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ALGORITHMIC ASPECTS
OF
TOPOLOGY CONTROL PROBLEMS
FOR
AD HOC NETWORKS
Rui "NMI" Liu, University
of
Delaware, Newark,
DE
Errol L. Lloyd, University
of
Delaware, Newark, DE
Madhav
V.
Marathe, D-2
Ram "NMI" Ramanathan, Internetwork Design Department, Cambridge, MA
S.
S.
Ravi, University
at
Albany
-
SUNY
The Third ACM International Symposium
on
Mobile Ad
Hoc
Networking
&
Computing
Lausanne, Switzerland
June
9-11,2002
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Form
836
(1
0196)
Algorithmic Aspects
of
Topology
Control Problems
for
Ad
hoc
Networks1
RUI LIU E
RROL
L, L
LOYD
Department of Computer and Information Science
University
of Delaware, Newark, DE 19716
Email:
{ruliu, e1loyd)Qcis .udel
.
edu.
MADHAV
V.
MARATHE
Los Alamos National Laboratory
MS
M997,
P.O.
Box 1663
Los
Alamos,
NM
87545
Email:
maratheQlan1.
gov
R
AM
RAMANATHAN
Internetwork Design Department
BBN
Technologies
(A
Division of GTE)
Cambridge,
MA
Email:
ramanathobbn
.
corn
S.
S.
RAVI
Department of Computer Science
University
at
Albany
-
SUNY
Albany, NY 12222
Email:
raviQcs albany
,
edu.
Note:
The research was supported by the Department of Energy under Contract W-7405-ENG-
36,
NSF
Grant CCR-97-34936 and by the collaborative participation in the Communications and
Networks Consortium sponsored by the
U.
S.
Army Research Laboratory under the Collaborative
Technology Alliance Program, Cooperative Agreement
DAAD19-01-2-0011. The
U.
S.
Government
is authorized to reproduce and distribute reprints for Government purposes notwithstanding any
copyright notation thereon.
Algorithmic
Aspects
of
Topology Control Problems
for
Ad
hoc
Networks
Abstract
Topology control problems are concerned with the
as
-
signment of power values to nodes of an ad hoc net
-
work
so
that the power assignment leads to
a
graph
topology satisfying some specified properties. This
paper considers such problems under several opti
-
mization objectives, including minimizing the max
-
imum power and minimizing the total power.
A
general approach leading to
a
polynomial algorithm
is presented for minimizing maximum power for
a
class of graph properties, called
monotone
prop
-
erties. The difficulty of generalizing the approach
to properties that are not
monoione
is
pointed out.
Problems involving the minimization of total power
are known to be NP
-
complete even for simple graph
properties.
A
general approach that leads to an ap
-
proximation algorithm for minimizing the total power
for some monotone properties is presented. Using
this approach,
a
new approximation algorithm for the
problem of minimizing the total power for obtaining
a
2-node-connected graph is obtained.
It
is shown that
this algorithm provides
a
constant performance guar
-
antee. Experimental results from an implementation
of the approximation algorithm are also presented.
1 Introduction
1.1
Motivation
An ad hoc network consists of
a
collection of
transceivers,
All
communication among these
transceivers is based on radio propagation. For each
ordered pair
(u,v)
of transceivers, there is
a
trans
-
mission power threshold,
denoted by
p(u,
w),
with
the following significance:
A
signal transmitted by
the transceiver
u
can be received by
v
only when the
transmission power of
u
is
at
least
p(u,
v).
The trans
-
mission power threshold for
a
pair of transceivers de
-
pends on
a
number of factors including the distance
between the transceivers, the direction of the antenna
at
the sender, interference, noise, etc. [RROO].
Given the transmission powers of the transceivers,
an ad hoc network can be represented by
a
directed
graph. The nodes
of
this directed graph are in one
-
to
-
one correspondence with the transceivers.
A
directed
edge
(u,
w)
is
in
this graph
if
and only
if
the trans
-
mission power of
u
is
at
least the transmission power
threshold
p(u,
w).
The main goal of
topology control
is
to assign
transmission powers to the transceivers
so
that the re
-
sulting directed graph satisfies some specified proper
-
ties. Since the battery power of each transceiver is an
expensive resource, it is important
to
achieve the goal
while minimizing
a
given function of the transmis
-
sion powers assigned to the transceivers. Examples
of
desirable graph properties are connectivity, small
diameter,
etc. Examples of minimization objectives
considered in the literature are the maximum power
assigned to
a
transceiver and the total power
of
all
transceivers (the latter objective is equivalent to min
-
imizing the average power assigned to
a
transceiver).
As
stated above, the primary motivation to study
topology control problems is to make efficient use of
available power
at
each node. In addition, using mini
-
mum amount of power
at
each node to achieve a given
task is also likely to decrease the MAC layer interfer
-
ence between adjacent radios. We refer the reader
to
[LHB+Ol,
RMMO1,
WL+Ol,
RROO, RM99, TK841
for
a
thorough discussion
of
the power control issues
for ad hoc networks.
1.2
Formulation
of
Topology
Control
Problems
Topology control problems have been studied un
-
der two graph models. The discussion above cor
-
responds to the
directed graph model
studied in
[RROO]. The
undirected graph model
proposed in
1
[KK+97] represents the ad hoc network
as
an undi
-
rected graph in the following manner. First, the di
-
rected graph model for the network is constructed.
Then, for any pair of nodes
u
and
w,
whenever both
the directed edges
(u,
w)
and
(w,
u)
are present, this
pair of directed edges is replaced by a single undi
-
rected edge
{u,w}.
All
of
the remaining directed
edges are deleted. Under this model, the goal of
a
topology control problem is to assign transmission
powers to nodes such that the resulting undirected
graph
has
a
specified property and
a
specified func
-
tion of the powers assigned to nodes is minimized.
Note that the directed graph model allows two
-
way
communication between some pairs
of
nodes and one
-
way
communication between other pairs of nodes. In
contrast, every edge in the undirected graph model
corresponds to
a
two
-
way communication.
In general,
a
topology control problem can be spec
-
ified by
a
triple of the form
(M,
P,
0).
In such
a
specification,
M
E
{D
IRECTED
,
U
NDIRECTED
}
rep
-
resents the graph model,
B
represents the desired
graph property and
0
represents the minimization
objective. For the problems considered in this
pa
-
per
0
E
{M
AX
P
OWER
,
T
OTAL
P
OWER
}.
For ex
-
ample, consider the
(D
IRECTED
,
STRONGLY CON
-
NECTED,
M
AX
POWER)
problem. Here, powers must
be assigned
to
transceivers so that the resulting di
-
rected graph
is
strongly connected and the maximum
power assigned to
a
transceiver is minimized. Simi
-
larly, the
(UNDIRECTED, %NODE
C
ONNECTED
,
TO
-
TAL
P
OWER
)
problem seeks to assign powers to the
transceivers
so that the resulting undirected graph
has
a
node connectivity
of
(at least)
2
and the sum of
the powers assigned to
all
transceivers is minimized.
2
Additional Definitions
This section collects together the definitions of some
graph theoretic and algorithmic terms used through
-
out this paper.
Given an undirected graph
G(V,
E),
an
edge
sub
-
graph
G'(V,E')
of
G
has all
of
the nodes of
G
and
the edge set
E'
is
a
subset
of
E.
Further, if
G
is an
edge weighted graph, then the weight of each edge in
G'
is the same as it is in
G.
The
node connectivity
of an undirected graph is
the smallest number of nodes that must be deleted
from the graph
so
that the resulting graph is discon
-
nected. The
edge connectivity
of an undirected
graph is the smallest number of edges that must be
deleted from the graph
so
that the resulting graph
is disconnected. For example,
a
tree has node and
edge
connectivities equal to
1
while
a
simple cycle has
node and edge
connectivities equal to
2.
When the
node (edge) connectivity of
a
graph is
k,
the graph is
said to be
k-node connected (k-edge connected).
Given an undirected graph, polynomial algorithms
are known for finding its node and edge
connectivi-
ties
[vago].
The main results of this paper use the following
definition.
Definition
2.1
A
property
P
of
the (directed or
undirected) graph associated with an ad hoc network
is
monotone
the property continues to hold even
when the powers assigned to some nodes are increased
while the powers assigned to the other nodes remain
unchanged,
Example:
For any
IC
2
1,
the property
IC-
N
ODE
C
ONNECTED
for undirected graphs is mono
-
tone since increasing the powers of some nodes while
keeping the powers of other nodes unchanged may
only add edges to the graph. However, properties
such as
A
CYCLIC
or
B
IPARTITE
are not monotone.
Some
of
the topology control problems considered
in this paper are NP
-
complete. For such problems,
we study approximation algorithms. In this con
-
text, an approximation algorithm provides
a
per-
formance guarantee
of
p
if for every instance
of
the problem, the solution produced by the approxi
-
mation algorithm
is
within the multiplicative factor
of
p
of the optimal solution.
A
polynomial time
approximation scheme
(PTAS) is an approxima
-
tion algorithm that, given
a
problem instance and an
accuracy requirement
e,
produces
a
solution that is
within
a
factor
1
+
E
of the optimal solution.
3
Previous
Work
and
Summary
of
Results
3.1
Previous
Work
The form of topology control problems consid
-
ered in this paper was proposed by Ramanathan
and Rosales-Hain (RROO]. They presented effi
-
cient algorithms for two topology control prob
-
lems, namely
(UN
DIR
ECTED,
NODE
CON-
2
NECTED,
M
AX
POWER)
and
(UNDIRECTED,
2-
N
ODE
C
ONNECTED
,
M
A
X
P
OWER
).
After deter
-
mining the minimum value for the objective, their
algorithms also reduce the power assigned to each
transceiver such that each power level is minimal
while maintaining the desired graph property. In ad
-
dition, they presented efficient distributed heuristics
for these problems.
Several groups of researchers have studied the
(U
NDIRECTED
,
NODE
C
ONNECTED
,
To-
TAL
P
OWER
)
problem [CH89, KK+97, CPS99,
CPSOO]. Reference [CH89] proves that the prob
-
lem is NP-hard and presents an approximation
al
-
gorithm with
a
performance guarantee of 2. The
other references consider
a
geometric version of the
problem along with
a
symmetry
assumption con
-
cerning transmission power thresholds. More pre
-
cisely, these references assume the following: (a) Each
transceiver
is
located
at
some point of &dimensional
Euclidean space. (b) For any pair of transceivers
u
and
w,
p(u,w)
=
p(w,u)
=
the Euclidean distance
between the locations of
71
and
w.
For
a
justifi
-
cation of this model, see Kirousis et
a1
[KK+97].
They show that the
(U
NDIRECTED
,
NODE
C
ON
-
N
E
CTED,
T
OTAL
P
OWER
)
problem is NP-hard when
transceivers are located in
3-dimensional space. They
also present an approximation algorithm with
a
per
-
formance guarantee of 2 for the problem
in
any met
-
ric space. In addition, they provide some results for
the
l-dimensional version of the
(U
NDIRECTED
,
1-
N
ODE
C
ONNECTED
,
T
OTAL
P
OWER
)
problem where
there is an additional constraint on the diameter
of the resulting undirected graph.
Clementi et
a1
[CPS99] show that the 2-dimensional version of
the
(U
NDIRECTED
,
NODE
C
ONNECTED
,
To-
TAL
P
OWER
)
problem remains NP-hard. They also
show that the
2-dimensional version with
a
diameter
constraint can be efficiently approximated
to within
some constant factor and that the
3-dimensional ver
-
sion does not have
a
polynomial time approximation
scheme.
Researchers have also addressed other versions of
topology control problems.
Hu [Hug31 proposed
a
distributed algorithm based on Delaunay triangula
-
tion to maintain connectivity. However, that paper
does not address the issue of assigning transmission
powers to nodes.
Radoplu and Meng [RM99] present
a
distributed protocol for maintaining strong connec
-
tivity in
a
network with mobile nodes. The networks
generated by their protocol include
minimum-energy
paths (i.e., paths that allow messages to be transmit
-
ted using
a
minimum amount of energy) from each
node to
a
designated master node. Wattenhofer et
a1
[WL+Ol] discuss
a
cone-based distributed algo
-
rithm for topology control; their algorithm generates
a
power assignment which ensures that the size of the
node set that remains connected under this power
as
-
signment is the same
as
the one in which every node
is assigned the full power.
Li and Halpern [LHOl]
improve upon the protocol of [RM99] by proposing
another protocol; given
a
network G, the new pro
-
tocol creates
a
subnetwork
G'
such that whenever
there is
a
path between
a
pair of nodes in
G,
there is
a
minimum-energy path between them in
G'.
Li et
a1
[LHB+Ol] provide
a
more detailed analysis of the
protocol
of
[WL+Ol] and establish
a
precise bound
on the angle of the cone that ensures connectivity.
They
also
establish several properties of the protocol
in
[WL+Ol].
3.2
Summary
of
Main
Results
The main results of this paper are the following.
1.
We show that for any monotone graph prop
-
erty
P
that can be tested in polynomial
time for undirected (directed) graphs, the
problem
(U
NDIRECTED
,
P,
M
A
X
P
OWER
)
((D
IRECTED
,
B,
M
AX
P
OWER
))
can be solved
in polynomial time. This generalizes some
of the results in
[RROO] where efficient algo
-
rithms were presented for two monotone prop
-
erties, namely
NODE
C
ONNECTED
and 2-
N
ODE
C
ONNECTED
.
2. We show that there are non-monotone and ef
-
ficiently testable properties (e.g. G
RAPH
I
S
A
TREE
)
for which the problem of minimizing the
maximum power is NP
-
complete. This result
shows that, in general, if the monotonicity con
-
dition is eliminated, then obtaining an efficient
algorithm for minimizing maximum power may
not be possible.
3.
As mentioned above, for any monotone and effi
-
ciently testable property
IF,
a
solution that min
-
imizes the maximum power can be obtained in
polynomial time,
However, if we introduce the
additional requirement that the number of nodes
that use the maximum power must also be min
-
imized, we show that there are monotone prop-
3
