Localized Topology Control for Heterogeneous
Wireless Ad-hoc Networks
Xiang-Yang Li
?
Wen-Zhan Song
?
Yu Wang
†
Abstract—We study topology control in heterogeneous wireless ad hoc
networks, where mobile hosts may have different maximum transmission
powers and twonodes are connected iffthey are within the maximum trans-
mission range of each other. We present several strategies that all wireless
nodes self-maintain sparse and power efficient topologies in heterogeneous
network environment with low communication cost. The first structure is
sparse and can be used for broadcasting. While the second structure keeps
the minimum power consumption path, and the third structure is a length
and power spanner with a bounded degree. Both the second and third
structures are power efficient and can be used for unicast. Here a struc-
ture is power efficient if the total power consumption of the least cost path
connecting any two nodes in it is no more than a small constant factor of
that in the original heterogeneous communication graph. All our methods
use at most O(n) total messages, where each message has O(log n) bits.
Keywords— Graph theory, wireless ad hoc networks, topology control,
heterogeneous networks, power consumption.
I. INTRODUCTION
An important requirement of wireless ad hoc networks is that
they should be self-organizing, i.e., transmission ranges and data
paths are dynamically restructured with changing topology. Lo-
calized ad hoc network topology control scheme is to let each
wireless node locally adjust its transmission power and select
proper neighbors to communicate according to certain strategy,
while maintaining a structure that can support energy efficient
routing and improve the overall network performance. Hence
it can efficiently conserve the transmission energy from soft as-
pects with low cost. In the past several years, topology control
algorithms have drawn significant research interest. Centralized
algorithms can achieve optimality or its approximation, which
are more applicable to static networks due to the lack of adapt-
ability to topology changes. In contrast, distributed algorithms
are more suitable for mobile ad hoc networks since the environ-
ment is inherently dynamic and they are adaptive to topology
changes at the cost of possible less optimality. Furthermore,
these algorithms only attempt to selectively choose some neigh-
bors for each node. The primary distributed topology control al-
gorithms for ad hoc networks aim to maintain network connec-
tivity, optimize network throughput with power-efficient rout-
ing, conserve energy and increase the fault tolerance.
Most prior art [1], [2], [3], [4], [5], [6] on network topology
control assumed that wireless ad hoc networks are modelled by
unit disk graphs (UDG), i.e., two mobile hosts can communicate
as long as their Euclidean distance is no more than a threshold.
However, practically, wireless ad hoc networks cannot be per-
fectly modelled as UDGs: the maximum transmission ranges of
wireless devices may vary due to various reasons such as the
device differences and the small mechanic/electronic errors dur-
?
Department of Computer Science, Illinois Institute of Technology. Email:
xli@cs.iit.edu, songwen@iit.edu. The work of Xiang-Yang Li is
partially supported by NSF CCR-0311174.
†
Department of Computer Science, University of North Carolina at Charlotte.
Email: wangyu@ieee.org.
ing the process of transmitting even the transmission powers of
all devices are set the same initially. In [7], [8], the authors ex-
tended UDG into a new model, called quasi unit disk graphs,
which is closer to reality than UDG. In this paper, we study a
more generalized model. Each wireless node u may have its own
transmission radius r
u
. Then heterogeneous wireless networks
are modelled by mutual inclusion graphs (MG): two nodes can
communicate directly only if they are within the transmission
range of each other, i.e., it has a link uv iff kuvk ≤ min(r
u
, r
v
).
Clearly UDG is a special case of MG. Few research efforts
addressed the topology control for heterogeneous wireless net-
works.
The main contribution of this paper is as follows. We propose
several localized strategies for heterogeneous wireless devices
to self-form a globally sparse power efficient network topol-
ogy: a power spanner, a sparse structure and a degree-bounded
length and power spanner respectively. Here an algorithm is said
to construct a topology H locally, if every node u can decide
which incident edge uv belong to H using only the informa-
tion of nodes within a constant number of hops of u. All our
algorithms have communication costs O(n), where each mes-
sage has O(log n) bits. Notice, to study the topology control
in heterogeneous networks, it would be helpful to extend the
ideas from the well-studied topologies, such as GG, RNG and
Yao, used in homogeneous networks. The topology control for
heterogeneous networks is not trivial, since many properties in
homogeneous networks disappear in heterogeneous networks.
The rest of the paper is organized as follows. In Section II we
introduce the background and review previous methods. Limita-
tions on heterogeneous network topology control are discussed
in Section III. We describe a strategy for all nodes forming a
sparse structure in Section IV, a sparse power spanner in Section
V, and a degree-bounded power and length spanner in Section
VI. We also analyze the communication complexities of these
methods. Our theoretical results are corroborated in the simu-
lations in Section VII. We conclude our paper in Section VIII
with the discussion of future works.
II. PRELIMINARIES
A. Heterogeneous Wireless Network Model
A heterogeneous wireless ad hoc network is composed of a
set V of n nodes v
1
, v
2
, ··· , v
n
, in which each node v
i
has
its own maximum transmission power p
0
i
. Let ²
i
be the me-
chanic/electronic error of a node v
i
in its power control. Then
the maximum transmission power considered in this paper is ac-
tually p
i
= p
0
i
− ². We adopt a common assumption in the
literature that the power needed to support the communication
between two nodes v
i
and v
j
is kv
i
v
j
k
β
, where β ∈ [2, 5] is
a real number depending on the environment and kv
i
v
j
k is the
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Euclidean distance between v
i
and v
j
. Consequently, the sig-
nal sent by a node v
i
can be received by all nodes v
j
with
kv
i
v
j
k ≤ r
i
, where r
β
i
≤ p
i
/p
0
, p
0
is the uniform threshold
that a signal with power p
0
can be recognized by a node. Thus,
for simplicity, we assume that each mobile host v
i
has its own
transmission range r
i
. The heterogeneous wireless ad hoc net-
work is then modelled by a mutual inclusion graph (MG), where
two nodes v
i
, v
j
are connected iff they are within the transmis-
sion range of each other, i.e., kv
i
v
j
k ≤ min(r
i
, r
j
). Previously,
no method is known for topology control when the networks are
modelled as mutual inclusion graphs.
B. Current State of Knowledge
Many structures were proposed for topology control in ho-
mogeneous wireless ad hoc networks. Due to limited spaces,
we will briefly review some of proximity geometric structures.
The relative neighborhood graph [9] RNG(V ) consists of all
edges uv such that the intersection of two circles centered at u
and v and with radius kuvk do not contain any vertex w from V .
The Gabriel graph [10] GG(V ) contains edge uv if and only if
disk(u, v) contains no other points of S, where disk(u, v) is the
disk with edge uv as a diameter. Both GG(V ) and RN G(V )
are connected, planar, and contain the Euclidean minimum span-
ning tree of V . The intersections of GG(V ), RNG(V ) with a
connected UDG(V ) are connected. Delaunay triangulation, de-
noted by Del(V ), is also used as underlying structure by several
routing protocols. Here a triangle 4uvw belongs to Del(V )
if its circumcircle does not contain any node inside. It is well
known that RNG(V ) ⊆ GG(V ) ⊆ Del(V ). The intersection
of Del(V ) with a connected UDG(V ) has a bounded length
spanning ratio [11].
The Yao graph [12] with an integer parameter k ≥ 6, denoted
by
−−→
Y G
k
(V ), is defined as follows. At each node u, any k equal-
separated rays originated at u define k cones. In each cone,
choose the shortest edge uv among all edges from u, if there
is any, and add a directed link
−→
uv. Ties are broken arbitrarily
or by ID. The resulting directed graph is called the Yao graph.
Let Y G
k
(V ) be the undirected graph by ignoring the direction
of each link in
−−→
Y G
k
(V ). Some researchers used a similar con-
struction named θ-graph [13], the difference is that it chooses
the edge which has the shortest projection on the axis of each
cone instead of the shortest edge in each cone.
The first effort for topology control in heterogeneous wire-
less networks was reported in [14] by the same authors of this
paper. In [14], we showed how to perform topology control
based on Yao structure for heterogeneous wireless networks.
The results presented in current paper have been available online
since around June 2003. Recently, several structures that extend
the relative neighborhood graph and local minimum spanning
tree were proposed in [15] for topology control in heteroge-
neous wireless networks. They build directed network topolo-
gies while the methods presented here build undirected topolo-
gies that are beneficial for routing. In addition, as the authors of
[15] acknowledged, their original methods cannot guarantee the
network connectivity. Then new methods were proposed to rem-
edy this in their online version of the paper. Two structures were
proposed by them: an extended relative neighborhood graph and
the extended local minimum spanning tree. It is unknown if their
structures are sparse, power efficient.
C. Spanners and Stretch Factors
When constructing a subgraph of the original communication
graph MG, we may need consume more power to connect some
nodes since we may disconnect the most power efficient path
in MG. Thus, naturally, we would require that the constructed
structure approximates MG well in terms of the power consump-
tion for unicast routing. In graph theoretical term, the structure
should be a spanner [16], [13]. Let G = (V, E) be a n-vertex
weighted connected graph. The distance in G between two ver-
tices u, v ∈ V is the length of the shortest path between u and
v and it is denoted by d
G
(u, v). A subgraph H = (V, E
0
),
where E
0
⊆ E, is a t-spanner of G if for every u, v ∈ V ,
d
H
(u, v) ≤ t · d
G
(u, v). The value of t is called the stretch
factor or spanning ratio. When the graph is a geometric graph
and the weight is the Euclidean distance between two vertices,
the stretch factor t is called the length stretch factor, denoted
by `
H
(G). For wireless networks, the mobile devices are usu-
ally powered by battery only. We thus pay more attention to the
power consumptions. When the weight of a link uv ∈ G is de-
fined as the power to support the communication of link uv, the
stretch factor of H is called the power stretch factor, denoted
by ρ
H
(G) hereafter. The power, denoted by p
G
(u, v), needed
to support the communication between a link uv in G is often
assumed to be kuvk
β
, where 2 ≤ β ≤ 5. Obviously, for any
weighted graph G and a subgraph H ⊆ G,
Lemma 1: [3] Graph H has stretch factor δ if and only if for
any link uv ∈ G, d
H
(u, v) ≤ δ · d
G
(u, v).
Thus, to generate a spanner H, we only have to make sure that
every link of G is approximated within a constant factor.
D. Sparseness and Bounded Degree
All well-known proximity graphs (GG(V ), RNG(V ),
Del(V ) and Y G(V )) have been proved to be sparse graphs
when network is modeled as a UDG. Recall that a sparse graph
means the number of edges is linear with the number of nodes.
The sparseness of all well-known proximity graphs implies that
the average node degree is bounded by a constant. Moreover, we
prefer the maximum node degree is bounded by a constant, be-
cause wireless nodes have limited resources and the signal inter-
ference in wireless communications. Unbounded degree (or in-
degree) at a node u will often cause large overhead at u, whereas
a bounded degree increases the network throughput. In addition,
bounded degree will also give us advantages when apply several
routing algorithms. Therefore, it is often imperative to construct
a sparse network topology with a bounded node degree while it
is still power-efficient. However, Li et al. [3] showed that the
maximum node degree of RNG, GG and Yao could be as large
as n −1. The instance consists of n −1 points lying on the unit
circle centered at a node u ∈ V . Then each edge uv
i
belongs to
the RNG(V ), GG(V ) and
−−→
Y G
k
(V ).
Recently, in homogeneous wireless ad hoc networks, some
improved or combined proximity graphs [17], [18] have been
proposed to build planar degree-bounded power spanner topol-
ogy, which meets all preferred properties for unicast. In het-
erogeneous networks, only a few research efforts [15], [14] are
285
reported so far. In the following, we will first discuss the dif-
ficulties and limitations for topology control in heterogeneous
networks, then present our localized strategies in detail.
III. LIMITATIONS
In heterogeneous wireless ad hoc networks, the planar topol-
ogy does not necessarily exist. Figure 1 (a) shows an example,
there are four nodes x, y, u and v in the network, where their
transmission range r
x
= r
y
= kxyk and r
u
= r
v
= kuvk, and
node u is out of the transmission range of node x and y, while
node v is in the transmission range of node y and out of the range
of x. The transmission ranges of x and y are illustrated by the
dotted circles. According to the definition of M G, there are only
three edges xy, vy and uv in the graph. Hence any topology con-
trol method can not make the topology planar while keeping the
communication graph connected. On the other hand, it is worth
to think whether we can design a new routing protocol on some
pseudo-planar topologies. As will see later, the pseudo-planar
topology GG(MG) and RNG(MG) proposed in this section has
some special properties which are different from other general
non-planar topologies. For instance, two intersecting triangles
can not share a common edge. We leave it as a future work.
x
v
u
y
w
w
2
w
p
w
0
1
v
(a) (b)
Fig. 1. Limitations on heterogeneous networks: (a) Planar topology does not
exist. (b) Degree of node v can not be bounded by constant.
We also can show that the node degree in heterogeneous net-
works can not be bounded by a constant if the radius ratio is un-
bounded. Figure 1 (b) shows such an example. In the example,
a node v has p + 1 incoming neighbors w
i
, 0 ≤ i ≤ p. Assume
that each node w
i
has a transmission radius r
w
i
= r
v
/3
p−i
and
kvw
i
k = r
w
i
. Obviously, kw
i
w
j
k > min(r
w
i
, r
w
j
), i.e., any
two nodes w
i
, w
j
are not directly connected in MG.
Obviously, none of those edges incident on v can be deleted,
hence there is no topology control method to bound the degree
by a constant without violating connectivity. Consider the ex-
ample illustrated by Figure 1 (b), edges vw
i
, 0 ≤ i ≤ p, are
all possible communication links. Thus, node v in any con-
nected spanning graph has degree p + 1. On the other hand,
the topology generated by our method in section VI can guran-
tee the maximum node degree bounded by O(log
2
γ), where
γ = max
v∈V
max
w∈I(v)
r
v
r
w
. Here, I(v) = {w | wv ∈ M G}.
This is optimal in the worst case. In previous example, recall
that 3
p
r
w
0
= r
v
, hence γ equals to 3
p
. Thus, v has degree
log
3
γ + 1 = Θ(log
2
γ). In the paper, we always assume γ
is a constant. It is practical, since it is trivial that two wireless
devices in same network have unbounded radius ratio.
IV. HETEROGENEOUS SPARSE STRUCTURE
In this section, we propose a strategy for all nodes to self-
form a sparse structure, called RNG(MG), based on the rela-
tive neighborhood graph structure, whose total number of links
is O(n). We add a link uv ∈ MG to RNG(MG) if there is
no another node w inside lune(u , v ) and both links uw and wv
are in MG. Here lune(u, v ) is the intersection of disk (u , kuv k)
and disk (v , kuv k). The algorithm will be similar to Algorithm
2, thus we omit it here. Notice that the total communication
cost of constructing RN G(M G) is O(n log n) bits, assuming
that the radius and ID information of a node can be represented
in O(log n) bits. In addition, the structure RNG(MG) is sym-
metric: if a node u keeps a link uv, node v will also keep the link
uv. Thus, a node u does not have to tell its neighbor v whether
it keeps a link uv or not.
It is not difficult to prove that structure RNG(MG) is con-
nected by induction. On the other hand, same as the case in
homogeneous networks (i.e., UDG mode), RNG(M G) does
not have a bounded length stretch factor, nor constant bounded
power stretch factor, and does not have bounded node degree. In
this paper, we will show that RNG(MG) is a sparse graph: it
has at most 6n links.
In the following, we define a new structure, called
ERNG(MG). Assume that each node v knows its maximum
transmission radius r
v
. Let B(u) = {v | r
v
≥ r
u
}. A node
u processes its incident link uv in MG only if r
v
≥ r
u
, i.e.,
v ∈ B(u). Node u removes a link uv, where v ∈ B(u), if there
is another node w ∈ B(u) inside lune(u, v ) with both links uw
and wv are in MG. All the links uv kept by all nodes form the
final structure ERNG(MG).
Algorithm 1: Constructing-ERNG
1. Each node u initiates sets E
MG
(u) and E
ERNG
(u) to be
empty. Here E
MG
(u) is the set of links of MG known to u so far
and E
ERNG
(u) is the set of links of ERNG known to u so far.
Then, each node u locally broadcasts a HELLO message with
ID
u
, r
u
and its position (x
u
, y
u
) to all nodes in its transmission
range. Note that r
u
= p
u
1/β
is its maximum transmission range.
2. At the same time, each node u processes the incoming mes-
sages. Assume that node u gets a message from some node v. If
kvuk ≤ min{r
u
, r
v
}, then node u adds a link uv to E
MG
(u). If
r
v
≥ r
u
, then node u performs the following procedures. Node
u checks if there is another link uw ∈ E
MG
(u) with the fol-
lowing additional properties: 1) w ∈ lune(u, v ), 2) r
w
≥ r
u
,
and 3) kwvk ≤ min{r
w
, r
v
}. If no such link uw, then add uv
to E
ERNG
(u). For any link uw ∈ E
ERNG
(u), node u checks
if the following conditions hold: 1) v ∈ lune (u, w ), and 2)
kw vk ≤ min{r
w
, r
v
}. If the conditions hold, then remove link
uw from E
ERNG
(u).
3. Node u repeats the above steps until no new HELLO mes-
sages received.
4. For each link uv ∈ E
ERNG
(u), node u informs node v to
add link uv.
5. All links uv in E
ERNG
(u) are the final links in
ERNG(MG) incident on u.
We then prove that the structure ERNG has at most 6n links.
Lemma 2: Structure ERNG(MG) has at most 6n links.
Proof: Consider any node u. We will show that u keeps
at most 6 directed links emanated from u. Assume that u keeps
more than 6 directed links. Obviously, there are two links uw
and uv such that ∠wuv < π/3. Thus, vw is not the longest link
in triangle 4uvw. Without loss of generality, we assume that
286
kuwk is the longest in triangle 4uvw. Notice that the existence
of link uw implies that kuwk ≤ min(r
u
, r
w
) = r
u
. Conse-
quently, kvwk ≤ kuwk ≤ min(r
u
, r
w
). Thus, from the fact
that r
u
≤ r
v
, we know kvwk ≤ min(r
v
, r
w
). Hence, link vw
does exist in the original communication graph MG, it implies
that link uw cannot be selected to ERNG.
From Lemma 2, we can prove the following lemma.
Lemma 3: Structure RNG(MG) has at most 6n links.
Proof: Imagine that each link uv has a direction as fol-
lows:
−→
uv if r
u
≤ r
v
. Then similar to Lemma 2, we can prove
that each node u only keeps at most 6 such imagined direct links.
Thus, total links are at most 6n.
Similarly, we can define a structure EGG(MG), which con-
tains an edge uv if r
u
≤ r
v
and there is no node w with
the following properties: 1) r
u
≤ r
w
, 2) w is inside the disk
disk (u, v ). However, we cannot prove that EGG(MG) has a
linear number of links.
V. HETEROGENEOUS POWER SPANNER
Then, we give a strategy for all nodes to self-form a power
spanner structure, called GG(MG), based on the Gabriel graph.
We add a link uv ∈ MG to GG(MG) if there is no another node
w inside disk (u, v ) and both links uw and wv are in MG. Our
localized construction method works as follows.
Algorithm 2: Constructing-GG
1. Let E
MG
(u) and E
GG
(u) are the set of links known to u
from MG and GG respectively. Each node u initiates both
E
MG
(u) and E
GG
(u) as empty. Then, each node u locally
broadcasts a HELLO message with ID
u
, r
u
and its position
(x
u
, y
u
) to all nodes in its transmission range.
2. At the same time, each node u processes the incoming mes-
sages. Assume that node u gets a message from some node v. If
kvuk ≤ min{r
u
, r
v
}, then node u adds a link uv to E
MG
(u).
Node u checks if there is another link uw ∈ E
MG
(u) with
the following two additional properties: 1) w ∈ disk (u, v ),
and 2) kwvk ≤ min{r
w
, r
v
}. If no such link uw, add uv
to E
GG
(u). For any link uw ∈ E
GG
(u), node u checks if
the following two properties hold: 1) v ∈ disk (u, w ), and
2) kw vk ≤ min{r
w
, r
v
}. If they hold, remove link uw from
E
GG
(u).
3. Node u repeats the above steps until no new HELLO mes-
sages received.
4. All links uv in E
GG
(u) are the final links in GG(MG) inci-
dent on u.
We first show that Algorithm 2 builds the structure GG(MG)
correctly. For any link uv ∈ GG(M G), clearly, we cannot re-
move them in Algorithm 2. For a link uv 6∈ GG(MG), assume
that a node w is inside disk (u, v ) and both links uw and wv be-
long to MG. If node u gets the message from w first, and then
gets message from v, clearly, uv cannot be added to E
GG
(u).
If node u gets the message from v first, then u will remove uv
from E
GG
(u) (if it is there) when u gets the information of w.
It is not difficult to prove that structure GG(MG) is con-
nected by induction. In addition, since we remove a link uv
only if there are two links uw and wv with w inside disk (u, v ),
it is easy to show that the power stretch factor of GG(MG) is 1.
In other words, the minimum power consumption path for any
two nodes v
i
and v
j
in MG is still kept in GG(M G). Remem-
ber that here we assume the power needed to support a link uv
is kuvk
β
, for β ∈ [2, 5].
On the other hand, same as the case in homogeneous net-
works (i.e., UDG mode), GG(MG) is not a length spanner, and
does not have bounded node degree. Furthermore, it is unknown
whether GG(MG) is a sparse graph. Recently, it was proven
in [19] that GG(MG) has at most O(n
8/5
log γ) edges where
γ = max r
u
/r
v
.
Notice that, the extension from Gabriel graph is non-trivial.
In [19], two structures defined as follows even cannot guaran-
tee the connectivity. In the first structure, called LGG
0
(MG),
they remove a link uv ∈ MG if there is another node w in-
side disk (u , v ). In the second structure, called LGG
1
(MG),
they remove a link uv ∈ MG if there is another node w inside
disk (u, v ), and either link uw or link wv is in MG.
VI. HETEROGENEOUS DEGREE-BOUNDED SPANNER
Undoubtedly, as described in preliminaries, we always pre-
fer a structure has more nice properties, such as degree-bounded
(stronger than sparse), power spanner etc. Naturally, we could
extend the previous known degree-bounded spanner, such as the
Yao related structures, from homogeneous networks to hetero-
geneous networks. Unfortunately, a simple extension of the
Yao structure from UDG to MG even does not guarantee the
connectivity. Figure 2 (a) illustrates such an example. Here
r
u
= r
v
= kuvk, r
w
= kuwk, r
x
= kvxk, and kuwk < kuvk,
kuwk < kvwk, kvxk < kuvk, and kvx k < kuxk. In addition,
v and w are in the same cone of node u, and nodes x and u
are in the same cone of node v. Thus, the original MG graph
contains links uv, uw and vx only and is connected. However,
when applying Yao structure on all nodes, node u will only have
information of node v and w and it will keep link uw . Simi-
larly, node w keeps link uw; node v keeps link vx ; and node x
keeps link xv. In other words, only link xv and uw are kept by
Yao method. Thus applying Yao structure disconnects node v, x
from the other two nodes u and w. Consequently, we need more
sophisticated extensions of the Yao structure to MG to guarantee
the connectivity of the structure.
u
w
v
x
b
b
b
b
v
a a a
a
c
c
c
c1
2
3
1
2
1
3
h
3
2
h
h
(a) (b)
Fig. 2. Extend Yao structure on heterogeneous networks: (a) Simple extension
of Yao structure does not guarantee the connectivity. (b) Further space partition
in each cone to bound in-degree.
A. Extended Yao Graph
Algorithm 3: Constructing-EYG
1. Initially, each node u divides the disk disk (u, r
u
) centered
at u with radius r
u
by k equal-sized cones centered at u. We
287
generally assume that the cone is half open and half-close. Let
C
i
(u), 1 ≤ i ≤ k, be the k cones partitioned. Let C
i
(u), 1 ≤
i ≤ k, be the set of nodes v inside the ith cone C
i
(u) with a
larger or equal
1
radius than u. In other words,
C
i
(u) = {v | v ∈ C
i
(u), and r
v
≥ r
u
}.
Initially, C
i
(u) is empty.
2. Each node u broadcasts a HELLO message with ID
u
, r
u
and
its position (x
u
, y
u
) to all nodes in its transmission range.
3. At the same time, each node u processes the incoming broad-
cast messages. Once it gets a HELLO message from some node
v, it sets C
i
(u) = C
i
(u)
S
{v}, if node v is inside the ith cone
C
i
(u) of node u and r
v
≥ r
u
.
4. Node u chooses a node v from each cone C
i
(u) so that the
link uv has the smallest ID(uv) among all links uv
j
with v
j
in
C
i
(u), if there is any.
5. Finally, each node u informs all 1-hop neighbors of its cho-
sen links through a broadcast message. Let
−−−→
EY G
k
(MG) be the
union of all chosen links.
Since the symmetric communications are required, let
EY G
k
(MG) be the undirected graph by ignoring the direction
of each link in
−−−→
EY G
k
(MG). Graph EY G
k
(MG) is the final
network topology. Since node u chooses a node v ∈ disk (u, r
u
)
with r
v
≥ r
u
, link uv is indeed a bidirectional link, i.e., u and
v are within the transmission range of each other. Additionally,
this strategy could avoid the possible disconnection by simple
Yao extension we mentioned before.
Obviously, each node only broadcasts twice: one for broad-
casting its ID, radius and position; and the other for broad-
casting the selected neighbors. Remember that it selects at
most k neighbors. Thus, each node sends messages at most
O((k + 1) · log n) bits. Here, we assume that the node ID and
its position can be represented using O(log n) bits for a network
with n wireless nodes.
Before we study the properties of this structure, we have to
define some terms first. Assume that each node v
i
of MG has a
unique identification number ID
v
i
= i. The identity of a bidi-
rectional link uv is defined as ID(uv) = (kuvk, ID
u
, ID
v
)
where ID
u
> ID
v
. Note that we use the bidirectional links
instead of the directional links in the final topology to guarantee
connectivity. In other words, we require that both node u and
node v can communicate with each other through this link. In
this paper, all proofs about connectivity or stretch factors take
the notation uv and vu as same, which is meaningful. Only in
the topology construction algorithm or proofs about bounded-
degree, uv is different than vu: the former is initiated and built
by u, whereas the latter is by node v. Sometimes we denote a di-
rectional link from v to u as
−→
vu if necessary. Then we can order
all bidirectional links (at most n(n − 1) such links) in an in-
creasing order of their identities. Here the identities of two links
are ordered based on the following rule: ID(uv) > ID(pq) if
(1) kuvk > kpqk or (2) kuvk = kpqk and ID
u
> ID
p
or (3)
kuvk = kpqk, u = p and ID
v
> ID
q
.
Correspondingly, the rank of each link uv, denoted by
rank(uv), is its order in sorted bidirectional links. Notice that,
1
This is the main difference between this algorithm and the simple extension
of Yao structure discussed before, in which it considers all nodes v that u can
get signal from.
we actually only have to consider the links in MG. We then show
that the constructed network topology is a length and power
spanner.
Theorem 4: The length stretch factor of EY G
k
(MG), k >
6, is at most ` =
1
1−2 sin(
π
k
)
.
Proof: Notice it is sufficient to show that for any nodes u
and v with kuvk ≤ min(r
u
, r
v
), i.e. uv ∈ M G , there is a path
connecting u and v in EY G
k
(MG) with length at most `kuvk.
We construct a path u ! v connecting u and v in EY G
k
(MG)
as follows.
Assume that r
u
≤ r
v
. If link uv ∈ EY G
k
(MG), then set the
path u ! v as the link uv. Otherwise, consider the disk (u , r
u
)
of node u. Clearly, node u will get information of v from v and
node v will be selected to some C
i
(u) since r
v
≥ r
u
. Thus,
from uv 6∈ EY G
k
(MG), there must exist another node w in
the same cone as v, which is a neighbor of u in EY G
k
(MG).
Then set u ! v as the concatenation of the link uw and the
path w ! v. Here the existence of path w ! v can be easily
proved by induction on the distance of two nodes. Notice that
the angle θ of each cone section is
2π
k
. When k > 6, then θ <
π
3
.
It is easy to show that kwvk < kuvk. Consequently, the path
u ! v is a simple path, i.e., each node appears at most once.
We then prove by induction that the path u ! v has total
length at most `kuvk.
Obviously, if there is only one edge in u ! v, d(u ! v) =
kuvk < `kuvk. Assume that the claim is true for any path with
l edges. Then consider a path u ! v with l + 1 edges, which
is the concatenation of edge uw and the path
2
w ! v with l
edges, as shown in Figure 3 where kwvk = kxvk.
ϕ
w
x
u
v
α
Fig. 3. The length stretch factor of EY G
k
(M G) is at most
1
1−2 sin(
π
k
)
.
Notice, from induction, d(w ! v) ≤ `kwvk. Then, let
ϕ = ∠wuv and α = ∠uvw, we have
kuwk
kuxk
=
sin(∠uxw)
sin(∠xwu)
=
sin(
π
2
+
α
2
)
sin(
π
2
+
α
2
+ ϕ)
=
1
cos ϕ − sin ϕ tan
α
2
≤
cos(
π
4
−
ϕ
4
)
cos(
π
4
+
3
4
ϕ)
≤
1
1 − 2 sin(
π
k
)
The first inequality is because 0 ≤ α ≤
π
2
−
ϕ
2
and the second
inequality is because 0 ≤ ϕ ≤
2π
k
. Consequently, d(u !
v) = kuwk + d(w ! v) < `kuxk + `kwvk = `kuvk, where
` =
1
1−2 sin(
π
k
)
. That is to say, the claim is also true for the path
u ! v with l + 1 edges.
2
In the procedure of induction, if r
w
≤ r
v
then we induct on path w ! v,
otherwise we induct on path v ! w. In fact, here w ! v is same as v !
w since the path is bidirectional for communication. Directional link is only
considered in building process and is meaningless when we talk about the path.
This induction rule is applied throughout the remainder of the paper.
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