A UNIFYING FRAMEWORK FOR TUNABLE TOPOLOGY CONTROL ...

A UNIFYING FRAMEWORK FOR TUNABLE TOPOLOGY CONTROLIN SENSOR NETWORKS

Sameera Poduri1 , Sundeep Pattem2, Bhaskar Krishnamachari1,2, Gaurav Sukhatme1

1 Department of Computer Science,2 Department of Electrical Engineering,University of Southern California,

Los Angeles, CA 90036, USA{sameera, pattem, bkrishna, gaurav}@usc.edu

ABSTRACT

Topology control, primarily concerned with ensuring de-sired levels of coverage and connectivity, is a vital self-configuration operation in unattended sensor networks. Wepresent a classification of sensor network topologies anddiscuss their implications for topology control. Our maincontribution is a unifying framework that forms a basis fortunable topology control in all classes of topologies. It isbased on a simple, local condition of ensuring a neighbor inevery theta (θ) angle sector of each node’s communicationrange. We present analysis to establish that varying this sin-gle parameterθ can indeed provide a wide range of coverageand connectivity tradeoffs. For specific values ofθ, we showthat the Neighbor-Every-Theta (NET) condition guaranteesvarious proximity graphs such as the relative neighborhoodgraph. The problem of maximizing coverage given such acondition is also addressed. Algorithms for controlled de-ployment are presented to demonstrate how the NET condi-tion can be integrated with positioning of nodes for tunabletopology control.

1. INTRODUCTION

In view of their distributed and unattended operation, self-configuration of nodes is essential for deploying large scalesensor networks. Topology control is a key operation forenabling self-configuring capability in these networks. Itconsists of mechanisms designed to provide desired levelsof coverage and connectivity and ensure power-efficient op-eration of the network. While power-efficiency is a basicrequirement given the nature of sensor networks, coverageand connectivity are closely tied to application-specific re-quirements. For scalability, topology control mechanisms

This work is supported in part by the National Science Founda-tion (NSF) under grants 0347621, 0325875, 0325875, IIS-0133947, EIA-0121141 and grants CCR-0120778, ANI-00331481 (via subcontract). Anyopinions, findings, and conclusions or recommendations expressed in thismaterial are those of the authors and do not necessarily reflect the views ofthe National Science Foundation.

that work well with local information and asynchronous nodeoperations are desirable.

Most prior work on topology control has dealt withun-controlleddeployments, where there is no explicit controlon positions of individual nodes [1]. The primary mech-anisms proposed are power control and sleep scheduling.These methods involve pruning an already existing, wellconnected communication graph in order to save powerwhile ensuring that the resultant sub-graph preserves con-nectivity. Given a network that is connected when all nodesare operating at maximum power, the aim of power con-trol is to use the minimum power level at each node forwhich the network remains connected [2]. Given an over-deployed network, sleep scheduling seeks to activate a min-imal subset of nodes to maintain connectivity and achieveother desired metrics [3].

In contrast,controlled deployments are feasible whenpositions of individual nodes can be altered. Such deploy-ments are interesting for two reasons. First, network topol-ogy with wireless communication relates directly to prox-imity relations and hence, position of nodes. Second, thereis increasing evidence that a large number of deploymentsare likely to involve careful, non-random placement of nodes[4, 5]. The positioning of nodes is controlled either by thenodes themselves or by external agents. Such networkspresent a different and interesting scenario for topology con-trol since it is possible to exploit control of the motion andplacement of the nodes to build efficient topologies.

Thus, sensor network topologies can be classified on thebasis of node mobility (static or mobile) and the nature oftheir deployment (controlled or uncontrolled):

1. Static nodes, uncontrolled deployment: This isthe assumption in a majority of work on sensor networks- resource-constrained, static nodes are randomly deployedin a given area. Power control (for sparse networks) andsleep scheduling (for over-deployed networks) are chieflyemployed for topology control. It is plausible that these twomechanisms are used in conjunction.

2. Mobile nodes, uncontrolled deployment:Such sys-

CRES Technical Report, June’05 1

Class ExampleTopology Control Mechanisms

Power Control Sleep Scheduling Position ControlUncontrolled Static 29 Palms Experiment [6]

√ √×

Uncontrolled Mobile ZebraNet [7]√

⊕ ×Controlled Static Structural Health Monitoring [5] ⊕ ⊕

√

Controlled Mobile NIMS [4] × ×√

√- primary mechanism,⊕ - secondary mechanism,× - not applicable

Table 1. Classes of sensor network topologies and corresponding applicable topology control mechanisms

tems are studied as mobile ad-hoc networks (MANETS).The challenge is to ensure network operation in spite of theautonomous and mutually independent motion of the nodes.Power control is the suitable mechanism for topology con-trol.

3. Static nodes, controlled deployment:In this class,static nodes are carefully positioned either by humans or byautonomous robots. Topology control is achieved by con-trolling the positions of nodes. Power-control and sleepscheduling might be employed, but largely as secondarymechanisms.

4. Mobile nodes, controlled deployment:This classinvolves mobile nodes which can position themselves withrespect to each other. Topology control mechanisms are in-tegrated with the motion and positioning strategy. Sincemotion is the dominant power consumer, power-efficiencyof communication is not a major consideration. Sleepscheduling is also not relevant because mobile deploymentsare not likely to be dense.

The classification is summarized in Table 1.We present a unifying framework for topology control in

sensor networks. This framework is a basic building blockfor designing mechanisms for completely distributed andtunable topology control for all four classes of topologies.It is based on the following simple, local condition: ensurethat each node has a neighbor in every theta angle sectorof its communication range. This Neighbor-Every-Theta(NET) condition results from our analysis of local node po-sitioning rules that guarantee global network properties.

The communication graph of a wireless network can bemodeled as a proximity graph because high quality linkstypically exist only between proximate nodes. We analyzeproximity graphs such as the relative neighborhood graph(RNG) which are known to have good coverage and con-nectivity properties and derive local conditions that guaran-tee formation of these graphs. The problem of maximiz-ing sensing coverage under the NET condition is addressed.For certain values of the sector angle, it is possible to placethe nodes such that the resultant communication graph is asymmetric tiling of the space. We show that such placementwill achieve the maximum coverage. These results are com-bined to present a graph construction mechanism which hastwo key features:

• the required conditions are simple, local and geomet-ric, and

• varying the single parameterθ provides a wide rangeof coverage and connectivity tradeoffs.

This general framework can support a number of exist-ing power control and sleep scheduling schemes. Nodes cancheck for neighbors satisfying the NET condition, integrat-ing it with the power adjustment and node activation policyrespectively. It can also support controlled deployment of astatic sensor network - the agent making a decision on thebest location of the next node based on the geometry of theexisting network. Self-deploying mobile nodes could usethis condition to decide their motion strategy. In this paper,the focus is on establishing position control as a topologycontrol mechanism. We present algorithms for controlleddeployment of both static and mobile nodes to concretelydemonstrate the use of the local geometric rules to imple-ment coverage and connectivity tradeoffs. Simulations willstudy relaxations of some idealized assumptions made in theproblem formulation for ease of analysis.

2. PROBLEM FORMULATION

Scalability is a major issue in deploying large scale sensornetworks. As a design principle, the need is for networkconfiguration mechanisms that depend solely on local in-formation and interactions and yet are capable of achievingsome global objective. In this work, we study this problemin the context of topology control by asking the followingquestion:

Do there exist local, geometric conditions that can guar-antee desired global topological properties?

If such conditions can indeed be found, it will be possi-ble to devise a general framework on which topology con-trol mechanisms can be based.

We analyze networks for the following desirable topo-logical properties:

• Coverage: The total area lying within the sensingrange of at least one node.


Rc>θ

X

Rc ≤θ

(a) (b)

Fig. 1. Illustration of the NET condition. In a), node X has one sector greater than a given angleθ with no neighbor. It doesnot satisfy the NET condition. In b), the largest sector with no neighbors is smaller thanθ, satisfying the NET condition

• Connectivity: A graphG is said to be k-(edge)-connected if there are at leastk node disjoint pathsbetween any two nodesu, v ∈ V . Connectivity is ameasure of fault tolerance or path diversity in the net-work. 1-connectivity is a fundamental requirementfor network operation.

• Sparseness:A graph is said to be sparse if the num-ber of edges isO(|V |) (linear in the number of nodes).A sparse topology is desired for power efficiency.

• Degree:The average number of neighbors across allnodes. A higher node degree can provide better sens-ing granularity. Bounded node degree leads to lowerinterference in wireless communication and hence in-creased throughput.

• Spanning Ratio: A graph G is called a c-spannerif for all u, v ∈ V there exists a pathp = (u =v0, v1, v2...., vl = v) from u to v in G with

|p| =l−1∑i=0

d(vivi+1) ≤ c · d(u, v),

whered(u, v) is the Euclidean distance. If G is a c-spanner, c is called its spanning ratio. The spanningratio is an indicator of the number of hops betweentwo nodes given their separation.

In the next section, we present analysis of these desiredproperties.

In order to obtain analytically tractable models, two keyidealized assumptions will be made - circular communica-tion range (in Section3) and availability of position (angleand distance) information (in Section4). Justifications andexplanations are provided at appropriate places. These as-sumptions will be relaxed in simulations (Sections 4.1 and4.2).

3. NEIGHBOR EVERY THETA (NET) GRAPHS

We define theNeighbor Every Theta(NET) condition [fig-ure 2] for a node as requiring at least one neighbor in everyθ sector of the communication range. For finite networks,boundary nodesor nodes on the network edge cannot sat-isfy such a condition. For a givenθ, we define boundarynodes as those which have no neighbor in at least aθ sectorof infinite radius. A NET graph is one in which every nodeexcept the boundary nodes satisfies the NET constraint fora givenθ.

An important feature of these graphs is that they canbe constructed from purely local information - each nodeonly needs to know the locations (distance and angle) of itsneighbors relative to itself.

NET graphs form a family of graphs based on a sin-gle parameterθ. As θ becomes smaller, the graphs becomedenser with increasing level of connectivity and decreasingcoverage. In this section we analyze the connectivity andcoverage of these graphs as a function ofθ.

3.1. Connectivity Analysis of NET Graphs

It has been shown that if every non-boundary node has atleast one neighbor in eachπ sector around it, the network isguaranteed to be connected [8]. This implies that forθ ≤ π,NET graphs are guaranteed to be 1-connected.

We now derive values ofθ for which NET graphs will beguaranteed to form each of three representative proximitygraphs, namely the Relative Neighborhood Graph (RNG),Gabriel Graph (GG) and the Delaunay Graph (DelG). Thesegraphs have been well studied and are known to be desirableglobal topologies for wireless communication networks.

The RNG, GG and DelG are all connected and are knownto be sparse when defined over adisk graph. The sparsenessof proximity graphs implies that the average node degree is


X Y

2∏/3

Rc Rc

X Y2∏/3Rc

β

d(X,Y)

(a) (b)

Fig. 2. RNG sector condition. a) The circle of radiusd(X, Y ) > Rc subtends an angle of2π3 at X. The lune contains an area

larger than a2π3 sector. b) The limiting case, whend(X, Y ) = Rc.

bounded by a constant. The RNG displays desirable prop-erties in terms of power consumption, average node degreeand fault tolerance [9]. GPSR [10] is a greedy geographicrouting protocol which derives its scalability from the RNGand GG as they allow routing decisions based on local stateonly. In [11], a restricted DelG is shown to improve GPSRperformance by virtue of its bounded spanning ratio. In[12] a solution to the rendezvous problem of robotic net-works is proposed that relies on forming proximity graphslike RNG, GG and DG.

We assume ideal, perfectly circular communicationrange. It is well known that the real communication rangeis very different from a circle [13]. However, using re-sults in [14], it is possible to obtain a range in which theprobability of packet receptions is close to1. These resultsprovide guidelines on how to leverage our analysis, whichis based on idealized assumption of circular communicationrange, for real deployments. In power control, a minimalsetting for the power and the corresponding circular regionof high-quality communication links can be obtained. Forsleep scheduling and deployment of static nodes using anexternal agent, a minimum density requirement based onnode power settings can be arrived at. Finally, mobile nodescan be expected to have high-power radios for which thereis a large circular neighborhood with high-quality links.

Let V be the set of nodes in a wireless communicationnetwork in R2. For u, v ∈ V let d(u, v) denote the Eu-clidean distance fromu to v. In what follows, the name andlocation of a node are used interchangeably.

The set of edgesE for various proximity graphs is de-fined as follows [15].

Disk graph (DG(V,E, r)): The undirected graph con-taining all edges not longer thanr.

E = {(u, v)|u, v ∈ V andd(u, v) ≤ r}

Given positiver ∈ R, letC(p, r) be the circle consistingof points whose distance from pointp is strictly less thanr.Define the lune, denotedL(p, q), to be the intersection oftwo circles, both of radiusd(p, q), centered at these points,that is,L(p, q) = C(p, d(p, q)) ∩ C(q, d(p, q)).

Relative Neighborhood Graph (RNG(V)): The undi-rected graph containing an edge(u, v) if there is no pointw ∈ V that is simultaneously closer to bothu andv. Equiv-alently,(p, q) is an edge ifL(p, q) ∩ V = ∅.

E = {(u, v)|u, v ∈ V and∃ now ∈ V 3

d(u, w) < d(u, v) andd(v, w) < d(u, v)}

Gabriel Graph (GG): The undirected graph containingan edge(u, v) if the disc whose diameter is edge(u, v) doesnot contain any other points of V, that is, ifC(u+v

2 , d(u,v)2 )∩

V = ∅.

E = {(u, v)|u, v ∈ V andC(u + v

2,d(u, v)

2) ∩ V = ∅}

Delaunay Graph (DelG): The undirected graph con-taining an edge(u, v) if the Voronoi regions ofu andv havenon-empty intersection. From the properties of Voronoi di-agrams it follows that


1. edges of triangle(u, v, w) are in DelG(V ) if theircircumcircle does not contain any other points ofV

2. edge(u, v) is in DelG(V ) if the disc whose diameteris the lineuv does not contain any other points ofV

These graphs are hierarchically related:RNG(V ) ⊆ GG(V ) ⊆ DelG(V ).

In the proofs that follow, we assume that every node inthe network has the same communication rangeRc. Thisis not a requirement and we discuss how the proofs willhold even when each node has a different communicationrange. However, this assumption makes the proofs simpler.Now, the communication graph of a wireless network is adisk graph with all edges being smaller than or equal to thecommunication rangeRc. Our next step is to find condi-tions under which the RNG of the network is contained inthe communication graph. This is equivalent to finding con-ditions under which no edge of an RNG is longer thanRc.The following theorem presents this condition. Later, weprove similar results for the GG and DelG.

Theorem 1 If each nodeX ∈ V has at least one neighborin every 2π

3 sector ofC(X, Rc), the communication graphis a supergraph ofRNG(V ). Moreover, 2π

3 is the largestangle that satisfies this property.Proof: Consider any nodex ∈ V . Supposex has at leastone neighbor in every2π

3 sector ofC(X, Rc). It is suffi-cient to show that for any nodeY outsideC(Y, Rc), theedge(X, Y ) /∈ RNG(V ). The luneL(X, Y ) will contain asector of at least2π

3 (figure 2). By premise,∃ a nodeZ thatlies inL(X, Y ).

This implies that the RNG does not have any edge lengthsgreater thanRc and thereforeRNG(V ) ⊆ DG(V,E, Rc).

Now suppose a nodeX ′ ∈ V ′ has two neighbors with asector angle of2π

3 + δ (δ > 0) between them (figure 3). Wecan place a nodeY ′ outsideC(X ′, Rc) such that the edge(X ′, Y ′) ∈ RNG(V ′). Therefore,2π

3 is the largest anglefor whichNET (V, θ) containsRNG(V ).2

Theorem 2 If each nodeX ∈ V has at least one neighborin everyθ = 2 arccos( r

Rc) sector ofC(X, r) (r ≤ Rc), the

communication graph is a supergraph ofGG(V ).

Proof: Consider a nodeX and a nodeY outsideC(X, Rc).For anyr ≤ Rc, let β be the angle subtended by the inter-section areaC(X, r) ∩ C(x+y

2 , d(X,Y )2 ) at X. From figure

4 we have,

r

2=

d(X, Y )2

cos(β

2)⇒ β = 2arccos(

r

d(X, Y )).

Sincearccos is strictly decreasing in[0, 1] andd(X, Y ) ∈(Rc,∞), the smallest angle subtended is

θ = inf β = 2arccos(r

Rc). (1)

P

Q

XY

2∏/3

+ δ

Rc

Fig. 3. θ = 2π3 bound for RNG -L(X, Y ) is empty

X Y

Rc r

β

d(X,Y)/2

r

β/2

β/2

Fig. 4. Sector condition for GG

Supposex has at least one neighbor in every

θ = 2arccos(r

Rc)

sector ofC(X, r). By definition, the edge(X, Y ) /∈ GG(V )if

C(d(X, Y )

2,d(X, Y )

2, ) ∩ V 6= ∅. (2)

(2) is satisfied for any choice ofY if there is a nodeZ ∈ Vin theθ sector ofC(X, r). By premise such aZ exists andhence the GG will not have any edge lengths greater thanRc. This implies thatGG(V ) ⊆ DG(V,E, Rc).2

Corollary: If a nodeX has at least one neighbor ineveryθ = 2arccos( r

Rc) sector ofC(X, r) (r ≤ Rc), the

communication graph is a supergraph ofDelG(V ).

Proof: Consider all circles passing throughX and Y .The smallest isC(X+Y

2 , d(X,Y )2 ) and the largest are two

circles of infinite radius of which the straight line throughX andY is an arc. From figure 5, it can be seen that the


Fig. 5. Sector condition for DelG

intersections of these circles withC(X, r) subtend an anglewhich is at leastθ at X, where

θ = inf β = 2arccos(r

Rc).

By premise, there is at least one node in everyθ sector ofC(X, r). Therefore, every circle passing throughX andYwill contain another node. So,

1. there is noZ ∈ V such that the4XY Z ∈ DelG(V )

2. trivially C(X+Y2 , d(X,Y )

2 ) also contains another node.

By 1,2 and definition of DelG, the edge(X, Y ) /∈ DelG(V ).Since the choice ofY is arbitrary, there are no edge lengthsgreater thanRc in the DelG. HenceDelG ⊆ DG(V,E, Rc).2

In all the proofs above, an arbitrary nodeX is consid-ered. Now assume that such a nodeX has a communicationrangeRX

c , which is different from the other nodes. It iseasy to see that the proofs can be slightly modified to showthat the assertions hold even in this case. For the RNG, the2π3 bound still holds. For the GG and DelG, each nodeX

will need at least one neighbor in everyθ = 2arccos( rRX

c)

sector ofC(X, r).In real networks, it is not possible for the boundary nodes

to satisfy the conditions required by the theorems. We showusing simulated deployments (in section 4) that the asser-tions can be validated in spite of these exceptions.

3.2. Coverage Analysis of NET Graphs

Having listed the conditions that guarantee global connec-tivity properties, we now turn to the problem of maximiz-ing coverage. Suppose that in order to satisfy the NET sec-tor conditions, a node must havek neighbors. From the

node’s local perspective, all neighbors must be located onthe perimeter of the communication range to maximize cov-erage. Intuitively, the nodes must also be placed symmetri-cally on the perimeter. Assume a circular sensing range ofradiusRs.

Theorem 3 For Rs = Rc, the area coverage is maximizedwhen thek ≥ 3 nodes are placed at the edges ofk disjoint2πk sectors ofC(X, Rc).

Proof: Consider a nodeX andk nodesY1, Y2, ...., Yk placedon the perimeter ofC(X, Rc). Let these nodes be placed inanticlockwise order at anglesβ1 = 0, β2, ...., βk respec-tively. Define

θi = βi+1 − βi, 1 ≤ i ≤ k − 1θk = 2π − βk (3)

We need to findθi such that the total area covered by thesek + 1 nodes is maximized. The open discs of all nodes liewithin C(2Rc, x) and are tangent to this disc at exactly onepoint (Ti) each. The discs of adjacent nodesi and i + 1intersect at pointIi (and at X). The total coverage liesbetweenπR2

c and 4πR2c and is maximized when the area

k∑i=1

TiIiTi+1 is minimum (figure 6).

TiIiTi+1

= TiCTi+1 − IiYiTi − IiYi+1Ti+1 − IiYiCYi+1

= (2Rc)2 ·θi

2− 2R2

c ·θi

2− 1

2(2Rc sin

θi

2)(2Rc cos

θi

2)

= R2c(θi − sin θi)


X

Ii

Ti

Yi+1

Yi

θi

θi

2R c R c

Ti+1

Fig. 6. Coverage withk nodes placed on the communicationperimeter of node X. The shaded area is the total coverage.

k∑i=1

TiIiTi+1 = R2c(

k∑i=1

θ −k∑

i=1

sin θi)

= R2c(2π −

k∑i=1

sin θi) (4)

The problem now reduces to findingmaxk∑

i=1

sin θi subject

tok∑

i=1

θi = 2π and0 ≤ θi ≤ 2π. Sincesin is non-negative

and concave in[0, π] and non-positive in(π, 2π], it followsthat the solution isθi = 2π

k , 1 ≤ i ≤ k.2

This result holds even for the case whenRs 6= Rc. Thiscan be shown by considering 3 cases separately: 1)Rc <Rs 2) Rs < Rc < 2Rs and 3)Rc > 2Rs. The expressionsobtained are involved and the proofs are not included herebecause of space constraints.This also implies that the resultholds when the communication range of all nodes is not thesame

The result from Theorem3 indicates how a node canlocally maximize coverage while satisfying a sector condi-tion. The symmetric nature of the locally optimal place-ment of neighbors has an interesting consequence. Whenk = 3, 4, 6, the symmetric arrangement of neighbors givesrise to tiling structures which are known to be globally opti-mal in terms of area coverage. The structures fork = 3, 4, 6can be equivalently described in terms of the sector angleθ = 2π

3 , π2 , π

3 respectively. It can be verified that the hexag-onal tiling (θ = 2π

3 ) is an RNG and in this case, GG≡RNG. This holds for the square tiling (θ = π

2 ) as well. Thetriangle tiling (θ = π

3 ) is a DelG.

Theorems1 and2 provide sector conditions for super-graphs of the corresponding proximity graphs. These condi-tions however do not predicate sparseness. The tiling graphshaveO(|V |) edges and are therefore sparse. While the RNGand GG have spanning ratios ofO(|V |) andO(

√|V |) in

general, the spatial arrangement of nodes in the tilings resultin constant spanning ratios. The exact ratios can be easilyobtained by symmetry arguments and are shown in Table 2.

The analysis of coverage and connectivity was limitedto certain values ofθ. As a first step towards characteriz-ing NET graphs for general values ofθ, in the next sectionwe study topologies resulting from deployment algorithmsbased on the NET condition.

4. TOPOLOGY CONTROL USING NET GRAPHS

We now describe the use of NET graphs and their proper-ties for topology control. In general, based on the coverageand connectivity requirements of an application, nodes caneither be pre-configured for a certain value ofθ or they cantune it dynamically. In order to satisfy the NET condition,nodes can then employ power control, sleep scheduling, po-sition control or a combination of these. For instance, eachnode can start with a low transmission power and increaseits transmission power till it has at least one neighbor ineveryθ sector around it. Similarly, nodes can decide theirsleep/wake schedules such that all the nodes that are awakesatisfy the NET condition. Power control and sleep schedul-ing mechanisms have been studied extensively in prior work[1]. Relatively, there has been very little work on positioncontrol [16, 17]. Position control is relevant for situationsin which deployment of static nodes by an external agentand for self-deployment of mobile nodes. In controlled de-ployment of a static sensor network, the agent can make adecision on the best location of the next node based on thegeometry of the existing network. Self-deploying mobilenodes can use this condition to decide their motion strategy.In all these cases, the coverage can be maximized by choos-ing, activating or positioning neighbors within eachθ sectorwhich are as far apart from each other as possible (i.e., bytrying to achieve a symmetric configuration)

It is evident that NET graphs naturally capture the fea-tures of all three primary topology control instruments: powerlevel, sleep cycling and position. As illustrated in Table1, combinations of these mechanisms can be employed tocover all classes of sensor network topologies. Thus, wehave arrived at aunifying framework for designing mech-anisms for completely distributed and tunable topology con-trol for all classes of topologies.

A key requirement in constructing NET graphs is an-gular information of neighboring nodes. In case of mobilenodes, this is a reasonable assumption because they are usu-ally equipped with ranging sensors for obstacle avoidance.


Name Degree K-Connectivity Number of edges Spanning Ratio Proximity RelationsLine Tiling (θ = π) 2 1 O(n) ≈ n 1 RNG, GGHexagonal Tiling (θ = 2π

3 ) 3 3 O(n) ≈ 32n 3

2 RNG, GGSquare Tiling (θ = π

2 ) 4 4 O(n) ≈ 2n√

2 RNG, GGTriangle Tiling (θ = π

3 ) 6 6 O(n) ≈ 3n 2√3

RNG, GG, DelG

Table 2. Properties of Tiling Graphs

The same is true in case of a static network being deployedby a mobile agent. For uncontrolled static networks, this in-formation can be obtained by estimating the angle-of-arrivalof messages [18]. While constructing NET graphs doesnot require global location information, in cases when suchinformation is available the angular information can be de-duced easily.

We now focus on establishing position control as a topol-ogy control mechanism. To this end, we demonstrate howthe NET condition can be integrated with the deploymentalgorithms. These algorithms were implemented in thePlayer/Stage software platform which simulates the behav-ior of real sensors and actuators with high fidelity [19].

4.1. Incremental Deployment of Static Nodes

Consider a naive incremental approach where a mobile robotdeploys static nodes one at a time. The robot looks at a localsnapshot of the already deployed nodes and examines themfor the NET condition. For each nodeX, it identifiesθ sec-tors that do not have a neighbor and deploys a new node inthat sector. Using theorem 3 as a guideline, it deploys thenew node at a distance equal to the communication rangefrom X so that the coverage is maximized fromX ′s localperspective. The robot does not have any knowledge of itsglobal location and deploys nodes relative to the positionsof already deployed nodes. To sense the positions of thesenodes, the robot uses a laser range finder which has a smallerror of±2o.

input : θoutput: Node Locationsplacenode1 at (0,0);DeployLocations← (NET locations ofnode1);for i=2 to Num. of nodesdo

l = pop(DeployLocations);placenodei at l;DeployLocations← (NET locations ofnodei);

end

Algorithm 1 : Incremental deployment

Figure 7a shows the topology obtained withθ = π3

for a deployment of 100 nodes. As expected, this topol-ogy is the triangle tiling discussed earlier where each (non-

boundary) node has exactly6 (= 2πθ ) neighbors. On the

other hand, a non-tiling angle likeθ = 2π5 (figure 8a) results

in over-deployment and the average node degree (figure 9c)is greater than5 (= 2π

θ ). The average coverage (figure 9a)obtained in this case is lesser than the coverage forθ = π

3 .This is because, the algorithm follows the condition for lo-cally optimizing coverage, which in case ofθ = 2π

5 does notresult in the best global coverage. For the tiling angles, thelocal condition results in globally optimal coverage, as dis-cussed earlier. Increasing the global coverage for the non-tiling angles will involve avoiding over-deployment result-ing from blindly following this local condition. Figure 9aalso compares the coverage of the incremental deploymentwith the asymptotic upper bound obtained from theorem 31

Figure 9b shows the edge connectivity values for thesegraphs. By Menger’s theorem, the edge-connectivity of anetwork equals the maximum number of edge-disjoint pathsbetween any two nodes in the network. To avoid edge ef-fects, we ignored all node pairs that contain a boundarynode. Connectivity was computed using the network flowroutines available inMathematica. It is our conjecture thata NET graph for sector angleθ, the edge-connectivity willbe bounded byb 2π

θ c. Figure 9c shows the number of sectorsper node that violate the NET condition. These occur onlyat the boundary nodes.

Figure 10 shows a comparison of the communicationgraph obtained from deployment withθ = π

3 with its corre-sponding RNG. In 10c, the dotted lines represent the edgesin the RNG that are absent in the communication graph. Itcan be seen that the communication graph differs from theRNG in exactly 3 edges at the boundary of the network.This is because, the boundary nodes do not satisfy the NETcondition for RNG, i.e., having at least one node in everyθ = 2π

3 sector. In practice, every network will have bound-ary nodes that cannot satisfy the NET condition. How-ever, this will only affect the edges incident on the boundarynodes and will not have a significant impact on the networkproperties.

We repeated the experiments with an additional error in-troduced in the angle and distance at which the robot de-

1Consider a nodeX with neighbors placed symmetrically on its com-munication range. By accounting for the overlaps within the sensing radiusof X, we can calculate the contribution ofX to the total coverage of thenetwork. This gives an upper bound on the per node coverage for an infinitenetwork


ploys nodes. The error was modeled a as zero-mean gaus-sians with standard deviations of3o in angle and5% of Rc

in position. The topologies obtained are shown in 7b and8b. Note that in case of the static network, this error isequivalent to the error in angle and position information ofneighbors. The topologies obtained are denser than thoseobtained without error. The coverage and connectivity per-formance is also close to the earlier deployment without theadditional error (figure 9).

Comparison of the communication graph forθ = 2π3

with the corresponding RNG (figure 11) shows that in addi-tion to the edges on the boundary, there are only a few edgesin the communication graph that are not in the RNG.

4.2. Distributed Deployment of Mobile Nodes

Potential field based algorithms have been widely used forthe deployment of mobile networks [20, 21]. These al-gorithms involve constructing local virtual forces betweenneighboring robots to encode their desired motion and/orplacement patterns. In our algorithm, we use two kinds offorces. The first,Frepel, causes the nodes to repel eachother to increase their coverage and the second,Fattract

constrains neighboring nodes to stay connected. By using acombination of these mutually opposing forces, each nodemaximizes its coverage while maintaining the NET condi-tion of having at least one neighbor in everyθ sector.

In a typical mobile deployment scenario all the nodesstart clustered together in random positions. In this initialstate the network is highly well connected and the NET con-ditions are satisfied. Every node applies a force ofFrepel +Fattract on each its neighbors. These forces have inversesquare law profiles -Frepel tends to infinity when the dis-tance between the nodes decreases to zero andFattract

tends to infinity when the distance between nodes increasesto Rc. As a result of these forces, each pair of nodes movesas far away from each other as possible without losing con-nectivity with each other. After every time step, each nodeevaluates the contribution of each of its edges towards sat-isfying theNET condition. An edge that is not necessaryto satisfy the condition of either of the nodes it connects isbroken by turning offFattract. As a result of these forcesthe nodes spread out as far as possible till no more edgescan be broken.

input : θoutput: Velocitywhile deployment not donedo

read locations of neighbors;for each edgedo

if edge is necessary for NET conditionthen

Fnet = Fattract + Frepel ;endelse

Fnet = Frepel;endacc+ = Fnet

mass ;endvelnew+ = acc ∗∆Time− ν ∗ velold;if no significant change in velocity overtimethen

deployment done;end

end

Algorithm 2 : Distributed deployment

In the simulator, each node is an omni-directional robotthat uses laser range finders for obstacle avoidance. Thelaser range finders return the positions of neighboring nodeswith an error of±2o in the angle and negligible error in dis-tance. In our experiments, communication between nodeswas abstracted out using shared variables in the simulationprogram.

In all our experiments the distributed algorithm was sig-nificantly faster than the incremental approach and convergedwithout any oscillations. Figures 7c and 8c show some ex-ample topologies. These topologies are significantly denser(figure 9c) than the topologies obtained in incremental de-ployment. This is expected because the algorithm is con-servative - it trades off sparseness in order to guarantee thatthe NET condition is satisfied. In our experiments, the NETcondition was satisfied at all the interior nodes. The dis-tributed algorithm has a lower coverage but greater connec-tivity as compared to the incremental approach (figure 9a).

Forθ = 2π3 , the communication graph contains the RNG

(figure 12). In contrast to the incremental approach, weobserve that there are no edges at the boundary that arepresent in the RNG and are not contained in the commu-nication graph. This is because in the distributed approach,the boundary nodes are constrained in their movement inorder to satisfy the NET condition.

Even in the presence of boundary nodes in real net-works, these results validate the assertion in Theorem 1 thatthe communication graph will contain the RNG forθ ≤ 2π

3 .Also, the communication graph contains only a few non-boundary edges that are not in the RNG. Hence these graphswill inherit the sparseness properties of RNG.


0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

Student Version of MATLAB

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60


0 10 20 30 40 50 60 70 800

10

20

30

40

50

60


(a) (b) (c)

Fig. 7. Topologies obtained forθ = π3 with 100 nodes andRs = Rc = 8m using a) Incremental deployment with small error

b) Incremental deployment with large error and c) Distributed deployment

0 10 20 30 40 50 60 700

10

20

30

40

50

60

70

80


0 10 20 30 40 50 60 70 0

10

20

30

40

50

60

70

80


0 10 20 30 40 50 60 700

10

20

30

40

50

60

70

80

x Student Version of MATLAB

(a) (b) (c)

Fig. 8. Topologies obtained forθ = 2π5 with 100 nodes andRs = Rc = 8m using a) Incremental deployment with small

error b) Incremental deployment with more error and c) Distributed deployment


(a) (b)

(c) (d)

Fig. 9. Performance of the Incremental and Distributed deployment Algorithms for a deployment of 100 nodes in terms of(a) Coverage, (b) Edge Connectivity (c) Average Degree and (d) NET condition satisfaction


Student Version of MATLAB Student Version of MATLAB Student Version of MATLAB

(a) (b) (c)

Fig. 10. Comparison of communication graph and corresponding RNG[Incremental Deployment, 100 nodes,θ = 2π3 ,

small error ] a) communication graph b) RNG c) Dashed lines indicate edges in RNG that are not in communication graph.Dark lines indicate edges in communication graph that are not in RNG(none in this case).


(a) (b) (c)

Fig. 11. Comparison of communication graph and corresponding RNG[Incremental Deployment, 100 nodes,θ = 2π3 , large

error ] a) communication graph b) RNG c) Dashed lines indicate edges in RNG that are not in communication graph. Darklines indicate edges in communication graph that are not in RNG.


(a) (b) (c)

Fig. 12. Comparison of communication graph and corresponding RNG[Distributed Deployment, 100 nodes,θ = 2π3 ] a)

communication graph b) RNG c) Dashed lines indicate edges in RNG that are not in communication graph. Dark linesindicate edges in communication graph that are not in RNG.


The algorithms presented serve the purpose of illustrat-ing the use of NET graphs for tunable topology control.They do not optimize the coverage due to over-deployment.However, the insights gained can be used to design moresophisticated algorithms.

5. RELATED WORK

There has been a lot of work on topology control of staticand ad-hoc mobile networks in the past [1, 22]. Our workis closest in spirit to the “bottom-up” approach to topologycontrol discussed in [8]. Most earlier approaches beginwith the assumption of a well connected graph and pruneit to obtain a sparse, connected subgraph. In contrast, weprove that satisfying certain local conditions can guaranteeglobal connectivity and coverage properties. In [8], it isshown that if each node has one neighbor in a everyπ sectoraround it, then the network is guaranteed to be 1-connected.

Power control techniques for topology control have beenstudied extensively for ad hoc wireless networks. Central-ized algorithms to minimize the maximal power used pernode while maintaining the (bi)connectivity of the networkhave been proposed in [23]. This work also introduceddistributed heuristics for mobile networks. However, theseheuristics cannot guarantee the preservation of network con-nectivity. In [24] a distributed algorithm for constructing anRNG has been proposed. Each node increases it’s transmitpower till its neighbors cover the2π region around it. Weshow that to obtain an RNG it is sufficient to have one neigh-bor in each2π

3 sector and present sector based conditions forother proximity graphs. Our work is similar to the CBTC al-gorithm [2]. In CBTC, every node divides the space aroundit into cones of angleα and increases its transmission powertill in each cone it can reach a neighbor or till it reaches itsmaximum transmission power. It has been shown that, if thenetwork is connected when all nodes operate at maximumtransmission power, then forα ≤ 5π

6 , CBTC is guaranteedto result in a connected graph. The key difference is thatCBTC assumes the existence of a connected graph and pro-vides ways to prune it while retaining connectivity. On theother hand, the NET condition is a necessary condition thatcan guarantee global network properties. In such a scenario,a sector angle ofπ is sufficient to guarantee connectivity[8].

Sleep scheduling or on/off control is the principal topol-ogy control technique for over-deployed static networks.SPAN [3] is an energy efficient algorithm for topologymaintenance, where nodes decide whether to sleep or jointhe backbone (a connected dominating set) based on con-nectivity information supplied by a routing protocol. In AS-CENT [25], each node measures the number of neighborsand packet loss locally and makes a decision to join thenetwork topology or to perform some form of adaptation

(e.g. reducing its duty cycle to save energy). In SPAN andASCENT, the primary goal is to maintain connectivity inthe network while we also seek to maintain a sparse topol-ogy and increase sensing coverage. CCP [26] is a sleepscheduling protocol to achieve tunable degrees of coverageand connectivity. The degree of coverage is defined as theminimum number of nodes that cover any point in networkregion. Therefore, achieving different degrees of coverageand connectivity are not opposing goals.

In the context of networks with controllable placement,the problem of topology control is closely linked to the prob-lem of deployment. Several deployment strategies have alsobeen proposed in the Robotics literature. An algorithm forincremental deployment team for mobile robots is presentedin [27]. It maximizes the sensing coverage while simulta-neously ensuring that the robots maintain line-of-sight con-nectivity. A potential field based self-deployment algorithmfor a group of mobile nodes is presented in [21]. It max-imizes coverage with the constraint that each node has atleastk neighbors, wherek is an input parameter to the al-gorithm. These algorithms maximize coverage but do notprovide guarantees on connectivity.

In [16] a robot deploys a network with a simple gridtopology using the deployed nodes as landmarks withoutusing a map of the environment or localization. In [17] astatic sensor network is deployed using an unmanned aerialvehicle (UAV). Typically, such algorithms have focussed onthe control strategies of the robot rather than the propertiesof the network. By providing rules that can be easily com-bined with robot controllers, our work provides a means ofintegrating the two objectives.

6. CONCLUSION AND FUTURE WORK

We present a classification of sensor network topologies onthe basis of the node mobility (static or mobile) and the na-ture of deployment (controlled or uncontrolled). The maincontribution of this paper is a unifying framework for de-signing mechanisms for completely distributed and tunabletopology control for all classes of topologies. It is based ona simple, local and geometric Neighbor-Every-Theta (NET)condition. We show that it is possible to obtain tunable cov-erage and connectivity tradeoffs by varying a single parame-ter - the sector angleθ. The NET condition results from ouranalysis of proximity graphs as representative of communi-cation graphs of wireless networks. Proofs of local, geomet-ric conditions that guarantee relative neighborhood, Gabrieland Delaunay graphs are presented. We also prove the sym-metric placement condition for maximizing coverage underthe NET condition. These findings could well be of inde-pendent interest. The NET based framework can supporta number of existing power control and sleep schedulingschemes suitable for uncontrolled deployments. The use


of distributed algorithms for controlled deployment of bothstatic and mobile nodes is simulated to concretely demon-strate use of the local geometric rules to implement globalcoverage and connectivity tradeoffs.

We are currently analyzing sparseness and spanner prop-erties of the topologies resulting from the deployment al-gorithms. Proof of the conjecture on the k-connectivity ofNET graphs is also of interest. Extensions of results for non-circular communication ranges will also be investigated.

Acknowledgements

The authors would like to thank Dr. Raissa D’Souza at Mi-crosoft Research Labs for reading the manuscript and givinghelpful feedback.

7. REFERENCES

[1] P. Santi, “Topology control in wireless ad hoc andsensor networks,” Technical Report IIT-TR-04/2003,Institute for Informatics and Telematics, Pisa, Italy,2003.

[2] R. Wattenhofer, L. Li, P. Bahl, and Y. M. Wang, “Acone-based distributed topology-control algorithm forwireless multi-hop networks,” IEEE/ACM Transac-tions on Networking, Feb 2005.

[3] B. Chen, K. Jamieson, H. Balkrishnan, and R. Morris,“Span: an energy-efficient coordination algorithm fortopology maintenance in ad hoc wireless networks,”ACM Wireless Networks Journal, vol. 8, no. 5, 2002.

[4] W. Kaiser, G. Pottie, M. Srivastava, G .S. Sukhatme,J. Villasenor, and D. Estrin, “Networked infomechani-cal systems (nims) for ambient intelligence,”AmbientIntelligence, Springer-Verlag, 2004.

[5] N. Xu, S. Rangwala, K. Chintalapudi, D. Ganesan,A. Broad, R. Govindan, and D. Estrin, “A wirelesssensor network for structural monitoring,” inACMConference on Embedded Networked Sensor Systems(Sensys’04), November 2004, pp. 13–24.

[6] UC Berkeley and MLB Co., “29 palms fixed/mobileexperiment,” http://robotics.eecs.berkeley.edu/-pister/29Palms0103, last visited Feb 2005.

[7] P. Juang, H. Oki, Y. Wang, M. Martonosi, L. Peh,and D. Rubenstein, “Energy-efficient computing forwildlife tracking: Design tradeoffs and early experi-ences with zebranet,” inTenth International Confer-ence onArchitectural Support for Programming Lan-guages and Operating Systems, San Jose, CA, Oct2002.

[8] R. D’Souza, D. Galvin, C. Moore, and D. Randall,“A local topology control algorithm that guaranteesglobal connectivity and geometric routing,” submittedfor publication, 2004.

[9] E. H. Jennings and C. M. Okino, “Topology controlfor efficient information dissemination in ad-hoc net-works,” in International Symposium on PerformanceEvaluation of Computer and Telecommunication Sys-tems, July 2002.

[10] B. Karp and H.T. Kung, “Greedy perimeter statelessrouting for wireless networks,” in6th Annual Inter-national Conference on Mobile Computing and Net-working, MobiCom’00, Boston, MA, August 2000,ACM/IEEE, pp. 243–254.

[11] J. Gao, L. J. Guibas, J. E. Hershberger, L. Zhang, andA. Zhu, “Geometric spanner for routing in mobilenetworks,” in2nd ACM international symposium onMobile ad hoc networking computing, MobiHoc’01,Long Beach, CA, 2001, pp. 45–55.

[12] J. Cortes, S. Martinez, and F. Bullo, “Robust ren-dezvous for mobile autonomous agents via proximitygraphs in d dimensions,”IEEE Transactions on Auto-matic Control, in submission.

[13] J. Zhao and R. Govindan, “Understanding packetdelivery performance in dense wireless sensor net-works,” in The First ACM Conference on EmbeddedNetworked Sensor Systems (Sensys’03), Nov 2003.

[14] M. Zuniga and B. Krishnamachari, “Analyzing thetransitional region in low power wireless links,” inFirst IEEE International Conference on Sensor and Adhoc Communications and Networks (SECON), SantaClara, CA, October 2004.

[15] R. Diestel, Graph Theory, Graduate Texts in Mathe-matics, second edition, vol. 173, Springer Verlag, NewYork, NY, 2000.

[16] M. Batalin and G. Sukhatme, “Coverage, explorationand deployment by a mobile robot and communica-tion network,” In Telecommunication Systems Jour-nal, Special Issue on Wireless Sensor Networks, vol.26, no. 2, pp. 181–196, 2004.

[17] P. I. Corke, S. E. Hrabar, R. Peterson, D. Rus, S. Sari-palli, and G. S. Sukhatme, “Autonomous deploymentand repair of a sensor network using an unmannedaerial vehicle,” inIEEE International Conference onRobotics and Automation, April 2004, pp. 3602–3609.

[18] K. Krizman, T. E. Biedka, and T. S. Rappaport, “Wire-less position location: fundamentals, implementation


strategies and source of error,” inIEEE 47th VehicularTechnology Conference, 1997, pp. 919–923.

[19] B. Gerkey, R. T. Vaughan, A. Howard, G Sukhatme,and M. J. Mataric, “Most valuable player: A robotdevice server for distributed control,” inIEEE/RSJ In-ternational Conference on Intelligent Robots and Sys-tems, IROS’02, 2002, pp. 299–308.

[20] A. Howard, M. J. Mataric, and G. S. Sukhatme, “Mo-bile sensor network deployment using potential fields:A distributed, scalable solution to the area coverageproblem,” in Distributed Autonomous Robotic Sys-tems, 2002, pp. 299–308.

[21] S. Poduri and G. Sukhatme, “Constrained coveragefor mobile sensor networks,” inIEEE InternationalConference on Robotics and Automation, May 2004,pp. 165–172.

[22] G. Srivastava, P. Boustead, and J. F. Chicharo, “Acomparison of topology control algorithms for ad-hocnetworks,” in Australian Telecommunications Net-works Applications Conference, ATNAC, Melbourne,2003.

[23] R. Ramanathan and R. Rosales-Hain, “Topologycontrol of multihop wireless networks using transmitpower adjustment,” inIEEE INFOCOM, 2000, pp.404–413.

[24] S. A. Borbash and E. H. Jennings, “Distributedtopology control algorithm for multihop wireless net-works,” in World Congress on Computational Intelli-gence, May 2002.

[25] A. Cerpa and D. Estrin, “Ascent: Adaptive self-configuring sensor networks topologies,”IEEE Trans-actions Journal on Mobile Computing Special Issueon Mission-Oriented Sensor Networks, vol. 3, no. 3,pp. 272–285, July-September 2004.

[26] X. Wang, G. Xing, Y. Zhang, C. Lu, R. Pless, andC. Gill, “Integrated coverage and connectivity con-figuration in wireless sensor networks,” inFirst ACMConference on Embedded Networked Sensor Systems,Sensys’03, Nov 2003.

[27] A. Howard, M. J. Mataric, and G. S. Sukhatme,“An incremental self-deployment algorithm for mobilesensor networks,”Autonomous Robots, Special Issueon Intelligent Embedded Systems, vol. 13, no. 2, pp.113–126, 2002.


A UNIFYING FRAMEWORK FOR TUNABLE TOPOLOGY CONTROL ...

Documents