1
Applications of k-Local MST for Topology
Control and Broadcasting in Wireless Ad Hoc
Networks
Xiang-Yang Li∗Yu Wang∗Wen-Zhan Song∗
∗Department of Computer Science, Illinois Institute of Technology, 10 W. 31st Street, Chicago, IL 60616. Emails:
xli@cs.iit.edu, wangyu1@iit.edu, songwen@iit.edu
DRAFT
Abstract
In this paper, we propose a family of structures, namely, k-localized minimum spanning tree (LMSTk) for
topology control and broadcasting in wireless ad hoc networks. We give an efficient localized method to construct
LMSTkusing only O(n)messages under the local-broadcast communication model, i.e., the signal sent by each
node will be received by all nodes within the node’s transmission range. We also analytically prove that the node
degree of the structure LMSTkis at most 6, LMSTkis connected and planar, and more importantly, the total edge
length of the LMSTkis within a constant factor of that of the minimum spanning tree when k≥2(called low
weighted hereafter). We then propose another structure, called Incident MST and RNG Graph (IMRG), that can
be locally constructed using at most 13nmessages under the local broadcast communication model. Test results
are corroborated in the simulation study. We study the performance of our structures in terms of the total power
consumption for broadcasting, the maximum node power needed to maintain the network connectivity. We theoret-
ically prove that our structures are asymptotically the best possible for broadcasting among all locally constructed
structures. Our simulations show that our new structures outperform previous locally constructed structures in terms
of the broadcasting and power assignment for connectivity.
Keywords
Localized algorithms, broadcasting, topology control, minimum spanning tree, wireless ad hoc networks.
I. INTRODUCTION
We consider a wireless ad hoc network composed of nwireless devices (called nodes here-
after) distributed in a two-dimensional plane. Assume that all wireless nodes have distinctive
identities, quasi-static, and each wireless node knows its geometry position information either
through a low-power Global Position System (GPS) receiver or through some other way. More
specifically, it is enough for our protocols when each node knows the distance to each of its
one-hop neighbors, which can be estimated by the strength of signal. We assume that each wire-
less node has an omni-directional antenna and a single transmission of a node can be received
by any node within its vicinity which, under a common assumption in the literature, is a unit
disk centered at this node. A wireless node can receive the signal from another node if it is
within the transmission range of the sender. Otherwise, they communicate through multi-hop
wireless links by using intermediate nodes to relay the messages. Consequently, each node in
the wireless ad hoc network also acts as a router, forwarding data packets for other nodes. By
one-hop broadcasting, each node ucan gather the location information of all nodes within its
transmission range. Consequently, all wireless nodes together define a unit-disk graph (UDG),
which has an edge uv iff the Euclidean distance kuvkis less than one unit.
A wireless ad hoc network needs some special treatment as it intrinsically has its own special
characteristics and some unavoidable limitations compared with wired networks. For exam-
ple, wireless nodes are often powered by batteries only and they often have limited memories.
So wireless ad hoc networks prefer localized algorithms and power-efficient network topolo-
gies. Unlike the wired networks, a transmission by a wireless device will be received by all
nodes within its vicinity. Thus, we model the communication characteristics as broadcasting by
assuming that the message sent by a node will always be received by all nodes within its trans-
mission range. We can utilize this broadcasting property to save the communications needed to
send some information. Throughout this paper, a local broadcast by a node means it sends the
message to all nodes within its transmission range; a global broadcast by a node means it tries
to send the message to all nodes in the network by the possible relaying of other nodes.
Due to the limited power and memory, a wireless node prefers to only maintain the informa-
tion of a subset of neighbors it can communicate, which is called topology control. In recent
years, there is a substantial amount of research on topology control for wireless ad hoc networks
[1], [2], [3], [4], [5]. These algorithms are designed for different objectives: minimizing the
maximum link length while maintaining the network connectivity [3]; bounding the node de-
gree [5]; bounding the spanning ratio [1], [2]; constructing planar spanner locally [1]. Here a
structure His a spanner of UDG if, for any two nodes, the length of the shortest-path connecting
them in His no more than a constant factor of the length of the shortest-path connecting them
in the original UDG. Planar structures are used by several localized routing algorithms [6]. In
[7], Wang and Li proposed the first localized algorithm to construct a bounded degree planar
spanner.
Recently, Li, Hou and Sha [8] proposed a novel MST-based method for topology control and
broadcasting. Each node uuses its one-hop neighbors to build a local minimum spanning tree
and an edge uv is kept if it belongs to this local minimum spanning tree. They proved that
the final graph, called local minimum spanning tree (LMST), is connected, and has a bounded
degree 6. However,we will show that LMST is not a low weight structure and the broadcasting
based on it can still consume power O(n2)times of the minimum in the worst case.
Minimum-energy broadcast/multicast routing in ad hoc networking environment has been
addressed in [9], [10]. Three centralized greedy heuristics algorithms were presented in [10]:
MST (minimum spanning tree), SPT (shortest-path tree), and BIP (broadcasting incremental
power). Wan et al. [11] showed that the approximation ratio of the MST-based approach is
between 6and 12 by assuming that the power needed to support a link uv is kuvkβ, where kuvk
is the Euclidean distance between uand v,βis a real constant between 2and 5dependent on
the wireless transmission environment. The best distributed algorithm [12] can compute MST
in O(n)rounds using O(m+nlog n)communications for a general graph with medges and
nnodes. Obviously, MST cannot be constructed in a localized manner, i.e., each node cannot
determine which edge is in the defined structure by purely using the information of the nodes
within some constant hops. Thus, several localized structures, such as RNG [13], have been
used for broadcasting. As shown in [14], the total energy used by RNG based approach could
be about O(nβ)times optimum.
The main contributions of this paper are as follows. Firstly, we propose a family of structures,
namely, k-localized minimum spanning tree (LMSTk) for topology control and broadcasting in
wireless ad hoc networks. We analytically prove that the node degree of the structure LMSTk
is at most 6, LMSTkis connected and planar, and more importantly, the total edge length of the
LMSTkis within a constant factor of that of the minimum spanning tree when k≥2. We give
an efficient localized method to construct the LMSTkusing only O(n)messages under a local
broadcast communication model, i.e., the message sent by a node is received by all nodes within
its transmission range. Secondly, we propose another structure, called Incident MST and RNG
Graph (IMRG), that can be constructed using at most 13nmessages under the local broadcast
communication model. Every node only uses its partial two-hop information to construct the
structure IMRG. Notice that it was shown in [14] that some two-hop information is necessary
to construct any low-weighted structure for UDG. Thirdly, we study the application of these
structures for efficient broadcasting in wireless ad hoc networks. Notice that Wan et al. [11]
proved that the broadcasting based on the MST consumes energy within a constant factor of the
optimum when only consider the energy consumed by the senders. However, in practice, the
receiver node also consumes energy to receive the signal. In this paper, we adopt the later model
and assume that the energy consumed by the receiver node is no more than the energy consumed
by the sender. We then prove that the approximation ratio of the MST-based approach is still
a constant when this more practical energy model is used. Since it is expensive to construct
MST in a distributed way, we will use our newly proposed structures LMSTkand IMRG to
approximate it. Although a low-weighted structure cannot guarantee that the broadcasting based
on it consumes energy within a constant factor of the optimum in the worst case, the energy
consumptions using our new structures LMSTk(k≥2), and IMRG are within O(nβ−1)of the
optimum theoretically in the worst case. This improves the previously known “lightest” structure
RNG and LMST by O(n)factor. We show that these structures are asymptotically optimum
for broadcasting among all locally constructed structures. Test results are corroborated in the
simulation study. Our extensive simulations show that the energy consumption of broadcasting
based on these structures is within a small constant factor of that based on the MST for randomly
deployed wireless networks.
The rest of the paper is organized as follows. In Section II, we review the related works
on network topology control and minimum energy broadcasting. In Section III, we present
our communication and computation efficient localized methods that can construct connected,
planar, bounded degree, low-weighted structures LMSTkand IMRG. The total communication
costs of our methods are O(n)(at most 13nfor IMRG). We then study the applications of
our structures in broadcasting and topology control by comparing the performances of these
structures with previously best-known structures in Section V. We conclude our paper in Section
VI.
II. RELATED WORK
Before reviewing the related works, we first introduce the formal definition of low weight.
Given a geometric structure Gover a set of points, let ω(G)be the total length of the links in
Gand ωβ(G) = Puv∈Gkuvkβ. Then, a structure Gis called low weight if ω(G)is within a
constant factor of ω(M ST ).
A. Topology Control
Recently, topology control for wireless ad hoc networks has attracted considerable attentions
[3], [15], [17], [18], [19], [20]. Rajaraman [21] conducted an excellent survey. Several geometri-
cal structures have been used in topology control, and broadcasting in wireless ad hoc networks,
whose definitions are reviewed as follows.
A disk centered at a point xwith a radius r, denoted by disk (x,r), is the set of points whose
distance to xis at most r. Let lune (u,v)defined by two points uand vbe the intersection of two
disks with radius kuvkand centered at uand vrespectively, i.e., lune(u,v) = disk (u,kuv k)∩
disk (v,kuv k). Let disk(u, v)be the disk with diameter uv. The relative neighborhood graph
[22], denoted by RNG, consists of all edges uv such that the interior of lune(u,v)contains no
node w∈V. The Gabriel graph (GG) [23] contains an edge uv if and only if disk(u, v)contains
no other node winside. It is easy to show that RNG is a subgraph of the Gabriel graph. For
unit disk graph, the relative neighborhood graph and the Gabriel graph only contain the edges in
UDG and satisfying the respective definitions.
Notice that, traditionally, the relative neighborhood graph will always select an edge uv even
if there is some node on the boundary of lune (u,v). Thus, RNG may have unbounded node
degree, e.g., considering n−1points equally distributed on the circle centered at the nth point
v, the degree of vis n−1. Notice that for the sake of lowing the weight of a structure, the
structure should contain as less edges as possible without breaking the connectivity. Li [14] then
extended the traditional definition of RNG as follows.
The modified relative neighborhood graph consists of all edges uv such that (1) the interior of
lune(u,v)contains no point w∈Vand, (2) there is no point w∈Vwith ID(w)< I D(v)on
the boundary of lune(u,v)and kwvk<kuvk, and (3) there is no point w∈Vwith ID(w)<
ID(u)on the boundary of lune (u,v)and kwuk<kuvk, and (4) there is no point w∈Von
the boundary of lune(u,v)with ID(w)< ID(u),I D(w)< ID(v), and kwuk=kuvk. See
Figure 1 for an illustration when an edge uv is not included in the modified relative neighborhood
graph. Li called such structure RNG’. Obviously, RNG’ is a subgraph of RNG and still can be
w
v
u
w
v
u
w
v
u
w
v
u
Fig. 1. Four cases when edges are not in the modified RNG.
constructed using nmessages. It was proved in [14] that RNG’ has a maximum node degree 6
and still contains a MST as a subgraph.
The Yao graph with an integer parameter k≥6, denoted by −−→
Y Gk, is defined as follows. At
each node u, any kequally-separated rays originated at udefine kcones. In each cone, choose
the shortest edge uv, if there is any, and add a directed link −→
uv. Ties are broken arbitrarily or by
the smallest ID. The resulting directed graph is called the Yao graph. Some researchers used a
similar construction named θ-graph [24]. Recently, the Yao structure has been re-discovered by
several researchers for topology control in wireless ad hoc networks of directional antennas.
Li et al. [25] extended the definitions of these structures on top of any given graph G. Wat-
tenhofer et al. [20] also proposed a two-phased approach that consists of a variation of the Yao
graph followed by a variation of the Gabriel graph.
Li et al. [18] proposed a structure that is similar to the Yao structure for topology control.
Each node ufinds a power pu,α such that in every cone of degree αsurrounding u, there is some
node that ucan reach with power pu,α. Notice that the number of cones to be considered in
the traditional Yao structure is a constant k. However, unlike the Yao structure, for each node
u, the number of cones needed to be considered in the method proposed in [18] is about 2n,
where each node vcould contribute two cones on both side of segment uv. Then the graph Gα
contains all edges uv such that ucan communicate with vusing power pu,α . They proved that, if
α≤5π
6and the UDG is connected, then Gαis a connected graph. On the other hand, if α > 5π
6,
they showed that the connectivity of Gαis not guaranteed by giving some counter-example [18].
Unlike the Yao structure, the final topology Gαis not necessarily a bounded degree graph.
Li et al. [25] also proposed another structure called YaoYao graph −−→
Y Y kby applying a reverse
Yao structure on −−→
Y Gk. They proved that the directed graph −−→
Y Y kis strongly connected if UDG
is connected and k > 6. In [5], Wang et al. considered another undirected structure, called
symmetric Yao graph YS k. An edge uv is selected if and only if both directed edges −→
uv and −→
vu
are in the −−→
Y Gk. Then it is obvious that its maximum node degree is k. They showed that the
graph YS kis strongly connected if UDG is connected and k≥6.
Recently, Li, Hou and Sha [8] proposed an MST-based method for topology control. Each
node ufirst collects its one-hop neighbors N1(u). Node uthen computes its minimum spanning
tree M ST (N1(u)) of the induced unit disk graph on its one-hop neighbors N1(u). Node ukeeps
a directed edge uv if and only if uv is an edge in M ST (N1(u)). They called the union of all
directed edges the local minimum spanning tree, denoted by G0. If only symmetric edges are
kept, then the graph is called G−
0, i.e., it has an edge uv iff both directed edge uv and directed
edge vu exist. If ignoring the directions of the edges in G0, the graph is called G+
0, i.e., it has
an edge uv iff either directed edge uv or directed edge vu exists. They proved that the graph is
connected, and has bounded degree 6.
Here, we prove that graph G−
0is also planar. For the sake of contradiction, assume that G−
0
is not planar and two edges uv and xy intersect each other. Assume that the clockwise order
of these four nodes are u,y,v,x. Obviously, one of the four angles
\
uxv,
\
xvy,
\
vyu, and
\
yux is at least π/2. Without loss of generality, assume that
\
uxv ≥π/2. Then, edge uv is the
longest edge among triangle △uvx. Thus, in the local minimum spanning tree M ST (N1(u)),
edge uv cannot appear since there is already a path uxv whose edges are all shorter than uv.
Similarly, graph G+
0is a planar graph (by replacing the undirected edges with directed edges in
the above proof).
We then construct an example such that the structures G−
0and G+
0are not low-weighted.
Figure 2 illustrates such an example. Since it uses only one-hop information, at every node, the
algorithm only knows that there are a sequence of nodes evenly distributed with small separation,
and another node which is one-unit away from current node. It is easy to show that the final
structure G+
0is exactly illustrated in Figure 2. The minimum spanning tree will only use one
horizontal link while LMST has n/2horizontal links. It is easy to show that the total edge length
of G0is O(n)times of that of MST for this example.
v
Fig. 2. G0could consumes arbitrarily large power for broadcasting compared with the optimum.
Inspired by the local minimum spanning tree structure in [8], in this paper, we propose a
sequence of structures called k-local minimum spanning tree (LMSTk). To improve the com-
munication cost, we further propose another structure, called IMRG. Our structures have an ad-
ditional property: they are low-weighted. We also show that our structures are always subgraphs
of the structures G+
0and G−
0constructed in [8]. Locally constructed low-weighted structure was
first proposed by us in [14]. We will show that our new structures are subgraphs of that structure
and our structures have less computational cost. We do rely on a main theorem proved in [14]
to show that our structures are low-weighted.
B. Power Assignment
A transmission power assignment on the vertices in Vis a function Pfrom Vinto real num-
bers representing the node power. The directed (or called asymmetric by some researchers) com-
munication graph, denoted by −→
GP, induced by a transmission power assignment P, is a directed
graph with Vas its vertices and has a directed edge −−→
vivjif and only if ||vivj||β≤ P(vi). The
undirected (or called symmetric by some researchers) communication graph, denoted by GP, in-
duced by a transmission power assignment P, is a undirected graph with Vas its vertices and has
an edge vivjif and only if ||vivj||β≤ P(vi)and ||vivj||β≤ P(vj). Given a graph H= (V, E),
we say the power assignment Pis induced by H, denoted by PH, if P(v) = max(v,u)∈E||vu||β.
In other words, the power assigned to a node vis the largest power needed to reach all neigh-
bors of vin H. The maximum-cost (and total-cost) of a transmission power assignment Pis
defined as mc(P) = maxvi∈VP(vi)(and sc(P) = Pvi∈VP(vi)respectively). The min-max
assignment (and min-total assignment) problem is to find a transmission power assignment
Pwhose cost mc(P)(and sc(P)respectively) is minimized while the induced communication
graph is connected.
Let EMST(V) be the Euclidean minimum spanning tree over a point set V. Both [3] and [26]
use the power assignment induced by EMST(V). It was proved in [3] that power assignment
induced by EMST(V) is optimum for the min-max assignment problem. Using the fact that
RNG, GG and Y Gkhave O(n)edges and contain EMST as a subgraph, min-max assignment
problem can be solved in O(nlog n)time complexity by a centralized algorithm and solved
using O(nlog n)messages in a distributed manner.
Kiroustis et al. [27] first proved that the min-total assignment problem is NP-hard when the
mobile nodes are deployed in a three-dimensional space. A simple 2-approximation algorithm
based on the Euclidean minimum spanning tree was also given in [27]. The algorithm guarantees
the same approximation ratio in any dimensions. Clementi et al. [28], [29] proved that the min-
total assignment problem is still NP-hard when nodes are deployed in a two dimensional space.
For the symmetric communication, several methods also guarantee a good performance. It
is easy to show that the minimum spanning tree method still gives the optimum solution for
the min-max assignment and a 2-approximation for the min-total assignment. Recently,
Cˇalinescu et al. [30] gave a method that achieves better approximation ratio 5
3by using idea
from the minimum Steiner tree. Like the minimum spanning tree method, it works for any
power definition.
Since it is expensive to construct the Euclidean MST in a distributed manner, we would like
to approximate the Euclidean MST efficiently in a distributed way. We thus will study the
performance of our structures for power assignment. Notice that our structures do approximate
the total edge length of the Euclidean minimum spanning tree. Our simulations show that our
locally constructed structures outperform the previous structures in terms of both the maximum
assigned power and the total assigned power while guarantee the network connectivity.
C. Minimum Energy Broadcasting
Minimum-energy broadcast/multicast routing in a simple ad hoc networking environment has
been addressed in [9], [10]. Any broadcast routing is viewed as an arborescence (a directed tree)
T, rooted at the source node of the broadcasting, which spans all nodes. Let PT(v)denote the
transmission power of the node vrequired by the tree T. For any leaf node vof T,PT(v) =
0. For any internal node vof T, let PT(v)denote the minimum power needed to reach its
farthest children in T. The total energy required by Tis Pv∈VPT(v). It is known [31] that
the minimum-energy broadcast routing problem cannot be solved in polynomial time if P6=
N P . Three greedy heuristics were proposed in [10] for the minimum-energy broadcast routing
problem: MST, SPT, and BIP. By assuming that the power needed to support a link uv is kuvkβ.
It was proved in [11] that, for any point set Vin the plane, the total energy required by any
broadcasting among Vis at least ωβ(M ST )/Cmst , where 6≤Cmst ≤12 is a constant related
to the Euclidean minimum spanning tree. In addition, they [11] showed that the approximation
ratio of MST based approach is between 6and 12 and the approximation ratio of BIP based
approach is between 13
3and 12; on the other hand, the approximation ratio of SPT is at least n
2,
where nis the number of nodes.
Unfortunately, all these structures cannot be constructed locally. Thus, several locally con-
structed structures have been proposed for broadcasting in wireless ad hoc networks, such as
RNG [13]. The ratio of the weight in RNG over the weight of MST could be O(n)for n
points set [25]. By assuming that the power needed to support a link uv is kuvkβ, an example
was given in [14] to show that the total energy used by broadcasting on RNG could be about
O(nβ)times of the minimum-energy used by an optimum method. The same example can be
used to show that the structure G0[8] could consumes power O(nβ)times of the optimum for
broadcasting. On the other hand, we will prove that ωβ(IM RG)≤O(nβ−1)·ωβ(M ST ), and
ωβ(LMSTk)≤O(nβ−1)·ωβ(M ST )for k≥2. In other words, the power consumption for
broadcasting based on our newly proposed structures are only O(nβ−1)times of the optimum in
the worst case, which improves the previously known structure RNG by O(n)factor. When we
assume that the receivers do consume power for receiving signal, all the statements still hold.
III. k-LOCAL MINIMUM SPANNING TREE (LMSTk)
In this section, we define a sequence of structures, namely, k-local minimum spanning tree
(LMSTk), which can be constructed locally using only O(n)messages. All these structures are
connected, low-weighted (when k≥2), planar and have a bounded degree.
A. k-local Minimum Spanning Tree (LMSTk)
We define a sequence of structures k-local minimum spanning tree (LMSTk) as follows. Let
Nk(u)be the set of nodes that are within khops of node uin UDG. Here Nk(u)includes node
uitself for the simplicity of notation later.
Definition 1: The k-local minimum spanning tree (LMSTk) contains a directed edge −→
uv if
edge uv belongs to M ST (Nk(u)). We further define two undirected variations LMST−
k, and
LMST+
k. Structure LMST−
kcontains an edge uv if both directed edge −→
uv and directed edge −→
vu
belong to LMSTk. Structure LMST+
kcontains an edge uv if either −→
uv or −→
vu belongs to LMSTk.
Notice that one way to construct MST is to add edges in the order of their lengths if it does
not create a cycle with previously added edges. If there are two edges with the same length, we
break the tie by comparing the larger ID of the two end-points then comparing the smaller ID of
the two-end points. We label an edge uv by (kuvk,max(ID(u), I D(v)),min(ID(u), ID(v))),
and an edge uv is ordered before an edge xy if the lexicographic order of the label of uv is less
than that of xy. In this paper, we only consider the minimum spanning tree constructed using
the above edge ordering.
Before we present our communication efficient method to construct them, we first study their
properties. First of all, it is easy to prove the following monotone property of the structures.
Lemma 1: LMSTk+1 ⊆LM STk,LM ST +
k+1 ⊆LM ST +
k, and LM ST −
k+1 ⊆LM ST −
k.
Lemma 2: LMST+
kis a subgraph of RNG, so does LMST−
k.
Proof. We prove it by contradiction. Assume that a node uadds an edge uv 6∈ RN G to LMSTk.
Since edge uv 6∈ RNG, there is a node winside the lune defined by segment uv. Remember
that the minimum spanning tree of the node set N1(u)can be constructed by adding edges in
ascending order whenever it does not create a cycle with previously added edges. Clearly, when
we process the edge uv, there is already a path connecting uand wand a path connecting wand
vsince uw and wv are not longer than uv. It implies that node ucannot add the edge uv to its
M ST (Nk(u)). Consequently, both graphs LMST+
kand LMST−
kare subgraphs of RNG.
Actually we can enhance Lemma 2 by showing that LMST+
kis a subgraph of RNG’. The
above lemma immediately implies that the structures LMSTk, LMST+
kand LMST−
kare planar.
Remember that k-local minimum spanning tree LMSTkis proposed to approximate Euclidean
minimum spanning tree MST. We then show that MST is a subgraph of LMSTkfor any k.
Lemma 3: Euclidean minimum spanning tree MST is a subgraph of LMSTkfor any k.
Proof. Consider any edge uv from MST. Assume that we add edges in ascending order of their
lengths to MST. Clearly, when we decide whether to add the edge uv, there is no path connecting
uand vusing edges added before uv. Obviously, this property still holds when node udecide
whether to add edge uv to the minimum spanning tree M ST (Nk(u)) of its k-hop neighbors
Nk(u). It implies that edge uv belongs to M ST (Nk(u)), and MST (Nk(v)). Consequently,
MST is a subgraph of all structures LMSTk, LMST+
kand LMST−
kfor any k.
The above lemma immediately implies that all these k-localized minimum spanning trees are
connected when the original communication graph UDG is connected.
Since every node in the Euclidean minimum spanning tree has a degree at most 6, the out-
degree of every node uin LMSTkis at most 6. Consequently, the degree of every node uin
LMST−
kis although at most 6since we keep an edge uv if both directed edges −→
uv and −→
vu belong
to LMSTk. We then show that the degree of every node in LMST+
kis also at most 6.
Lemma 4: Each node in LMSTkhas at most 6neighbors in LMST+
k.
Proof. We prove it by contradiction. Assume that one node vhas more than 6total in-neighbors
and out-neighbors. From the pigeonhole principle, there must have two neighbors, say u1and
u2, of vsuch that
\
u1vu2< π/3. There are three cases: 1) both u1and u2are in-neighbors; 2)
both u1and u2are out-neighbors; 3) one is out-neighbor and one is in-neighbor.
We first consider the case that both u1and u2are in-neighbors. Obviously,
\
u1vu2cannot
be the largest angle in the triangle u1vu2. Assume that
\
vu1u2is the largest, i.e., u2vis the
longest edge in triangle u1vu2. Thus, node u2cannot have u2vin its minimum spanning tree
M ST (Nk(u2)) since there is already a path (using node u1∈N1(u2)) connecting u2and vwhen
we try to add edge u2v. It is a contradiction to the fact that u2is an in-coming neighbor of v.
Similarly, we can prove that the other two cases are also impossible. This finishes the proof.
The above lemma immediately implies that every node in graphs LMST+
kand LMST−
khas
a degree at most 6. To show that the final structures LMSTk, LMST+
kand LMST−
kare low
weighted when k≥2, we first review a result proved in [14].
Lemma 5 ([14]) A subgraph Gof RNG’ is low-weighted if for any two edges uv ∈Gand
xy ∈G, neither uv nor xy is the longest edge of the quadrilateral uvyx.
We then prove the main result of this paper.
Lemma 6: All structures LMST+
kare low weighted when k≥2.
Proof. Since we showed that LMST+
kis a subgraph of modified RNG for any k, we will only
need prove that there are no two edges uv ∈LMS T +
kand xy ∈LM ST +
k, such that one of
them is the longest edge of the quadrilateral uvyx. We prove this by contradiction. Assume
that we have two edges uv ∈LM ST +
kand xy ∈LM ST +
k, and uv is the longest edge of the
quadrilateral uvyx. Clearly, x,vand yare at most 2-hops away from uin the unit disk graph.
Then when we decide whether to add edge uv to the minimum spanning tree M S T (Nk(u)) of
the k-hop neighbors Nk(u)for k≥2, edges xu,xy, and yv have already been processed, i.e.,
there are paths using shorter edges to connect uto x,xto y, and yto v. Thus, the edge uv will
not be added to M ST (Nk(u)) when k≥2. It is a contradiction to uv ∈LM ST +
k. This finishes
the proof.
B. Efficient Construction of k-local Minimum Spanning Tree (LMSTk)
We then discuss in detail how to construct the k-local Minimum Spanning Tree (LMSTk)
efficiently, i.e., using only O(n)messages under the local broadcasting model. Since LMST2
is already a low weighted structure, we will only describe our method for constructing LMST2
although the same method works for general LMSTk.
Algorithm 1: Construct LMST2Locally
1. Every node ucollects the location information of N2(u)based on an efficient method de-
scribed in [32] (reviewed in detail later).
2. Every node ucomputes the Euclidean minimum spanning tree M ST (N2(u)) of its 2-hop
neighbors N2(u), including uitself.
3. A node uproposes to add a directed edge −→
uv if uv ∈M ST (N2(u)) and ||uv|| ≤ 1.
4. If LMST +
2is needed, node ukeeps an edge uv when either uor vproposed to add it. If
LM ST −
2is needed, node ukeeps an edge uv when both uand vproposed to add it.
We then review the communication efficient method proposed in [32] to collect N2(u)for
every node uwhen the geometry information is known. Computing the set of 1-hop neighbors
with O(n)messages is trivial: every node broadcasts a message announcing its ID. Computing
the 2-hop neighborhood is not trivial, as the UDG can be dense. The approach in [32] is based
on the specific connected dominating set introduced in [33], which again is based on a max-
imal independent set (MIS). In the algorithm, each node uses its adjacent node(s) in the MIS
to broadcast over a larger area relevant information. Listening to the information about other
nodes broadcast by the MIS nodes enables a node to compute its 2-hop neighborhood. The al-
gorithm uses heavily the nodes in the connected dominating set, an example in [32] shows that
overloading certain nodes might be unavoidable.
We start from the moment the virtual backbone is already constructed, and every node knows
the ID and the position of its neighbors. The idea of the algorithm is for every node to efficiently
announce its ID and position to a subset of nodes which includes its 2-hop neighbors. The
responsibility for announcing the ID and position of a node vis taken by the MIS nodes adjacent
to v. Each such MIS node assembles a packet containing: <ID; position;counter>, with the
ID and position of v, and a counter variable being set to 2. The MIS node then broadcasts the
packet.
A connector node is used to establish a link in between several pairs of virtually-adjacent MIS
nodes, and will not retransmit packets which do not travel in between these pairs of MIS nodes.
Here two MIS nodes are said to be virtually-adjacent if they are within 2or 3hops of each other.
The connector node will rebroadcast packets with nonzero counter originated by one of the
nodes in a pair of virtually-adjacent MIS nodes, thus making sure the packet advances towards
the other MIS node in the pair. Recall that the path in between a pair of virtually-adjacent MIS
nodes has one or two connector nodes.
When receiving a packet of type <ID; position;counter>, an MIS node checks whether this
is the first message with this ID, and if yes decreases the counter variable and rebroadcasts the
packet. A node listens to the packets broadcast by all the adjacent MIS nodes and, using its
internal list of 1-hop neighbors, checks if the node announced in the packet is a 2-hop neighbor
or not - thus constructing the list of 2-hop neighbors.
The above approach can be extended to find the k-hop neighbors of every node using total
O(n)communications: the initial counter is set to k. The total communications used by this
approach is at most (6k+ 3)2·nafter a backbone based on MIS is constructed [4].
IV. STRUCTURES WITH IMPROVED COMMUNICATION COST
In the previous section, we defined a sequence of structures that are guaranteed to be low
weighted and can be constructed in a localized manner using only O(n)messages. However,
the hidden constant in the communication cost could be large although it is a constant. In this
section, we define several structures that can be constructed using at most 13nmessages. All
these structures are connected, low-weighted, bounded degree, planar graphs.
A. Sparse Structure From RNG’
In [14], Li gave the first localized method to construct a structureLRNG with weight O(ω(M ST ))
using total O(n)local-broadcast messages, but the computation at each node is expensive. For
the completeness of presentation, we first review the localized algorithm given in [14] that con-
structs a low-weighted structure using only some two hops information.
Algorithm 2: [14] Construct Low Weighted Sparse Structure LRNG
1. All nodes together construct the graph RNG’ in a localized manner.
2. Each node ulocally broadcasts its incident edges in RNG’ to its one-hop neighbors. Node u
listens to the messages from its one-hop neighbors.
3. Assume node ureceived a message informing the existence of an edge xy from its neighbor
x. For each edge uv in RNG’, if uv is the longest among uv,xy,ux, and vy, node uremoves
the edge uv. Ties are broken by the label of the edges. Here we assume that uvyx is the convex
hull of u,v,x, and y.
4. Let LRNG denote the final structure formed by all remaining edges in RNG’.
Obviously, if an edge uv is kept by node u, then it is also kept by node v, i.e., the edges kept
by all nodes are symmetric. It was shown in [14] that the structure LRNG has total edge length
Θ(ω(M ST )).
Clearly, the communication cost of Algorithm 2 is at most 7n: initially each node spends one
message to tell its one-hop neighbors its position information, then each node utells its one-hop
neighbors all its incident edges uv ∈RN G′(there are at most total 6nsuch messages since
RNG′has at most 3nedges). The computational cost of Algorithm 2 could be high since for
each link uv ∈RNG′, node uhas to test whether there is an edge xy ∈RN G′and x∈N1(u)
such that uv is the longest among uv,xy,ux, and vy. We continue to present our new algorithms
that improve the computational complexity of each node while still maintain low communication
costs.
B. Incident MST and RNG Graph (IMRG)
Although the structures LM ST −
2and LM ST +
2have several nice properties such as bounded
degree, planar, and low-weighted, the communication cost of constructing them could be very
large to save the computational cost of each node compared with structure LRNG. The large
communication costs are from collecting the two hop neighbors information N2(u)for each
node u, although the total communication of the protocol described in [32] is O(n), the hidden
constant is large.
We could improve the communication cost by using a subset of two hop information without
sacrificing any properties. For any node u, we define the partial two hop of uas
NRNG′
2(u) = {w|vw ∈RN G′and v∈N1(u)} ∪ N1(u).
Definition 2: The Incident MST and RNG Graph (IMRG) contains a directed edge −→
uv if edge
uv belongs to M ST (NRN G′
2(u)), the Euclidean minimum spanning tree of nodes NRNG′
2(u).
We further define two undirected variations IMRG−, and IMRG+. Structure IMRG−contains
an edge uv if both directed edge −→
uv and directed edge −→
vu belong to IMRG. Structure IMRG+
contains an edge uv if either −→
uv or −→
vu belongs to IMRG.
We then describe a communication efficient algorithm to build these structures as follows.
Algorithm 3: Construct Low Weighted Structure IMRG
1. Each node utells its position information to its one-hop neighbors N1(u)using a local broad-
cast model. All nodes together construct the graph RNG’ in a localized manner.
2. Each node ulocally broadcasts its incident edges in RNG’ to its one-hop neighbors. Node u
listens to the messages from its one-hop neighbors.
3. Each node ucollects NRNG′
2(u)and computes the Euclidean minimum spanning tree, denoted
by M ST (NRN G′
2(u)), of all nodes NRN G′
2(u), including uitself.
4. Node uproposes to add an edge uv ∈MST (NRNG′
2(u)) if kuvk ≤ 1.
5. If IMRG−is needed, node ukeeps an edge uv if both node uand node vproposed to add
edge uv. If IMRG+is needed, node ukeeps an edge uv if either node uor node vproposed to
add edge uv.
As will seen later (Lemma 7), the constructed structures are subgraphs of the modified RNG
graph. Thus, these structures are planar and have at most 3nedges. In addition, the total com-
munication cost of Algorithm 3 is at most 13nwhen either structure IMRG−or IMRG+is
needed; the total communication cost is at most 7nif the directed structure IMRG is needed.
We first show that these two structures IMRG+and IMRG−are still planar, bounded degree,
and low-weighted.
Lemma 7: Structure IMRG is a subgraph of modified RNG.
Proof. Consider any edge uv 6∈ RNG′. We show that node uwill not propose uv. From
the definition of RNG’, we know that there is a node winside the lune defined by segment
uv and edge uw and wv has a label less than uv. Considering the process of constructing
M ST (NRN G′
2(u)), when we decide whether to add edge uv after processing edges with smaller
labels, there is already a path connecting uand w, and a path connecting wand v. Thus, edge
uv cannot be added by node uto M ST (NRN G′
2(u)). This finishes the proof.
The above lemma immediately implies that all structures IMRG+and IMRG−are planar , and
have a bounded node degree at most 6. We then show that IMRG+and IMRG−are connected
by proving the following lemma.
Lemma 8: MST is a subgraph of IMRG+and IMRG−.
Proof. We prove this by induction on the length of the edges from MST.
Consider the shortest edge uv in the original unit disk graph. Clearly, the edge uv belongs to
MST, and uv belongs to M ST (NRNG′
2(u)) and M ST (NRN G′
2(v)). Thus, uv belongs to IMRG−.
Assume that the first kth shortest edges from MST are in IMRG−. Then consider the (k+1)th
shortest edge uv from MST. For the sake of contradiction, assume that node uremoves edge uv
since uv 6∈ M ST (NRN G′
2(u)). Consequently, there is a path in the unit disk graph formed on
NRNG′
2(u)connecting uand vusing edges with length at most kuvk(ties are broken by rank).
It is a contradiction to the fact that uv belongs to MST. Thus, edge uv is also kept IMRG−.
Therefore, MST is a subgraph of IMRG−and MST is a subgraph of IMRG+.
We then show that the structures IMRG−and IMRG+are low-weighted.
Lemma 9: The structures IMRG−and IMRG+are low-weighted.
Proof. The proof is similar to the proof that LMSTkis low weighted. We can show that there are
no two edges uv and xy from IMRG such that one of them is the longest edge in the quadrilateral
uvyx, which can be proved easily by contradiction. Notice that we already proved that IMRG−
and IMRG+are subgraphs of RNG’. Thus, we can use Lemma 5.
We then summarize the properties of the structure IMRG by the following theorem.
Theorem 10: Algorithm 3 constructs structures IMRG−and/or IMRG+using at most 13n
messages. The structures IMRG−or IMRG+are connected, planar, bounded degree (at most 6),
and low-weighted.
It is easy to show that the structure LMST2is always a subgraph of IMRG since IMRG uses
only a partial information to construct the minimum spanning tree. If an edge uv is removed
from M ST (NRN G′
2(u)), it means that there is a path connecting uand vusing shorter edges
when we process uv. By a simple induction, we can show that there is also a path connecting
uand vwhen we process uv in constructing M ST (N2(u)). We further show that the structure
IMRG is a subgraph of LMST1. Consider any directed edge −→
uv that is not proposed by node u
in constructing M ST (N1(u)). It means that there is a path connecting uand vin the induced
unit disk graph on N1(u), whose edges have length less than kuvk. Clearly, this path is still
in the induced unit disk graph on NRNG′
2(u)since N1(u)⊂NRNG′
2(u). Consequently, edge
uv cannot appear in the Euclidean minimum spanning tree M ST (NRNG′
2(u)). It then implies
that the structure IMRG is always a subgraph of LMST1. Consequently, the structure IMRG+
is always a subgraph of the structure G+
0and the structure IMRG−is always a subgraph of the
structure G−
0constructed in [8].
Lemma 11: Structure IMRG is a subgraph of LMST1and a supergraph of LMST2.
C. Fault-Tolerance
We have presented algorithms to build structures that are connected, planar, low-weighted and
have a bounded node degree. However, none of these structures are fault-tolerant in the worst
case. Here we say that a structure is node fault-tolerant if the graph is still connected when one
node breaks down. In [34], Li et al. discussed how to build a k-fault-tolerant structure such that
each node has a degree at most 6kand is a spanner. In this subsection, we present a method that
transforms any structure into a fault-tolerant structure by at most doubling the total edge length.
Notice that also this method has been used previously for various purposes [15], [16], we will
show that it keeps the low-weight and bounded degree properties. Assume that we are given a
topology structure Gthat is connected.
Algorithm 4: Transform Structure Gto Fault-Tolerant
1. Each node ucollects all incident edges uv ∈G.
2. Node usorts all its incident neighbors from Gin a clockwise order and let v1,v2···,vdbe its
neighbors. Node uinforms node vito add links vi−1viand vivi+1 . Here v0=vdand vd+1 =v1.
Let F(G)be the final structure formed by all edges, including the edges from G.
Lemma 12: If structure Ghas bounded degree ∆, then graph F(G)has degree at most 3∆.
Proof. Consider any node vi. Notice that, only the neighbors of node vican add edges vi−1vi
and vivi+1 incident on vi. Node vihas at most ∆neighbors in G. Thus, there are at most 2∆
newly added edges to node vi. Considering the previous incident edges (at most ∆), the total
number of edges incident on viis at most 3∆.
Lemma 13: If structure Ghas low weight, then graph F(G)has low weight.
Proof. We show that ω(F(G)) ≤3ω(G). Consider any node uand the added edges vivi+1 .
Clearly, kvivi+1k ≤ kuvik+kuvi+1k. Thus, Pd
i=1 kvivi+1k ≤ 2Pd
i=1 kuvik. Clearly, ω(F(G))
is at most ω(G)plus the summation of all newly added edges vivi+1, which is at most 2ω(G).
Thus, ω(F(G)) ≤3ω(G).
Lemma 14: Structure F(G)is fault-tolerant.
Proof. Consider any path that uses node uand assume that node ubreaks down. Assume that
viuand uvjare the two links in that path. Then we can use the path vivi+1 ···vjto connect vi
and vj. Thus, there is still another path connecting the source and the target without node u.
This finishes the proof that F(G)is fault-tolerant.
It is not difficult to show that the total communications of transforming a structure into a
fault-tolerant one uses messages at most 2m, where mis the number of edges in the original
structure G. Since the structures discussed in this paper all have at most 3nedges, the total
communication cost of this transforming is at most 6n. The price of this transforming is that the
new structure F(G)is not guaranteed to be a planar graph even if the original graph Gis planar.
Lemma 15: Structures F(LMSTk) and F(IMRG) have bounded node degree at most 18, have
total edge length at most O(ω(E M ST )), are connected, fault-tolerant, and can be constructed
using O(n)messages under local broadcast communication model. Structure F(IMRG) can be
constructed using at most 19nmessages. Each of the messages has at most 2 log nbits.
Notice that, here we implicitly assumed that the maximum transmission power of each node
can support the additional links added by Algorithm 4. In addition, instead of connecting the
neighbors of a node uin a clockwise order, we can connect the neighbors of each node uusing
the minimum spanning tree of these nodes, which will further decrease the total edge length of
the final structure.
D. Impossibility Results
Power assignment and topology control have been well studied recently by various researchers.
Although most questions can be solved exactly or approximated within a constant factor using
a centralized approach, it is still unknown whether we can solve or approximate some questions
using localized approaches. For example, using centralized methods, we can minimize the max-
imum transmission power while the resulting network topology has some properties that can be
tested in polynomial time. Such property includes the network is connected, or the network is
k-connected, or the network topology is a spanner of the original communication graph UDG. In
addition, using centralized methods, we can approximate the minimum total transmission power
of all nodes within a constant factor, while the resulting network is connected, or k-connected, or
consumes the minimum energy for broadcasting. However, centralized methods are expensive
to implement in wireless ad hoc networks due to their possible massive communications. Thus,
it is natural to ask what kind of questions we can approximate within a constant factor using
localized approaches, and what kind of questions we cannot.
We have shown that we can construct a bounded degree planar spanner, or a bounded de-
gree planar low-weighted structure, or a bounded degree k-fault tolerant spanner, or a bounded
degree fault tolerant low-weighted structure, in a localized manner using only O(n)messages.
In the following, we will show that several questions in wireless ad hoc networks cannot be
approximated within a constant factor in a localized manner at all.
The first such example is the min-max assignment problem. It was proved in [3] that the
longest edge of the Euclidean minimum spanning tree EMST(V) is always the optimum solution
to the min-max assignment problem. Since it is communication expensive to construct MST in
a distributed manner, we would like to know whether we can construct a structure in a localized
manner such that the longest edge of this structure is within a constant factor of that of MST. We
show by example that there is no such deterministic localized algorithm unfortunately. Assume
that there is such a deterministic localized algorithm Athat uses k-hop information. Figure 3
illustrates an example that algorithm Acannot approximate the longest edge of the MST within
a constant factor. In the example, ||ux|| > k and kuvk= 1. Then algorithm Awill have the
y
u v
xy
u v
x
(a) (b)
Fig. 3. No localized algorithm approximates the minimum of the maximum node power while the resulting struc-
ture is connected.
same information at the node ufor both configurations illustrated in Figure 3 (a) and (b). If A
decides to keep edge uv, then the longest edge kept by Acould be arbitrarily larger than that of
MST for configuration (a). If Adecides not to keep edge uv, then the structure constructed by
Ais not connected for configuration (b).
Figure 3 also shows that there is no deterministic localized algorithm that can find a structure
that approximates the total energy consumption of broadcasting within a constant factor1of the
1Weactually can show that no deterministic localized algorithm can find a structure such that the energy consumed by broad-
optimum, or that approximates the total node power within a constant factor of the optimum
while the network topology is connected. Similarly, we can show that there is no deterministic
localized algorithm that can find a structure minimizing the total node power while the structure
is node fault-tolerant.
V. APPLICATIONS OF OUR STRUCTURES IN BROADCASTING AND TOPOLOGY CONTROL
After we proved some properties of our structures, we then study how our structures can be
used to improve the performances of broadcasting and topology control compared with some
previously developed structures.
A. Worst Case Performances
We first assume that the energy needed to support the communication between a link uv is
kuvkβ. Li proved in [14] that, if His a low-weighted structure, then ωβ(H)≤O(nβ−1)·
ωβ(M ST ). Here ωβ(G) = Puv∈Gkuvkβ. It is easy to show that the total power consumption
of broadcasting based on any connected structure His at most 2ωβ(H). Let T⊆Hbe the tree
used for broadcasting. The power consumption of each node uis at most ||uv||β, where uv is
the longest link incident on uin T. The claim follows from that any such link uv will be used at
most twice to define the power for a node. It is also known the minimum power consumption of
broadcasting is at least ωβ(M ST )/12. Consequently, we have the following theorem.
Theorem 16: If His a low-weighted structure, then the power consumption of broadcasting
based on His at most O(nβ−1)times of the optimum.
We then show that there is a configuration of nodes such that the broadcastings based on the
low-weighted structures LMSTkand IMRG do consume power Θ(nβ−1)times of the optimum.
Consider the example illustrated by Figure 3 (a). Clearly, our structures will keep the link
uv. Thus, the total power consumptions based on our structures are O(1), while the optimum
structure (without link uv) has power consumption only 1/nβ−1. Notice that, this example shows
that the broadcasting based on any locally constructed structure has power consumption at least
Θ(nβ−1)times of the optimum in the worst case.
casting based on this structure is within o(nβ−1)of the optimum. Here assume that the power needed to support a link uv is
kuvkβ.
It has been shown in [14] that the broadcastings based on RNG could consume power Θ(nβ)
times of the optimum. The same example can also show that the broadcastings based on one hop
local minimum spanning tree G0[8] could consume power Θ(nβ)times of the optimum. Thus,
our low-weighted structures improve the performances for broadcasting of previously proposed
structures by Θ(n)factor in the worst case.
We then consider the scenario when the receiver node does consume a power to receive the
signal, and we assume that this power is no more than the power consumed by the sender al-
ways. Notice that in all our structures, there are at most 6receivers. Thus, the total power
consumed by both senders and receivers in this new energy model is no more than 7times of
the total power consumption of all senders in the previous energy model. We also show that the
broadcasting based on MST is still a good approximation. Let E0(G)be the energy consumption
of the broadcasting based on a structure Gwhen assume that the power needed to support the
communication between a link uv is kuvkβand the receiver does not consume power. Let OP T0
be the optimum structure for broadcasting in this model. Let E1(G)be the energy consumption
of the broadcasting based on a structure Gwhen the power consumed by each receiver is con-
sidered and this power is assumed to be no more than the power used by the sender. Let OP T1
be the optimum structure for broadcasting in this model. Wan et al. [11] essentially proved
that E0(M ST )≤12E0(OP T0). We argue that E1(MS T )≤cE1(OP T1)for some constant
cas follows. Since MST has a node degree bounded by 6,E1(M ST )≤7E0(M ST ). Notice
that E1(OP T1)≥E0(OP T1)≥E0(OP T0), which implies our statement. Consequently, our
structures consume powers no more than O(nβ−1)times of the optimum.
We summarize the worst case performances of out structures LMSTkand IMRG.
Theorem 17: The power consumption of broadcasting, and the total node power needed to
achieve network connectivity, based on the structure LMSTkor IMRG is at most O(nβ−1)times
of the optimum. Our structures are asymptotically the optimum among all locally constructed
structures.
B. Performances for Random Wireless Ad Hoc Networks
We then conduct extensive simulations to study the performances of our structures in terms
of the maximum transmission power used by all nodes, the total transmission power used by all
nodes, and the total length of links. Although network throughput is an important performance
metric, it is influenced by many other factors such as the MAC protocol, routing protocol and
so on. Therefore, most related works do not test the throughput performance. To study various
aspects of our structures, we will use the following metrics to compare the performances:
1. Total Messages: In wireless networks, less messages to construct the topology will save
energy consumption. We showed that the total messages of constructing IMRG is at most 13n.
2. Max Messages: We also test what is the maximum number of messages a node will send
in building the structure. A large number of messages sent by a node will delay the topology
updating and drain out its battery power quickly.
3. Average Node Degree: A smaller average node degree often implies less contention and
interference for signal and thus a better frequency spatial reuse, which in turn will improve the
throughput of the network.
4. Max Node Degree: We also test the maximum node degree. A larger node degree will cause
more contention and interference for signal, and also may drain out its battery power quickly.
5. Max Node Power: Each node uwill set its transmission range equal to the length of the
longest edge incident on u. A smaller node power will always save the power consumption. The
max-node-power captures the maximum power used by all nodes. Here, in all our simulations,
we set the constant β= 2, so that the power needed to support a link uv is kuvk2.
6. Total Node Power: The total node power approximates the total power used by all nodes to
keep the connectivity of the network.
7. Total Node Power for Broadcasting: This measures the total node power of all nodes that
have a degree at least 2, i.e., internal nodes. This approximates the total power used by a broad-
casting based on this structure. Notice that the nodes with degree 1(except the possible source
node) do not relay the message in a broadcasting.
8. Total Edge Length: We proved that all our structures have a total edge length within a
constant factor of that of MST. We want to see the actual approximation performances.
9. Total Link Power: It was proved in [11] that the minimum total power needed for broad-
casting is within a constant factor of the total link power in MST. We thus compare the total link
power used by our structures with previously known structures and especially that of MST.
In the simulations, we will only test the performances of structures LRNG, LMST−
2and
IMRG−, and compare them with previously known structures LMST1(called G0in [8]), and
RNG in terms of the above metrics. The reason for only selecting G−
0and RNG is that in [8],
their simulations already showed that G−
0out-performs other previously known structures in
terms of the node degree, max node power, and the total node power. Hereafter, we use the term
LMST, LMST2and IMRG instead of G−
0, LMST−
2and IMRG−in the experiments, if it is clear.
UDG MST RNG LMST
LMST2LRNG IMRG
Fig. 4. Different structures from a UDG.
In the first simulation, we randomly generate 100 nodes uniformly in a 1000m×1000mregion.
The maximum transmission range of each node is set as 250m. The topology derived using the
maximum transmission power (UDG), MST, RNG, LMST1(or called G−
0), LMST2, LRNG,
and IMRG (actually IMRG−) are shown in Figure 4 respectively. To make the performance
testing precise, we generate 100 sets of nodes, each of which has 100 nodes, and compute the
performance metrics accordingly. The average degree of UDG is 15.37 and the maximum degree
is 26. The corresponding performances are illustrated in the following Table V-B. Here for max
node degree, max message and max node power, we show both the maximum and average values
over the 100 sets. We found that structure LMST2outperforms all other structures in all metrics
significantly (except the number of messages used). In addition, structure IMRG performs better
than LMST with slightly high communication cost to construct it. For example, structure LMST
uses about 5% percent more total node power than the structure IMRG for broadcasting, while
RNG consumes about 50% percent more total node power for broadcasting than the structure
IMRG. We did not count the messages used to find the two hops neighbors for all nodes when
computing the total messages used to construct LMST2(such messages number is marked by a
star in our results).
TABLE I
THE PERFORMANCES COMPARISON OF SEVERAL STRUCTURES.
MST RNG LMST LMST2LRNG IMRG
MaxMaxMsg - 1.00 5.00 5.00⋆5.00 9.00
AvgMaxMsg - 1.00 4.50 4.50⋆4.92 8.42
TotMsg - 100.00 305.72 299.88⋆334.76 538.68
MaxMaxDeg 4.00 4.00 4.00 4.00 4.00 4.00
AvgMaxDeg 3.50 3.92 3.50 3.50 3.92 3.50
AvgDeg 1.98 2.35 2.06 2.00 2.30 2.04
MaxMaxNPow 4.13 5.40 4.69 4.13 5.40 4.69
AvgMaxNPow 2.93 4.17 3.77 3.03 4.17 3.55
TotNPow 79.85 122.80 92.79 82.56 119.69 90.10
TotNPowBrdcst 66.48 118.21 83.26 70.08 114.74 79.43
TotLength 132.79 183.59 144.86 135.55 175.52 141.99
TotLPow 112.47 187.37 131.85 116.56 177.29 127.13
We then vary the number of nodes in the region from 50 to 500. The transmission range of
each node is still set as 250m. We plotted the performances of all structures in Figure 5. Finally,
we fix the number of nodes in the region as 500 and grow the transmission range of each node
from 100mto 300m. We plotted the performances of all structures in Figure 6.
All the results show that IMRG has better performances than LMST and RNG: IMRG has the
least total link length and least total node power for broadcasting; it has the least node power
to keep the connectivity. The number of messages used for constructing IMRG is slightly more
than the number of messages used to construct LMST. The simulation results confirm all our
theoretical analysis. Remember that, in the worst case, IMRG may spend O(nβ−1)times the
total power used by the optimum broadcasting. However, our simulations show that the energy
consumption of broadcasting based on IMRG is within a small constant factor (about 15% more)
of that based on the MST and is much better than that based on RNG. In summary, IMRG is the
best among all these known local structures; additionally, it can approximate MST theoretically
and be used for energy efficient broadcasting.
50 100 150 200 250 300 350 400 450 500
50
100
150
200
250
300
350
400
450
number of nodes
total link length
MST
RNG
LMST
LMST2
LRNG
IMRG
50 100 150 200 250 300 350 400 450 500
100
120
140
160
180
200
220
number of nodes
total link power
MST
RNG
LMST
LMST2
LRNG
IMRG
50 100 150 200 250 300 350 400 450 500
1.9
2
2.1
2.2
2.3
2.4
2.5
number of nodes
avg node degree
MST
RNG
LMST
LMST2
LRNG
IMRG
total-link-length total-link-power avg-node-degree
50 100 150 200 250 300 350 400 450 500
3.4
3.5
3.6
3.7
3.8
3.9
4
4.1
4.2
number of nodes
avg max node degree
MST
RNG
LMST
LMST2
LRNG
IMRG
50 100 150 200 250 300 350 400 450 500
0
500
1000
1500
2000
2500
3000
number of nodes
total messages
RNG
LMST
LMST2
LRNG
IMRG
50 100 150 200 250 300 350 400 450 500
1
2
3
4
5
6
7
8
9
10
number of nodes
avg max node messages
RNG
LMST
LMST2
LRNG
IMRG
avg-max-node-degree total-message avg-max-message
50 100 150 200 250 300 350 400 450 500
70
80
90
100
110
120
130
number of nodes
total node power
MST
RNG
LMST
LMST2
LRNG
IMRG
50 100 150 200 250 300 350 400 450 500
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
number of nodes
avg max node power
MST
RNG
LMST
LMST2
LRNG
IMRG
50 100 150 200 250 300 350 400 450 500
60
70
80
90
100
110
120
130
number of nodes
total node power without leaves
MST
RNG
LMST
LMST2
LRNG
IMRG
total-node-power avg-max-node-power total-node-power-brdcst
Fig. 5. Results when the number of nodes in the networks are different (from 50 to 500). Here the transmission
range is set as 250m.
VI. CONCLUSION
We defined a sequence of low-weighted sparse structures LMSTk, and presented an efficient
method to construct them locally using only O(n)messages. Here a structure is called low-
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
280
300
320
340
360
380
400
420
440
460
transmission ranges
total link length
MST
RNG
LMST
LMST2
LRNG
IMRG
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
100
120
140
160
180
200
220
transmission ranges
total link power
MST
RNG
LMST
LMST2
LRNG
IMRG
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
1.9
2
2.1
2.2
2.3
2.4
2.5
transmission ranges
avg node degree
MST
RNG
LMST
LMST2
LRNG
IMRG
total-link-length total-link-power avg-node-degree
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
3.94
3.96
3.98
4
4.02
4.04
4.06
4.08
4.1
transmission ranges
avg max node degree
MST
RNG
LMST
LMST2
LRNG
IMRG
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
500
1000
1500
2000
2500
3000
transmission ranges
total messages
RNG
LMST
LMST2
LRNG
IMRG
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
1
2
3
4
5
6
7
8
9
10
transmission ranges
avg max node messages
RNG
LMST
LMST2
LRNG
IMRG
avg-max-node-degree total-message avg-max-message
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
70
80
90
100
110
120
130
transmission ranges
total node power
MST
RNG
LMST
LMST2
LRNG
IMRG
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
transmission ranges
avg max node power
MST
RNG
LMST
LMST2
LRNG
IMRG
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
60
70
80
90
100
110
120
130
transmission ranges
total node power without leaves
MST
RNG
LMST
LMST2
LRNG
IMRG
total-node-power avg-max-node-power total-node-power-brdcst
Fig. 6. Results when the transmission range are different (from 100mto 300m). Here the number of nodes is 500.
weighted if its total link length is within a constant factor of that of the Euclidean minimum
spanning tree. We further defined a bounded degree planar low weighted connected structure
IMRG that can be constructed more efficiently. The total communication cost of our localized
method is at most 13n. We showed that both structures are asymptotically the best structures that
can be constructed locally for broadcasting. We conducted extensive simulations to study the
performances of our structures and compared them with previously known localized structures.
Our structures out-perform all previously known structures and structure IMRG only incurs a
small message overhead.
The constructed structures are planar, bounded degree, and low-weighted. Li et al. [35]
recently gave an O(nlog n)-time centralized algorithm to construct a bounded degree, planar,
and low-weighted spanner. However, it is still unknown how to make that a distributed algorithm
using O(n)communications without sacrificing the spanner property. On the other hand, Li et
al. [7] showed how to construct a planar spanner with bounded degree in a localized manner
(using O(n)messages) for unit disk graph. However, the constructed structure does not seem to
be low-weighted. It remains open how to construct a bounded degree, planar, and low-weighted
spanner in a distributed manner using only O(n)communications under the local broadcasting
communication model.
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