2.1 Network Model
The wireless ad hoc network concerned in this paper consists of a set V of n wireless nodes distributed on a deployment region ℵ, which is a subset of the two-dimension plane ℜ2. We assume that 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 transmission ranges of all nodes to be 1 for simplicity. In addition, each node u can obtain its location Loc(u) through a lower-power GPS or some other ways [14], and an unique id(u) is also available to each node u.
This network can be modeled as a unit disk graph, UDG(V). In this graph, an edge uv exists if and only if the Euclidean distance between u and v, denoted as ||uv||, is at most 1.
The least power required to transmit immediately between u and v is modeled as p(u, v) = ||uv||α, where α 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 as below.
Let π(u, v) = v0v1…vh-1vh be a unicast path connecting nodes u and v, where v0 = u and
Let be the least-energy path connecting u and v in graph G(V). Given a controlled topology S(V) of UDG(V), tthe power stretch factor of S(V) with respect to UDG(V)is defined as,
This factor indicates the worst ratio of the least energy required to relay on S(V) in compared to that of a uncontrolled topology for all possible communication pairs.
Clearly, a smaller ratio is preferable. On the other hand, the maximum node degree of the topology S(V) is defined as
In addition, the following symbols will be used throughout this article.
z D(u, d): the closed disk centered at Loc(u) with radius d.
z C(u, d): the circled centered at Loc(u) with radius d.
z Nu(G(V)): the set of neighbor of u in a graph G(V).
2.2 Stationery Topology Control
In the field of topology control for stationary nodes, a majority of researches were conducted by designing the proximate graph. A proximate graph is a geometric structure in which each node determines its neighbors based on the positions of nodes in its province. In other words, a topology approach based on such structure can be carried out in a fully distributed and localized way. A number of instances can be found in the literature [15, 16, 18, 26]. These works are diverse in their sparseness and the energy efficiency of preserved routes. We discuss the most well-know structures below. Most of them or their extensions are purely localizable:
The constrained Relative Neighborhood Graph [28], denoted by RNG(V), has an edge uv if and only if ||uv|| ≤ 1 and the intersection of two open disks1 centered at u, v with radius ||uv|| contains no node w ∈V, see Figure 2.1 (a),
The constrained Gabriel Graph [6], denoted by GG(V), has an edge uv if and only if ||uv||≤ 1 and the open disk using ||uv|| as diameter contains no node w ∈V, see Figure 2.1 (b).
The constrained Yao Graph [33] with a parameter k ≥ 6, denoted by YGk(V) 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
||uv|| ≤ 1 and the cone contains no vertex w ∈V such that ||uw|| < ||uv||. Ties are broken arbitrarily. YGk(V) is denoted as the underlying undirected graph of
) (V
YGk , see Figure 2.1 (c).
A Delaunay Triangulation, denoted by Del(V), is a triangulation of V in which the interior of the circumcircle of each Δuvw contains no node w ∈ V. The unit
1 An open disk centered at point x with radius d is the collection of points with distance less than d from Loc(x).
Delaunay Triangulation, denoted by UDel(V), has all edges of Del(V) except those longer than 1 [8, 18], see Figure 2.1 (d).
(a) (b)
(c)
(d) Figure 2.1: (a) RNG(V) (b) GG(V) (c) YGk(V), k = 8 (d) UDel(V).
Let us discuss the properties of these structures and their extensions. We say a objective f(.) of a structure S(V) is bounded if there is a constant C such that f(S(V)) ≤ C, for any set V of n nodes. Li et al. [15] showed that dmax(RNG(V)) is unbounded if there is a node u ∈ V having an unbounded number of neighbors adjacent to u at exactly the same distance in the underlying UDG(V). To overcome this problem, Wattenhofer and Zollinger [32] proposed an algorithm to find a structure, denoted by XTC(V). They showed that that XTC(V) is a subgraph of RNG(V) and the dmax(XTC(V)) is at most 6. Especially, if there is no node having two or more neighbors at exactly
the same distance in V, XTC(V) is identical to RNG(V) [24]. Their results infer the following theorem.
THEOREM 2.1: Given a set V of nodes on ℜ2, if there is no node having two or more neighbors at exactly the same distance, then dmax(RNG(V)) ≤ 6.
We denote the condition in Theorem 2.1 as assumption AS. That is,
AS : There is no node in V having two or more neighbors at exactly the same distance.
This theorem reveals that even RNG(V) has no constant bound on its node degree, it is still useful since the distances of nodes in real world are rarely exactly the same. The constrained Gabriel Graph GG(V) has the least power stretch factor 1, in comparison with the unbounded power stretch factor n – 1 of RNG(V) [15]. However, dmax(GG(V)) could be as large as n – 1. An extended structure, Enclosure graph [16, 14, 24], denoted by EG(V) is generalized from GG(V). It can always result in a subgraph of GG(V) [16]. Even so, its maximum node degree is still unbounded [20, 24].
To overcome the tradeoff 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. YGk(V)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
(
1−(
2sinπ/k)
α)
[15]. We say an objective f(.) of an adjustable structure Sk(V) with parameter k is partially bounded if there is at least one k0 such that is bounded. According this definition, the maximum out-degree and power stretch factor of))
YGk are partially bounded since for some ranges of k, k and 1
(
1−(
2sinπ/k)
α)
are constants. However, the asymmetric edges of YGk(V) may lead to large in-degrees even when k is very small [15]. So, can be neither bounded nor partially bounded. To improve this, an extension of))
limit the maximum node degree in (k +1)2 – 1 and result symmetric edges.
Unfortunately, in this structure the neighbors of some node should be recursively determined by one another so that it can not be purely localizable. The unit Delaunay triangulation UDel(V) has bounded power stretch factor. However, neither Del(V) nor UDel(V) can be computed locally. So, Li et al. [18] suggested a localized version of the Delaunay graph, denoted by LDel(h)(V), where h means that each node uses at most k-hop information. The power stretch factor of LDel(k)(V) 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 on Figure 2.2, where EMST(V) is the Euclidean minimum spanning tree of UDG(V). With these relations, their connectivity can planarity can be easily inferred.
Figure 2.2: The relations of the pure localizable structures and their extensions.
Regarding the connectivity: we know that EMST(V) is connected if UDG(V) is itself a connected component of V. Therefore, when UDG(V) is connected, all graph containing EMST(V) are connected. That is, RNG(V), GG(V), EG(V), UDel(V),
LDe(k)l(V), YGk(V) are all connected. The connectivity of XTC(X) was proven by different way [24].
Regarding the planarity: LDel(k)(V) is planar for any k ≥ 2 [18]. Therefore, all subgraphs of LDel(2)(V) are planar. That is, UDel(V), GG(V), EG(V), RNG(V), XTC(V), EMST(V) are all planar. On the contrary, YGk(V), and LDe(1)l(V) can not avoid producing the crossed link, so they are not planar [15, 18]. Table 2.1 summarizes above discussion.
From above table, 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 GG(V) and RNG(V).
Table 2.1: The properties of the four main purely localizable structures.
Power stretch factor Maximum node degree Planar Symmetric Connected
RNG(V) Unbounded
Bounded (with AS)
Unbounded (without AS) Yes Yes Yes
GG(V) Bounded Unbounded Yes Yes Yes
) (V
YGk Partially bounded Unbounded No No Yes
LDel(k)(V) Partially bounded Unbounded
No ( k = 1)
Yes ( k ≥ 2) Yes Yes
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 OrdYao(V) [1], is a variation of . It has the partially bounded maximum node degree and length stretch factor. However, the planarity can not be guaranteed. Therefore, Wang and Li [19, 31] applied OrdYao(V) onto LDel(2)(V) to avoid the crossed edges produced by OrdYao(V); Song et al. [25] improves it by applying the OrdYao(V) on GG(V), using only one-hop information. Their Result are summary in Table 2.2. However, the construction of OrdYao(Y) requires exchanging the computed status as well as partial results between nodes. Consequently, none of them is purely localized or purely localizable.
)
*( V YGk
Table 2.2: The properties ofrepresentative adjustable structures.
Parameter Power stretch factor Maximum node degree
YGk*
2.3 Mobile Topology Control Protocols
Distributed protocols for proximate graphs can be also found in the literature [34]. However, existent results are all applied to stationary network only. There is no approach explicitly designed for mobile nodes based on such structure. The reason is probably that the construction depends on node positions, so that even a slight change in nodes placement could trigger a reconstruction process to handle the broken link or
deteriorated link quality.
There are relatively fewer works considering nodes mobility. The LINT (and its extension LILT) is perhaps the first topology control protocol explicitly designed for mobile network [36, 37]. In this protocol, each node continually adjusts its transmission power such that the number of covered neighbors is within a lower and high threshold. Accordingly, the energy can be saved by declining the high threshold, and the network connectivity can be achieved by uplifting the low threshold. It however has no guarantee on connectivity if the low threshold is underestimated. To improve that, Blough et al [38] proposed a similar approach, named the K-NEIGH.
The protocol connects each node with its k-closest neighbors and removes all asymmetric links, where k is a predefined parameter. The most interesting result is that if n nodes are uniformly distributed at random and k is taken as Θ(logn), then the connectivity can be held with high probability. These protocols are called the neighbor-based approach, since a node’s construction relays on the ability of ordering or measuring distances of nodes in its province [34]. The direction-based approach is another stem. It uses the angles among nodes for the construction. An example is the Cone Based Topology Control (CBTC) [39]. The basic idea is to let each node transmits with the minimum power that covers at least one neighbor in every cone of an angle ρ centered at it. The authors show that ρ ≤ 2π/3 is a sufficient condition to ensure connectivity. Li et al. [40] proposed a reconfiguration procedure to deal with node mobility by detecting changing events from received beacons.
The most important features of these protocols are that their constructions are based on either nodes distance or nodes directions. Compared with the proximate graph, both the neighbor-based and direction-based approaches can be more accommodating to nodes movement. The reason is that the changing on nodes
distances or directions will be relatively small with respect to nodes positions.
Therefore, by using either of the two less precise information, a fewer number of topology reconstruction will be required when nodes move.
Even thought our protocols are based on a proximate graph. We will show that such disadvantage can be easily mitigated in an elegant way. In addition, Compared with K-NEIGH, LINT (LILT) and CBTC, our protocol guarantees the network connectivity in any stabilized status, without any assumption on nodes distribution, or parameter setting. Furthermore, both CBTC and K-NEIGH attend symmetricity by exchanging linking status among nodes. This will incur additional control overhead.
Our protocol ensures that any established link is inherently bidirectional.
Chapter 3
Graphic Structures
In this chapter, we will introduce a new adjustable structure, called the r-neighbor graph. It can be adjusted between the maximum node degree and power stretch factor through the parameter r. The structure can also produce connected planar with symmetric edges. However, its maximum node degree will be unbounded in certain cases. To comprehend the theoretic property, we will then propose an enhanced version, called the extended r-neighborhood graph to deal with the special circumstance.
To apply the proposed structure to our mobile protocol, extensive investigations on the r-neighborhood graph will be given. First of all, we define a generalized structure, called the (r, α)-neighborhood graph. The generalization can gain better quantitative results. Next, an equivalent structure, called the (r, α)-Enclosed graph will be given. Its diverse representation enables the design of a shrinking power mechanism in Chapter 4. Then, we further generalized the structure such that each node having its own r, named the (fr, α)-neighborhood Graph. This graph provides essential properties for the self-configuration process in Chapter 6.
3.1 r-Neighborhood Graph
In this section, we introduce the adjustable structure. First, we define a region on
ℜ2. It will be used to compose our structure.
DEFINITION 3.1: Given a nodes pair (u, v) on ℵ, the r-neighborhood region of (u, v), denoted as NRr(u, v), is defined as:
Figure 3.1: The r-neighborhood region of nodes u and v.
When no confused, we use m and l instead of muv and luv respectively. In Figure 3.1, the shaded region intersected by the three open disks sketches an example of the r-neighborhood region. This region is obviously equivalent to the following point set:
}
For any node w located on NRr(u, v), this region limits the power consumed by path uwv. This property is shown in Lemma 3.1 and derived in Appendix.
LEMMA 3.1: Given two nodes u and v on ℵ, for any node w such that Loc(w) ∈ NRr(u, v), p(uwv) < ||uv||α(2 + rα), for all α ≥ 2.
This lemma explains why we call such plane a neighborhood region: For any node w located in the region NRr(u, v), 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 + rα times of the immediate transmission. Based on this region, we structure is defined below.
DEFINITION 3.2: Given a set V of nodes on ℵ, the r-neighborhood graph of V, denoted as NGr(V), has of an edge uv if and only if ||uv|| ≤ 1 and NRr(u, v) contains no node w ∈ V, where 0 ≤ r ≤ 1.
By Definition 3.2, if edge uv is not in UDG(V) or a node w is inside NRr(u, v), there is no direct link connecting u and v in NGr(V), which mean that all transmissions between u and v should be relied through some other node(s) in NGr(V). Now, we explore the desired properties in our structure. Before this, we shall discussion the following relations.
LEMMA 3.2: For any set V of nodes on ℵ, RNG(V) ⊆ NGr(V) ⊆ GG(V), for all 0≤ r≤1.
Proof. Consider the open disk D(m, ||uv||/2), defining GG(V). Suppose uv ∈ NGr(V), the region NRr(u, v) has no node inside. Since D(m, ||uv||/2) is obviously a subregion of NRr(u, v), for any 0 ≤ r ≤ 1, there is also no node in D(m, ||uv||/2). Therefore, according to the definition of GG(V), we get uv ∈ GG(V). On the other hand, consider the two open disks D(u, ||uv||) and D(v, ||uv||), defining RNG(V). Suppose uv ∈ RNG(V), no node is inside the intersection of D(u, ||uv||) and D(v, ||uv||), which obviously covers the region NRr(u, v), for any 0 ≤ r ≤ 1. Therefore, no node can be
inside NRr(u, v) and we get uv ∈ NGr(V). □
Specifically, as r = 0, NR0(u, v) ≡ D(m, ||uv||/2), which is the disk defining GG(V). On the contrary, as r = 1, NR1(u, v) ≡ D(m, ||uv||), which is the disk defining RNG(V).
Therefore, GG(V) ≡ NG0(V) and RNG(V) ≡ NG1(V). So, we can conclude the following theorem.
THEOREM 3.1: 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 GG(V) and connectivity of RNG(V),
we can infer the following two theorems.
THEOREM 3.2: For any set V of nodes on ℵ, NGr(V) is planar, for all 0 ≤ r ≤ 1.
THEOREM 3.3: For any set V of nodes on ℵ, if the underlying UDG(V) is connected, NGr(V) is connected, for all 0 ≤ r ≤ 1.
Now we consider the energy efficiency and node degree of NGr(V). We will show that the upper bound of ρ(NGr(V)) is increased by r and contrarily the upper bound of dmax(NGr(V)) is decreased by r. In other words, the r-neighborhoodgraph 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 first. It can be used to simplify our proof.
LEMMA 3.3 [15]: Given a subgraph G’(V) ⊆ UDG(V) and a constant C, ρ(G’(V)) ≤ C if and onlyiffor any edge uv in G(V), there is a path π(u, v) in G’(V) such that
uvα
C v u
pG'(V)( , )≤ .
This lemma indicates that to derive an upper bound for ρ(NGr(V)), it is sufficient to the consider only those nodes pairs having direct links in UDG(V). So, we aim to derive a strictly decreasing function F(r), such that for any uv in UDG(V), a path
) , ( vu
π is in NGr(V) such thatp
(
π( vu, ))
≤ F(r)||uv||α. To achieve this, we investigate an algorithm EXPANSION with an input of any two nodes (u, v) and outputs subgraph S of NGr(V) related to (u, v). Let P(S) be the total transmission power of edges in S. i.e.P(S) = ∑st∈Sp(s, t). We can show that there is some path in S connecting (u, v) and P(S)
≤ F(r)||uv||α.
In this algorithm, S’ is a set of nodes pairs, in which an edge st in NGr(V) can be a part of S only if its two ends (s, t) are in S’ as described at step 3. So, to determine S, we have discuss the S’ first. Initially, S’ contains only (u, v). Then, it will be recursively
expanded as follows: for each (s, t) in S’, if a node w is in NRr(s, t) and not considered before, replace (s, t) with (s, w) and (w, t); if a node w is in NRr(s, t) but considered before, replace (s, t) with (s, w); Otherwise, keep (s, t) unchanged. We use the set Q to record the considered nodes.
ALGORITHM EXPANSION
Input: A nodes pair (u, v) in V
Output: A subgraph S and a positive value P.
Step 1: S = {}, S’ ={(u, v)}, Q = {u, v}, P = ||uv||α;
Step 2: When some node pair (s, t) is in S such that a node w ∈ NRr(s, t) S’ = S’ – (s, t);
If w ∉ Q then
S’ = S’ ∪ (s, w) ∪ (w, t);
Q = Q ∪{w};
P = P + (||st||r)α; Otherwise,
S’ = S’ ∪ (s, w);
Step 3: S = {xy ∈ NGr(V) | (x, y) ∈ S’};
Step 4: Stop and output E and P.
When some (s, t) is in S’ such that a node w ∈ NRr(s, t), no matter w is considered or not, by (4.1), the replaced nodes pair(s) must be shorter than ||st||. i.e.
||sw|| < ||st|| and ||wt|| < ||st||. Thus after finite iterations, each node pair in S’ can be replaced by another node pair with shortest distance. So, the algorithm is terminable.
Now we show that (u, v) is connected by some path in the subgraph S when termination.
LEMMA 3.4: Given any set V of node on ℵ, for any two nodes u and v in V, if edge uv is in UDG(V) and UDG(V) is connected, there is some path in S connecting (u, v).
Proof: Since Q includes u and v, we can prove this lemma by showing that all nodes in the Q are connected in S. For each expansion of S’, we define a dummy graph S” in which an edge st exists if and only if (s, t) is in S’ (Note that any edge in S” is not necessarily in either UDG(V) or NGr(V)). First, we show that at any iteration, all considered nodes in Q are connected by S”. Initially, Q is connected by S”, since S’ = {(u, v)} and Q ={u, v}. We assume for induction that all nodes in Q are connected by S” at k-th iteration. Then, we show that it is true for the next iteration. At k+1-th iteration, if there is no pair in S’ satisfies the entrance condition of step 2, the claim is correct, since Q and S” are unchanged; Otherwise, a node pair (s, t) ∈ S’ is expended.
In this case, if the chosen w ∉ Q, w is connected with all nodes in Q via dummy edges sw and wt; otherwise, w ∈ Q, which implies all nodes in Q are still connected by S” as the previous iteration. As described above, the distance of any expended nodes pair is no longer than the previous one. So, if uv is in UDG(V), all edges in S” are also in UDG(V). Then, as the algorithm processes to step 3, no nodes can be in the r-neighborhood region of any nodes pair in S’. With these two facts, all dummy edges in S” are also in NGr(V) when termination. So S is equivalent to the last S”.
Consequently, if UDG(V) is connected, by Theorem 4.4, all nodes in the last Q are
connected S. □
Then we derive a strictly decreasing function F(r) using the value P in this algorithm.
LEMMA 3.4: Given any set V of n nodes on ℵ, for any two nodes u and v in V, uvα
r F S
P( )≤ ( ) and F(r)=1+(n−2)rα for all 0 ≤ r ≤ 1 and α ≥ 2.
Proof: Let P(S’) = ∑(s,t)∈S’p(s, t). We show that P(S’) ≤ P at each iteration of step 2.
Initially, S’ = {(u,v)}. We can get P(S’) = ||uv||α = P. Then at the first iteration, if no node w is in NRr(u, v), the claim remains true since neither P nor S is changed;
Initially, S’ = {(u,v)}. We can get P(S’) = ||uv||α = P. Then at the first iteration, if no node w is in NRr(u, v), the claim remains true since neither P nor S is changed;