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A Minimum-Energy Path-Preserving Topology-Control Algorithm

Li (Erran) Li Joseph Y. HalpernDept. of Computer Science Dept. of Computer Science

Cornell University Cornell UniversityIthaca NY 14853 Ithaca NY 14853

lili@cs.cornell.edu halpern@cs.cornell.edu

Abstract— The topology of a wireless multi-hop network canbe controlled by varying the transmission power at each node.In general, it is not energy efficient to use the communicationnetwork

�������where every node transmits with maximum

power. For energy efficient operations, it is desirable to have asubnetwork that preserves a minimum-energy path between everypair of nodes (where a minimum-energy path is one that allowsmessages to be transmitted with a minimum use of energy). Inthis paper, we first identify conditions that are necessary andsufficient for a subnetwork

�of

�������to preserve this property.

Using this characterization, we then propose an efficient topology-control algorithm that, given a communication network

������,

computes a subnetwork�

that it preserves at least one minimum-energy path between every pair of nodes. We also proposean energy-efficient reconfiguration protocol that maintains thisminimum-energy path property as the network topology changesdynamically. We demonstrate the performance improvements ofour algorithm over other existing topology-control algorithmsthrough simulation.

Index Terms— Topology control, connectivity, minimal energypath, ad hoc networks.

I. INTRODUCTION

Multi-hop wireless networks, especially sensor networks,are expected to be deployed in a wide variety of civil andmilitary applications. Minimizing energy consumption hasbeen a major design goal for wireless networks. As pointedout by Chandrakasan et al. [3], network protocols that reduceenergy consumption are key to low-power wireless sensornetworks.

We can characterize a communication network using a graph��� ��where the nodes in

��� ��represent the nodes in the

network, and two nodes � and � are joined by an edge if itis possible for � to transmit a message to � if � transmits atmaximum power. Transmitting at maximum power requires agreat deal of energy. To reduce energy usage, we would like asubgraph

of

�� ��such that (1)

consists of all the nodes

in �� ��

but has fewer edges, (2) if � and � are connectedin

��� ��, they are still connected in

, and (3) a node �

can transmit to all its neighbors in

using less power thanis required to transmit to all its neighbors in

��� ��. Indeed,

what we would really like is a subnetwork

of��� ��

withthese properties where the power for a node to transmit to its

Work supported in part by NSF under grants grants IRI-96-25901, IIS-0090145, and NCR97-25251, and ONR under grants N00014-00-1-03-41,N00014-01-10-511, and N00014-01-1-0795. A preliminary version of thispaper, with the title “Minimum-energy mobile wireless networks revisited”appeared in the Proceedings of the IEEE Conference on Communications,2001,

neighbors in��� ��

(possibly over several hops) is minimal. Asubnetwork

is said to have the minimum-energy property if

the following holds: given �� ��

, it guarantees that betweenevery pair ��������� of nodes that are connected in

�� ��, the

subgraph

has a minimum-energy path between � and � ,one that allows messages to be transmitted with a minimumuse of energy among all the paths between � and � in

�� ��.

In this paper, we first identify conditions that are neces-sary and sufficient for a graph to have this minimum-energyproperty. We use this characterization to construct a topology-control algorithm called SMECN (for small minimum-energycommunication network). In the SMECN algorithm, a node �tries to find the minimum power ������� such that transmittingwith ������� ensures that, it takes no less power to transmitdirectly to any node � outside the range of ������� than torelay through a node inside ������� ’s range. We prove that thesubnetwork constructed by SMECN contains no 2-redundantedges in

if broadcasts at a given power setting are able to

reach all nodes in a circular region around the broadcaster.(An edge ����� �!� in

is 2-redundant if there exists a 2-hop

path between � and � such that transmitting a message overthe 2-hop path consumes energy no greater than transmittingthe message directly from � to � does.)

The work most closely related to ours is that of Rodopluand Meng [22]. They provide a protocol that, given a commu-nication network, computes a subnetwork that preserves theminimum-energy property. We call their protocol MECN (forminimum-energy communication network). The code for theSMECN algorithm is simpler than that of MECN and it runsfaster (although we have not tried to quantify the extent towhich it runs faster). In addition, the subnetwork constructedby SMECN is provably smaller than that constructed byMECN if broadcasts at a given power setting are able toreach all nodes in a circular region around the broadcaster.We conjecture that, in practice, SMECN will construct smallersubnetworks even when this assumption does not hold. Oursimulations show that, by being able to use a smaller network,SMECN has lower link maintenance costs than MECN and canachieve a significant saving in energy usage.

There are a number of other papers in the literature ontopology control. Hu [9] describes an algorithm that doestopology control using heuristics based on a Delauney trian-gulation of the graph. There seems to be no guarantee thatthe heuristics preserve connectivity. Ramanathan and Rosales-Hain [20] describe a centralized spanning tree algorithm forachieving connected and biconnected static networks, while

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minimizing the maximum transmission power. (They alsodescribe distributed algorithms that are based on heuristicsand are not guaranteed to preserve connectivity.) Li Li etal. [13] considered a cone-based topology-control algorithmwhich requires only the availability of directional information.The cone-based algorithm takes as a parameter an angle " . Anode � then tries to find the minimum power ��#%$ & such thattransmitting with �'#%$ & ensures that in every cone of degree "around � , there is some node that � can reach with power ��#($ & .It is shown that taking "*),+.-�/10 is necessary and sufficientto preserve connectivity. Several connectivity-preserving opti-mizations are also proposed. The basic algorithm is referredto as CBTC. The optimized algorithm is referred to as OPT-CBTC.

Other researchers working in the field of packet radionetworks, wireless ad hoc networks, and sensor networks havealso considered the issue of power efficiency and networklifetime, but have taken different approaches. For example,Hou and V.O.K. Li [8] analyze the effect of adjusting trans-mission power to reduce interference and hence achieve higherthroughput as compared to schemes that use fixed transmis-sion power [26]. Heinzelman et al. [7] describe an adap-tive clustering-based routing protocol that maximizes networklifetime by randomly rotating the role of per-cluster localbase stations (cluster head) among nodes with higher energyreserves. Chang and Tassiulas[4] propose a multicommoditymaximum-flow based approach to maximize the time until thefirst node is depleted of energy. Qun Li et al.[14] and Singh etal.[24] exploit the use of different routing metrics to increasenetwork lifetime. Chen et al. [5] and Xu et al. [29] proposemethods to conserve energy and increase network lifetime byturning off redundant nodes. Finally, Wu et al. [28] and Monkset al. [28], [15] describe their power-controlled MAC protocolsto reduce energy consumptions and increase throughput. Theydo this through power control of unicast packets. However,they make no attempt at reducing the power consumption ofbroadcast packets. Since topology-control algorithms do notsolve the routing problem, power-aware routing protocols [7],[4], [24], [14] can be used in conjunction with topology-control algorithms. In general, the basic idea of powering-off receivers to reduce energy consumption is orthogonal toour approach, so the benefits of the two approaches couldpotentially be combined. Power-controlled MAC can also beusefully combined with our approach to further improve thenetwork performance.

The rest of the paper is organized as follows. Section IIgives the network model (which is essentially the same as thatused in [22]). Section III identifies a condition necessary andsufficient for achieving the minimum-energy property. Thischaracterization is used in Section IV to construct the SMECNprotocol and prove that it constructs a network smaller thanMECN if the broadcast region is circular. In Section V,we show how SMECN can be used to deal with topologychanges using only local message exchanges. Rodoplu andMeng deal with topology changes by having each node reruntheir MECN algorithm, which will typically involve muchmore communication. In Section VI, we give the results ofsimulations showing the energy savings obtained by using the

network constructed by SMECN. Section VII concludes ourpaper.

II. THE MODEL

We assume that a set 2 of nodes is deployed in a two-dimensional area, where no two nodes are in the same physicallocation. Each node has a GPS receiver on board, so knowsits own location. It does not necessarily know the location ofother nodes. Moreover, the location of nodes will in generalchange over time.

A transmission between node � and � takes power ������� �!�),354'����� �!�56 for some appropriate constant 3 , where 798;: isthe path-loss exponent of outdoor radio propagation models[21], and 4'����� �!� is the distance between � and � . A receptionat the receiver takes power < . This power expenditure at thereceiver is referred to as the receiver power. Computationalpower consumption is ignored.

Suppose that there is some maximum power � �� ��at which

the nodes can transmit. Thus, there is a graph=�� �� )�>2?��@ �� �� � where ����� �!�9AB@ �� ��

if it is possible for � totransmit to � at maximum power. Clearly, if ����� �!��A9@ �� ��

,then 354C��������� 6ED � �� ��

. However, we do not assume that anode � can transmit to all nodes � such that 354'����� �!� 6FD � �� ��

.One reason is that there may be obstacles between � and �that prevent transmission. Even without obstacles, if a unittransmits using a directional transmit antenna, then only nodesin the region covered by the antenna (typically a cone-likeregion) will receive the message. Rodoplu and Meng [22]implicitly assume that every node can transmit to every othernode. Here we take a first step in exploring what happensif this is not the case. However, we do assume that the graph��� ��

is connected, so that there is a potential communicationpath between any pair of nodes in 2 .

Because the power required to transmit between a pair ofnodes increases as the 7 th power of the distance betweenthem, for some 7G8H: , it may require less power to relayinformation than to transmit directly between two nodes. Asusual, a path IJ)K����L(�NMOMOMN� ��P.� in a graph

)Q�>2?��@�� isdefined to be an ordered list of nodes such that ���SR5� ��RUT�V���AW@ .The length of IW)X��� L �OMNMOM���� P � , denoted Y IZY , is [ . The totalpower consumption of a path IF)\��� L ����V.�O]N]O]���� P � in

�� ��is the sum of the transmission and receiver power consumed.It is clear that the transmission power is ^ P`_ VRba L �����'R � ��RbT�VO� .With regard to receiver power, if only nodes along the pathexpend energy on receiving messages, then the total receiverpower consumed is [!< , making the total energy used in thetransmission c ��I.�d)J[!<fe P`_ Vg RUa L ����� R � � RbT�V ��M (1)

However, in general, not just nodes on the path can hear thetransmissions sent by nodes on the path. Any node in therange of a transmission can hear the transmission if its radioreceiver is powered on. If h nodes within range are poweredon, then the receiver power consumed is <�h , which may besignificantly greater than <�7 . We refer to nodes that are noton the path but are in range of a transmission made by a

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node on the path as overhearing nodes. In practice, it is hardto power-off the radio receivers of all the overhearing nodesduring a transmission. However, techniques are available toalleviate the problem [23]. For example, if a node hears thatone neighbor is transmitting and another is receiving, the nodeshould power off even if it has packets to send. The durationof power off is determined by MAC control messages suchas RTS/CTS that the node has overheard. In light of this, forsimplicity, our analysis will focus on the power consumptionmodel where there is no overhearing, so that (1) describespower consumption. We remark that most other papers in theliterature have also ignored the overhearing problem (see, e.g.,[24], [22], [7], [14]).

A path Ii)H��� L �NMOMNM�� � P � is a minimum-energy path from�CL to ��P ifc ��I.� D c ��I.j�� for all paths I1j in

��� ��from �'L

to ��P . For simplicity, we assume that <lk*m . (Our results holdeven without this assumption, but it makes the proofs a littleeasier.) A subgraph

)\�n2f�o@�� of��� ��

has the minimum-energy property if, for all �����pAW2 , there is a path I in

that

is a minimum-energy path in��� ��

from � to � .

III. A CHARACTERIZATION OF MINIMUM-ENERGYCOMMUNICATION NETWORKS

Our goal is to find a minimal subgraph

of �� ��

thathas the minimum-energy property. Note that a graph

with

the minimum-energy property must be connected since, bydefinition, it contains a path between any pair of nodes. Givensuch a graph, the nodes can communicate using the links in

.For this to be useful in practice, it must be possible for each

of the nodes in the network to construct

(or, at least, therelevant portion of

from their point of view) in a distributed

way. In this section, we provide a condition that is necessaryand sufficient for a subgraph of

��� ��to be minimal with

respect to the minimum-energy property. In the next section,we use this characterization to provide an efficient algorithmfor constructing a graph

with the minimum-energy property

that, while not necessarily minimal, still has relatively fewedges.

Clearly if a subgraph ) �n2f�o@�� of

�� ��has the

minimum-energy property, an edge ����� �!�qAr@ is redundantif there is a path I from � to � in

such that Y IZY�kBs andc ��I1� D c ����� �!� , i.e., if a multihop path can provide a lower-

energy path. Let=tfuwv )x�n2f�o@ tfuyv � be the subgraph of

��� ��such that ����� �!�pA;@ tfuwv

iff there is no path I from � to �in

��� ��such that Y IZY�k,s and

c ��I1� D c ����� �!� . As the nextresult shows,

�tfuwvis the smallest subgraph of

��� ��with the

minimum-energy property.Theorem 3.1: A subgraph

of

�� ��has the minimum-

energy property iff it contains tfuyv

as a subgraph. Thus,�tfuyvis the smallest subgraph of

��� ��with the minimum-

energy property.Proof: We first show that

=tfuyvhas the minimum-energy

property. Suppose, by way of contradiction, that there arenodes ��� �WAi2 and a path I in

�� ��from � to � such thatc ��I1��z c ��I j � for any path I j from � to � in

tfuyv. Suppose

that I{)|��� L �NMOMNMO� � P � , where �})B� L and �~)�� P . Without

loss of generality, we can assume that I is the longest minimal-energy path from � to � . Note that I has no repeated nodesbecause any cycle can be removed to give a path that requiresstrictly less power. Since

tfuyvhas no redundant edges, for

all ��)�mZ�OMNMOM��[p�;s , it follows that ����R ���'RUT�V���A}@ tfuyv. For

otherwise, there is a path I�R in �� ��

from ��R to �'RUT�V such thatY I R Y�k�s andc ��I R � D c ��� R ��� RUT�V � . But then it is immediate

that there is a path I(� in��� ��

such thatc ��I(��� D c ��I1� andI(� is longer than I , contradicting the choice of I .

To see that�tfuwv

is a subgraph of every subgraph of��� ��

with the minimum-energy property, suppose that there is somesubgraph

of

��� ��with the minimum-energy property that

does not contain the edge ����� �!�{A�@ tfuwv. Thus, there is a

minimum-energy path I from � to � in

. It must be the casethat

c ��I1� D c ��������� . Since ����� �!� is not an edge in

, wemust have Y IZYZkEs . But then ����� �!��/AW@ tfuyv

, a contradiction.

This result shows that in order to find a subgraph of

with the minimum-energy property, it suffices to ensure thatit contains

�tfuwvas a subgraph.

IV. A POWER-EFFICIENT ALGORITHM FOR FINDING AMINIMUM-ENERGY COMMUNICATION NETWORK

Checking if an edge ����� �!� is in @ tfuwvmay require checking

nodes that are located far from � . This may require a great dealof communication, possibly to distant nodes, and thus requirea great deal of power. Since power-efficiency is an importantconsideration in practice, we consider here an algorithm forconstructing a communication network that contains

�tfuwvand

can be constructed in a power-efficient manner rather thantrying to construct

=tfuyvitself.

Say that an edge ����� �!�lA�@ �� ��is [ -redundant if there is

a path I in��� ��

such that Y IZYd)�[ andc ��I1� D c ��������� .

Figure 1 gives an example of a 3-redundant edge.

r =( u, 2u,1 )v,

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u

������������ ����

u

C(r) > C(u,v)

v

u u

Fig. 1. Edge �b�Z���N� is 3-redundant.

Notice that ����� �!��A�@ tfuyviff it is not [ -redundant for all[�kEs . Let @�� consist of all and only edges in @ �� ��

that arenot 2-redundant. In our algorithm, we construct a graph

)�>2?��@=� where @,�}@ � ; in fact, under appropriate assumptions,@�)E@ � . Clearly @ � �}@ tfuyv, so

has the minimum-energy

property.There is a trivial algorithm for constructing @ � . Each node� starts the process by broadcasting a “Hello” message at

maximum power � �� ��, stating its own position. If a node� receives this message, it responds to � with a Ack message

stating its location. Let ������� be the set of nodes that respondto � and let �=�(����� denote � ’s neighbors in @�� . Clearly���%�����i �������� . Moreover, it is easy to check that ���(�����

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consists of all those nodes �{A~������� other than � such thatthere is no ��A¡������� such that

c ��������� �!� D c ��������� . Since� has the location of all nodes in ������� , ���%����� is easy tocompute.

The problem with this algorithm is in the first step, whichinvolves a broadcast using maximum power. While this expen-diture of power may be necessary if there are relatively fewnodes, so that power close to � �� ��

will be required to transmitto some of � ’s neighbors in @�� , it is unnecessary in densernetworks. In this case, it may require much less than � �� ��to find � ’s neighbors in @�� . We now present an algorithmfor finding these neighbors in a power-efficient way. The ideais to start broadcasting with relatively low power, increasingthe power gradually until “enough” neighbors are found. Bydefining “enough” appropriately, we get an algorithm thatprovably maintains connectivity and is more power-efficientthan the MECN algorithm proposed by Rodoplu and Meng[22].

For this algorithm, we assume that if a node � transmits withpower � , it knows the region ¢������>�'� around � which can bereached with power � . If there are no obstacles and the antennais omni-directional, then this region is just a circle of radius4.£ such that 354 6£ )¤� . We are implicitly assuming that evenif there are obstacles or the antenna is not omni-directional, anode � knows the terrain and the antenna characteristics wellenough to compute ¢���������� .

Before presenting the algorithm, it is useful to define a fewterms.

Definition 4.1: Given a node � , let Loc ���!� denote the phys-ical location of � . The relay region of the transmit-relay nodepair ��������� is the physical region ¥ #�¦�§ such that relayingthrough � to any point in ¥ #.¦�§ takes less power than directtransmission. Figure 2 illustrates the concept of relay region.

vu v

u

R

Fig. 2. Relay region ¨�©Nª�« .

Formally,¥¬#.¦�§�)¤­���®�� ¯Z�d° c ����� �'�N��®���¯!��� D c �����N��®�� ¯Z� ��±%�where we abuse notation and take

c �����N��®���¯!��� to be the costof transmitting a message from � to a virtual node whoselocation is ��®���¯!� . That is, if there were a node �!j such thatLoc ����jU�²)³��®���¯Z� , then

c �����N��®�� ¯Z���²) c ����� ��j´� ; similarly,c ����� �'�N��®�� ¯Z� �²) c �������C���µj¶� . Note that, if a node � is inthe relay region ¥ #.¦�· , then the edge ����� �!� is 2-redundant.Moreover, since <lk²m , ¥ #.¦�# );¸ .

Given a region ¢ , let�=¹~)º­N�pA{2�° Loc ���!��AW¢�±%»if ¢ contains � , let¥l¹������f) ¼·�½(¾À¿ �>¢�������� �� �� �À�~¥ #.¦�· ��M (2)

� ¹ denotes all the nodes in region ¢ ; ¥ ¹ ����� denotes theregion such that transmitting directly to any point in the regiontakes less power than relaying through any node in ¢ . ¥ ¹ �����is the direct-transmission region of � induced by the nodes in¢ . Figure 3 shows � ¹ and ¥ ¹ ����� for a circular region ¢ .

w

FR (u) {u,v,w,t}NFt

vu

=

F

Fig. 3. Á? and ¨�Â��U�%� for a circular region à .

Note that if ¢Ä�x¥ ¹ ����� then all the nodes outside ¢ arenot in the direct-transmission region, so that transmitting toany node in 2r�{� ¹ takes more power than relaying througha particular node in � ¹ . The following proposition uses thisobservation to give a useful characterization of � � ����� .

Proposition 4.2: Suppose that ¢ is a region containing thenode � . If ¢��;¥l¹������ , then ��Å ¿�Æb#�Ç �¤���%����� . Moreover, if¢ is a circular region with center � and ¢È�B¥ ¹ ����� , then��Å ¿�Æb#�Ç )E� � ����� .Proof: Suppose that ¢É�Ê¥ ¹ ����� . We show that �=Å ¿ ÆU#`Ç�Ë� � ����� . Suppose that �ÌAG� � ����� . Then clearly Loc ���!�/AxÍ ·�½µÎ ¥ #.¦�· and Loc �����WAº¢�������� �� �� � . Thus, Loc ���!�WA¥l¹������ , so �iA9��Å ¿SÆb#�Ç .

Now suppose that ¢ is a circular region with center � and¢,�*¥ ¹ ����� . The argument in the preceding paragraph showsthat �=Å ¿�Æb#�Ç �;� � ����� . We now show that ��Å ¿�Æb#�Ç  ;� � ����� .Suppose that �EAÏ�=Å ¿ ÆU#`Ç . If ��/Ax� � ����� , then there existssome � such that

c ����� ������� D c ��������� . Since transmissioncosts increase with distance, it must be the case that 4C�������¬� D4'����� �!� . Since �¡A��=Å ¿ ÆU#`Ç  ¤� ¹ and ¢ is a circular regionwith center � , it follows that �HA;� ¹ . Since

c ����� ������� Dc ����� �!� , it follows that Loc ������A²¥ #.¦�· . Thus, �E/A²¥l¹������ ,contradicting our original assumption. Thus, �pAÐ�q�(����� .

The SMECN algorithm for node � constructs a set ¢ suchthat ¢Ñ�Q¥ ¹ ����� , and tries to do so in a power-efficientfashion. By Proposition 4.2, the fact that ¢,�*¥ ¹ ����� ensuresthat ��Å ¿ Æb#�Ç �}� � ����� . Thus, the nodes in �=Å ¿ Æb#�Ç other than� itself are taken to be � ’s neighbors. By Theorem 3.1, theresulting graph has the minimum-energy property.

Essentially, the algorithm for node � starts by broadcastinga “Hello” message with some initial power � L , getting Acks

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from all nodes in ¢������>� L � , and checking if ¢�������� L �Ì�¥ ¹ ÆU#($ £OÒoÇ ����� , that is, checking whether the initial region¢�������� L � probed for neighbors encloses the direct-transmissionregion of � induced by the nodes in ¢������>� L � . If not, it trans-mits with more power. It continues increasing the power � until¢��������'���\¥ ¹ ÆU#($ £OÇ ����� . It is easy to see that ¢������>� �� �� �F�¥ ¹ ÆU#($ £OÓ�Ô�Õ�Ç ����� , so that as long as the power increases to � �� ��eventually, then this process is guaranteed to terminate. In thispaper, we do not investigate how to choose the initial power�ÖL , nor do we investigate how to increase the power at eachstep. We simply assume some function Increase such thatIncrease P �w�ÖL��W)×� �� ��

for sufficiently large [ . An obviouschoice is to take Increase �w�'�F)Ø:�� . If the initial choice of�ÖL is less than the power actually needed, then it is easy tosee that this guarantees that � ’s estimate of the transmissionpower needed to reach a node � will be within a factor of 2of the minimum transmission power actually needed to reach� . In practice, a node may control a number of directionaltransmit antennae. Our algorithm implicitly assumes that theyall transmit at the same power. This was done for ease ofexposition. It would be easy to modify the algorithm toallow each antenna to transmit using different power. Allthat is required is that after sufficiently many iterations, allantennae transmit at maximum power. All our claims are validregardless of the type of antenna used.

Thus, the protocol run by node � is simply��)Ù�ÖL ;while ¢����������¬Ú�*¥ ¹ Æb#%$ £OÇ ����� do Increase �y��� ;�Û�����f)J� Å ¿µÜyÝ`Þ ßà

A more careful implementation of this algorithm is given inFigure 4. Note that we also compute the minimum power �������required to reach all the nodes in �Ù����� . In the algorithm,á

is the set of all the nodes that � has found so far inthe search and � consists of the new nodes found in thecurrent iteration. In the algorithm, â is the direct-transmissionregion of � induced by the currently known nodes

áin the

region ¢������>�'� . In the computation of â in the second-last lineof the algorithm, we take ãä§`½%å���¢������>� �� �� �?�Ù¥¬#.¦�§1� to be¢�������� �� �� � if �æ)J¸ . For future reference, we note that it iseasy to show that, after each iteration of the while loop, wehave that âp)Jã §`½(ç ��¢������>� �� �� �À�~¥ #�¦�§ � .

Define the graph )è�n2f�o@�� by taking ���������ÛAB@ iff�¤A��Ù����� , as constructed by the algorithm in Figure 4. It

is immediate from the earlier discussion that @Q�|@¬� and@Ä)B@�� if ¢�������� �� �� � is circular with center at � for eachnode � . Thus, the following theorem holds.

Theorem 4.3: )�>2?��@�� has the minimum-energy prop-

erty. Moreover, for each node � , if ¢�������� �� �� � is circular withcenter at � , then @ does not contain any 2-redundant edges.

We stress that we do not require the assumption thatbroadcasts at a given power setting are able to reach all nodesin a circular region to prove the correctness of our algorithm;we need this assumption only to show that the subnetworkconstructed by SMECN has no 2-redundant edges. While theassumption is certainly unreasonable in practice, we believethat this result does give some insight into the propertiesof our algorithm. And, to the extent that the assumption is

Algorithm SMECN��)~� L ;á );¸ ;é�ê1ëµé=ì�íî )E¸Z»âq)E¢�������� �� �� � ;while ¢������>�'�¬Ú�²â do��) Increase �w�'� ;

Broadcast “Hello” message with power � and gatherAcks;�ï)x­N��Y Loc ���!��AW¢������������ �{ÚA á � �{Ú)J��± ;á ) áið � ;

for each �ñAÐ� doforeach �xA á

doif Loc ���!��AÐ¥ #�¦�· thené=ê.ëµé=ì�íî ) é�ê.ë�é�ìOíî ð ­`�Ö± ;else if Loc ���¬��AW¥ #.¦�§ thené=ê.ëµé=ì�íî ) é�ê.ë�é�ìOíî ð ­`��± ;â�)râ�ãÐò §`½%å ��¢�������� �� �� �À�~¥ #.¦�§ );�Ù�����f) á � é=ê.ë�é�ìOíî

;�������f)róqôbõ�­��W°(¢����������d�*â'±Fig. 4. Algorithm SMECN running at node � .

approximately true, the result also gives some insight into whatour algorithm will do in practice.

We next show that SMECN dominates MECN. MECN isdescribed in Figure 5. For easier comparison, we have madesome inessential changes to MECN to make the notation andpresentation more like that of SMECN. The main differencebetween SMECN and MECN is the computation of the regionâ . As we observed, in SMECN, â�)öã §`½(ç �>¢�������� �� �� ���¥ #�¦�§ � at the end of every iteration of the loop. On the otherhand, in MECN, âÙ)Gã §`½(ç�_S÷�ø v ÷�ùûúûü �>¢�������� �� �� �d�*¥ #.¦�§ � ,where

é�ê1ëµé=ì�íîis the set of nodes not chosen by (S)MECN

to engage in direct transmission. Moreover, in SMECN, a nodeis never removed from

é=ê.ë�é�ìOíîonce it is in the set, while in

MECN, it is possible for a node to be removed fromé�ê1ëµé�ìOíî

by the procedure ý?þyÿ � . Roughly speaking, if a node �XA¥l#�¦�· , then, in the next iteration, if �BA9¥�#.¦�� for a newlydiscovered node 3 , but �Ï/A;¥�#.¦�� , node � will be removedfrom

é=ê.ëµé=ì�íîby ý?þyÿ �?���!� . In [22], it is shown that MECN is

correct (i.e., it computes a graph with the minimum-energyproperty) and terminates (and, in particular, the procedureý?þyÿ � terminates). Here we show that, if they search the sameregions, then SMECN is guaranteed to perform at least as wellas MECN.

Theorem 4.4: SMECN terminates earlier than MECN doesif they search the same region in each iteration before eitheralgorithm terminates. In addition, the communication graphconstructed by SMECN is a subgraph of the communicationgraph constructed by MECN as long as either of the followingholds: (1) The overal search region of SMECN is the same asMECN; (2) The overall search region of SMECN is circularwith the center at the given node.Proof: For each variable ® that appears in SMECN, let ® P�denote the value of ® after the [ th iteration of the loop;similarly, for each variable in MECN, let ® På denote the

6

Algorithm MECN��)~� L ;á )E¸ ;é�ê.ë�é�ìOíî )J¸!»âq)E¢�������� �� �� � ;while ¢������>�'��Ú�*â do�ñ) Increase �y��� ;

Broadcast “Hello” message with power � and gatherAcks;�ï)º­`��Y Loc ���!��AF¢������>�'��� �ÐÚA á ���ÐÚ)J��± ;á ) áið � ;é=ê.ë�é�ìOíî ) é�ê1ëµé�ìOíî ð � ;

for each �ñAW� doý?þyÿ �?����� ;â�)Eò §`½!Æbç�_S÷�ø v ÷�ùûúûü5Ç �>¢�������� �� �� ����¥ #.¦�§ � ;�Ù�����f) á � é�ê1ëµé=ì�íî;�������f)Jó�ôUõ�­o�F°%¢��������'�d�²â'±

Procedure ý?þyÿ �f���!�if �¡ÚA é=ê.ëµé=ì�íî

thené=ê.ë�é�ìOíî ) é�ê1ëµé=ì�íî ð ­N�C± ;for each �xA á

such that Loc ���¬��AF¥�#.¦�§ doý?þyÿ �?���¬� ;else if Loc ������/AWÍä·�½(ç�_�÷�ø v ÷�ùnúûü�¥l#.¦�· thené=ê.ë�é�ìOíî ) é�ê1ëµé=ì�íî �*­N�C± ;

for each �xA ásuch that Loc ���¬��AF¥ #.¦�§ doý?þyÿ �?���¬� ;

Fig. 5. Algorithm MECN running at node � .

value of ® after the [ th iteration of the loop. It is almostimmediate that SMECN maintains the following invariant:��A é�ê1ëµé�ìOíî P� iff ��A á P� and Loc ���!�~A�Í ·�½(ç��� ¥ #.¦�· .Similarly, it is not hard to show that MECN maintains thefollowing invariant: �ÛA é�ê1ëµé�ìOíî På iff �ÙA á P� and Loc ���!�A}Í ·�½(ç���l_S÷�ø v ÷�ùûúûü��� ¥¬#.¦�· . (Indeed, the whole point of theýfþwÿ � procedure is to maintain this invariant.) Since it is easyto check that

á P� ) á På , it is immediate thaté=ê.ë�é�ìOíî P�� é�ê1ëµé�ìOíî På . Suppose that SMECN terminates after [ �

iterations of the loop and MECN terminates after [�å iterationsof the loop. Hence â P�  Eâ På for all [ D ó�ôUõ��n[ � �[%åW� . Sinceboth algorithms use the condition ¢�������������â to determinetermination, it follows that SMECN terminates no later thanMECN; that is, [ � D [µå .

Clearly, the communication graph constructed by SMECN isa subgraph of the communication graph constructed by MECNif SMECN goes through the same iterations as MECN. Nowwe prove the last claim of the theorem. Suppose the searchregion used by SMECN is circular with the center at node � ,by Proposition 4.2,

á P �� � é=ê.ë�é�ìOíî P �� )�� � ����� . Moreover,even if we continue to iterate the loop of SMECN (ignoringthe termination condition), then ¢������>�'� keeps increasingwhile â keeps decreasing. Thus, by Proposition 4.2 again, wecontinue to have

á P� � é�ê1ëµé=ì�íî P� )º�=�(����� even if [Ð8;[ � .That means that if we were to continue with the loop afterSMECN terminates, none of the new nodes discovered wouldbe neighbors of � . Since the previous argument still applies

to show thaté�ê.ë�é�ìOíî P �� � é=ê.ë�é�ìOíî P �å , it follows that� � ������) á P �� � é=ê.ë�é�ìOíî P ��   á P �å � é=ê.ëµé=ì�íî P �å . That

is, the communication graph constructed by SMECN has asubset of the edges of the communication graph constructedby MECN.

In the proof of Theorem 4.4, we implicitly assumed thatboth SMECN and MECN use the same value of initial value� L of � and the same function Increase. In fact, this assumptionis not necessary if the overall search region of SMECN iscircular, since the neighbors of � in the graph computed bySMECN are given by �=�(����� independent of the choice of�ÖL and Increase, as long as ¢��������'L`�ÌÚ�É¢�������� �� �� � andIncrease P �y�ÖL.��8ï� �� ��

for [ sufficiently large. Similarly,the proof of Theorem 4.4 shows that the set of neighborsof � computed by MECN is a superset of � � ����� , as longas Increase and �'L satisfy these assumptions. In the proofof Theorem 4.4, it is shown that SMECN terminates earlierthan MECN. Because of this and the fact that MECN usesthe recursive ¢��¶� procedure, we expect that SMECN willtypically run faster than MECN.

Theorem 4.4 shows that the neighbor set computed byMECN is a superset of �=�(����� . As the following exampleshows, it may be a strict superset (so that the communicationgraph computed by SMECN is a strict subgraph of thatcomputed by MECN).

Example 4.5: Consider a network with 4 nodes 3��������C��� ,where Loc ���!�JA|¥l#�¦�· , Loc ���l�JAH¥l#.¦�� , and Loc ���!��/A¥l#�¦� . As shown in Figure 6, it is not hard to choose powerfunctions and locations for the nodes which have this property.It follows that � � �����?)º­N3± . (It is easy to check that Loc ��3 ��/A¥l#�¦�§�Í9¥¬#.¦�· .) On the other hand, the neighbor set of �computed by MECN includes � and 3 . Rodoplu and Meng[22] show that the final neighbor set of MECN is independentof the ordering of the nodes considered.

V. RECONFIGURATION

In a multi-hop wireless network, nodes can be mobile. Evenif nodes do not move, nodes may die if they run out of energy.In addition, new nodes may be added to the network. Weassume that each node uses a Neighbor Discovery Protocol(NDP), a periodic message that provides all its neighbors withits current position (according to the GPS) in order to detectchanges in the topology of the network. A node � sends outthe message with just enough power to reach all the nodesthat it currently considers to be its neighbors (i.e., the nodesin � � ����� ). Once a node detects a change, it may need toupdate its set of neighbors. This is done by a reconfigurationprotocol.

Rodoplu and Meng [22] do not provide an explicit re-configuration protocol. Rather, they deal with changes innetwork topology by running MECN periodically at everynode. While this will work, it is likely to be inefficient. Ifmobility is low, topology changes are rare and localized. Thus,rerunning MECN periodically is more wasteful than a low-frequency beacon mechanism. When mobility is high enoughsuch that topology changes occur frequently in almost everypart of the network, each node will probably have to do

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t

w

v

u

t

w

v

u

(a) MECN (b) SMECN

Fig. 6. A network where SMECN dominates MECN.

some reconfiguration within every beacon interval. In this case,using a beacon is likely to be more wasteful than periodicallyrerunning MECN (unless the NDP beacon is needed by otherprotocols of the network, in which case it can be used “forfree”). However, topology control is not really intended to dealwith frequently topology changes. Indeed, using maximumtransmission power might work better in this case.

We now present a reconfiguration protocol where, in aprecise sense, we run SMECN only when necessary (in thesense that it is run only when not running it may result in anetwork that does not satisfy the minimum-energy property).

There are three types of events that trigger the reconfigura-tion protocol: leave events, join events, and move events:� A leave #����!� event happens when a node � that was

in � ’s neighborhood is detected to no longer be inthe neighborhood (since its beaconing message is notreceived). This may happen because � is faulty or diesor because it has in fact moved away.� A join # ���!� event happens when a node � is detected tobe within � ’s neighborhood by the NDP.� A move #'���C� � � event happens when � detects that � hasmoved from the previous location to the current location� . ( � ’s location � is relative to � ’s location, so the eventcould be due to � ’s own movement.)

It is straightforward to see how to update the neighborset if � detects a single change. Suppose ��� is � ’s currentpower setting (that is, the final power setting used in the lastinvocation of SMECN by � ); let ¢=��)B¢����������O� be the lastregion searched by � . let

á � consist of all the nodes in ¢=�(that is, the set of all nodes discovered by the algorithm).� If a single leave #����!� or a move #C���C� � � is detected, letá j') á �!��­N�C± if leave #C����� is detected and let

á j') á � ifmove ���'��� � is detected. Let ¥�j¹ ) ò ·�½(ç�� ��¢������>� �� �� ���¥l#�¦�·À� , where the new location for � is used in the com-putation if �¡A á j . (Note that ¥�j¹ is defined essentiallyin the same way as ¥�¹������ in Equation (2).) If ¢����*¥�j¹ ,then take � ’s updated neighbor set to be � Å � ¿ ; otherwise,run SMECN taking � L )9��� .

� If a single join # ����� is detected, recompute the neigh-bor set as follows. Let

á j ) á � Í;­N�C± . Let ¥ j¹ )ò ·�½(ç � �>¢�������� �� �� �1�=¥ #.¦�· � . Take � ’s updated neighborset to be � Å � ¿ . Then let �'j~) óqôUõ�­�� °F¢������>�'�º�ò ·�½(ç � �>¢�������� �� �� �À�~¥ #.¦�· �± .

The following proposition is almost immediate from thedescription of our protocol.

Proposition 5.1: Suppose that a graph

has the minimum-energy property. If the nodes in

observe a sequence of single

changes and update their edge sets as above, the resultinggraph

�(�>2?��@=�`� still has the minimum-energy property forthe new topology. Moreover, if ¢���������� is a circular region forall � , then @=�¬)r@ � .

In general, there may be more than one change event that isdetected at a given time by a node � . (For example, if � moves,then there will in general be several leave and move eventsdetected by � .) If more than one change event is detected by� , we consider the events observed in some order. If we canperform all the updates without rerunning SMECN, we do so;otherwise, we rerun SMECN starting from �S� . By rerunningSMECN, we can deal with all the changes simultaneously.

Up to now we have assumed that no topology changes aredetected while SMECN itself is being run. If changes are infact detected while SMECN is run, then it is straightforward toincorporate the update into SMECN. For example, if � detectsa join # ���!� event, then � is added to the set

áin the algorithm,

while if � detects a leave #'���!� event, � is dropped fromá

andâ is recomputed.As we mentioned earlier, there is no reconfiguration pro-

tocol given in [22]. However, it is easy to modify the recon-figuration protocol given above for SMECN so that it worksfor MECN. If a leave #'���!� or move #C���C� � � is detected, then thesame approach works (except that MECN rather than SMECNis called with �CL¬)Ù��� ). Similarly, if a join # ���!� is detected, weupdate the neighbor set using the approach of MECN ratherthan SMECN.

Note that we have assumed a perfect MAC layer in ourreconfiguration discussion. Our reconfiguration works fineeven with a MAC layer that drops packets. The reason is

8

as follows. If the Ack message of some nodes get dropped,then the final power setting � � using an imperfect MAClayer will be bigger than the corresponding �S� using a perfectMAC layer. Since NDP beaconing with �S� reaches all nodesin �=�(����� , beaconing with a bigger power � � will still reachall nodes in �=�%����� . Eventually all the nodes in ���1����� whoseAcks are lost will be detected by � through NDP beacons.Thus, the neighbor set computed using an imperfect MAClayer converges to a superset of � � ����� . If the final searchregion is circular, then the neighbor set converges to the set� � ����� .

VI. SIMULATION RESULTS AND EVALUATION

How can using the subnetwork computed by (S)MECN helpperformance? Clearly, sending messages on minimum-energypaths is more efficient than sending messages on arbitrarypaths, but the algorithms are all local; that is, they do notactually find the minimum-energy path, they just construct asubnetwork in which it is guaranteed to exist.

There are actually two ways that the subnetwork constructedby (S)MECN helps. First, in the route discovery process ofmany routing protocols, e.g. [18], [10], flooding is used. Inthe flooding process, each node broadcasts the first copy of theroute discovery message it receives. Rather than broadcastingwith maximum power, it suffices for � to broadcast with power������� , the final power computed by (S)MECN. In addition,when sending periodic beaconing messages, it suffices for �to use power ������� . Second, the routing algorithm is restrictedto using the edges Íä#(½µÎ?�Ù����� . While this does not guaranteethat a minimum-energy path is used, it makes it more likelythat the path used is one that requires relatively little energyconsumption. For example, suppose there are 3 nodes � L , �SVand ��� . Route ��� L ����V.� ���N� takes less energy than ��� L ���'�`�does. However, if ��� hears from � L first, route reply will besent to �'L directly and the direct route will be used. If ����L(��� � �is not present, then � � will discover the two-hop route thattakes less energy. Therefore, with a smaller subgraph, AODVis more likely to find a route that takes less energy.

We would like to compare SMECN with other topology-control algorithms that preserve connectivity and are energyefficient. As observed in Section I, only MECN and CBTCalgorithm satisfy this requirement. In addition to comparingSMECN to MECN and CBTC, we also make the comparisonwith the no-topology-control case, where each node alwaysuses the maximum transmission radius when broadcasting apacket. (We refer to this approach as MaxPower. We choosemaximum power because it guarantees that there will be nonetwork partitions due to insufficient transmission power. Ofcourse, for unicast packets, only the minimum power to reacha given next hop is used.)

A. Simulation Environment

The topology-control algorithms—CBTC, MECN, SMECNand MaxPower are implemented in ns-2 [19], using the wire-less extension developed at Carnegie Mellon [6]. We generated20 random networks, each with 200 nodes. The nodes wereplaced uniformly at random in a rectangular region of 1500 by

1500 meters. (There has been a great deal of work on realisticplacement, e.g. [2]. However, this work has the Internet inmind. Since the nodes in a multihop network are often bestviewed as being deployed in a somewhat random fashion andmove randomly, we believe that the uniform random placementassumption is reasonable in many large multihop wirelessnetworks.) Each node has a maximum transmission range of500 meters. The corresponding � �� ��

is only large enoughto reach 1/4 of the network diameter, not the whole network.(Due to hardware constraints, � �� ��

can not be arbitrarily large.Even if � �� ��

is large enough to reach a distant nodes, it wouldbe silly to use so much transmission power that one messagewould exhaust the whole battery power.)

We assume a s./.4�� transmit power roll-off for radio prop-agation. The carrier frequency is 914 MHz and transmissionraw bandwidth is 2 MHz. We further assume that each nodehas an omni-directional antenna with 0 dB gain and is placedat 1.5 meter above the node. The receive threshold is -94 dBW, the carrier sense threshold is -108 dBW, and thecapture threshold is 10 dB. These parameters simulate the 914MHz Lucent WaveLAN DSSS radio interface. Given theseparameters, the 3 parameter in Section II is -101 dBW. InWaveLAN radio, it has been measured that radio receiverpower can be quite significant [25] and accounts for �(+�� ofthe fixed transmission power. However, techniques on reducingthe power consumption of radio electronics are fast improving.A radio typically consists of transmitter electronics, receiverelectronics and transmit amplifier. Low-power circuit designsand signal processing reduce the power expended in thetransmitter and receiver electronics. As a result, the receiverpower of future radios is likely to be quite small. However,the power needed by the transmit amplifier is constrained bythe rapid radio attenuation in space. Therefore, it is expectedthat transmission power will dominate receiver power in thefuture. Because radio receiver power varies from radio to radioand has an impact on the computation of the minimal-energypath, we would like to vary the receiver power < to study itseffect on MECN and SMECN.

Each node in our simulation has an initial energy of 1Joule. We would like to see how our algorithm affects networkperformance. To do this, we need to simulate the network’sapplication traffic. We used the following application scenario.All nodes periodically send UDP traffic to a sink node situatedat the boundary of the network. The sink node is viewedas the master data collection site. The application traffic isassumed to be CBR (constant bit rate); application packetsare all 256 bytes. The sending rate is one packet per second;each node has a buffer size of 64. This application scenario hasalso been used in [7]. Although this application scenario doesnot seem appropriate for telephone networks and the Internet(cf. [16], [17]), it does seem reasonable for ad hoc networks,for example, in environment-monitoring sensor applications.In this setting, sensors periodically transmit data to a datacollection site, where the data is analyzed. There are manytechniques to improve the performance of this type of data-gathering application. For example, it is typically beneficialto spread the routing load over several nodes close to sink,so as to avoid overloading any particular neighbor. However,

9

our goal is to use this type of application to evaluate therelative merit of the topology-control algorithms rather than tomaximize the performance of this type of application. Thus,we assume that the data-gathering algorithm and/or routingprotocols are given, and focus on topology control.

To find routes along which to send messages, we useAODV [18]. However, as mentioned above, we restrict AODVto finding routes that use only edges in Íä#%½µÎ?�Ù����� . Thereare other routing protocols, such as LAR [12], GSPR [11],and DREAM [1], that take advantage of GPS hardware; therealso exist routing protocols that are power aware [4], [14]. Weused AODV because it is readily available in our simulatorand it is well studied. Since we would like to optimize withrespect to the minimum-energy path metric, we modify thens-2 AODV implementation to use the minimum-energy pathmetric instead of using the current shortest-path metric. Al-though other routing protocols (or AODV modified with otherpower-aware routing metrics [24]) may give better networkperformance than our modified AODV, our focus in this paperis the performance comparison of different topology-controlalgorithms.

In order to simulate the effect of power control, we madechanges to the physical layer of the ns-2 simulation code.Specifically, for every neighbor broadcast packet, when usingalgorithm (S)MECN, node � transmits using the final trans-mission power ������� used in the neighbor-search process ofthe algorithm. (For example, with SMECN, node � transmitswith the minimum power � such that ¢��������'�¤�ïâ .) Forevery unicast packet, node � transmits with the minimumpower needed to reach the destination, as determined duringthe neighbor-search process. A node’s energy reserve is thendecreased by the appropriate amount for every transmissionand reception.

We assumed that each node in our simulation had an initialenergy of 1 Joule and then ran the simulation for 1600simulation seconds. Each data point represents an average of20 randomly generated networks. For the sake of fairness,identical traffic scenarios are used for MECN, SMECN, CBTCand MaxPower. For simplicity, we simulated only a staticnetwork (that is, we assumed that nodes did not move),although some of the effects of mobility—that is, the triggeringof the reconfiguration protocol—can already be observed withnode deaths. Also for simplicity, we did not actually simulatethe execution of MECN, SMECN, and CBTC. Rather, weassumed the neighbor set �Û����� and power ������� computed byMECN, SMECN, and CBTC each time it is run were given byan oracle. (Of course, it is easy to compute the neighbor setand power in the simulation, since we have a global pictureof the network.) Thus, in our simulation, we did not take intoaccount one of the benefits of SMECN over MECN, that itstops earlier in the neighbor-search process.

Since a node’s available energy is decreased after eachpacket reception or transmission, nodes in the simulation dieover time. After a node dies, the network must be reconfigured.In [22], this is done by running MECN periodically. In oursimulation, the NDP triggers the reconfiguration protocol.(When running MECN, we use the same reconfigurationprotocol as the one we use for SMECN, with the appropriate

modifications, as discussed in Section V.) The NDP beacon forMECN, SMECN, and CBTC is sent with a period of 1 secondand uses the power ������� of MECN, SMECN and CBTCrespectively. Since reconfiguration is local and happens onlywhen nodes run out of energy, the performance of MaxPoweris not penalized much by not simulating the power-adjustmentprocess in MECN, SMECN and CBTC.

In this setting, we are interested in network lifetime, asmeasured by two metrics: (1) the number of nodes that arestill alive over time and (2) the number of nodes that are stillconnected to the sink. In the absence of detailed applicationrequirements, the definition above is reasonable in capturingthe essential aspects of network lifetime. Of course, if we havemore knowledge of the application, the definition of networklifetime can be made even more application-specific. Forexample, in a sensing application of sensor networks, networklifetime can be defined as the time that the overlapping sensingrange of sensors can cover a deployment region.

B. Performance Evaluation

In this Section, we compare the performance of SMECNwith MECN, CBTC and MaxPower. We compare only to theoptimized CBTC algorithm with parameter "�)Ê:.-�/�� . Wefirst report the experimental results when the receiver poweris m . We then vary the receiver power and discuss how theperformance results change. Finally we discuss the effect ofMAC layer.

1) Performance Results When the Receiver Power is Zero:Before describing the performance, we consider some featuresof the subnetworks computed by MECN, SMECN, and OPT-CBTC. Since the search regions will be circular with an omni-directional antenna, Theorem 4.4 assures us that the networkused by SMECN will be a subnetwork of that used by MECN,although it does not say how much smaller the subnetwork willbe. The initial network in a typical execution of the MECN,SMECN, and OPT-CBTC is shown in Figure 7. The numberof neighbors of a node, average over the 20 networks, is 3.21when running MECN, and 2.71 for SMECN. Thus, each noderunning MECN has roughly 19% more links than the samenode running SMECN. This makes it likely that the final powersetting computed will be higher for MECN than for SMECN.In fact, our experiments show that it is roughly 38% higher,so more power will be used by nodes running MECN whensending messages. Moreover, AODV is unlikely to find routesthat are as energy efficient with MECN. The average numberof neighbors of OPT-CBTC is initially �ZM � , 40% more thanSMECN. As a reference, each node in MaxPower initially hasan average of 51 neighbors.

As nodes die (a result of running out of power), the networktopology changes due to reconfiguration. Nevertheless, asshown in Figure 8, for MECN, SMECN, and OPT-CBTC, theaverage number of neighbors of a node stays roughly the sameover time, thanks to the reconfiguration protocol.

Turning to the network-lifetime metrics discussed above,as shown in Figure 9, SMECN performs consistently betterthan MECN for both. The number of nodes still alive and thenumber of nodes still connected to the sink decrease much

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Fig. 7. Initial network computed by MECN, SMECN, OPT-CBTC and MaxPower with ����� mW. Note that the big circle in the middle of the left boundarydenotes the sink node.

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Fig. 9. Network lifetime with respect to two different metrics with ����� mW.

more slowly in SMECN than in MECN. For example, inFigure 9(b), at time :(m(m(m , �.0�� of the nodes are disconnectedfrom the sink for MECN while only �µ:�� of the nodes aredisconnected from the sink for SMECN. SMECN performsslight better than OPT-CBTC. MaxPower performs muchworse than MECN, SMECN, and OPT-CBTC.

Finally, we collected data on average energy consumptionper node at the end of the simulation, on the total numberof packets delivered, and on end-to-end delay. MECN uses

36% more energy per node than SMECN. SMECN delivers23% more packets than MECN by the end of the simulation.SMECN’s delivered packets have an average end-to-end delaythat is only 77% of the delay of MECN. Overall, it is clearthat the performance of SMECN is significantly better thanMECN if the receiver power is negligible.

SMECN delivers almost as many packets as OPT-CBTC(only 3% less) while its end-to-end delay is only 63% of thatof OPT-CBTC. OPT-CBTC and SMECN are able to deliver

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:�� times more packets than MaxPower does throughout thesimulation. The packet-delivery and network-lifetime statisticsfor MECN, SMECN, OPT-CBTC, and MaxPower show thatit is undesirable to transmit to a large radius; it increasesenergy consumption and causes unnecessary interference, andconsequently decreases throughput.

2) Varying the Receiver Power: We now vary the receiverpower < to study its impact on MECN and SMECN. As wementioned earlier in this section, the receiver power of a radiois expected to be small in the future. Hence, we set < to asmall value ( :(m mW). A typical network topology maintainedby MECN and SMECN is shown in Figure 10. As we cansee, these topologies (generated with <r)Q:(m mW) tend tomaintain more direct links than those shown in Figure 7. Theaverage number of neighbors and broadcast power ������� usingMECN and SMECN are quite similar to each other (MECNonly maintains 3% more links than SMECN). As a result, itis not surprising that the performance of the two algorithmsis quite similar in this case. This is further substantiated byexperimental results using the average number of neighborsmetric (shown in Figure 11) and the two metrics of networklifetime (shown in Figure 12). Note that, although OPT-CBTCperforms better than MECN and SMECN in terms of thenumber of nodes that remain alive, it performs similar interms of the number of nodes that remain connected to thesink and the amount of packets delivered. In fact, SMECNdelivers 7% more packets than OPT-CBTC. The number ofnodes that remains alive decreases much faster in the case of<�)�:1m mW than in the case of <�)xm mW. This is specific ofour application scenario where nodes close to the sink receivemuch more packets than nodes are further away.

Based on the results of Section VI-B.1 and VI-B.2, we seethat the performance improvement of SMECN decreases overMECN as the receiver power increases. This is because thedifference of ������� for the two subnetworks decreases. Notethat this observation is based on static networks with randomnode placement. It may not be true in other network setting,e.g. networks with significant mobility. More experimentationneeds to be done to see the extent to which SMECN dominatesMECN in other network settings. (By Theorem 4.4, SMECNwill do at least as well as MECN regardless of the setting.)

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3) The Effect of MAC layer: We have used CSMA MACprotocols that are subject to the hidden-terminal problem[27]. The standard IEEE 802.11 does not work correctly withtopology-control algorithms. The reason is that the mechanismin IEEE 802.11 that deals with the hidden-terminal problem—RTS/CTS/ACK—assumes that every node has the same trans-mission range. This is not the case for networks using topologycontrol as each node determines its own transmission range.Of course, the hidden-terminal problem can be solved bysending RTS/CTS with maximum power instead of withbroadcast power ������� . Since the number of hops between agiven source and destination in the network using topologycontrol is likely to be longer than in the network withoutusing topology control, the channel efficiency of the networkusing topology control is likely to be lower. Similarly, thepower consumed by the nodes along a path between a givensource and destination is likely to be higher in the networkusing topology control. This clearly defeats the purpose oftopology control, which is to improve channel efficiency andenergy efficiency. Therefore, a simple modification of IEEE802.11 does not work well with topology control. Of thetwo recently-proposed power-controlled MAC protocols [28],[15], the PCMA protocol [15] is particularly promising. Ifit is combined with topology-control algorithms, both energyefficiency and channel efficiency can be achieved. It will beinteresting to investigate how much power-controlled MACprotocols such as PCMA improve network performance overthe simple CSMA protocol used in this paper. We leave thisto future research.

VII. SUMMARY

We have proposed a protocol SMECN that computes anetwork with the minimum-energy property. In the case ofa circular search space, SMECN computes the set @ � con-sisting of all edges that are not 2-redundant. In addition,we have proposed an energy-efficient reconfiguration protocolthat maintains the above minimum-energy path property asthe network topology changes dynamically. Finally, we haveshown the performance improvements of SMECN over MECNand OPT-CBTC through simulation.

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Fig. 10. Initial network computed by MECN and SMECN with ������� mW.

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Acknowledgments

We thank Volkan Rodoplu at Stanford University for hiskind explanation of his MECN algorithm, and for making hiscode publicly available. We thank Victor Bahl, Yi-Min Wang,and Roger Wattenhofer at Microsoft Research for helpfuldiscussions.

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Li (Erran) Li received a B.E. in Automatic Controlfrom Beijing Polytechnic University in 1993, a M.E.in Pattern Recognition from the Institute of Automa-tion, Chinese Academy of Sciences, in 1996, and aPh.D. in Computer Science from Cornell Universityin 2001 where Joseph Y. Halpern was his advisor.During his graduate study at Cornell University, heworked at Microsoft Research and Bell Labs Lucentas an intern, and at AT&T Research Center at ICSIBekerley as a visiting student. He is presently amember of the Networking Research Center in Bell

Labs. His research interests are in networking with a focus on wirelessnetworking and mobile computing.

Joseph Y. Halpern received a B.Sc. in mathematicsfrom the University of Toronto in 1975 and a Ph.D.in mathematics from Harvard in 1981. In between,he spent two years as the head of the MathematicsDepartment at Bawku Secondary School, in Ghana.After a year as a visiting scientist at MIT, he joinedthe IBM Almaden Research Center in 1982, wherehe remained until 1996. He served as manager of theMathematics and Related Computer Science Depart-ment at IBM from 1988-1990 and was a consultingprofessor at Stanford from 1984-1996. In 1996, he

moved to Cornell University, where he is a professor in Computer Science.His major research interests are in reasoning about knowledge and uncer-

tainty, qualitative reasoning, belief revision, (fault-tolerant) distributed com-putation, game theory, decision theory, and security. Together with his formerstudent, Yoram Moses, he pioneered the approach of applying reasoning aboutknowledge to analyzing distributed protocols and multi-agent systems. He hascoauthored 5 patents, a book (”Reasoning About Knowledge”), and well over100 technical publications.

Halpern was program chairman and organizer of the first conference onTheoretical Aspects of Reasoning about Knowledge, program chairman ofthe fifth ACM Symposium on Principles of Distributed Computing, the 23rdACM Symposium on Theory of Computing, and the 16th IEEE Symposiumon Logic in Computer Science. He received the Publishers’ Prize for BestPaper at at the International Joint Conference on Artificial Intelligence in1985 (joint with Ronald Fagin) and in 1989, the 1997 Godel Prize (joint withYoram Moses), and two IBM Outstanding Innovation Awards. He is a Fellowof the American Association of Artificial Intelligence and the Association forComputing Machinery, and in 2001-02 was the recipient of a Guggenheim anda Fulbright Fellowship. He is editor-in-chief of Journal of the ACM, and alsoserves on the editorial board of Journal of Logic and Computation, ChicagoJournal of Theoretical Computer Science, and Artificial Intelligence.