We use the set Q to record the considered nodes.
When some ðs; tÞ is in S0such that a node w 2 NR rðs; tÞ,
no matter whether w is considered or not, by (1), the replaced node pair(s) must be shorter than kstk, i.e., kswk < kstk and kwtk < kstk. Thus, after finite iterations, each node pair in S0 can be replaced by another node pair with
the shortest distance. So, the algorithm is terminable. Now, we show that ðu; vÞ is connected by some path in the subgraph S when termination occurs.
Lemma 5.Given any set V of node on <2, for any two nodes u
and v in V , if edge uv is in UDGðV Þ and UDGðV Þ is connected, there is some path in S connecting ðu; vÞ.
Proof.Since Q includes u and v, we can prove this lemma by showing that all nodes in Q are connected in S. For each expansion of S0, we define a dummy graph S00 in
which an edge st exists if and only if ðs; tÞ is in S0(note
that any edge in S00 is not necessarily in either UDGðV Þ
or NRrðV Þ). First, we show that, at any iteration, all
considered nodes in Q are connected by S00. Initially, Q is
connected by S00, since S0¼ fðu; vÞg and Q ¼ fu; vg. We
assume for induction that all nodes in Q are connected by S00at the kth iteration. Then, we show that it is true for the next iteration. At the k þ 1th iteration, if there is no pair in S0 that satisfies the entrance condition of Step 2,
the claim is correct, since Q and S00 are unchanged; otherwise, a node pair ðs; tÞ 2 S0 is expended. In this
case, if the chosen w =2 Q, w is connected with all nodes in Qvia dummy edges sw and wt; otherwise, w 2 Q, which implies that all nodes in Q are still connected by S00as in the previous iteration. As described above, the distance of any expended node pair is no longer than the previous one. So, if uv is in UDGðV Þ, all edges in S00 are also in
UDGðV Þ. Then, as the algorithm proceeds to Step 3, no nodes can be in the r-neighborhood region of any node pair in S0. With these two facts, all dummy edges in S00
are also in NRrðV Þ at termination. So, S is equivalent to
the last S00. Consequently, if UDGðV Þ is connected, by
Theorem 4, all nodes in the last Q are connected to S. tu Then, we derive a strictly decreasing function F ðrÞ using the value P in this algorithm.
Lemma 6.Given any set V of n nodes on <2, for any two nodes u
and v in V ,
PðSÞ F ðrÞkuvk and FðrÞ ¼ 1 þ ðn 2Þr; for all 0 r 1 and 2.
Proof.Let P ðS0Þ ¼Pðs;tÞ2S0pðs; tÞ. We show that P ðS0Þ P at each iteration of Step 2. Initially, S0¼ fðu; vÞg. We can get
PðS0Þ ¼ kuvk
¼ P . Then, at the first iteration, if no node w is in NRrðu; vÞ, the claim remains true since neither P nor S
is changed; otherwise, a node w is in NRrðu; vÞ. Since there
is no node except u and v in Q so far, ðu; vÞ is replaced by ðv; wÞ and ðw; vÞ. By Lemma 2,
PðvwÞ þ P ðwuÞ P ðuvÞð1 þ rÞ ¼ P þ ðkuvkrÞ
: Consequently, the new P remains an upper bound of PðS0Þ. We assume for induction that P ðS0Þ P at the
kthiteration. Then, we prove that the claim is true at the next iteration. If the entrance condition of Step 2 is not satisfied or the chosen w =2 Q, it can be proved by the same reasons as in the first iteration. Otherwise, ðs; tÞ is replaced by ðs; wÞ only. By (1), P ðstÞ P ðswÞ, which implies that the unchanged P is still an upper bound of P ðS0Þ. Besides,
(1) further implies that the distance between any two nodes in S0 is no greater than kuvk. So, another upper
bound P0can be found by replacing P ¼ P þ ðkstkrÞ
by P0¼ P0þ ðkuvkrÞ
. Moreover, we can observe that the
situation where w is chosen from some NGrðs; tÞ that is not
in Q never happens more than n 2 times, since, in this case, the size of Q must be increased by 1. Consequently, PðS0Þ P P0 P ðuvÞ þ P ðuvÞrðn 2Þ. Finally, we get
FðrÞ ¼ ð1 þ rÞðn 2Þ. tu
With Lemmas 4, 5, and 6, we can conclude the following theorem:
Theorem 5.For any set V of n nodes on <, for all 0 r 1 and 2,
ðNGrðV ÞÞ 1 þ rðn 2Þ ¼ F ðrÞ:
Although this bound is related to the node size n so that ðNRrðV ÞÞ cannot be bounded, it can still be constant when
r is 0 or sufficiently small, i.e., ðNRrðV ÞÞ is bounded in
some range of r. So, we can make the following conclusion: Corollary 1. The power stretch factor of the r-neighborhood
graph is partially bounded.
Consider the maximum node degree of the r-neighborhood graph. Since NRrðV Þ consists of all edges in RNGðV Þ, the
maximum node degree of NRrðV Þ is no less than that of
RNGðV Þ. In Section 3, we know that dmaxðRNGðV ÞÞ is not
always bounded in any case of V . Thus, dmaxðNGrðV ÞÞ is also
unbounded. Fortunately, Theorem 1 indicates that dmaxðRNGðV ÞÞ is bounded in most cases of V , where AS is
assumed. Therefore, in the following theorem, we analyze the maximum node degree of the r-neighborhood graph under assumption AS.
Theorem 6.For any set V of nodes on <2with assumption AS,
for all 0 r 1,
dmaxðNGrðV ÞÞ = sin1ðr=2Þ
:
Proof.To prove this statement, it is sufficient to show that, in NGrðV Þ, there are no adjacent edges enclosing an
angle less than 2 sin1ðr=2Þ. Assume for contradiction that two edges uv and uw in NGrðV Þ enclose an angle
< 2 sin1ðr=2Þ at node u, where w, v 2 V . Without loss of generality, we assume that kuwk < kuvk. With assumption AS, all nodes are placed on different positions, i.e., P ðxÞ 6¼ P ðyÞ, for any two nodes x, y 2 V .
Consider the length of vw: If ffuwv is obtuse, it is clear that kvwk < kuvk (note that kvwk cannot be equal to kuvk, since P ðuÞ 6¼ P ðwÞ); see Fig. 4b. Otherwise, if ffuwv is not obtuse, kvwk is less kvw0k, where kuw0k ¼ kuvk; see
Fig. 4a. By the law of cosines, we have kvw0k2
¼ kuw0k2
þ kuvk2 2kuw0kkuvk cos
¼ 2kuvk2 2kuvk2cos
< 2kuvk2 2kuvk2cos 2 sin 1ðr=2Þ:
ð2Þ
If 0¼ 2 sin1ðr=2Þ, we get sinð0=2Þ ¼ r=2. Then, one of
the corresponding right-angled triangles is as shown in Fig. 4c. In this case, cos 0¼ ð2 r2Þ=2. Thus, we can get
that 2 sin1ðr=2Þ ¼ 0¼ cosðð2 r2Þ=2Þ. Consequently,
Equationð2Þ ¼ 2kuvk2 2kuvk2cos cos 2 r2=2
Consequently, we have that, for any case of ffuwv, kvwk < max kuvkr; kuvkf g ¼ kuvk: ð4Þ Consider the length of um: If ffuwm is obtuse, kwmk < kuvk=2; see Fig. 4b. Otherwise, kmwk is less kmw0k; see Fig. 4b. By the law of cosine, we have
kmw0k2
¼ kuw0k2
þ kum0k2
2kuw0kkum0k cos
<kuvk2þ kuvk2=4 kuvk2cos
< 5kuvk2=4 kuvk2ð2 r2Þ=2¼ kuvk2
ð1 þ 2r2Þ=4
: ð5Þ
Similarly, we have, for any case of ffuwm,
kmwk < max kuvkn pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2r2=2;kuvk=2o¼ l: ð6Þ
By (4), (6), and the assumption of kuwk < kuvk, w is included in the set of points specified in (1). Therefore, PðwÞ 2 NRrðu; vÞ. However, it contradicts the assumption
that uv is in NRrðV Þ. Thus, we conclude this theorem. tu
However, for those instances of V without AS, Theorem 6 cannot hold anymore. See the instance in Fig. 5, where all nodes except viare placed on the outlier of NRrðvi; v1Þ. This
will result n 1 neighbors adjacent to vi in NRrðV Þ.
So, in the next section, we propose an extended version of the r-neighborhood graph. As the readers will see, the extended structure has the partially bounded maximum node degree for all cases of V and inherits almost all of the desired features in NRrðV Þ.
5
T
HEE
XTENDEDr-N
EIGHBORHOODG
RAPHIn this section, an extended structure of the r-neighborhood graph is given. The main goal is to avoid the unbounded maximum node degree in NRrðV Þ. In this extension,
Assumption AS is not required anymore. Instead, a unique identifier idðuÞ is available to each node u in V . The structure is defined as follows:
Definition 4. Given a set V of nodes <2, the extended
r-neighborhood graph of V , denoted as NG
rðV Þ, has an
edge uv if and only if kuvk 1 and there exists no node w2 V satisfying one of the following three conditions:
. D1: PðwÞ 2 NRrðu; vÞ;
. D2: PðwÞ 2 Dðmuv; luvÞ \ Cðv; kuvkÞ and
idðuÞ > idðwÞ;
. D3: PðwÞ 2 Dðmuv; luvÞ \ Cðu; kuvkÞ and
idðvÞ > idðwÞ.
Without D2 and D3, NGrðV Þ is clearly equivalent to the
original r-neighborhood graph. In conditions D2and D3, the
two subregions of Dðmuv; luvÞ intersected by Cðv; kuvkÞ and
Cðu; kuvkÞ are, as depicted in Fig. 6, the solid left arc and the right arc along the outlier of NRrðu; vÞ, respectively. When a
node w is located in these two arcs, the existence of edge uv should be further determined by their identifiers.
Fig. 4. (a)ffuwv and ffuwm are not obtuse. (b) ffuwv and ffuwm are obtuse. (c) A right-angled triangle with angle ¼ 2 sinðr=2Þ.
Fig. 5. dmaxðNGrðV ÞÞ is not bounded if Assumption AS does not hold.
Fig. 6. The r-neighbor region of nodes u and v and the two intersections defined in D2and D3.
Hereafter, we say that a node w 2 V blocks an edge uv in UDGðV Þ if and only if w satisfies one of the three conditions in Definition 4.
In NG
rðV Þ, an edge uv of UDGðV Þ will not only be
blocked by some node w in NRrðu; vÞ, but may also be
blocked when either D2or D3 happens. Therefore, NGrðV Þ
constitutes a subgraph of NRrðV Þ, which means that the
maximum node degree of NG
rðV Þ is no worse than its
original version. In the following theorem, we show that the upper bound of dmaxðNGrðV ÞÞ in Theorem 6 remains correct
in dmaxðNGrðV ÞÞ and the correctness is, for any case of V ,
not subject to Assumption AS.
Theorem 7.For any set V of nodes on <2, for all 0 r 1,
dmaxðNGrðV ÞÞ
sin1ðr=2Þ
:
Proof. Using the same argument as Theorem 6, we assume for contradiction that two edges uv and uw in NG
rðV Þ
enclose an angle 0< 2 sin1ðr=2Þ at node u. Without loss of
generality, we assume that kuwk kuvk. If kuwk < kuvk, the argument of Theorem 6 has proved the contradiction. Consider kuwk kuvk: Let w0 be a point crossed by
Cðu; kuvkÞ and the outlier of Dðmuv; luvÞ, as shown in
Fig. 7. The two edges w0uand uv enclose an angle 0. By the
law of cosines, we have cos 0¼kuw 0k2 þ kuvk=2ð Þ2kmuvw0k2 kuw0kkuvk ¼kuvk 2 þ kuvk=2ð Þ2l2 uv 2 kuvkkuvk ¼ 1 þ r 2=2:
Then, one corresponding right-angle triangulation is as Fig. 4c. In this case, sinð0=2Þ ¼ r=2. Thus, we can get that < 0¼ 2 sin1ðr=2Þ. Since kuwk kuvk, both P ðwÞ
and P ðvÞ are on Cðu; kuvkÞ. The fact that < 0 further
limits P ðwÞ on the arc intersected by Dðmuv; luvÞ.
Similarly, P ðvÞ is limited on the arc intersected by Dðmuv; luvÞ for the same reason. Therefore, P ðwÞ and
PðvÞ are on the regions defined in D2, with respect to
edges uw and uv, respectively.
Next, the existence of uv and uw should be deter-mined by their identifiers. If idðvÞ > idðwÞ, uv is blocked by w. Otherwise, if idðvÞ < idðwÞ, uw is blocked by v. As a sequel, no matter what the values of idðvÞ and idðwÞ are, at least one of the edges enclosing 0 cannot be in
NGrðV Þ. Thus, we proved this theorem. tu From Theorem 7, we can see that dmaxðNGrðV ÞÞ is
constant when r is sufficiently large. Therefore, there is some setting of r such that dmaxðNGrðV ÞÞ is bounded by
some constant for any set V of n nodes. So, we reach the following conclusion:
Corollary 2. The maximum node degree of the extended r-neighborhoodgraph is partially bounded.
In the rest, we show that NG
rðV Þ inherits all desired
properties achieved by NRrðV Þ, except the generality
for RNGðV Þ. The fact that NG
rðV Þ NGrðV Þ confirms
the planarity of NGrðV Þ, since NRrðV Þ is planar for
any r. Moreover, when r ¼ 0, the two arcs defined in D2 and D3 are empty. Thus, whether an edge is in
NGrðV Þ is solely dependent on D1, which means that
NG0ðV Þ NG0ðV Þ GGðV Þ. Therefore, NGrðV Þ remains
a general structure of GGðV Þ.
However, as shown in Theorem 7, some adjacent edges having the same length in RNGðV Þ would be avoided in NG
rðV Þ. Thus, RNGðV Þ is not always a subgraph of
NGrðV Þ. This means that NG
1ðV Þ is not essentially
equivalent to RNGðV Þ. Even more, NG
1ðV Þ could be a
subgraph of RNGðV Þ. Therefore, NGrðV Þ is no longer a
general structure of RNGðV Þ.
About the connectivity, because RNGðV Þ is not always a subgraph of NG
rðV Þ, we cannot ensure the connectivity of
NGrðV Þ directly from that of RNGðV Þ. Therefore, we apply an entirely different logic to prove this property. The idea is based on comparing the lexicographic orders of nodes pairs. This idea has been successfully used to prove the connectivity of XT CðV Þ [32], another subgraph of RNGðV Þ. We define a three-field tuple ðkuvk; idðuÞ; idðvÞÞ for each node pair ðu; vÞ. The lexicographic order of ðu; vÞ is smaller than that of another node pair ðs; tÞ if one of the follow-ing three cases happens: 1) kuvk < kstk, 2) kuvk ¼ kstk,
Fig. 7. If < 2 sin1ðr=2Þ and kuwk ¼ kuvk, either uw or uv cannot be in NG rðV Þ.
and idðuÞ < idðsÞ, or 3) kuvk ¼ kstk, idðuÞ ¼ idðsÞ and idðvÞ < idðtÞ. Now, we prove the connectivity of NG
rðV Þ
in Theorem 8.
Theorem 8. For any set V of nodes on <2, if the underlying
UDGðV Þ is connected, NG
rðV Þ is connected, for all
0 r 1.
Proof.Suppose UDGðV Þ is connected. Let UðV Þ be the set of unconnected nodes pairs in NGrðV Þ. We assume for
contradiction that some nodes pairs in NG
rðV Þ are not
connected, i.e., UðV Þ is not empty. Let ðu; vÞ be the node pair with smallest lexicographic order in UðV Þ. Assume that edge uv is not in UDGðV Þ, i.e., kuvk > 1. Since UDGðV Þ is connected, there must be some path longer than one hop connecting u and v. Let ðu; vÞ be such a path in UDGðV Þ. Since kuvk > 1, the lengths of each edge on ðu; vÞ is less than kuvk. When this path is mapped to NGrðV Þ, there are some node pairs on ðu; vÞ
uncon-nected in NGrðV Þ. Thus, some unconnected node pair on
ðu; vÞ has length shorter than kuvk, which, however, contradicts that ðu; vÞ has the smallest lexicographic order in UðV Þ. Therefore, edge uv must be in UDGðV Þ.
Since edge uv is in UDGðV Þ and not in NG
rðV Þ, there
must be some node w satisfying one of the three conditions in Definition 4. Besides, either ðu; wÞ or ðw; vÞ is in UðV Þ; otherwise ðu; vÞ can be connected by path uwv. We consider the three cases:
1. If D1 happens, P ðwÞ 2 NRrðu; vÞ. So, we have
kuwk < kuvk and kwvk < kuvk, which means that the lexicographic orders of ðu; wÞ and ðw; vÞ are less than that of ðu; vÞ.
2. If D2 happens, we have kwvk ¼ kuvk and
idðuÞ > idðwÞ, which means that the lexicographic order of ðw; vÞ is less than that of ðu; vÞ;
3. If D3 happens, we have kuwk ¼ kuvk and
idðvÞ > idðwÞ, which means that the lexicographic order of ðu; wÞ is less than that of ðu; vÞ.
Therefore, we cannot find any node pair in UðV Þ having the smallest lexicographic order. In other words, UðV Þ is empty, which, however, is a contradiction. Thus, we have proven this algorithm. tu
Due to the fact that NGrðV Þ NGrðV Þ, there may be
some paths in NGrðV Þ not in NGrðV Þ. Therefore,
ðNG
rðV ÞÞ is no better or even worse than ðNGrðV ÞÞ.
Even so, the upper bound of NG
rðV ÞðUDGðV ÞÞ can be as good as that proven in Theorem 5. We briefly explain this: All arguments in Theorem 5 are not related to the two additional conditions D2and D3, except those referred from
Lemma 2. Whichever D1, D2, or D3 happens, kuwk kuvk,
kvwk ¼ kuvk, and kmvk < l, which means that all inequal-ities in the proof of Lemma 2 are unchanged. Consequently, Theorem 5 is still correct, even if all conditions of Definition 4 are considered. So, ðNG
rðV ÞÞ is also partially
bounded.
Below, we show that the bound 1 þ rðn 2Þ in
Theorem 5 is not only correct, but also asymptotically tight to the worst possible value of NG
rðV ÞðUDGðV ÞÞ. In other words, it is very hard to find another upper bound of NG
rðV ÞðUDGðV ÞÞ better than ours. We apply the same argument as that used to verify the tightness of the length stretch factor [3] and the power stretch factor [15] of RNGðV Þ
Theorem 9.For any n 2 and 0 r 1, there is a set V of nnodes such that
sup
jV j¼n
NG
rðV ÞðUDGðV ÞÞ > 1 þ r
ðn 2Þ ";
for any sufficient small " > 0.
Proof.Let 1¼ 2 sin1ðr=2Þ2 and 2¼ =2 sin1ðr=2Þþ,
where > 0. We construct a set of n nodes V ¼ fv1; v2; . . . ;
v2m1; v2m; . . . ; vng, where n 2 is even and m ¼ n=2 as
follows:
1. kv1v2k 1 and kviviþ1k ¼ kv1v2k, for
i¼ 2; 3; . . . ; 2m 1; 2. ffviviþ1viþ2¼ 1, for i ¼ 1; 2; . . . ; 2m 2;
3. ffviþ2viviþ1¼ ffviviþ2viþ1¼ 2, for i ¼ 1; 2; . . . ; 2m2;
4. idðviÞ ¼ n i þ 1, for i ¼ 1; 2; . . . ; n.
One corresponding UDGðV Þ is as shown in Fig. 8a. For i¼ 1; 2; . . . ; 2m 2, since ffviviþ1viþ2¼ 1< 2 sin1ðr=2Þ
and kviviþ1k ¼ kviþ1viþ2k, by the argument in Theorem 7,
we get P ðviþ2Þ 2 Dðvi; viþ1Þ \ Cðmviviþ1; lviviþ1Þ. That is,
Fig. 8. A worst-case instance V of n nodes in NG
Pðviþ2Þ is in the regions with respect to edge viviþ1, defined
in D2. Moreover, idðviÞ > idðviþ2Þ. Thus, edge viviþ1is not
in NGrðV Þ. Then, the remaining edges are exactly a path
(spanning tree) v1v3v5. . . v2m3v2m1v2mv2m2. . . v6v4v2 of
V, connecting all nodes, as the bold links in Fig. 8a. Therefore, we can get that
p NG rðV Þðv1; v2Þ ¼ X 2m2 i¼1 rkviviþ2kþ kv2m1v2mk:
As ! 0, 1! 2 sin1ðr=2Þ, which implies that kviviþ2k !
rkviviþ2k ¼ rkv1v2k according to (3). Consequently, as
! 0, we get that X 2m2 i¼1 rkviviþ2kþ kv2m1v2mk !X 2h2 i¼2 rkv1v2kþ kv1v2k ¼ kv1v2kððn 2Þrþ 1Þ:
On the other hand, since kv1v2k 1, we get
pðUDGðV Þðu; vÞÞ ¼ kuvk:
Therefore, as ! 0, NGrðV ÞðUDGðV ÞÞ ! 1 þ r
ðn 2Þ.
That is, supjV j¼nNG
rðV ÞðUDGðV ÞÞ > 1 þ r
ðn 2Þ ",
for any sufficient " > 0. For any odd n 2, the result can be obtained by applying the same argument to the instance as shown in Fig. 8b. So, we proved this
theorem. tu
Actually, an equivalent structure of NGrðV Þ, without an
original version like NRrðV Þ, was mentioned in our
previous paper3 [9]. In that preliminary work, however, only qualitative results were given. To prove the quantita-tive results, we separate NRrðV Þ from NGrðV Þ in this paper
because NRrðV Þ has a clearer form in definition that can be
used to highlight the main tricky parts in our derivations. Besides, all qualitative results in [9] are reevaluated here using different arguments.
6
P
URELYL
OCALIZEDA
LGORITHMIn this section, we propose an efficient purely localized algorithm, named PLA, to construct the r-neighborhood graph. This algorithm consists of two main procedures, GETINFand FINDNB. First, GETINFcollects a set of nodes’ information within one-hop distance, denoted as INu. Then,
the collected information will be fed into FINDNB to determine a set of neighbors in NRrðV Þ, denoted as NBu.
ALGORITHM PLA Input: A ratio 0 r 1.
Output: A set of neighbors adjacent to u. Step 1: INu:¼ GETINFðu; rÞ;
Step 2: NBu:¼ FINDNBðu; r; INuÞ;
Step 3: Stop and output NBu;
To collect the one-hop information, the simplest way is to let each node broadcast its information at the maximum transmission range 1 and gather the information from others. However, the severe path loss and the frequent change in topology may cause considerable power in such transmission. Therefore, in GETINF, we aim to reduce the transmission range during construction. The main idea is to incrementally raise the transmission power from a small range and then use some rule to stop the increment earlier before the transmission range 1 is reached. The detailed steps are explained as follows: The transmission range is initiated at a small distance d0, and then it will be
incrementally raised for several rounds. Let d1 and d2 be
the previous and the current transmission ranges of a round, respectively. In each round, a node broadcasts a request to distance d2 and waits for the responses from the
receiving nodes to gather the nodes’ information. To avoid replying to a node for the second time, the request of a node u contains the position P ðuÞ and the previous distance d1. As a node v receives this request, it calculates
the euclidean distance kuvk. Then, if kuvk > d1, v responds
with its information, P ðvÞ, to u at distance kuvk; otherwise, it just neglects the request. In each round, the range is increased by multiplyingpffiffiffi2
, which means the transmission power is multiplied by 2 each time. The process is continued until the following stopping criterion is satisfied. Let v1 and v2 be two crossed points intersected by
Cðu; kuvkÞ and Cðm; lÞ; see Fig. 9a. We define SCðu; vÞ to be the semicircle enclosed by uv1 and uv2 with radius ",
where " > 0 is a small value less than the distance between any pair of nodes in V . Then, given a distance d, a semicircle ðu; dÞ is defined as follows:
ðu; dÞ ¼ [
kuvkd
SCðu; vÞ:
We can prove that, if ðu; dÞ is exactly the circle Cðu; "Þ, like Fig. 9b, then a disk centered at u with d radius can cover all neighbors of u in NRrðV Þ. In other words, GETINFcan be
halted as ðu; d2Þ Cðu; "Þ. Let NuðGðV ÞÞ be the set of
neighbors of node u in a graph GðV Þ. This property is proven in Lemma 7.
Lemma 7. Given a node u 2 V and distance d 2 <, if ðu; dÞ Cðu; "Þ,
NuðNGrðV ÞÞ v 2 V jP ðvÞ 2 Dðu; dÞf g:
Proof. We assume for contradiction that some node s in NuðNGrðV ÞÞ is not in fv 2 V jP ðvÞ 2 Dðu; dÞg. Since " is less
than the distance between any pair of nodes in V , we get kusk > ". Thus, edge us intersects a point on the circle Cðu; "Þ. Due to the fact that ðu; dÞ Cðu; "Þ, us must intersect at least one semicircle that composes ðu; dÞ; see Fig. 9b. Let SCðu; vÞ be one of the semi-circles intersected by us. Then, us is enclosed by uv1and uv2in SCðu; vÞ. In
other words, ffsuv ffv1uvor ffsuv ffvuv2. According to
the argument in Theorem 7, we can get that ffv1uv¼ ffvuv2
¼ 2 sin1ðr=2Þ. Therefore, we have ffsuv 2 sin1ðr=2Þ. Moreover, since s is not in fv 2 V jP ðvÞ 2 Dðu; dÞg, s must be farther than v from u. So, P ðvÞ 2 NRrðu; sÞ. According to
Definition 4, us is not in NRrðV Þ, which, however,
contradicts that s is a neighbor of u in NRrðV Þ. Thus, we
conclude this lemma. tu
3. The term “r-neighborhood graph” in [9] does not refer to the original version in Definition 3, but to the extended version in Definition 4. In this paper, we reuse the same term to name the original version and rename the previous structure in [9] the extended version.
The total transmission power used by GETINFcould be as large as d
0ð1 þ 21þ 22þ þ 2IÞ, where I is the number
of rounds. This result could be worse than the maximum transmission power 1 as I is large. Fortunately, when n is large, nodes are closer to and evenly surrounded by each other so that ðu; dÞ has more chance to be quickly shaped as Cðu; "Þ. So, we can benefit from GETINF in higher probability as the number of nodes increases.
The steps of GETINFare described below. Neglecting the communication overhead at step 2, the execution time of GETINF is dominated by the union operation at step 4. This step can be implemented by some search-and-merge algorithm. Thus, the time complexity of GETINF is Oðn log nÞ.
GGET INFðu; rÞ
Step 1: d1:¼ 0, d2:¼ d0, IN :¼ , ðu; d2Þ :¼ ;
Step 2: Broadcast a request ðP ðuÞ; d1Þ to distance d2and
gather a set R of responses from nodes within d1 and d2;
Step 3: For each v 2 R do
ðu; d2Þ :¼ ðu; d2Þ [ SCðu; vÞ;
Step 4: IN :¼ IN [ R;
Step 5: If d2 1 and ðu; d2Þ is not the circle Cðu; "Þ do
d1¼ d2;
d2:¼ d2 21=;
Return to step 2; Step 6: Stop and output IN;
Now, we discuss the communication cost of GETINF. As d0 is multiplied by
ffiffiffi 2 p
over log2ð1=d0Þ times, it is larger
than 1. Therefore, the number of rounds to increase the transmission range d2 is dominated by log2ð1=d0Þ þ 1.
Assume a node’s position can be encoded by log2nbits. Each
node has to broadcast at most ðlog2nÞð log2ð1=d0Þ þ 1Þ bits
for the request messages. In addition, a node will reply to the same node no more than once. Thus, a node needs at most ðlog2nÞðn 1Þ bits to reply to all requests. Combining these results, the communication cost of a node is no more than ðlog2nÞð log2ð1=d0Þ þ nÞ bits.
Once the information INuis collected, node u can start to
determine its neighbors in NRrðV Þ. One institutive way is
to apply Definition 2 on INu directly, as in the following
procedure: Step 1: N :¼ INu;
Step 2: For each node v in N do For each node w 2 INudo
If P ðwÞ 2 NRrðu; vÞ do
N :¼ N fvg; Step 3: Output N and stop;
In this procedure, the existence of a neighbor v in INuis
determined by checking whether some node w is located in NRrðu; vÞ. The correctness is obvious, while, in the worst
case, it should take Oðn2Þ time on each node. This time is
usually not tolerable when topology changes frequently. Therefore, we aim to reduce the time complexity in this part. In FINDNB, the main idea is to reverse the original procedure. That is, instead of checking whether some node w can block an edge uv, for each uv, we check whether some edge uv can be blocked by a node w for each w. The procedure is below.
This checking is begun from the farthest to the closet nodes in INu. So, we index all elements of INu in the
nondecreasing order of kuwk in Step 2. The set NB contains all candidates that could be a neighbor of u during the process. As a node w is given, we remove from NB all failed candidates that that are already blocked by w. After that, w is added into NB to be a new candidate of NuðNGrðV ÞÞ. The
process continues until all ws in INuare considered. Now,
we prove the correctness of FINDNB. F
F INDNBðu; r; INuÞ
Step 1: NB :¼ ;
Step 2: Index the elements of INuin nonincreasing order of
kuwk;
Step 3: For each node w 2 INuwith smallest index do
For each node v 2 NB do If P ðwÞ 2 NRrðu; vÞ do
NB :¼ NB fvg; NB :¼ NB þ fwg; Step 4: Stop and output NB;
Theorem 10.For any set V of nodes on <2, NB
u¼ NuðNGrðV ÞÞ,
for any u 2 V .
Proof. We prove this by showing that, for any v 2 V , v 2 NBu if and only if edge uv is in NRrðV Þ. Suppose an
edge uv is in NRrðV Þ. By Definition 2, there is no w 2
NuðUDGðV ÞÞ such that P ðwÞ 2 NRrðu; vÞ. This implies
that, once v is added in NB, there is also no w 2 INu
such that v can be removed at Step 3. Since v 2 NuðNGrðV ÞÞ INuand each node in INucan be added
to NB, v must be in NB at least one time. So, we can get that v is in the final output of NBu. Contrarily, we
suppose uv =2 NRrðV Þ. Some node w 2 NuðUDGðV ÞÞ is
located in NRrðu; vÞ. If v =2 INu, the result clearly follows
by Lemma 7. Otherwise, v 2 INu. In this case, all nodes
blocking uv are in INu. Besides, every node w blocking
uv is always considered after v in GETNB. Therefore, even if v can be added to NB, there must be a node w2 INu such that v can be removed from NB at the
successive iteration. So we get v =2 INu. tu
Lemma 7 also implies that, if uv 2 NRrðV Þ, then v 2 Nu
and u 2 Nv and that, if uv 2 NRrðV Þ, then v =2 Nu and
u =2 Nv. So, the neighbors (links) determined by GETNB are
symmetric.
Corollary 3.Any topology resulted by PLA is symmetric. Consider the time complexity of FINDNB. Step 2 can be done by some sorting algorithm in Oðn log nÞ. Before a node w 2 INuis added to NB, any v 2 IN blocked by w is
removed from NB. Therefore, for any two nodes in NB, neither of them can be blocked by the other. Let s and t be two nodes in NB. The argument of Theorem 7 indicates that, if ffsut < 2 sin1ðr=2Þ, then either s blocks t or t block s. Since neither s blocks t nor t blocks s, we get that ffsut 2 sin1ðr=2Þ. Therefore, during the process, the size
of NB can be never greater than dmaxðNGrðV ÞÞ.
Conse-quently, FINDNB can be done in Oðn maxflog n; dmaxðNGrÞgÞ
time. We can observe that this time complexity depends on the parameter r. When r is equal or close to 0 (the worst cases), the time complexity of FINDNB is still Oðn2Þ. However, when r is sufficiently large, such that
dmaxðNGrðV ÞÞ is a constant, FINDNB can be done in
Oðn log nÞ.
With a slight modification, PLA can be easily applied on the extended r-neighbors graph and all results can be preserved. We omit the detailed explanation here.
7
C
ONCLUSIONIn this paper, we proposed a purely localized structure to control the topology in wireless networks. We showed that the worst case of the power stretch factor is an increasing function of r and the worst cast of the maximum node degree is contrarily a decreasing function of r. So, the two objectives can be adjusted in our structure. Although the
power stretch factor is related to n so that our structure is not really a spanner, ðNGrðV ÞÞ can still be bounded for
some range of r. Therefore, the power stretch is partially bounded in our structure. About the maximum node degree, we proposed an upper bound derived for dmaxðNGrðV ÞÞ. However, this result is correct only when
no node has two or more neighbors at exactly the same distance. For this reason, an extended structure NG
rðV Þ
was given to comprehend this theorem.
Besides, the proposed structure can always result in a connected topology with symmetric edges. Any resulting topology is always planar. The relations between the r-neighborhoodgraph and existent structures are summar-ized in Fig. 10. Specially, NRrðV Þ is a general structure of both GGðV Þ and RNGðV Þ.
To construct our structure, we proposed a 1-hop purely localized algorithm, PLA. It can avoid long-distance transmission when collecting information and can be efficiently done in Oðn log nÞ time when dmaxðNGrðV ÞÞ is
constant.
For further research, a localized topology control approach enables the design of localized routing protocols. For instance, the greedy route discovery in CFG [26] and GPSR [11] are based on GG. We anticipate that the r-neighborhood graph could provide a concrete basis for many interesting extensions due to the sound theoretical results. Moreover, the parameter r can be turned to find the best settings for different scenarios. Another interesting issue for possible further work is to evaluate the stability of the proposed structure when perfect position (range) information is not available or when the accuracy of position information differs from node to node.
A
PPENDIXProof of Lemma 2. Without loss of generality, we assume that kuwk kvwk. Let y be the projection of w on uv so that yw is perpendicular to uv. We can derive that
kymk ¼ kvwk 2 2kmvk kwmk2 2kmvk kmvk 2 and kyxk ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kwvk2 kvwk 2 2kmvk kwmk2 2kmvkþ kmvk 2 !2 v u u t :
Thus,
kuwk2¼ kwyk2þ ðkumk kymkÞ2 ¼ kwyk2þ ðkmvk kymkÞ2 ¼ kvwk2 kwvk 2 2kmvk kwmk2 2kmvkþ kmvk 2 !2 þ 3kmvk 2 kvwk2 2kmvkþ kwmk2 2kmvk !2 ¼ 2kmvk2 kvwk2þ 2kwmk2:
Then, the power consumed by path uwv is as follows: pðuwvÞ ¼ kuwkþ kvwk ¼ ð2kmvk2 kvwk2þ 2kwmk2Þ2þ kvwk: From (1), we get kwmk < l ¼ kuvkpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2r2=2¼ kmvkpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2r2 and kvwk < kuvk, so 2kmvk2 kvwk2þ 2kwmk2 2 þkvwk ð4 þ 4r 2Þkmvk2 kvwk2 2 þkvwk ð4 þ 4r 2Þ 4 kuvk 2 kuvk2 2 þkuvk ¼ r 2kuvk2 2 þkuvk¼ kuvkð1 þ rÞ:
Thus, we have that pðuwvÞ kuvkð1 þ rÞ. tu
A
CKNOWLEDGMENTSThis paper was supported in part by the National Science Council of the ROC, under grants NSC 95-2219-E-009-008 and NSC-95-2752-E-009-005-PAE, and in part by the New Generation Broadband Wireless Communication Technolo-gies and Applications Project of the Institute for Informa-tion Industry, ROC.
R
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Andy An-Kai Jeng received the BS degree in statistics from Tamkang University and the MS degree in management information systems from National Chi Nan University in 2001 and 2003, respectively. He is currently pursuing the PhD degree in the Department of Computer and Information Science at National Chiao-Tung University, Taiwan, Republic of China. His research interests include wireless networks, algorithm design and analysis, scheduling theo-ry, and operations research.
Rong-Hong Jan received the BS and MS degrees in industrial engineering and the PhD degree in computer science from National Tsing Hua University, Taiwan, in 1979, 1983, and 1987, respectively. He joined the Department of Computer and Information Science, National Chiao Tung University, in 1987, where he is currently a professor. From 1991-1992, he was a visiting associate professor in the Department of Computer Science, University of Maryland, College Park, Maryland. His research interests include wireless networks, mobile computing, distributed systems, network reliability, and operations research. He is a member of the IEEE.
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