1 Using Local Geometry for Tunable Topology Control in...

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... 1 Using Local Geometry for Tunable Topology Control in Sensor Networks Sameera Poduri 1 , Member, IEEE, Sundeep Pattem 2 , Member, IEEE, Bhaskar Krishnamachari 1,2 , Member, IEEE, and Gaurav S. Sukhatme 1,2 , Senior Member, IEEE Abstract—Neighbor-Every-Theta (NET) graphs are such that each node has at least one neighbor in every theta angle sector of its communication range. We show that for θ<π, NET graphs are guaranteed to have an edge-connectivity of at least b 2π θ c, even with an irregular communication range. Our main contribution is to show how this family of graphs can achieve tunable topology control based on a single parameter θ. Since the required condition is purely local and geometric, it allows for distributed angle-only topology control. For a static network scenario, a power control algorithm based on the NET condition is developed for obtaining k-connected topologies and shown to be significantly efficient compared to existing schemes. In controlled deployment of a mobile network, control over positions of nodes can be leveraged for constructing NET graphs with desired levels of network connectivity and sensing coverage. To establish this, we develop a potential fields based distributed controller and present simulation results for a large network of robots. Lastly, we extend NET graphs to 3D and provide an efficient algorithm to check for the NET condition at each node. This algorithm can be used for implementing generic topology control algorithms in 3D. Index Terms—topology control, k-connectivity, distributed deployment of mobile nodes, 3D networks 1 I NTRODUCTION Topology control for ad-hoc wireless and sensor net- works has traditionally only dealt with uncontrolled de- ployments, where there is no explicit control on positions of nodes [1]–[3]. The primary mechanisms proposed are power control and sleep scheduling. These methods involve removing edges from an existing, well connected communication graph in order to save power while ensuring that the resultant sub-graph preserves connec- tivity. Controlled deployments, feasible when positions of individual nodes can be altered, present a different and interesting scenario for topology control. Since con- nectivity properties directly depend on the positions of nodes, position control can be leveraged for effective topology control. There is increasing evidence that such deployments will be possible in, as well as be required by, a number of future sensor network applications. Con- sider the following examples - sensors implanted in civil structures at select, unobtrusive locations e.g. SHM [4], sensors in water bodies either anchored to stay in place or mounted on boats e.g. NAMOS [5], sensors that are not inexpensive enough to afford high redundancy e.g. MASE [6], small and medium scale networks main- tained by a robot or human e.g. energy harvesting using robots [7], and mobile sensor networks e.g. NIMS [8]. Traditional approaches are ill-suited in such scenarios 1 Department of Computer Science, 2 Department of Electrical Engineering, University of Southern California, Los Angeles, CA 90089. Partial support for this research was provided by the NSF through the STC (CCR-0120778), ITR (CNS-0325875), CAREER (CNS-0347621 and IIS-0133947), and NeTS- NOSS programs (CNS-0627028 and CNS-0520305). Sukhatme acknowledges support from the Okawa Foundation, and Poduri was supported by a merit fellowship from the USC Women in Science and Engineering (WiSE) Program. since they are not designed to exploit control of the motion and placement of the nodes. In this paper, we develop a general approach that supports traditional methods like power control and also allows new designs for controlled deployments. Our main contributions are: a k-connectivity result, based on local conditions and independent of communication model topology control algorithms based on the k- connectivity result for a) power control with static nodes, and b) distributed deployment of mobile nodes 3D extensions of 2D results and an efficient algo- rithm for topology control in 3D. We define Neighbor-Every-Theta (NET) graphs in which each node (except those at the boundary) have at least one neighbor in every θ sector of its communication range. Our key theoretical result is that for θ<π,a NET graph has an edge-connectivity of at least b 2π θ c irrespective of the communication model. This implies that the condition only depends on the angles between the neighbors of each node and holds for arbitrary edge-lengths. This feature is particularly relevant for sensor networks using low-power radios that have ir- regular communication range [9], [10]. For the special case with an idealized disk communication model, we derive conditions for maximizing the sensing coverage area (defined as the total area sensed by at least one node) while satisfying the k-connectivity constraint, and for obtaining proximity graphs such as the Relative Neighborhood Graph. NET graphs are naturally suited for distributed power control. We implement a typical power control proto-

Transcript of 1 Using Local Geometry for Tunable Topology Control in...

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Using Local Geometry for Tunable TopologyControl in Sensor Networks

Sameera Poduri1, Member, IEEE, Sundeep Pattem2, Member, IEEE,Bhaskar Krishnamachari1,2, Member, IEEE, and Gaurav S. Sukhatme1,2, Senior Member, IEEE

Abstract—Neighbor-Every-Theta (NET) graphs are such that each node has at least one neighbor in every theta angle sector of itscommunication range. We show that for θ < π, NET graphs are guaranteed to have an edge-connectivity of at least b 2π

θc, even with

an irregular communication range. Our main contribution is to show how this family of graphs can achieve tunable topology controlbased on a single parameter θ. Since the required condition is purely local and geometric, it allows for distributed angle-only topologycontrol. For a static network scenario, a power control algorithm based on the NET condition is developed for obtaining k-connectedtopologies and shown to be significantly efficient compared to existing schemes. In controlled deployment of a mobile network, controlover positions of nodes can be leveraged for constructing NET graphs with desired levels of network connectivity and sensing coverage.To establish this, we develop a potential fields based distributed controller and present simulation results for a large network of robots.Lastly, we extend NET graphs to 3D and provide an efficient algorithm to check for the NET condition at each node. This algorithm canbe used for implementing generic topology control algorithms in 3D.

Index Terms—topology control, k-connectivity, distributed deployment of mobile nodes, 3D networks

F

1 INTRODUCTION

Topology control for ad-hoc wireless and sensor net-works has traditionally only dealt with uncontrolled de-ployments, where there is no explicit control on positionsof nodes [1]–[3]. The primary mechanisms proposedare power control and sleep scheduling. These methodsinvolve removing edges from an existing, well connectedcommunication graph in order to save power whileensuring that the resultant sub-graph preserves connec-tivity. Controlled deployments, feasible when positionsof individual nodes can be altered, present a differentand interesting scenario for topology control. Since con-nectivity properties directly depend on the positions ofnodes, position control can be leveraged for effectivetopology control. There is increasing evidence that suchdeployments will be possible in, as well as be requiredby, a number of future sensor network applications. Con-sider the following examples - sensors implanted in civilstructures at select, unobtrusive locations e.g. SHM [4],sensors in water bodies either anchored to stay in placeor mounted on boats e.g. NAMOS [5], sensors thatare not inexpensive enough to afford high redundancye.g. MASE [6], small and medium scale networks main-tained by a robot or human e.g. energy harvesting usingrobots [7], and mobile sensor networks e.g. NIMS [8].Traditional approaches are ill-suited in such scenarios

1Department of Computer Science, 2Department of Electrical Engineering,University of Southern California, Los Angeles, CA 90089. Partial supportfor this research was provided by the NSF through the STC (CCR-0120778),ITR (CNS-0325875), CAREER (CNS-0347621 and IIS-0133947), and NeTS-NOSS programs (CNS-0627028 and CNS-0520305). Sukhatme acknowledgessupport from the Okawa Foundation, and Poduri was supported by amerit fellowship from the USC Women in Science and Engineering (WiSE)Program.

since they are not designed to exploit control of themotion and placement of the nodes.

In this paper, we develop a general approach thatsupports traditional methods like power control andalso allows new designs for controlled deployments. Ourmain contributions are:

• a k-connectivity result, based on local conditionsand independent of communication model

• topology control algorithms based on the k-connectivity result for a) power control with staticnodes, and b) distributed deployment of mobilenodes

• 3D extensions of 2D results and an efficient algo-rithm for topology control in 3D.

We define Neighbor-Every-Theta (NET) graphs in whicheach node (except those at the boundary) have at leastone neighbor in every θ sector of its communicationrange. Our key theoretical result is that for θ < π, aNET graph has an edge-connectivity of at least b 2πθ cirrespective of the communication model. This impliesthat the condition only depends on the angles betweenthe neighbors of each node and holds for arbitraryedge-lengths. This feature is particularly relevant forsensor networks using low-power radios that have ir-regular communication range [9], [10]. For the specialcase with an idealized disk communication model, wederive conditions for maximizing the sensing coveragearea (defined as the total area sensed by at least onenode) while satisfying the k-connectivity constraint, andfor obtaining proximity graphs such as the RelativeNeighborhood Graph.

NET graphs are naturally suited for distributed powercontrol. We implement a typical power control proto-

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col using satisfaction of NET condition at each non-boundary node as the termination criterion and showthat a power-efficient network with edge connectivityb 2πθ c can be achieved even with realistic, irregular links.This is in contrast to the sector based topology con-trol algorithms in literature including CBTC [11], [12],that rely on an idealized disk communication model toguarantee connectivity properties. In CBTC based powercontrol [13], for a network to be k-connected, each nodemust either have a neighbor in every θ = 2π

3k sector oroperate at full power. Our results imply that a sectorangle of θ = b 2πk c is sufficient for k-connectivity. Athree times larger value of sector angle implies a muchlower power requirement. We present simulation resultsto establish this in Section 6.

Scenarios where it is possible to control node posi-tions are ideal for obtaining topologies that are NETgraphs. Given the properties guaranteed by the NETcondition, an external agent deploying a static sensornetwork can make decisions on the best positions fordeployment of new nodes based on the geometry of theexisting network. Self-deploying mobile nodes can usethis condition to decide their motion strategy. We presenta distributed algorithm for self-deployment of mobilenodes to concretely demonstrate the use of the localand geometric conditions to implement coverage andconnectivity tradeoffs. The algorithm involves achiev-ing NET condition satisfaction through purely pair-wisenegotiations between neighbors. The performance ofthis algorithm is studied through an implementationon the Player/Stage simulation platform [14]. Resultsillustrate the tunable tradeoff between connectivity andcoverage and show that the distributed algorithm canachieve approximations of some coverage-maximizingtiling structures that also satisfy NET conditions.

It has been argued [15] that topology control in threedimensions (3D) is much more complex. There are sev-eral operations that are common to a wide range oftopology control algorithms that have increased andsometimes prohibitively high complexity in 3D. As aresult, there is very little work on topology controlalgorithms in 3D. We argue that controlled deploymentand NET graphs are well suited for sensor networks in3D and present extensions to our 2D results. We proposean efficient algorithm for identifying the largest emptycone of a node’s communication range. This algorithmcan be used in conjunction with several deployment andtopology control mechanisms for implementing their 3Dextensions.

The paper is organized as follows. The next sectiondefines the problem and section 3 presents NET graphsand analysis of their connectivity properties which areextended to 3D in section 4. Sections 5 and 6 presentapplications of NET graphs for distributed deploymentof mobile robots and distributed power control respec-tively. Finally we put our work in context with a dis-cussion of related work in section 7 and conclude insection 8.

2 PROBLEM FORMULATION

Local conditions that guarantee global network proper-ties are the key to designing distributed topology controlalgorithms. In particular, we seek local conditions thatguarantee global k-edge-connectivity of the network.Once such conditions are found, they can be integratedwith controls available in order to design topology con-trol algorithms. Further, it is important to understand ifthe results can be extended to 3D networks in an efficientmanner. A detailed description of these three problemsfollows.

Problem 1: k-connectivity certificates

Given a network, find local geometric conditions between nodepositions that can guarantee global k-edge-connectivity.

A graph is said to be k-edge-connected if at least k edgesmust be removed to disconnect it [16]. By Menger’stheorem, this is equivalent to saying that there exist atleast k edge-disjoint paths between any two nodes in thegraph. The graph that we consider is the communicationnetwork where two nodes are said to have an edgebetween them if they can communicate. In this case, highedge-connectivity is desirable because it implies highfault-tolerance and path diversity.

By ‘local’ we mean that each node only has infor-mation about positions of its communicating neighborsrelative to its own position. The global coordinates ofnodes are not available. The conditions we seek mustbe based only on this local information so that eachnode can independently decide whether it satisfies thecondition.

Problem 2: Topology Control Algorithms

How can the local geometric conditions for k-connectivity beintegrated with the controls available, like (1) communicationpower and (2) node positions, for efficient topology control?

We have the following sub-problems.

a) Distributed deployment of mobile nodes: GivenN mobile nodes, design a distributed control law such thatthey move to maximize sensing coverage while maintainingglobal k-connectivity.

b) Power control in a static network: Given astatic network of N nodes, how should they choose theircommunication power such that the network is k-connectedand the energy is minimized?

Lastly, we study the extension to networks in threedimensions.

Problem 3: Extensions to networks in three dimensions

Do the local geometric conditions for 2D networks extend to3D networks?

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(a) (b)

Fig. 1. Illustration of the NET condition. Red circles areneighbors. In a) node does not satisfy NET conditionbecause it has an empty sector greater than θ. In b) nodesatisfies NET condition.

3 NEIGHBOR-EVERY-THETA (NET) GRAPHS

In this section, we will define Neighbor-Every-Thetagraphs and show that they give us k-connectivity cer-tificates.

Definition 3.1: The Neighbor-Every-Theta (NET) con-dition (Fig. 1) for a node embedded in the 2D plane isdefined as requiring at least one symmetric neighbor inevery θ sector of its communication range.Nodes A and B are symmetric neighbors if A cancommunicate to B and B can communicate to A.

For finite networks, nodes on the network boundarycannot satisfy such a condition. The boundary of a net-work can be defined as a cycle of nodes such that everyother node lies inside the cycle [17]. A node that does notbelong to the boundary is called an interior node.

Definition 3.2: A NET graph is one in which everyinterior node satisfies the NET condition for a given θ.

We now analyze the connectivity and coverage prop-erties of NET graphs. For large networks where thenumber of boundary nodes is small compared to thenetwork size, we show that NET graphs have an edgeconnectivity of at least b 2πθ c, independent of the com-munication model. With the stronger assumption of aidealized disk communication model, NET graphs, forspecific values of θ, contain proximity graphs such as theRelative Neighborhood Graph (RNG). An upper boundon the sensing coverage is shown to be obtained from asymmetric arrangement of nodes and can be computedas a function of θ. NET graphs form a family of graphsbased on the single parameter θ - as θ becomes smaller,the graphs become denser with an increasing level ofconnectivity.

3.1 Connectivity Analysis of NET Graphs

The edge-connectivity of any graph is at most its min-imum node degree and can in general be arbitrarilylow irrespective of the node degree [16]. For somespecial graphs, higher node degree implies higher edge-connectivity. One example of such a graph is the ran-dom geometric graph, where an average node degree

(a) ‖ V1 ‖= 1 (b) ‖ V1 ‖= 2

Fig. 2. The red nodes and edges represent G1, dottededges form the cut set C.

of O(logN) (where N is the network size) guaranteesan edge-connectivity of 1 with high probability [18].It turns out that for NET graphs also there is relationbetween minimum node degree and edge-connectivity.In NET graphs every interior node has a degree of atleast b 2πθ c. We will show that the edge-connectivity isalso at least b 2πθ c. We will now formally establish thisproperty by first considering the simple case of graphsthat are very large so that edges close to the boundaryare not significant. An extreme case of such graphs arethe “boundary-less” graphs defined below. Later, we willanalyze the impact of boundary nodes on connectivity.

Definition 3.3: A NET graph is boundary-less if everynode satisfies the NET condition. For θ < π, such a graphmust span the entire 2D plane.

Definition 3.4: Given a graph G = {V,E}, a cut set is aset of edges E′ ⊆ E such that the graph G′ = {V,E−E′}has more than one component. The edge-connectivitya graph, λ, is the size of its smallest cut set, C. Notethat if G is connected, C must divide it into exactly twocomponents.

Definition 3.5: For a graph embedded in the Euclideanplane a cut is a curve that partitions the graph into twoor more components.

Theorem 3.6: For θ < π, if a boundary-less NET graphis connected, then it has an edge-connectivity λ ≥ b 2πθ cProof: Consider a NET graph G = {V,E}. Let C ⊆ E bethe smallest cut set of G so that λ =‖ C ‖. Let the twocomponents of G′ = {V,E − E′} be G1 = {V1, E1} andG2 = {V2, E2}. WLOG, ‖ V1 ‖≤‖ V2 ‖.

If ‖ V1 ‖= 1 then ‖ C ‖≥MinDegree ≥ b 2πθ c (Fig 2(a)).If ‖ V1 ‖= 2 then ‖ C ‖≥ 2

(b 2πθ c − 1

)≥ b 2πθ c since θ < π

(Fig. 2(b)).Suppose ‖ V1 ‖> 2. Construct the convex hull, H1 of

V1. Then H1 ⊆ V1. Define angle φi at hi ∈ H1 as shownin Fig 3. There will exist at least 3 vertices1 hi such thatφi > π i.e. , ∠hi < π. WLOG, assume that φ1, φ2 andφ3 > π. Since G = {V,E} is a NET graph, for i = 1, 2, 3,φi must contain ≥ bπθ c edges ∈ C. Therefore, ‖ C ‖≥3bπθ c ≥ b

2πθ c. �

1. Since H is a convex polygon, ∠hi ≤ π,∀i. Suppose all except2 internal angles are = π, say 0 < ∠h1,∠h2 < π and ∠hi = π fori = 3, 4, ...k, where k is number of vertices of H , then

Pk1 ∠hi = ((k−

2)π+∠h1 +∠h2) > (k−2)π. Contradiction sincePk

1 ∠hi = (k−2)π.

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Fig. 3. ‖ V1 ‖≥ 3 The red nodes and edges represent G1,the dotted edges form the cut set C. The correspondingconvex hull H is shown with solid black lines.

The above result holds for the general case wherethe graph has boundary nodes provided the cut iscompletely in the interior of the network, i.e. if all thenodes in V1 satisfy the NET condition. If on the otherhand, the cut mostly consists of boundary edges thenit is not interesting because it is unlikely to impactnetwork performance. Now consider cuts that intersectthe network boundary twice - when entering and leavingthe graph (fig. 4(a)). WLOG, assume that the cut isminimal in length i.e. it is the shortest curve corre-sponding to its cut set. The case when the cut intersectsitself is covered by the above theorem - the number ofedges cut inside the loop alone must be at least b 2πθ c.Assume that the cut does not intersect itself and byway of contradiction, that it disconnects the networkby cutting less than b 2πθ c edges. We will now analyzethe nature of such a cut and prove that away from thenetwork boundary, the distance along the cut betweentwo consecutive edges must be less than Rc.sec(θ/2),where Rc is an upper bound on the edge-length. Thisimplies that away from the boundary the cut cannot belonger than b 2πθ c · Rc.sec(θ/2). If θ is not close to π thisexpression is bounded. For example, for θ = 0.9π it is≈ 13 ·Rc and for θ = 2π

3 it is 6 ·Rc. For large networks, a“short” cut like this can only exist close to the boundary(fig. 5(a)) or if the boundary is “pinched” (fig. 5(b)).

Lemma 3.7: For θ < π, the length of a minimal cutbetween two consecutive edges in a boundary-less NETgraph is less than Rc · sec(θ/2).Proof: Let L be the cut and l its lengthbetween two consecutive edges it cuts, say(u1, v1) and (u2, v2) (figure 4(b)). Then l ≤max{d(u1, u2), d(u1, v2), d(v1, u2), d(v1, v2)}, whered() is the euclidean distance function. Let the nodepair that leads to maximum distance be P1, P2. NowP1 must have at least one neighbor in each of the twoθ sectors adjoining

−−−→P1P2. Let these be Q1 at an angle

0 ≤ α1 ≤ θ and Q′1 at an angle 0 ≤ α′1 ≤ θ with−−−→P1P2.

0 ≤ α1 + α′1 ≤ θ < π. WLOG, α1 < π/2. Similarly Q1

must have a neighbor Q2 at an angle α2 ≤ θ, and soon till some Qm−1Qm intersects

−−−→P1P2. Note that the

(a) (b)

Fig. 4. (a) A finite graph where all non-boundary nodessatisfy the NET condition. The think line L is a cut throughthe graph (b) L cuts polygons

point of intersection cannot be in between P1 and P2

because otherwise L would cut Qm−1Qm contradictingthe assumption that (u1, v1) and (u2, v2) are consecutiveedges.

We have,

l = ‖P1P2‖≤ r1 · cos(α1) + r2 · cos(α1 + α2 − π) + · · ·

+rm · cos(α1 + α2 + · · ·+ αm − (m− 1)π)(where π/2 > α1 > α1 + α2 − π > · · ·> α1 + α2 + · · ·+ αm − (m− 1)π) > −π/2)

≤ Rc(cos(α1) + cos(α1 + α2 − π) + · · ·+cos(α1 + α2 + · · ·+ αm − (m− 1)π))(where Rc is an upper bound on edge length)

≤ Rc((cos(α1) + cos(α1 + θ − π) + · · ·+cos(α1 − (m− 1)(π − θ)))(since cos is a concave function in [−π/2, π/2])

= Rcsin(m·(θ−π)

2

)· cos

(α1 + (m−1)·(θ−π)

2

)sin(θ−π

2

)≤ Rc

1cos(θ/2)

�It must be emphasized that the connectivity result only

needs the largest edge in the network to be bounded andholds even for a non-ideal and irregular communicationmodels. In the following subsections, we present furtherinteresting properties of NET graphs with a strongerassumption of an idealized disk communication model.

3.2 Proximity GraphsProximity graphs such as the Relative NeighborhoodGraph (RNG), Gabriel Graph (GG) and the DelaunayGraph (DelG) have several properties such as connec-tivity, sparseness, efficient network routes, etc. that makethem desirable network topologies. We will now analyzeconditions under which NET graphs will contain theseproximity graphs. Here we assume an idealized diskcommunication model but allow each node to have adifferent communication range. Further, the number of

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(a) (b)

Fig. 5. Degenerate cases of cuts through NET graphsthat result in poor connectivity. (a) A cut close to theboundary (b) pinched boundary

boundary nodes is assumed to be small compared to thenetwork size. The results are non-trivial since the edge-lengths in NET graphs are restricted by the communi-cation range of nodes, while in proximity graphs theydepend only on the relative positions of nodes and verylong edges are possible.

The set of edges E for various proximity graphs isdefined as follows [16]. In what follows, we use the nameand location of a node interchangeably.

Disk graph (DG(V,E,R)): The directed graph con-taining all outgoing edges of a node, u ∈ V , not longerthan R(u).

E = {(u, v)|u, v ∈ V and d(u, v) ≤ R(u)}

Given positive r ∈ R, let C(p, r) be the circle consistingof points whose distance from point p is strictly less thanr. Define the lune, denoted L(p, q), to be the intersectionof two circles, both of radius d(p, q), centered at thesepoints, 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 both u and v.Equivalently, (p, q) is an edge if L(p, q) ∩ V = ∅.

E = {(u, v)|u, v ∈ V and ∃ no w ∈ V 3

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

We assume that each node, u, in the network has acommunication range R(u) so that the communicationgraph of the network is a disk graph. Our next step is tofind conditions under which the RNG of the network iscontained in the communication graph. This is equiva-lent to finding conditions under which no edge (u, v) ofan RNG is longer than either R(u) or R(v). The followingtheorem presents this condition.

Theorem 3.8: If each node x ∈ V has at least one neigh-bor in every 2π

3 sector of C(x,R(x)), the communicationgraph is a supergraph of RNG(V ). Moreover, 2π

3 is thelargest angle that satisfies this property.Proof: Consider any node x ∈ V . Suppose x has at leastone neighbor in every 2π

3 sector of C(x,R(x)). We firstshow that for any node y outside C(x,R(x)), the edge(x, y) /∈ RNG(V ). The lune L(x, y) will contain a sectorof at least 2π

3 (Fig.6). By premise, ∃ a node in L(x, y). This

Fig. 6. RNG sector condition. The circle of radiusd(x, y) ≥ Rc subtends an angle ≤ 2π

3 at x.

implies that RNG(V ) does not have any edges incidenton x that are longer than R(x).

Next we show that for any node z ∈ V insideC(x,R(x)) such that d(x, z) > R(z) i.e. (x, z) ∈DG(V,E,R) but (z, x) /∈ DG(V,E,R), the edge (x, z) /∈RNG(V ). Since z also has a neighbor in every 2π

3 sectorof C(z,R(z)) and d(x, z) > R(z), from the above argu-ment (x, z) /∈ RNG(V ).

Therefore RNG(V ) ⊆ DG(V,E,R(x)).Now suppose a node x′ ∈ V ′ has two neighbors with

a sector angle of 2π3 +δ (δ > 0) between them (Fig.??). We

can place a node y′ outside C(x′, Rc) such that the edge(x′, y′) ∈ RNG(V ′). Therefore, 2π

3 is the largest angle forwhich this condition holds.�

We have shown that for θ ≤ 2π3 , NET graph contains

the RNG assuming an idealized disk communicationmodel. A similar property holds for GG and DelG forθ = 2arccos( r

R(x) ) where r is the distance between thenode and its neighbor. We skip the proof here due tospace restrictions.

In real networks, it is not possible for the boundarynodes to satisfy the conditions required by the theorems.We show using simulated deployments (in section 5)that the assertions can be validated in spite of theseexceptions.

3.3 Coverage Analysis of NET Graphs

Having listed the conditions that guarantee global con-nectivity properties, we now turn to the problem ofmaximizing coverage. We assume that all nodes have aidealized disk sensing model with a sensing radius of Rs.Maximizing sensing coverage is equivalent to packingproblem with a constraint placed on the communicationneighbors. This is a very hard problem given an irregularcommunication model and even harder to solve locally.Therefore, we make a simplifying assumption that thecommunication model is also an idealized disk witha radius Rc for all nodes. Suppose that in order tosatisfy the NET sector conditions, a node must havek = b 2πθ c neighbors. From the node’s local perspective,all neighbors must be located on the perimeter of thecommunication range to maximize coverage. Intuitively,the nodes must also be placed symmetrically on the

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perimeter. We prove this result for the special case whenRc = Rs.

Theorem 3.9: For Rs = Rc, the area coverage is maxi-mized when the k ≥ 3 nodes are placed at the edges ofk disjoint 2π

k sectors of C(x,Rc).Proof: Consider a node x and k nodes y1, y2, ...., yk placedon the perimeter of C(x,Rc). Let these nodes be placedin anticlockwise order at angles β1 = 0, β2, ...., βk respec-tively. Define

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

We need to find θi such that the total area covered bythese k + 1 nodes is maximized. The open disks of allnodes lie within C(x, 2Rc) and are tangent to this disk atexactly one point (Ti) each. The disks of adjacent nodesi and i + 1 intersect at point Ii (and at X). The totalcoverage lies between πR2

c and 4πR2c and is maximized

when the areak∑i=1

TiIiTi+1 is minimum (Fig.7).

TiIiTi+1

= TiXTi+1 − IiYiTi − IiYi+1Ti+1 − IiYiXYi+1

= (2Rc)2 ·θi2− 2R2

c ·θi2− 1

2(2Rc sin

θi2

)(2Rc cosθi2

)

= R2c(θi − sin θi)

k∑i=1

TiIiTi+1 = R2c(

k∑i=1

θi −k∑i=1

sin θi)

= R2c(2π −

k∑i=1

sin θi) (2)

The problem now reduces to finding maxk∑i=1

sin θi sub-

ject tok∑i=1

θi = 2π and 0 ≤ θi ≤ 2π. Since sin is non-

negative and concave in [0, π] and non-positive in (π, 2π],it follows that the solution is θi = 2π

k , 1 ≤ i ≤ k.�For k = 3, 4, 6, it is possible to place nodes such that

each node has its neighbors placed symmetrically onits communication range. The resulting communicationgraph will be a tiling of the space: hexagonal, square andtriangle for k = 3, 4 and 6 respectively. In these cases,since each node maximizes coverage locally, the totalcoverage of the network will also be the maximized. Forvalues of k other than 3, 4 and 6 such an arrangement isnot possible since the corresponding tiling structures donot exist. Therefore in these cases, the local condition foroptimizing coverage will not necessarily optimize globalcoverage. Due to their symmetry, tiling graphs possesssome other desirable properties. It can be verified thatthe hexagonal tiling (θ = 2π

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

2 ) aswell. The triangle tiling (θ = π

3 ) is a DelG.Based on the above theorem, we can compute an

upper bound on the maximum achievable coverage of a

Fig. 7. Coverage with k nodes placed on the communi-cation perimeter of node x. The shaded area is the totalcoverage.

NET graph as a function of θ. Consider the arrangementin Fig. 7 with all the neighbors placed symmetricallyaround x. If each overlap area within the sensing rangeof x is divided by the number of nodes that cover it,then the sum of these weighted areas will give the max-imum possible per-node coverage. In this computationwe ignore the boundary nodes which cannot have asymmetric placement of neighbors. Therefore, this is anasymptotic bound. This upper bound on coverage isplotted in Fig. 11 along with the lower bound for edge-connectivity derived in theorem 3.9.

In the next section we extend NET graphs to 3D andstudy their connectivity and coverage properties.

4 NET GRAPHS IN THREE DIMENSIONS

Network configuration in 3D is significantly more com-plex than in 2D [15]. There are several operations that arecommon to a wide range of sensor network configura-tion algorithms in 2D that have increased and sometimesprohibitively high complexity in 3D. We argue that con-trolled deployment and NET graphs are well suited forsensor networks in 3D and present extensions to our 2Dresults. We propose an efficient algorithm for identifyingthe largest empty cone of a node’s communication range.This algorithm can be used in conjunction with a numberof generic deployment and topology construction mech-anisms for implementing their 3D extensions.

A node satisfies the NET3D condition if it has at leastone symmetric neighbor in every cone of solid angle θ. ANET3D graph is one in which every node except those onthe boundary satisfy the NET3D condition. For specificvalues of θ, NET3D graphs contains proximity graphssuch as RNG, GG, and DelG.

Lemma 4.1: If each node x ∈ V has at least oneneighbor in every θ = π cone of S(X,R(x)), the com-munication graph is a supergraph of RNG(V ).

Lemma 4.2: If each node x ∈ V has at least oneneighbor in every θ = 2π(1 − r

Rc) cone of S(X,R(x)),

the communication graph is a supergraph of GG(V ) and

... 7

Fig. 8. NET graph: each cone of angle θ must have atleast one neighbor.

DelG(V ). Moreover, 2π(1− rR(x) ) is the largest angle that

satisfies this property.The proofs follow from the corresponding proofs in 2D

and the fact that a cone with apex angle α will containa solid angle of θ = 2π(1 − cos(α)). For example in theproof for RNG, the 3D lune will contain a cone with apexangle α = 2π

3 and solid angle of θ = π.We expect that the edge-connectivity result will extend

as follows.Conjecture 4.3: For θ < 2π, a boundary-less NET3D

graph has an edge-connectivity λ ≥ b 2πθ c �

4.1 Integrating geometric conditions with topologycontrolThe local conditions described above can be integratedwith node placement to construct efficient topologies.A key requirement is an algorithm to check for emptycones larger than a given θ. For instance, to check forformation of RNG, θ = π. This step is non-trivial in 3Dbecause there exists no natural “order” of neighbors. Wepropose the following algorithm for finding the largestempty cone around a given node.

Algorithm 1: largestCone(G = (V,E),v ∈ V )let S be the unit sphere centered at vfor each u ∈ Neighbor(v), let −→vu be the

direction vector from v to ulet cu be the intersection of −→vu with Slet DT be spherical delaunay triangulation cu∀ufind ai,j,k = area of circumcircle of triangle

(ui, uj , uk) ∈ DTreturn max(ai,j,k)

Algorithm 1: Find largest empty cone around a node

Theorem 4.4: For a given graph G = (V,E) and nodev, largestCone returns the largest empty cone around v.Proof: The circumcircle of every (spherical) triangle in thespherical delaunay triangulation is empty [19]. Thereforethe cone returned by largestCone is certainly empty.

Suppose there exists an empty cone (whose image onthe unit sphere is the circle c) that is larger than theone returned by largestCone. Then the center of c liesin some triangle t of the delaunay triangulation. Since cis empty, none of the vertices of t must lie inside c. Thisimplies that the circumcircle of t will be larger than cwhich is a contradiction. Therefore largestCone correctlyreturns the largest empty cone.�.

The computational complexity of largestCone isO(dlog(d)) where d is the number of neighbors of a node.This is because, the complexity of spherical triangulationis O(dlog(d)), the number of triangles generated is O(d),and sorting them will also take O(dlog(d)) time.

Several topology control algorithms [1], [11], [20]–[22] in 2D rely on directional information and in par-ticular use the angle between adjacent neighbors. Thisalgorithm can be used as a primitive for extending suchalgorithms to 3D.

5 DEPLOYMENT USING NET GRAPHS

Controlled deployments are well suited to take advan-tage of the global properties of NET graphs resultingfrom local geometric conditions. This applies to scenar-ios of deployment of static nodes by an autonomousagent and self-deployment of mobile nodes. In controlleddeployment of a static sensor network, the agent canmake decisions about the best locations for new nodesbased on the local geometry of the existing network. Self-deploying mobile nodes can use this local condition todecide their motion strategy. The coverage can be maxi-mized by positioning neighbors within adjacent θ sectorsas far apart from each other as possible. In general,based on the coverage and connectivity requirements ofan application, nodes can either be pre-configured for acertain value of θ or they can tune it dynamically.

This section presents a virtual potential fields basedself-deployment algorithm for mobile nodes. We assumethat all nodes have idealized disk models for commu-nication and sensing with radii of Rc and Rs respec-tively. This ensures that negotiations between neighbor-ing nodes are symmetric and simplifies the problem.

5.1 Distributed Deployment of Mobile NodesPotential field based algorithms have been widely usedfor the deployment of mobile networks [23]–[25]. Thesealgorithms involve constructing local virtual forces be-tween neighboring robots to encode their desired motionand/or placement configuration. In our algorithm, weuse two kinds of forces. The first, Frepel, causes thenodes to repel each other to increase their coverage andthe second, Fattract constrains neighboring nodes to stayconnected. These forces have inverse square law profiles- Frepel tends to infinity when the distance between thenodes decreases to zero and Fattract tends to infinitywhen the distance between nodes increases to Rc. Theyare tuned such that when two nodes apply a force ofFrepel +Fattract on each other, they settle exactly at Rc.

... 8

The convergence of this controller is well known [25]. Byusing a combination of these mutually opposing forces,each node maximizes its coverage while maintaining theNET condition of having at least one neighbor in everyθ sector.

The algorithm involves exchange of purely pairwiseinformation between nodes. Each individual node thencombines this information to check for NET conditionsatisfaction. The result of this check is then translated toindividual decisions for each of its neighbors.

In a typical mobile deployment scenario, all nodesstart in positions close to each other so that the initialnetwork is highly well connected and the NET conditionis trivially satisfied. Each node begins by repelling allneighbors to increase its sensing coverage. In the process,it loses communication with some neighbors that movefarther than Rc. When the number of neighbors is closeto the number required to satisfy the NET condition,the node assigns priority values to each of its neighborsbased on their contribution towards satisfying its NETcondition. This is done by computing sector angles be-tween adjacent neighbors. The neighbors contributing tolarger sector angles have a higher priority and the nodeapplies the attractive force to hold them within its com-munication range. Nodes on the boundary designate allneighbors as low priority and hence allow the decisionto be made based on the requirement of their neighbors.The pseudocode for this algorithm is shown below. c1, c2are parameters that can be tuned for how strictly theNET condition is required to be satisfied.

Algorithm 2: assignPriority(θ, node, neighborList)if node is on boundary

for q ∈ neighborListpriority[node][q] = 1

elsefor q ∈ neighborList

sectorV oid[q] = angle(qprev, q) + angle(q, qnext)if sectorV oid[q] < c1 · θ

priority[node][q] = 0 (redundant)else if c1 · θ ≤ sectorV oid[q] < c2 · θ

priority[node][q] = 1 (low priority)else priority[node][q] = 2 (high priority)

To ensure that the forces are symmetric, nodes ex-change pair-wise priority information and apply forcesbased on the average priority.

Algorithm 3: distributedDeployment(θ)find neighborList

degree = sum(neighborList)if (degree > ( 2π

θ + 1))repel all neighbors

else if (degree > 2πθ or boundary)

priority = assignPriority(θ, self , neighborList)for q ∈ neighborList,

if (priority[self ][q] + priority[q][self ]) ≥ 1repel + attract q

else repel qelse repel + attract all neighbors

The condition used for classifying boundary nodesplays an important role in determining the final net-work structure. During deployment, the boundary of thenetwork grows and nodes that were previously interiornodes become boundary nodes. Once a node becomesa boundary node, it is not required to satisfy the NETcondition. If interior nodes are allowed to easily switchto becoming boundary nodes, the coverage will increasebecause nodes will spread out but the connectivity prop-erties will suffer. On the other hand, if interior nodes areconstrained to always satisfy NET condition and are notable to switch to the boundary then the network cannotspread out and coverage will be poor. We use a heuristicbased on the observation that boundary nodes have largeempty sectors. Initially all nodes are designated as non-boundary. If a node has an empty sector greater thanαb for time τ , then it declares itself as a boundary node.Similarly, if a boundary node has no empty sector greaterthan αb for time τ , then it becomes an interior node.

5.2 Simulation ResultsThe algorithms were implemented in the Player/Stagesoftware platform which simulates the behavior of realsensors and actuators with high fidelity [14]. It does notprovide support for realistic communication models andhence we use a simple, idealized disk communicationmodel. In our simulations, Rs was chosen equal to Rc.This is not a requirement for the algorithm and thecoverage results are intended to be illustrative. We notethat the connectivity results are independent of Rs [24].If Rs < Rc

2 , then the per node coverage is very highand in fact close to πR2

s because nodes can satisfyedge constraints without any overlap of sensing areas.If on the other extreme, Rs is significantly larger than2 · Rc then the per node coverage is low compared toπR2

s because of large overlaps. The interesting behaviorhappens when Rs and Rc are comparable.

The parameters c1 and c2 in assignPriority can betuned depending on how strictly NET satisfaction isdesired. For the results presented in this section, aconservative choice was sought and the empiricallydetermined values are shown in Table. 1. The sectorparameter αb is used by individual nodes for boundarydetection. Choosing a large value for αb will result inmany boundary nodes considering themselves interior

... 9

θ 2π3

π2

2π5

π3

2π7

π4

c1 1.15 1.15 1.25 1.0 1.1 1.1c2 1.45 1.5 1.6 1.3 1.3 1.3αb

21π20

11π12

5π6

5π6

5π6

5π6

TABLE 1Parameter settings

nodes trying to ensure NET satisfaction and impedes thespreading. On the other hand, choosing a small valueleads to interior nodes switching to boundary nodes anddecreases network connectivity. The appropriate settingof boundary sector αb depends on the θ value.

Results are shown for deployments of 100 robotswith each experiment repeated 10 times. Fig.9 show thefinal network configuration from sample runs of thedistributed algorithm for varying θ values. For θ = 2π

3 ,hexagonal tiling is difficult to achieve in a distributedmanner and combined with the conservative choice ofc1 and c2, the algorithm pushes the deployment to asquare tiling (Fig.9a). For the non-tiling angle θ = 2π

5 (atleast 5 neighbors required), in most cases it is actuallybeneficial to settle for 6 neighbors and tile triangularly.Fig. 9b shows that the algorithm correctly allows forthe highly stable tiling in large portions of the finalconfiguration. This is also obtained for the tiling angleθ = π

3 , as shown in figure Fig. 9c. Fig.10 shows thatthe deployment algorithm adapts well to obstacles. Onencountering an obstacle, the nodes continue to spreadwhile avoiding it and surround it with little impact onthe NET graph. Obstacles have the effect of increasingthe number of boundary nodes in the network and as aresult, the network structure is not as uniform as before.

Fig. 11a compares the coverage obtained from thedeployment algorithms to an asymptotic upper boundon coverage obtained from Lemma 3.9 by consideringnode overlaps as described in Section.5. The algorithm’scoverage is close to the bound for smaller values ofθ and the difference grows with θ. The difference canbe attributed to a conservative choice of parameters c1and c2. This is reflected in the fact that the averagenode degree resulting from the deployment algorithmis always greater than b 2πθ c (Fig. 11c) even though theboundary nodes have smaller node degrees. Also notethat the coverage upper bound is tight only in case oftiling angles. For non-tiling angles, it is not clear whatthe optimal deployment is.

Fig. 11b shows the edge connectivity values. For thek-connectivity calculation, boundary nodes and one hopneighbors of boundary nodes are not considered. Forthe chosen c1 and c2 values a connectivity of k = b 2πθ cis not guaranteed, but it is very often achieved andconnectivity b 2πθ c − 1 is almost always assured. Usinglower values for c1 and c2 can provide a guaranteedlevel of connectivity at the cost of some coverage. Fig.11d shows the number of sectors of interior nodes thatviolate the NET condition. A 10 % leeway was allowed

for deviation from θ, i.e., a violation occurs if sector angle> 1.1 · θ. There is a drop in violations for θ = 2π

5 sinceit defaults to the 6-neighbor triangular tiling while thereis a sudden increase for θ = 2π

7 and θ = π4 since there

are no tiling angles in sight making it difficult for thepair-wise negotiations to attain symmetrical placementof neighbors. The asymmetry and small value of θ resultin increased violations. Note that this does not adverselyaffect the k-connectivity.

Fig. 12 shows a comparison of the communicationgraph obtained from deployment with θ = π

3 with itscorresponding RNG. In Fig. 12c, the dark lines representthe edges in the RNG that are absent in the commu-nication graph. The communication graph differs fromthe RNG in exactly 3 edges at the boundary of thenetwork. Even in the presence of boundary nodes in realnetworks, these results validate the assertion in Theorem3.8 that the communication graph will contain the RNGfor θ ≤ 2π

3 . Also, the communication graph contains onlya few non-boundary edges that are not in the RNG andwill therefore inherit the sparseness properties of theRNG.

6 POWER CONTROL WITH NET GRAPHS

In this section we illustrate how NET graphs can beused for topology control of a static network by varyingthe communication power. The result is a power con-trol mechanism that can guarantee k-edge-connectivitywithout any restriction on the communication model. Incontrast, most existing power control algorithms, includ-ing the well-known CBTC [11], [13] algorithm, require anidealized binary disk communication model.

In the power control problem, we are given a networkdeployed with a sufficieltly large density so that connec-tivity (k-connectivity) is guaranteed when nodes operateat full power. The objective is to find an assignment oftransmission powers to nodes so that the reduced graphretains connectivity (k-connectivity) while expendingminimum energy. In some applications it is also desirableto maintain other properties such as planarity, sparse-ness, spanner, etc. NET graphs are naturally suited fordistributed power control because• they are based on local geometric conditions that

guarantee global k-connectivity,• communication range for the low power radios

used in sensor network applications is highly ir-regular [9], [10]. The connectivity properties ofNET graphs are independent of the communicationmodel.

Consider a sensor network such that when nodesoperate at full power, they satisfy the NET condition fora given θ. Using a typical power control protocol andsatisfaction of NET condition at each internal node asthe termination criterion, a power-efficient k-connectednetwork can be achieved.

We have implemented a completely distributed powercontrol algorithm as follows. Starting with a minimum

... 10

(a) θ = 2π3 (b) θ = 2π

5 (c) θ = 2π6

Fig. 9. Sample deployments with 100 nodes and Rc = Rs = 8.0.

(a) θ = 2π3 (b) θ = 2π

5 (c) θ = π3

Fig. 10. Sample deployments in the presence of obstacles with 100 nodes and Rc = Rs = 8.0.

value, each node incrementally increases its transmissionpower till the NET condition is satisfied. We assume thatthe orientation of neighbors can be computed either fromangle-of-arrival of messages or localization information.Each node computes its empty sectors and incrementspower if 1) NET condition is not satisfied or 2) if it hasan asymmetric incoming link from a node that does notsatisfy NET condition. Since boundary nodes (detectedusing an algorithm such as [17]) are not required to sat-isfy NET condition, they only increase power in responseto incoming asymmetric links. This is a simple imple-mentation with scope for further power optimizationwhich we plan to pursue in future. Our objective here isto demonstrate that for a given θ the resulting topologiespreserve b 2πθ c edge-connectivity. In simulations, we usedrealistic statistical models for wireless links developed in[10].

Fig. 13 shows typical topologies resulting with 500nodes distributed uniformly at random on a 80m x 80msquare. Nodes that cannot satisfy the NET condition atfull power are identified as boundary nodes (shown asred squares). The initial transmission power was set to−10dB and incremented in steps of 1dB. Links that hada packet reception rate of at least 90% were consideredactive. Note that the figures only show bidirectionallinks. The simulations were terminated when all non-boundary nodes satisfied NET condition.

Fig. 13 (a) and (b) show the comparison the topolo-gies resulting from NET based power control andCBTC(α) [13] for 2-edge-connectivity. According toCBTC(α) a node must either have a neighbor in every

θ = 2π3k or operate at full power to guarantee k-edge-

connectivity. We show that having a neighbor in everyθ = 2π

k is sufficient to guarantee k-edge-connectivity. Asa result, NET condition results in a significantly sparser,and hence efficient, topologies compared to CBTC(α).

Fig. 14 shows the edge-connectivity and averagepower over 100 iterations. Edge-connectivity was com-puted as the minimum number of paths consisting ofsymmetric links between any two internal nodes. In allexperiments, the edge-connectivity was always greaterthan b 2πθ c. This validates our key theoretical result, The-orem 3.4. The average power used increases quickly asθ decreases (fig. 14(b)). This implies that for efficienttopology control it is important to choose the largestvalue of θ possible depending on the application require-ments. Because NET graph requires an angle (θ = 2π/k)that is three times bigger than the angle required byCBTC(α) (θ = 2π/3k), the power saved will very large.For example, for 2-edge-connectivity, CBTC(α) requiresan average power of 4dB while NET graph requires−6dB.

These initial results are promising because we areable to achieve power-efficient topologies that guaranteek-connectivity through distributed power control evenwith realistic, irregular link models.

7 RELATED WORK

Sector based graphs have existed for a long time. The Yaograph [26] (also called θ-graph) introduced by A. C. Yaoin 1977 is constructed by dividing the area around each

... 11

(a) (b)

(c) (d)

Fig. 11. Performance of the Sequential and Distributed deployment Algorithms for a deployment of 100 nodes in termsof (a) Coverage, (b) Edge Connectivity (c) Average Degree and (d) NET condition satisfaction

(a) (b) (c)

Fig. 12. For θ ≤ 2π3 , the communication graph is a supergraph of RNG. (a) Communication Graph (b) RNG (c)

Difference for θ = 2π3 .

node into equal sectors of θ each and adding an edgeto the closest node in each sector if there exists one. Thesymmetric Yao graph [12] is a subgraph of the Yao graphcontaining only the edges chosen by both of the two endnodes. Every NET graph is a symmetric Yao graph butthe converse is not true. This is because in a Yao graph,the angle between two adjacent neighbors of a node canbe greater than θ depending on how the sectors wereinitially defined but in a NET graph this angle cannotbe greater than θ. As a result, the properties we provefor NET graphs do not hold for Yao graphs. θ ≤ π

3guarantees that the Yao graph contains the RNG whereas

a much larger angle of θ ≤ 2π3 is sufficient to guarantee

that the NET graph contains the RNG. Similarly, forθ = π, NET graphs are connected [27] but the Yao graphis not necessarily connected.

Sector based conditions were introduced for topologycontrol of wireless ad-hoc networks by Li, Wang, Bahland Wattenhofer [11] and Li, Wan, and Wang [12]. In theCone Based Topology Control mechanism (CBTC) [11],each node either has a neighbor in every θ sector oroperates at full power. Under the assumption of anidealized disk communication model, it is shown thatthe graph is connected if θ ≤ 2π

3 . Further, if the full

... 12

(a) θ = π (b) θ = 2π6

Fig. 13. Comparison of NET graphs and CBTC(α) for a uniform random network of 500 nodes. For 2-edge-connectivity,(a) NET graph requires sector angle θ = π and average power −5.91dB while (b) CBTC requires θ = 2π

6 and averagepower of 4.10dB.

(a) edge-connectivity (b) average power

Fig. 14. Simulation results of power control based on NET graphs for a uniform random network of 500 nodesaveraged over 100 runs. The edge-connectivity is greater than the 2π

k lower bound derived in Theorem 3.4. Theaverage communication power for achieving 2-connectivity using CBTC(α) is 4dB (red square) compared to −6dBusing NET graphs.

power graph is k-connected then for θ ≤ 2π3k , the reduced

graph retains k-connectivity [13]. In comparison, NETgraphs achieve k-connectivity with a much larger angle(and hence significantly lower power) of θ = 2π

k and donot require idealized disk communication. The differencebetween the two techniques is established in Section 6.Li et al proposed using geometric graphs such as RNG,GG, Delaunay and Yao graphs for topology control andto address the non-planarity and high node degree ofYao graph extended it to symmetric Yao graph, YaoGG,YaoYao graph, reverse Yao graph, and SθGG [12]. In[28] a modified Yao structure is proposed for k-vertexconnectivity, where each node must have at least k + 1neighbors in each sector of angle less than π

3 around it.This implies that each node must have at least 6 · (k+1)neighbors which is 6 times the corresponding numberrequired for k-edge-connectivity using NET graphs. Forscatternet formations, Stojmenovic proposed protocolsthat apply geometric graphs such as Yao graphs, RNG,and GG on the scatternet graph to limit node degreeand ensure planarity while retaining connectivity [29].Recently, Xue and Kumar [30] have defined θ-coveragecondition which is equivalent to the NET condition andanalyzed the critical radius for asymptotic θ-coveragefor a randomly deployed 2D network. Further, theyprove using geometric arguments that for θ ≤ π, θ-

coverage implies 1-connectivity. The same result wassimultaneously established by D’Souza et al [27] for aweak monotonicity communication model which is moregeneral than the idealized disk communication model.Our paper extends this result to address k-connectivityas a function of θ, for any arbitrary communicationmodel. In addition to sector based techniques, there is arich body of literature on topology control with a widevariety of approaches (see [1]).

Sleep scheduling seeks to activate the smallest frac-tion of nodes in a densely deployed network that cansimultaneously achieve complete coverage and connec-tivity. Here coverage is defined as every point in thenetwork domain being within the sensing range of atleast one active node. Under this definition, coverageand connectivity are not opposing goals; in fact, if thesensing range is at least twice the communication range,then complete coverage implies connectivity [21], [22]. Incontrast we focus on the problem of maximizing sensingcoverage given a fixed number of nodes so that highercoverage, in most cases, implies poorer connectivity.Even though we do not consider sleep-scheduling mech-anisms in depth, it is possible to think of designs wheredensely deployed nodes make sleep/wake decisions in adistributed manner by using local, pair-wise negotiationsbased on combinations of NET condition satisfaction and

... 13

other criteria. Self deployment of networks of robots,where the key focus is on maximizing sensing coverage,has been well studied. Cortes et al [31], present aVoronoi partition based algorithm that is distributed andguaranteed to maximize coverage. Potential field baseddeployment algorithms also maximize coverage [23] andhave been extended to impose local constraints suchas minimum degree of each node [24]. Simulationsin [24] show that constraining the degree can resultin the deployed network being connected with highprobability. The above algorithms, like the NET baseddistributed deployment, require information about angleand distance to neighbors. They maximize coverage butdo not guarantee network connectivity.

Topology control in 3D is significantly more complexand is relatively less studied in the literature. In [13],an algorithm has been presented to extend CBTC [11]to 3D. The computational complexity at each node isO(d3log(d)), d being the average node degree. The spher-ical Delaunay triangulation based algorithm that wepresent has complexity O(dlog(d). Our algorithm hasbeen implemented in [32] along with another techniquewhere CBTC is applied on projections of 3D points ona 2D plane. Simulation results presented show that bothtechniques result in retaining connectivity with highprobability. In XTC [33] links with poor quality that canbe substituted with multi-hop paths with better qual-ity, are incrementally deleted. It does not use the diskassumption or angular information; given an initiallyconnected network in 3D, it can retain connectivity.

8 SUMMARY AND CONCLUSIONS

This paper addresses distributed topology control usinga construct called Neighbor-Every-Theta (NET) graphs.These graphs have the following two properties thatallow for simple and practical algorithm design:• tunable connectivity based on a single parameter,

the sector angle θ• connectivity guarantees do not depend on commu-

nication modelNET graphs are such that each node has at least one

neighbor in every θ sector around it. We prove that fora given θ < π, NET graphs have an edge connectivityof at least b 2πθ c, except in pathological cases where thenetwork can be partitioned close to its boundary. Thisproperty holds even in cases when the communicationmodel is irregular. For the special case of an idealizeddisk communication model, it is shown that for specificvalues of θ, NET graphs contain proximity graphs suchas the relative neighborhood graphs that are well knownto be desirable communication topologies. We also provethe symmetric neighbor placement condition for maxi-mizing sensing coverage under the NET condition.

To concretely demonstrate the use of NET graphs fortunable topology control, we consider two scenarios:• For deployment of mobile nodes, we develop a

distributed controller that maximizes sensing cover-

age while maintaining local sector conditions. Thiscontroller is based on virtual potential fields, wherea combination of attractive and repulsive forcesdecides the motion, and pair-wise negotiations be-tween communicating neighbors. Simulations onthe Player/Stage platform provide controller perfor-mance evaluation and also serve to substantiate theanalysis.

• For power control in a static network, we imple-ment a typical protocol using satisfaction of NETcondition at each internal node as the terminationcriterion and show that a power-efficient networkwith edge connectivity 2π

θ can be achieved evenwith realistic, irregular links.

Lastly, we consider NET graphs in three dimensionsand study their connectivity properties. It is our con-jecture that for a given solid angle θ < 2π, NET3Dgraphs have an edge connectivity of at least b 2πθ c whenpartitions close to the boundary are ignored. Thereis very little earlier work on topology control in 3D.Several geometric computations become complex andeven intractable when extended from 2D to 3D. Wehave developed an efficient algorithm for determiningthe largest empty cone around a node in 3D based onspherical Delaunay triangulations. The running time isO(dlogd) for average node density d, which is a signif-icant improvement over the earlier O(d3logd) algorithmproposed in the context of CBTC [13]. The new algorithmcan be used to extend several sector based topologycontrol algorithms to 3D. Topology control in 3D is animportant area for further research.

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Sameera Poduri received the BTech degree inMechanical Engineering and MTech degree inComputer Aided Design and Automation fromthe Indian Institute of Technology Bombay, Indiain 2002. She is currently a PhD candidate in theComputer Science Department at the Universityof Southern California, Los Angeles. Her currentresearch interest is in developing distributed al-gorithms for control and coordination of large-scale sensor/actuator networks.

Sundeep Pattem received the B.Tech. degreein Electrical Engineering from the Indian Instituteof Technology Bombay, India in 2002. He is cur-rently a PhD candidate in Electrical Engineeringat the University of Southern California. His re-search is in the area of routing and compressionfor wireless sensor networks.

Bhaskar Krishnamachari received the BE de-gree in electrical engineering from the CooperUnion, New York, in 1998 and the MS andPhD degrees from Cornell University in 1999and 2002, respectively. He is currently an Asso-ciate Professor in the Department of ElectricalEngineering, University of Southern California.His primary research interest is the design andanalysis of efficient mechanisms for operatingwireless networks.

Gaurav S. Sukhatme (SM’05) received the M.S.and Ph.D. degrees in computer science fromUniversity of Southern California (USC), LosAngeles. He is an Associate Professor of Com-puter Science (joint appointment in ElectricalEngineering) at USC. He is the Codirector ofthe USC Robotics Research Laboratory and theDirector of the USC Robotic Embedded Sys-tems Laboratory, which he founded in 2000. Hisresearch interests include multirobot systems,sensor/actuator networks, and robotic sensor

networks. He has published extensively in these and related areas. Prof.Sukhatme was a recipient of the NSF CAREER Award and the OkawaFoundation Research Award. He has served as PI on several NSF,DARPA, and NASA grants. He is a Co-PI on the Center for EmbeddedNetworked Sensing (CENS), an NSF Science and Technology Center.He is a member of AAAI and the ACM and is one of the founders ofthe Robotics: Science and Systems conference. He is program chair ofthe 2008 IEEE International Conference on Robotics and Automationand the Editor-in-Chief of Autonomous Robots. He has served as anAssociate Editor of the IEEE TRANSACTIONS ON ROBOTICS ANDAUTOMATION, the IEEE TRANSACTIONS ON MOBILE COMPUTING,and on the editorial board of IEEE Pervasive Computing.