energy-efficient topology control

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with cooperative communication if vi transits the signalwith a set of helper nodes Hij and the summation of theirtransmission power satisfiesX

vk2vi[Hij

Pk dkj ! Pk PMAX: 2

Fig. 2a shows such an example. Node v1 wants to transmit

data to node v2. Using cooperative communication, it canfirst transmit to its nearby neighbors v3 and v4, then thethree nodes together transmit to v2. If the combined SNR islarger than , v2 can successfully decode the data. Noticethat, when node v1 transmits data to its helpers v3 and v4 inthe first phase, v2 also receives a partial data from v1. Thus,in some CC models, such partial data are used by v2 tojointly decode the original data during the second phase ofCC transmission. In this paper, we ignore such partial dataduring the first phase for simplicity (as [2] did). However,the proposed algorithms can be easily adopted to the modelwhere the partial data are considered during decoding.Physical layer techniques for implementation of CC could

be found in [23]. Carefully selecting the helper set and usingcooperative communication can reduce the transmissionpower and extend the transmission coverage. For example,if v1 cannot directly reach v2 (i.e., PMAX d12

< ), its

transmission coverage can be extended with v3 and v4shelp. On the other hand, even v1 can directly reach v2, if thetransmission power it needs is larger than the sum oftransmission power consumes when using CC, i.e.,P1 P3 P4 < d12

, CC transmission can still saveenergy from the direct transmission.

3.2 Network Model

We consider a wireless ad hoc network with n nodes whichare capable of receiving and combining partial receivedpackets in accordance with the CC model. The networktopology is modeled as a 2D directed graph: G V ; E,where V v1; . . . ; vn denotes the set of wireless nodes andEdenotes a set of directed communication links. A directedlink vivj 2 E denotes that node vi can transmit data to nodevj either directly or using CC. Nvi is the set of directneighbor nodes of vi within its maximum transmissionrange Rmax, i.e., for all vk 2 Nvi, there exists Pi PMAXsuch that Pi dik

! . In other words, node vi is able tocommunicate with its neighbor vk directly. We assume thateach node has a unique ID and knows its own locationinformation. Node ID and location information are ex-

changed among all nodes. Then, we can define severalimportant concepts.

Definition 1 (Direct Link). A direct link vivj is a link in Erepresenting that node vi can transit the information to node vj

directly (i.e., without CC). We use a solid black line to denote adirect link.

Definition 2 (Helper Node Set). Hij symbolizes the helpernode set including all helper nodes of node vi for its CCtransmission to vj. In this paper, we assume all helper nodes ofvimust be a direct neighbor ofvi, i.e., Hij & Nvi. In other words,all elements in Nvi are the candidates of helper nodes of vi.

Definition 3 (Helper Link). A helper link is a direct link vivkbetween a source node vi and its helper node vk. For example,the link v1v3 is a helper link in Fig. 2a.

Definition 4 (Cooperative Communication Link (CC-link)). A CC-link gvivj is a link in E representing node vican transit the information to node vj cooperatively with a setof helper nodes Hij. We use a solid blue line to denote a CC-link. As shown in Fig. 2b, link gv1v2 is a CC-link.The unions of all direct links and CC-links are E and

eE, respectively. Similarly, we define the direct commu-nication graph and CC communication graph as G

V ; E and eG V ; eE, respectively. Notice that E E eE and G G eG.

In traditional topology control problem, the directcommunication graph G is assumed to be strongly connected(i.e., any two nodes vi; vj 2 V are mutually reachable bydirect links). Similarly, we can assume the stronglyconnectivity of the network G under CC model for ournew cooperative topology control problem.

Definition 5 (Strongly Connected under CC Model). Thecooperative network G is strongly connected under CCmodel if and only if for any two nodes vi and vj there exists adirected path to connect them in G. In other words, by using

the combination of direct links and CC-links the packet fromany source node can reach any other node in the network.

3.3 Cooperative Energy Spanners

In [1], given a strongly connected direct communicationgraph G, the authors studied how to use a few CC-linksreplacing some direct links such that the resulting graph G0

is strongly connected under CC model and the total energyconsumption is minimized. In [2], given a stronglyconnected communication graph G under CC model (thecorresponding G may be disconnected), the authors studiedhow to build a sparse subgraph G0 such that G0 is stronglyconnected under CC model and the total energy consump-

tion is minimized. Both existing works on topology controlunder CC model [1], [2] do not consider the energyefficiency of paths inside the generated topologies. Noticethat the method in [1] can take the output of a traditionalenergy-efficient topology control (without using CC) as itsinput and construct an energy spanner of the directcommunication graph G, however, it may not be an energyspanner of the overall communication graph G. In theexample of Fig. 1, if the link v1vn is a CC-link, the method in[1] will not add it in the final topology, thus, it will cause anarbitrarily large energy stretch factor between v1 and vn. Toaddress the energy efficient problem, we formally definethe energy consumption of each link and each path in

cooperative ad hoc networks.Definition 6 (Link Weight). For each link vivj 2 E, we define

the link weight wvivj represents the minimum total energyconsumption(i.e., total transmission power) of all nodes

ZHU ET AL.: ENERGY-EFFICIENT TOPOLOGY CONTROL IN COOPERATIVE AD HOC NETWORKS 1483

Fig. 2. Cooperative Communication: (a) v1 together with its helpers v3and v4 transmit to v2 using cooperative communication; (b) a CC linkgv1v2 represents such cooperative communication.


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participated in maintaining a directed link from vi to vj. For adirect link vivj, wvivj dij. For a CC-link gvivj, to select aset of helper nodes Hij for vi from its one-hop neighbors Nvi,such that the total energy consumption of the constructed CC-link is minimized, is a challenging problem itself. Therefore, weuse an estimated total energy cost as follows, for gvivj with aselected helper node set Hij

w

gvivj

maxvk2Hij dik

jHijj 1

Pvk2vi[Hij

dkj : 3

Here, jHijj denotes the number of elements in set Hij. Thelink weight of a CC-link includes two parts: the first one isthe energy consumption for node vi to communicate withits helper node set Hij directly, and the second part is thetotal energy consumption for vi and its helpers in Hij tocooperatively communicate with vj. Notice that here wesimplify the CC-model by assuming the transmissionpowers of node vi and its helper node set Hij during CCtransmission are the same. In addition, we only consider thetransmission power at each sender here. However, it is easyto extend our model to take the receivers power intoconsideration. If the receiving power at each receiver is

denoted by Pr, we can simply add Pr and jHijj 1Pr towvivj and wgvivj, respectively. This will not affect ourproposed algorithms and analysis.

Consider a unicast path vi; vj in a cooperative networkG from node vi to node vj under CC model, the totaltransmission energy consumed by this path vi; vj ispvi; vj

Pe2vi;vj

we. Here, e can be either a directlink or a CC-link. Let PGvi; vj be the path consuming theleast energy among all paths connecting vi and vj in G. Wecall PGvi; vj the least-energy path in G for vi and vj. LetpPGvi; vj be the total energy of the least-energy path.Then, we can define the stretch factor as follows:

Definition 7 (Energy Stretch Factor). Let G0 be a subgraph1 of

G. The energy stretch factor of a node pair vi; vj in G0 underCC model with respect to G is defined as G

0

G vi; vj pPG0 vi;vjpPGvi;vj

. See Fig. 3 for illustration. The energy stretchfactor ofG0 under CC model with respect to G is the defined as

GG0 max

vi;vj2VG

0

G vi; vj maxvi;vj2V

pPG0 vi; vj

pPGvi; vj: 4

Definition 8 (Cooperative Energy t-Spanner). A subgraphG0 of the cooperative network G is a cooperative energy t-spanner ofG if its stretch factor under CC model with respectto G is no larger than a constant t, i.e., GG

0 t.

If a topology is a cooperative energy spanner of theoriginal communication graph G, then we guarantee there isa path between each pair of nodes whose energy consump-tion is similar to the original optimal one when all possibledirect and CC links are used. This will benefit energyefficient routing performance on the network topology.

3.4 Problem Formulation

Now we can define the new topology control problem wewill study in this paper.

Energy-Efficient Topology Control Problem with CC.Given a wireless multihop network G V ; E which isstrongly connected under CC model, assign transmissionpower Pi to every node vi such that 1) the induced topologyG0 from this power assignment is a cooperative energy t-spanner of G, i.e., GG

0 t; and 2) the sum of transmis-sion power of all nodes,

Pvi2V

Pi, is minimized.Notice that the spanner property also guarantees that the

induced topology G0 is strongly connected under CC model.The above topology control problem is equivalent to the

following one: given the strongly connected G, construct asubstructure G0 spanning all nodes such that 1) G0 is acooperative energy t-spanner of G; and 2) the inducedpower assignment from G0 minimizes the total transmissionenergy

Pvi2V

Pi. Here, the induced transmission power ofnode vi from G

0 is the maximum power to support all of itsoutgoing links in G0.

Hardness of ETCC. In [1], [2], the topology controlproblem under cooperative communication model (TCC),which aims to minimize the total energy while maintainingthe connectivity, has been proved to be NP-complete. Nowwe also prove that ETCC is also NP-complete. Given thepower assignment for each node in the network, it is easily

to verify in polynomial time whether the energy stretchfactor of the induced topology G0 is less than or equal to tand whether the total cost of this assignment is less than afixed value. Thus, ETCC belongs to the NP-class. Inaddition, TCC problem is a special case of ETCC, wherethe stretch factor constrain t is set to an arbitrarily largeconstant. In such case, the spanner property reduces to thebasic connectivity requirement. As TCC is NP-complete,ETCC is also NP-complete. Therefore, in this paper, we willfocus on heuristic algorithms to guarantee the spannerproperty while minimizing the transmission energy.

Table 1 lists all the symbols used in the paper.

4 ENERGY-EFFICIENT TOPOLOGY CONTROL WITHCOOPERATIVE COMMUNICATION

In this section, we propose two topology control algorithmswhich build energy-efficient cooperative energy spanners.To keep the proposed algorithms simple and efficient, weonly consider its one-hop neighbors as possible helper nodesfor each node when CC is used. Thus, theoriginalcooperativecommunication graph G contains all direct links and CC-links with one hop helpers, instead of all possible direct linksand CC-links. In addition, for each pair of nodes vi and vj, weonly maintain one link with least weight if there are multiplelinks connecting them. Here, all links are directional links.

Both proposed algorithms are greedy algorithms. The majordifference between them is the processing order of links. Thefirst algorithm deletes links from the original graph Ggreedily, while the second algorithm adds links into G00

1484 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 23, NO. 8, AUGUST 2012

Fig. 3. Energy stretch factor: the energy stretch factor of a node pair

vi; vj in G0 with respect to G, G0

G vi; vj pPG0 vi ;vj pPG vi ;vj

, represents the

difference between the costs of least-energy paths in G0 and G.

1. Here, a subgraph G0 has the same vertex set V with the graph G butwith less links than G.


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greedily. Here, G00 is a basic connected subgraph of G. Bothalgorithms can guarantee the cooperative energy spannerproperty of the constructed graph G0.

4.1 Greedy Algorithm 1Deleting Links

We first propose a simple greedy algorithm for energy-efficient topology control, which is inspired by the classical

greedy algorithm [41], [42] for low-weight spanner. The

general idea is to start with the original cooperativecommunication graph G as described above. Then, gradu-

ally delete the least energy-efficient link (the link with

highest weight) from G if doing so does not break the

energy stretch factor requirement. Finally, the transmission

power of each node is decided from the constructed

topology G0. The details of these three steps are as follows:Step 1: Construction of G. Initially, G is an empty graph.

First, add every direct links vivj into G, if node vi can reach

node vj when it operates with PMAX. Then, for every pair ofnodes vi and vj, we select a set of helper nodes Hij for nodevi from its one-hop neighbors Nvi, such that the linkweigh wgvivj of the constructed CC-link is minimized.Notice that this helper node decision problem is challengingeven under our assumption that the transmission powers ofvi and its helper node set to maintain CC-link are the same.If we try all combinations of the helper sets to find theoptimal helper set which minimizes the total energyconsumption of vi and its helpers, the computationalcomplexity is exponential to the size of the one-hopneighborhood Nvi. It is impractical to do so in case of a

large number of neighbors. Therefore, we directly use thegreedy heuristic algorithm1 in [2], Greedy Helper SetSelectionvi; Nvi; vj, to select the helper set Hij. Fordetails, please refer to [2]. Then, we compare wgvivj with

pPGvi; vj which is the current shortest path from node vito node vj in G. If wgvivj < pPGvi; vj and

Pvk2vi[Hij

dkj PMAX;

add this CC-link

gvivj into G. If there already exists a direct

link vivj, delete it after the new CC-link

gvivj is added (since

it costs more energy than the CC-link). Notice that ifP

vk2vi[Hijdkj

! PMAX;

node vi cannot communicate with node vj within one-hopeven in CC model.

Step 2: Construction of G0. Copy all links in G to G0, andsort them in the descending order of their weights. Start toprocess all links one by one and delete the link vivj from G

0

if G0 vivj is still a cooperative energy t-spanner of G.Hereafter, we use G e or G e to denote the graphgenerated by removing link e from G or adding link e intoG, respectively. In addition, when a CC-link gvivj is kept inG0, all its helper links must be kept in G0 too.

Step 3: Power Assignment from G0. For each node vi, itstransmission power is decided by the following equation:

Pi max maxvivj2G0

Pdi j; maxfvivj 2G0 Pcci j( )

: 5

Here, Pdi j

dij

and

Pcci j P

vk 2vi[Hijdkj

are the energy consumption at vi for a direct link vivj and a

CC-link gvivj, respectively.Algorithm 1 gives the detailed topology control algo-

rithm. The guarantee of the energy t-spanner property isstraightforward, since links are deleted from G0 only whenthe deletion does not break the t-spanner requirement.The time complexity of Algorithm 1 is On2log mnn log n m Omnn log n m, where n, m, and are the number of nodes, the number of links (both directlinks and CC-links), and the maximum node degree of theoriginal cooperative communication graph G, respectively.Detailed time complexity of each part of this algorithm is asfollow: Lines 2-6 cost On2; Lines 7-15 cost On2log;

Line 17 needs perform ordering with cost of Om log m;and Lines 18-23 cost Omnn log n m for testing thestretch factor requirement.

Algorithm 1. Greedy Algorithm 1 - Deleting LinksInput: A set ofn wireless nodes Vand nodes locations; themaximum transmission power PMAX; and the energystretch factor requirement t.Output: A topology G0 and its induced power assignmentPi for each node vi.

1: G V ; ;2: for all vi and vj 2 V do

3: if PMAXdij

! then4: Add direct link vivj into G;5: end if6: end for


TABLE 1Summary of Notations


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7: for all vi and vj 2 V do8: Hij Greedy Helper Set Selection vi; Nvi; vj;9: if wgvivj < pPGvi; vj and P

vk2vi [Hijdkj

PMAXthen

10: Add CC-link gvivj into G;11: if direct link vivj 2 G then12: Delete vivj from G;13: end if14: end if15: end for16: Copy G to G0;

17: Sort all linksv

iv

j 2G0

in the descending order of theirweights and mark them deletable;18: for all link vivj 2 G0 processed in the sorted order do19: if GG

0 vivj t and vivj is deletable then20: Delete link vivj from G

0;21: else if link vivj is a CC-link then22: Make all helper links vivk for vk 2 Hij undeletable;23: end if24: end for25: for all vi 2 V do26: Pi maxfmaxvivj2G0 P

di j; max

fvivj2G0

Pcci jg;27: end for

Figs.4a,4b,4c,and4dshowtheprocessofAlgorithm1onasimpleCC network. Fig. 4a gives allpossiblelinks (bothdirectlinks and CC links) in the network G. Algorithm 1 tries toremove links with high costs without hurting the stretchfactor beyond the requirement. In this example, assume thatthe stretch factor requirement t 3. Removing twolinkswithcost 5 and 7 (shown in Figs. 4b and 4c) does not affect thestretch factor. Removing link cd(shown in Fig. 4d) raises thestretchfactorto 7

3:1% 2:26 whichstill satisfies therequirement.

However, further removing link ab will lead to too largestretch factor ( 10

3:1% 3:23). Therefore, the output topology of

Algorithm 1 is Fig. 4d.

4.2 Greedy Algorithm 2Adding LinksThe second topology control algorithm starts with a sparsetopology G00 which is strongly connected under CC model.We can use the output of the algorithm in [2] as the initial

topology. Then, we gradually add the most energy-efficientlink into G00. Here, the energy-efficiency of a link is definedas the gain on reducing energy stretch factors by addingthis link. Our algorithm will terminate until the constructedgraph G0 satisfies the energy stretch factor requirement. Thedetail steps are summarized as follows:

Step 1: Construction of G and G00. The step of

constructingG

is the same as the one in Algorithm 1. Then,we call the algorithm in [2] to generate G00, a connectedsparse subgraph of G.

Step 2: Construction of G0. Initialize G0 G00, for everylink vivj 2 G but 62 G

0, compute its stretch-factor-gaingG

0

G vivj as follows:

gG0

G vivj X

vp;vq2V

G

0

G vp; vq G0vivjG vp; vq

: 6

In other words, the total gain of a link vivj is the summationof the improvement of stretch factors of every pair of nodesin G0 after adding this link. In each step, we greedily add

the link with the largest stretch-factor-gain into G0

. If thereis a tie, we use the link weight to break it by adding the linkwith the least weight. We repeat this procedure until G0

meets the stretch factor requirement t.Step 3: Power Assignment from G0. For every node vi,

assign its power level Pi using (5).Algorithm 2 gives the detailed topology control algo-

rithm. The guarantee of the energy t-spanner property isstraightforward since the algorithm terminates adding linksuntil the t-spanner requirement is satisfied. In the worstcase, every links are added into G0, then the t-spannerrequirement must be satisfied. The time complexity ofAlgorithm 2 is also Omnn log n m. Detailed time

complexity of each part of this algorithm is as follows:Lines 2-6 cost On2; Lines 7-15 cost On2log; Line 16costs Om n log n for building the minimum spanningtree; and Lines 18-23 cost Omnn log n m for testing thestretch factor gain.

Algorithm 2. Greedy Algorithm 2 - Adding LinksInput: A set ofn wireless nodes Vand nodes locations; themaximum transmission power PMAX; and the energystretch factor requirement t.Output: A topology G0 and its induced power assignmentPi for each node vi.

1: G V ; ;2: for all vi and vj 2 V do3: if PMAXdij

! then4: Add direct link vivj into G;5: end if6: end for7: for all vi and vj 2 V do8: Hij Greedy Helper Set Selection vi; Nvi; vj;9: if wgvivj < pPGvi; vj and P

vk2vi [Hijdkj

PMAXthen

10: Add CC-link

gvivj into G;

11: if direct link vivj 2 G then

12: Delete vivj from G;13: end if14: end if15: end for


Fig. 4. Examples of two greedy algorithms: (a) communication graph G;(b)-(d) deleting links from G using Algorithm 1; (e)-(f) adding links fromthe connected sparse topology G00 using Algorithm 2. Link weights arelabeled on links, and the stretch factor requirement t 3.


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16: Call the algorithm in [2] to generate a connected sparsetopology G00 of G;

17: Copy G00 to G0;18: while GG0 > t do19: for all link vivj 2 G but 62 G

0 do

20: gG0

G vivj P

vp;vq2VG

0

G vp; vq G0vivjG vp; vq;

21: end for

22: Add link vivj with the largest stretch-factor-gaingG

0

G vivj into G0; If there is a tie, break it by adding

the link with the least weight wvivj; If link vivj isa CC-link, also add all helper links vivk for vk 2 Hijin G0.

23: end while24: for all vi 2 V do25: Pi maxfmaxvivj2G0 P

di j; maxfvivj2G0 Pcci jg;

26: end for

Consider the same network in Fig. 4a. Algorithm 2 firstconstructsa connected sparse topology G00 asshowninFig.4eand copy it to G0. Obliviously, G0 cannot satisfies the stretch

factor requirement. Thus, Algorithm 2 greedily chooses newlinks to add to decrease the stretch factor. Based on thestretch-factor-gain defined in (6), we can calculate the gain ofadding link ab or cdas follows:

gG0

G a; b

G0

G a; b G0abG a; b

G0

G c; d G0abG c; d

6

3 1

10

3:1

7

3:1

% 1:97

and

gG0

G c; d

G0

G a; b G0cdG a; b

G0

G c; d G0cdG c; d

6

3

6

3 103:1 1 % 2:23:Therefore, Algorithm 2 adds cd into the topology G0 with alarger gain. Since this results stretch factor of 2 whichsatisfies the requirement, the algorithm exists with Fig. 4f asits output topology.

In Algorithm 2, we consider the summation of allimprovements on stretch factors of every pair of nodes inG0 as the stretch-factor-gain of a link. However, forcertain pairs of nodes, if their stretch factors are alreadybelow t, further improvement for these pairs may not bevery useful. Therefore, we can change the definition ofstretch-factor-gain to

gG0

G vivj X

vp;vq2B

G

0

G vp; vq G0vivjG vp; vq

: 7

Here, B is the set of node pairs in G0 whose stretch factor isstill larger than t. By doing so, we hope that the algorithmcan converge faster. However, this method may also leadto a situation where no link can provide further gain onstretch-factors, meanwhile the entire networks stretchfactor is still larger than t. In this case, we can either usethe original Algorithm 2 as the backup or directly add oneleast-energy path in G to G0 whose stretch factor in G0 isthe largest. Similarly, another alternative way to computethe stretch-factor-gain is

gG0

G vivj GG0 GG

0 vivj; 8

in which the gain is defined as the improvement of overallstretch factor.

5 LOCALIZED ENERGY-EFFICIENT TOPOLOGYCONTROL WITH LOCAL INFORMATION

Thetwoproposed topology control algorithms (Algorithms 1and 2) are both centralized algorithms, since they need theglobal information of the network. However, such solutionsarenot very practical for certain applications, since thead hocnetworks are usually self-organized and without centralizedcontrol. Thus, we are also interested in extending theproposed methods into distributed or localized algorithms.Here, a distributed topology control algorithm (to construct agraph G0) is a localized algorithm if every node vi can exactlydecide all edges incident on vi based only on the informationof all nodes within a constant hops of vi.

Fortunately, both of our cooperative topology controlalgorithms can be easily extended to localized algorithms

with only local information. For each node vi 2 V, we defineits k-hop local neighborhood Nkvi, which includes all nodeswithin k-hop in graph G under CC model. Let Gkvi be thelocalgrapharound vi whichincludesalllinksamongNkvi inG. Then, our localized topology control algorithm works asfollows: First, each node vi collects the local information fromits k-hop local neighborhood (i.e., Gkvi). Then, vi applieseither Algorithm 1 or Algorithm 2 on its local graph Gkviand generates its local energy spanner G0kvi which is asubgraph of Gkvi and has its stretch factor bounded by twith respect to Gkvi. Node vi notices its k-hop neighbors alledges in G0kvi. Thefinal topology G

0 istheunionof all G0kvi

for allnodes, i.e., G0

Tni1 G0kvi.Wecanprovethatthefinalconstructed graph G0 is still a t-spanner ofG.Theorem 1. The constructed graph G0 by our localized topology

control algorithm is a cooperative energy t-spanner of theoriginal communication graph G.

Proof. To prove the theorem, we only need to prove that forany two nodes vi; vj 2 V there is a path in G

0 such thatp t pPGvi; vj. As shown in Fig. 5, we nowconsider the least energy path PGvi; vj in G and partitionit into serval segments v0 vi; v1; v1; v2 . . . ; vm2;vm1; vm1; vm vj, where each segment is less than orequal to k-hop. Notice that each segment vq; vq1 is

basically PGvq; vq1, since PGvi; vj is the least energypath. Because PGvq; vq1 is less than or equal to k-hop,vq1 and this segment are inside the local graph Gkvq ofvq. Based on our localized algorithm, the least cost path


Fig. 5. Localized algorithms can still guarantee the energy stretch factor

requirement t, i.e., for any vi and vj, there exists a path connecting

them in the constructed graph G0 such that p t pPGvi; vj.


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PG0k

vqvq; vq1between vq and vq1 in G0kvq costs at most t

time of energy than the least cost path PGkvqvq; vq1 (i.e.,PGvq; vq1) does. In other words, there is a pathPG0

kvqvq; vq1 in the constructed local graph G

0kvq such

that pPG0k

vqvq; vq1 pPGvq; vq1. N o ti c e t h a tPG0

kvqvq; vq1 is also in G

0 since G0 takes the union of alllocal graphs. Now consider the path which connects allpaths pPG0

kvqvq; vq1 for q 0 to m 1 in G

0 (see Fig. 5for illustration), we have

p Xm1

p0

pPG0k

vq

vq; vq1

Xm1p0

t pPGvq; vq1 t PGvi; vj:

This finishes the proof. tu

In summary, our localized topology control algorithmsonly need local information to construct the cooperativeenergy t-spanner, even though the spanner property is aglobal requirement. This makes them applicable even forlarge-scale ad hoc networks. However, compared with thecentralized versions, they may keep more communication

links in G

0

due to the lack of global information of thenetwork, thus may lead to larger energy cost in G0. Noticethat with the increasing of k, the constructed G0 of ourlocalized algorithm will converge toward the output of thecentralized algorithm. Therefore, there is a clear tradeoffbetween the communication cost of collecting informationduring construction and the transmission energy consump-tion of the constructed topology. We will see suchobservations in our simulation results.

6 SIMULATIONS

In this section, we will evaluate our proposed topology

control algorithms, namely, Greedy Algorithm 1 (Greedy-DelLink) and Greedy Algorithm 2 (GreedyAddLink), bycomparing their performances with Cooperative Bridges-based Method (Coop. Bridges) [2]. We implement all these

three algorithms in a simulator developed by our group.The underlying wireless networks are randomly generatedin an area of 100 100. For convenience, we set the pathloss factor 2 and the SNR threshold 1. The value ofPMAX is set to 400 so that the maximum transmission rangeof a direct link is 20. Then, we perform proposed topologycontrol algorithms on G. Fig. 6 shows two sets of topologiesconstructed by all three topology control algorithms for twodifferent networks, respectively, with energy stretch factorrequirement t 1:2 and t 1:4. In Fig. 6 the number ofnodes n 50. Clearly, our methods keep more links than

the Coop. Bridges does, since our topologies guarantee theenergy stretch factor bound while Coop Bridges doesnt. Inaddition, the larger t is, the sparser our topologies are.

In the following simulations, we take three metrics as theperformance measurement of topology control algorithms:

. Total Link Cost: the total link cost of the constructedtopology G0, i.e., wG0

Pe2G0 we.

. Average Node Transmission Power: the averagenode transmission power of the constructed topologyG0, i.e., PiG

0 1n

Pvi2V

Pi where Pi is the transmis-sion power of node vi assigned from G

0 by (5).. Maximum Energy Strength Factor: the maximum

energy strength factor of graph G0 under CC modelwith respect to G, i.e., GG0 maxvi;vj 2V

pPG0 vi;vjpPGvi;vj

.

For all the simulations, we repeat the experiment formultiple times and report the average values of thesemetrics. It is clear that a desired topology should have smalltotal link cost wG0 and small node transmission powerPiG

0 while the energy strength factor GG0 should be

smaller than the requirement t.For the first set of simulations, we increase the network

density by rising the number of nodes n from 10 to 100, andkeep energy stretch factor requirement t at 1.6. Fig. 7a showsthe ratios between the total link cost of the generated graphG0 and that of the original cooperative communication graph

G when n increases. This ratio implies how much cost savingachieved by the topology control algorithm, compared withthe original network without topology control. Fig. 7b showsthe ratios between average node transmission power


Fig. 6. Topologies generated by different topology control algorithms over the same random network: black/blue links are direct/cooperative links,

respectively. Upper row (a-d): t 1:2; Lower row (e-h): t 1:4.


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assigned by the topology control algorithm and thatassigned from G. Here, for each topology control algorithm,we also apply the DTCC algorithm in [1] to further reduce

the transmission power with CC. From these results, alltopology control algorithms can significantly reduce the totalcost of maintaining the connectivity and save the nodetransmission energy. When the network is denser, the savingof topology control is larger. Compared with Coop Bridges,our two methods generate a denser topology. However, theircosts and transmission power are just slightly larger thanthat of Coop Bridges. DTCC algorithm can indeed furtherreduce the transmission power. Fig. 7c shows the energystretch factor GG

0 of all methods. Clearly, Coop Bridgescannot satisfy the energy stretch factor requirement t 1:6,and its stretch factor increases with the number of nodes.When the network is dense, it could be more than 10, whichmeans some paths could have huge energy consumption!From observing above, we can conclude that both of ourproposed methods can achieve similar small energy stretchfactors, all of them are bounded by the requirement t, withthe costs of very slightly larger total link costs andtransmission power. This demonstrates the energy efficiencyof our energy cooperative spanners.

In the second set of simulations, we fix the number ofnodes n 50 and run our algorithms with different energystretch factor requirement t increasing from 1.0 to 2.0. Fromresults shown in Figs. 8a and 8b, we can observe that tighter(i.e., smaller) stretch factor requirement results in higher costand larger node transmission power of all topology algo-rithms. The reason is when the stretch factor requirement

becomes larger, the restriction on removing links becomesweaker. In the extreme case with infinite t, our energy-efficient topology control problem converges to TCC whichonly preserves the connectivity. Clearly, there is a tradeoff

between stretch factor and link cost, transmission power. Wecan choose a suitable stretch factor bound to match theparticular application needs. From the results in Fig. 8c, we

can see that the stretch factor of Coop Bridges does not affectby the stretch factor requirement, however both of ourmethods can perfectly satisfy the stretch factor requirement.Again it proves our methods can efficiently satisfy the stretchfactor requirement while Coop Bridges cannot.

In the last set of simulations, we implement our localizedalgorithmswith k-hoplocalinformation.Wefixthenumberofnodes n 100, the maximum power PMAX 144, and theenergy stretch factor requirement t 1:6. Fig. 9 shows theresults when k 1 to 6. It is clear that with larger neighbor-hood information (larger value of k) our algorithms canachievebetterenergyconservation(i.e.,withsmallertotalcostandlower average node power) butwith larger stretch factor.However, all of our localized algorithms can still satisfy thestretch factor requirement.As we expect, largervalueofk alsoleads to higher number of messages need to collect theneighborhood information (as shown in Fig. 9d). Therefore,there is a tradeoff between the communication cost ofcollecting information during construction and the transmis-sion energy consumption of the constructed topology. Fordifferent applications, we may have different setting ofk.

7 CONCLUSION

In this paper, we introduced a new topology controlproblem: energy-efficient topology control problem with co-operative communication, which aims to keep the energy-

efficient paths in the constructed topology. We proved thatthis problem is very challenging (NP-complete), andproposed two new topology control algorithms usingcooperative communications. Both algorithms can build a


Fig. 7. Simulation results on random networks with fixed stretch factor requirement t 1:6 and varying number of nodes n.

Fig. 8. Simulation results on random networks with fixed number of nodes n 50 and varying stretch factor requirement t.


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cooperative energy spanner in which the energy efficiency of

individual paths are guaranteed even with only local

information. Simulation results confirm the nice perfor-

mance of both proposed algorithms.Possible future works include: 1) design more efficient

algorithms with lower complexity to achieve cooperative

energy spanner with CC model; 2) investigate how to adapt

the constructed spanners to unexpected changes in the

network such as link and node failures. Notice that even

though the proposed methods mainly work for static

cooperative ad hoc networks, they do have potential to

handle changes in dynamic networks. Especially, for loca-

lized versions of the proposed algorithms, any node or link

change will only affect the final topology within its k-hop

neighborhood. However, with the existence of cooperative

communication links, such neighborhood could be large.

ACKNOWLEDGMENTS

This work is supported in part by the US National Science

Foundation (NSF) under Grant No. CNS-0915331 and CNS-

1050398.

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Ying Zhu received the BEng degree in auto-matic control and MEng degree in patternrecognition from the University of ElectronicScience and Technology of China in 2002 and2005, respectively. She is currently workingtoward the PhD degree in the University ofNorth Carolina at Charlotte, majoring in compu-ter science. Her current research focuses onwireless networks, ad hoc and sensor networks,delay tolerant networks, and algorithm design.

Minsu Huang received the BEng degree inmechanical engineering from Central SouthUniversity in 2003 and the MS degree incomputer science from Tsinghua University in2006. He is currently working toward the PhDdegree in the University of North Carolina atCharlotte, majoring in computer science. Hiscurrent research focuses on wireless networks,ad hoc and sensor networks, delay tolerantnetworks, and algorithm design.

Siyuan Chen received the BS degree fromPeking University, China in 2006. He is currentlyworking toward the PhD degree in the Universityof North Carolina at Charlotte, majoring incomputer science. His current research focuseson wireless networks, ad hoc and sensornetworks, delay tolerant networks, and algorithmdesign.

Yu Wang received the BEng degree (1998), theMEng degree (2000) in computer science fromTsinghua University, China, and the PhD degree(2004) in computer science from Illinois Instituteof Technology. He is an associate professor ofComputer Science at the University of NorthCarolina at Charlotte. His research interest

includes wireless networks, ad hoc and sensornetworks, delay tolerant networks, mobile com-puting, and algorithm design. He has published

more than 100 papers in peer-reviewed journals and conferences. Hehas served as program chair, publicity chair, and program committeemember for several international conferences (such as IEEE INFOCOM,IEEE IPCCC, IEEE GLOBECOM, IEEE ICC, IEEE MASS, etc.). He is arecipient of Ralph E. Powe Junior Faculty Enhancement Awards fromOak Ridge Associated Universities in 2006 and a recipient of Out-standing Faculty Research Award from College of Computing andInformatics at UNC Charlotte in 2008. He is a senior member of theACM, the IEEE, and the IEEE Communications Society.

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