theoretic analysis of unique localization for wireless sensor networks

Ad Hoc Networks 10 (2012) 623–634

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Ad Hoc Networks

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Survey Paper

Theoretic analysis of unique localization for wireless sensor networks

Yuan Zhang a,b,⇑, Shutang Liu b, Xiuyang Zhao a, Zhongtian Jia a

a Shandong Provincial Key Laboratory of Network Based Intelligent Computing, Jinan 250022, Chinab Shandong University, School of Control Science & Engineering, Jinan 250014, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 8 September 2010Received in revised form 11 June 2011Accepted 21 June 2011Available online 13 July 2011

Keywords:Unique localizationWireless sensor networkFrameworkRigid

1570-8705/$ - see front matter � 2011 Elsevier B.Vdoi:10.1016/j.adhoc.2011.06.016

⇑ Corresponding author at: Shandong ProvinciaNetwork Based Intelligent Computing, Jinan 250022

E-mail address: [email protected] (Y. Zhang).

Node self-localization has become an essential requirement for realistic applications overwireless sensor networks (WSNs). Although many distributed localization algorithms havebeen proposed, fundamental theoretic analysis of unique localization is still in its earlystage of development. This paper aims at a synthetic and homogeneous survey of the the-oretical basis on WSN localization problem carried out thus far. Specifically, subsequent toestablishing a technological context of relevant terms, we construct a graph and then a for-mation for each WSN to present current state-of-the-art by analyzing possible conditionsfor unique localization, as well as corresponding verification algorithms, by drawing on thepowerful results from rigidity theory, distance geometry, geometric constraints in CAD, andcombinatorial theory. We show that the unique localization problem is well understood intwo-dimension, however, only partial analogous results are available in three-dimension.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction location information also enables a myriad of applications

Dramatic advances in radio frequency (RF) and micro-electromechanical systems integrated circuit (MEMS IC)design have made possible the use of WSNs for a varietyof monitoring and control applications [1–3]. Typicallysuch networks involve a large number of small, battery-powered, and wireless connected sensor nodes densely de-ployed over physical space. This means that certain inher-ent constraints on the network’s design can be distilledincluding:

� limited processing capability,� limited communication range,� restricted battery lifetime,� coarse sensing potential, and� low reliability in supporting miniature nodes [4].

Since the position of sensor nodes may not be predeter-mined in some scenarios, a localization system is requiredfor the sensed data to be meaningful. The availability of

. All rights reserved.

l Key Laboratory of, China.

such as inventory management, intrusion detection, roadtraffic monitoring, health monitoring, reconnaissance andsurveillance. In the future ubiquitous computing environ-ment [5], implementing positioning services is still a funda-mental issue [6,7]. All of these factors make localizationsystems a key technology for the development and opera-tion of WSNs.

Sensor network localization is not just trivial extensionsto the traditional localization techniques like GPS [8] andLPS [9]. They involve further challenges in several aspects:(1) the environments in which sensor networks are de-ployed are often complicated, involving urban environ-ments, indoor environments and non-line-of-sightconditions; (2) WSNs are usually composed of and low-cost sensors with limited computational capabilities; (3)WSN localization techniques are often required to beimplemented using available measurements and withminimal hardware investment; (4) sensor network locali-zation techniques are often required to be suitable fordeployment in large scale multi-hop networks; and (5)the choice of sensor network localization techniques tobe used often involves consideration of the trade-offamong cost, size and localization accuracy to suit therequirements of a variety of applications. It is these

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624 Y. Zhang et al. / Ad Hoc Networks 10 (2012) 623–634

challenges that make localization in WSNs unique andintriguing.

The more reasonable solution is to assume that only asmall portion of sensor nodes (called anchors) obtain theirposition information via GPS or the system administration,while the remaining nodes without positioning informa-tion (called ordinary nodes) can estimate their locationsaccording to the coordination information of the anchors.

A large number of algorithms have been proposed in thepast decade. Based on the techniques they use for distanceestimation and position computation, we classify the cur-rent localization algorithms as either range-based orrange-free systems.

Range-based systems, or fine-grained localization sys-tems, utilize timing-based, directionality-based, or signal-strength-based techniques for distance estimation, andthe position of the node is computed using multilaterationor triangulation. Systems that fall into this category are[10–18]. As distance estimates are fairly accurate, refine-ment may not yield a great deal of improvement in the ini-tial position estimates as shown in [19]. Due to thehardware limitations of sensor devices, range-free localiza-tion algorithms are a cost effective alternative to the range-based approaches. They usually utilize hop-based tech-niques for distance estimation that infer the proximity ofa node to some reference points and use centroid calcula-tions or area-based triangulation to estimate the nodespositions [20–25]. What we should note is that no singlealgorithm performs best. Which algorithm is to be pre-ferred depends on the condition like range errors, connec-tivity, anchor fraction and so on.

Although the designs of the previous schemes havedemonstrated clever engineering ingenuity and their effec-tiveness in certain settings, critical theoretical foundationis still in its early stage of development. Many fundamentalquestions lack satisfying answer. For instance, a naturalquestion is: what are the properties of a sensor networkwhich ensure unique solvability of the localization prob-lem? The network, with the given locations and measure-ments, is said to be uniquely localizable if there is aunique set of locations consistent with the given data. Cor-respondingly, given an instance of the network localizationproblem with unique localizability, can we have polyno-mial-time algorithm to compute a realization? What isthe computational complexity involved in a solution?

Related issues have been raised and studied in the pastfew years, most of which make the use of rigidity and glo-bal rigidity in frameworks, the coordination of formationsof autonomous agents, and geometric constraints in CAD.All these works belong to the fine-grained category that re-lies on measurements. The type of measurement employedand the corresponding precision determine the estimationaccuracy of a localization system.

Measurement techniques in WSN can be broadly classi-fied into two categories: direction or bearing measure-ments and distance related measurements. Based onthese two kinds of measurements, this paper aims at a syn-thetic and homogeneous presentation of the theoreticalanalysis of WSN localization problem carried out thus far.Our contributions lie in three aspects. Firstly, although afew survey articles on WSN localization have been

published previously [26–28], all of them took the visionof algorithm design. To the best of our knowledge, this isthe first time that a survey paper focuses on the fundamen-tal mathematical analysis domain, which we deem a basisfor specific algorithms. Secondly, nearly all the previousworks apply results from the distance-measurement basedgraph rigidity literature to the network localization prob-lem, while few of them took bearing constraint as an alter-native. Our paper combines both distance measurementand bearing measurement to provide systematic answers.Thirdly, there always seems to be a disconnection betweena theoretical analysis and a specific localization algorithm.In this paper we remedy the situation and try to provideboth the efficient verification and realizations wheneverpossible.

The rest of the paper is organized as follows. In Section2, we establish a technological context of relevant conceptsand formulations. In Sections 3–5, we analyze various pos-sible conditions for unique localization, as well as corre-sponding verification algorithms, based on distanceconstraints, based on bearing constraints, and based ontheir combination, respectively. Finally we draw our con-clusions and briefly discuss future research trends inSection 6.

2. Definitions and problem formulation

In this paper, we are concerned with the WSN localiza-tion problem with distance and/or bearing constraints.Such problem can be studied in the framework of graphtheory. Let d P 1 (usually d = 2 or 3) be the dimension inwhich the sensor nodes reside. We first give formal defini-tions of the correlative terms and concepts.

Definition 1. Sensor node si: Also known as ordinary orfree node. This term refers to the node whose positionneeds to be settled. These nodes hold the majority of aWSN.

Definition 2. Anchor node ai: Also known as landmark orbeacon. These are the nodes that do not need a localizationsystem in order to estimate their physical positions. Theircoordinate is obtained by manual placement or externalmeans such as GPS.

These nodes form the base of many localization systemsfor WSNs, but may be unfeasible in some applicationenvironments.

Definition 3. Neighbor: Each node is located at a fixedposition in Rd and has associated with it a specific set ofneighboring nodes. Two nodes are called neighbors if theyhave communication links between them.

We assume that the neighbor relationship is symmetricin the sense that node i is a neighbor of node j if and only ifnode j is also a neighbor of node i. Such an assumption willlead to an undirected graph, which will be defined below.

All fine-grained localization strategies require rangeestimation techniques either based on distance measure-ment or based on direction/bearing measurement. Distancerelated measurements include propagation time based

Y. Zhang et al. / Ad Hoc Networks 10 (2012) 623–634 625

measurements, i.e., one-way propagation time measure-ments, round trip propagation time measurements andtime-difference-of-arrival (TDOA) measurements, and RSSmeasurements. By adopting an appropriate model for thepropagation speed of the communication signal node ican estimate the distance to its neighbor j as dij. By provid-ing information about the direction to neighboring sensorsrather than the distance to neighboring sensors, angle of ar-rival (AOA) measurements [15,29] provide localizationinformation complementary to the TOA and RSS measure-ments. It can be further divided into two categories: thosemaking use of the receiver antenna’s amplitude responseand those making use of the receiver antenna’s phase re-sponse. In our case, the AOA capability provides for eachnode bearings to neighboring nodes with respect to anode’s own axis. The accuracy of the measurements is notunder discussion in this paper though it is widely knownthat localization is susceptible to measurement noise andadditional problems.

Definition 4. G = (V, E): We associate a graph, whichconsists of a vertex set V = {1, 2, . . . , n} and an edge set Ewith a sensor network. The vertexes of G are partitionedinto two subsets: the set Vs = {1, 2, . . . , m} of sensors, andthe set Va = {m + 1, m+2, . . . , n} of anchors.

Definition 5. Bearing [30]: A bearing is the angle betweenthe x-axis of the local coordinate system of node i and the linesegment joining node i with its neighbor node j. See Fig. 1.

The angle is measured in counter-clockwise rotationdirection from x-axis of its own local coordinate system.By a node’s local coordinate system is meant a coordinatesystem which is chosen by each node based on some crite-ria [15]. It is also evident that direction information can beeasily converted to bearing information, and vice versa. Forsimplicity’s sake and without loss of generality, we assumethat the given distance and direction data are exact.

WSN localization problem with distance and bearinginformation is to find an assignment of coordinates to allsensor nodes in Rd given the graph of the network G, thepositions of the anchor nodes, and the distance and/orbearing data between enough neighboring pairs. Readersmay contrast this description with that of Aspnes’s [31].We draw on the bearing measurement but do not requiredistance and direction information from each pairwise

θ ij

θ ji

i

j

Fig. 1. Bearing constraints for node i and node j are denoted by hij and hji,respectively.

neighbor. In general, there could be multiple realizationsconsistent with the given data. However, a given instancecould have a unique realization, i.e. generic solvable [32],if the underlying graph G = (V, E) of a framework maintainenough distance constraints and/or bearing constraints.

Definition 6. Distance constraint: A distance constraint isa requirement that the distance between two nodes i and j,depicted with dij, be maintained through a communicationlink and some control strategy.

Distance between all node pairs can be held fixed withonly enough distance constraints between certain pairs ofnodes. Such property is from rigidity theory, which castthe main ideas of this work using this perspective. Notethat a given graph may not be realizable under a particularset of edge weights. In the context on WSN localizations,realization of a graph G = (V, E) with distance constraintfunction d(i, j) can be formulated as a global optimizationover vectors of points {p1, p2, . . . pn} of the following form:

minimizeX

ij2E

ðdði; jÞ � kpðiÞ � pðjÞkÞ2

Definition 7. Bearing constraint [30]: For two neighboringnodes i and j, bearing constraints, denoted in Fig. 1 by hij

and hji respectively, are the angles between the x-axis ofeach node’s own coordinate system and the link (i, j).

Our aim is to obtain a relation between the coordinatesof node i and j given the bearing constraint between them.Note that, when distance/bearing refers to the measure-ment obtained via signals, it is distance/bearing informa-tion. When this information is treated as a constraint tobe satisfied in the computation of positions of nodes inthe network localization problem, it is called a distance/bearing constraint.

To study the solvability of WSN localization problem,we apply in an in-depth way the concept of framework inmathematics [33,34].

Definition 8. Framework: Let G = (V, E) be an undirectedgraph with vertices V and edges E. We then embed G intoRd by assigning to each vertex i a location pi e Rd. Thelength of an edge ij e E is given by the Euclidean distancebetween the points p(i) and p(j). Then a framework F p is apair (G, p), where p is a map from V to Rd such thatkpðiÞ � pðjÞk ¼ dij and p(i) – p(j) for all ij e E. See Fig. 2.

It is clear that a framework F p is uniquely determinedby its graph and distance function at most up to a congru-ent transformation. The idea of framework is essentiallythe same as the concept of point formation studied in some

Fig. 2. A graph and the corresponding framework.


works [30,31,34]. For our purpose, a framework provides anatural high-level model for an n-node WSN in real two orthree-dimensional space. It is a collection of geometric ob-jects such as points (representing the positions of bothsensor nodes and anchors), line segments (representingthose specific node pairs whose internode distances are gi-ven), and circular arcs (representing bearing informationbetween neighbors) in Rd, together with constraints onand between these objects. In the present context, theWSN localization problem can be restated in terms of itsassociated framework F p. That is, to determine F p, giventhe graph, distance and bearing functions of F p, as wellas the anchor positions.

A framework also refers to a graph G along with a con-figuration of G in Rd. A configuration, which is an assign-ment of the vertices V to points in Rd, is called generic ifthe coordinates are algebraically independent over the ra-tional [35]. Intuitively speaking, a generic configurationhas no degeneracy, i.e., no three points staying on the sameline, no three lines go through the same point, etc. For ageneric framework we can ‘wiggle’ the vertices a little bitwithout altering any of its rigidity properties. The studyof generic configurations is important because in practice,we can assume that the sensor locations are generic.

Two frameworks ( G, p) and ( G, q) are equivalent ifkpðiÞ � pðjÞk ¼ kqðiÞ � qðjÞk holds for all pairs i, j with ij e E.Two frameworks ( G, p) and ( G, q) are congruent ifkpðiÞ � pðjÞk ¼ kqðiÞ � qðjÞk holds for all pairs i, j with i,j e V. This is the same as saying that (G, q) can be obtainedfrom (G, p) by an isometry of Rd, i.e., a combination of trans-lations, rotations and reflection. It is clear that a frameworkis uniquely determined by its graph and a measurementfunction.

A framework is called flexible if there exists a continu-ous deformation from the given configuration to another,such that edge lengths are preserved [33]. If no such defor-mation exists, it is called rigid, which we give more formaldefinition below.

Definition 9. Rigid: A framework (G, p) is rigid if thereexists a sufficiently small positive e such that if (G, q) isequivalent to (G, p) and kpðiÞ � qðiÞk < e for all i e V then(G, q) is congruent to (G, p). See Fig. 3.

Rigidity is related but not equal to the isomorphism offrameworks [36]. Two frameworks are isomorphic if theyare related by some sequence of translations and rotationsin Rd. We can thus describe a framework as rigid if the onlylocal deformation of the framework is isomorphic to theoriginal frame work. Note that there exist rigid frameworks

(a) (b)Fig. 3. Two-dimensional illustration of rigidity and global rigidity. (a) flexible; (rigid.

(G, p) and (G, q) which are equivalent but not congruent,see for example Fig. 3b and c. Such frameworks are notglobally rigid.

Definition 10. Globally rigid: A framework is called glob-ally rigid in Rd if every framework, which is equivalent to(G, p) is congruent to (G, p) [35]. See Fig. 3d.

A globally rigid framework is exactly determined up tocongruence by its graph and distance or bearing con-straints. Obviously the notion of global rigidity is morerestrictive than that of rigidity, and it forms a necessaryand sufficient condition for the solvability of WSNlocalization.

Supplied with this rigidity theory background, we areready to explore and answer the questions raised in Sec-tion 1. We categorize the state-of-the-art as either basedon distance constraint or based on bearing constraint,which will be analyzed systematically in Sections 3 and4, respectively.

3. Distance constraint based localizability analysis

The solvability problem for WSNs can be thought of asfollows. Suppose a framework is constructed which is arealization, i.e., the edge lengths corresponding to the in-ter-sensor distance constraints. The framework may ormay not be rigid. Even if it is rigid, there may be a secondand differently shaped framework which also forms a real-ization (constructable with the same vertex, edge and dis-tance constraints). If up to congruence there is a uniquerigid realizing framework consistent with the distance con-straints, i.e., the framework is globally rigid, then the WSNcan be thought of as like a rigid entity of known structure,and one only needs to know the Euclidean position of sev-eral sensors in it to locate the whole framework, in two orthree dimensional space. Most of the research in this spe-cific topic, although in its early process, is based on dis-tance constraint methodology.

We observe that the theory of globally rigid frameworksis the mathematical background which is needed to inves-tigate the unique localizability of WSNs. However, it is ahard problem to decide if a given framework is globallyrigid. Indeed Saxe [37] has shown that this problem isNP-hard even for one-dimensional frameworks. Furtherhardness results can be found in [31,38].

But the problem becomes more tractable if we suitablyrestrict the input to avoid certain coincidences. In particu-lar, one strong way of restricting the problem is to assumethat the framework is generic, i.e., the coordinates of

(c) (d) b) rigid but not globally rigid; (c) rigid but not globally rigid; (d) globally


sensor and anchor nodes do not satisfy any non-trivialalgebraic equation with rational coefficients. Restrictingto generic frameworks give us two important ‘stabilityproperties’. The first is that, if (G, p) is a globally rigid d-dimensional generic framework then there exists an e > 0such that all frameworks ( G, q) which satisfykpðiÞ � qðiÞk < e for all i e V are also globally rigid [39].The second is that either all generic realizations of a graphG are globally rigid, or none of them are.

In what follows we shall consider the unique realizationproblem for generic frameworks. We will see that we cancharacterize the graph G with the property that every genericrealization of G in d-space is globally rigid, when d = 1, 2. Theproblems of characterizing when a d-dimensional genericframework is rigid or globally rigid are unsolved for d P 3.Hence this survey focuses on the most relevant cases oftwo-dimensional frameworks, but stating results of three-dimension wherever possible to match the real applications.

With the previous definitions, we can now summarizeone of the main results.

Theorem 1. [32]. Let N be a network in Rd consisting of mordinary sensor nodes located at positions p1, p2, . . . , pm andn �m anchor nodes located at pm+1 , . . . , pn. Suppose thatthere are at least d + 1 anchors in general position. Let G bethe grounded graph of N and let p = (p1, p2, . . ., pm). Then thenetworks is uniquely localizable if and only if (G, p) is globallyrigid.

3.1. Conditions for rigidity

Solvability of the WSN localization problem demandsthat F p be rigid, for if F p were not rigid, it would be impos-sible to determine F p up to equivalent, let alone to deter-mine it to be congruent and unique. The framework isrigid if it has no non-trivial continuous deformations. Withthe concept of generic in mind, we look at rigidity as aproperty of the connectivity but not of the geometry ofthe formation. We can thus use ‘a rigid graph’ and ‘a rigidframework’ interchangeably without ambiguity.

Informally, we need to answer the following question:how many edges are necessary for a graph of n nodes tobe generically rigid? In two-dimension, the n nodes have2n degrees of freedom. Each edge that we add to the graphcan remove at most one degree of freedom. Global rota-tions and translations are always going to be possible, soat least 2n � 3 edges are necessary for a graph to be rigid.

n=5, 2n-3=7n=4, 2n-3=5n=3, 2n-3=3

(a) (b) (c)Fig. 4. (a) rigid; (b) rigid; (c) flexible.

However, 2n � 3 edges are not always sufficient, becausesome edges might be redundant. See Fig. 4.

Clearly, we need 2n � 3 well-distributed edges. To put itmore formally, if a subgraph has more edges than neces-sary, some edges are redundant. Non-redundant edges areindependent, in the sense that their corresponding rowsof the rigidity matrix are linearly independent. Each inde-pendent edge always removes a degree of freedom. There-fore 2n � 3 independent edges will guarantee rigidity.

Definition 11. Rigidity matrix [40]: Let (G, p) be ad-dimensional realization of a graph G = (V, E). The rigiditymatrix of the framework (G, p) is the matrix M(G, p) of size|E| � d|V|, where, for each edge e=vivj e E, in the rowcorresponding to e, the entries in the two columnscorresponding to vertices i and j contain the d coordinatesof (p(vi) � p(vj)) and (p(vj) � p(vi)), respectively, and theremaining are zeros.

Gluck [41] characterized rigid graphs in terms of theirrank.

Theorem 2. Let G = (V, E) be a graph. Then G is rigid in Rd ifand only if either |V| 6 d + 1 and G is complete, or |V| P d + 2and rank MðG; pÞ ¼ djV j � ðdþ1Þd

2 .Therefore a graph with n vertices which is generically

rigid in Rd has at least nd� ðdþ1Þd2 edges. Note that there is

no known polynomial algorithm for calculating the rankof a matrix in which the entries are liner functions of alge-braically independent numbers. For the plane, Laman con-dition provides a strong combinatorial characterization ofthe generically rigid graphs.

Theorem 3. [42]. A graph G = (V, E) with |V| = n is genericallyrigid in R2 if and only if it has 2n � 3 edges and no subgraph E0

of k vertices has more than 2k � 3 edges.One may interpret the first condition of Theorem 3 as

requiring that E contain enough edge to be rigid, and thesecond condition as requiring that none of these edges be‘wasted’ by packing too many between the vertices of anysubset of V. We can use Theorem 3 to determine if a graphis rigid without embedding the vertices into R2 and calcu-lating the rank of the rigidity matrix. Moreover, from acomputational point of view, the Laman condition providesa county algorithm to test the rigidity of a graph. Unfortu-nately, Laman’s conditions are not easy to check for largegraphs because we are required to check all subgraphsE0 # E of the correct cardinality, and then all subgraphsthereof. Imai presented an O(n2) algorithm for rigidity test-ing using a network flow approach [43]. This complexitywas matched by Hendrickson using bipartite matching[44] and by Jacobs using construction technique to grow amaximal set of independent edges one at a time [45].

There is a similarity between Theorem 3 and the follow-ing theorem, which also leads to an O(n2) algorithm forrigidity testing, called the Henneberg construction [46].

Theorem 4. [47]. A graph is the edge disjoint union of twospanning trees if and only if |E| = 2n�2 and, for all E0# E,|E0| 6 2k�2, where k is the cardinality of the set of endpoints ofE0.


Various polynomial algorithms for decomposing agraph were proposed [48–50] and we can thus easily getpolynomial algorithms for testing generic rigidity.

In consideration of the design of a geometric WSN rout-ing algorithm, we further present the following character-ization of rigid graphs in a slight reformulation form of [51].

Definition 12. Cover: Give a graph G = (V, E), a cover of G isa family v = {X1, X2, . . . , Xt} of subsets of V such that |X| P 2for all X e v and UXevE(X) = E. A cover v is 1-thin if|Xi\Xj| 6 1 for all Xi, Xj e v.

Theorem 5. Let G = (V, E) be a graph. Then G is rigid if andonly if for all covers of G we have

PXev (2|X| � 3) P 2|V| � 3,

where the equality can be attained over all 1-thin covers of G.Theorem 5 is illustrated in Fig. 5. The authors [51] used it

to show that every 6-connected graph is rigid, but there ex-ist a family of 5-connected graphs which are not rigid in R2.They conjectured that every 12-connecnted graph is rigid inR3. To date no one has been able to show that there even ex-ists a finite k such that all k-connected graphs are rigid in R3.

In 3-space, each node has three degrees of freedom.Additionally every body has six ‘internal’ degrees of free-dom corresponding to rigid motions. It is thus clear thata graph on n vertices should have at least 3n � 6 edges(in order to destroy all external degrees of freedom) to berigid. Similar to the planar case (see Theorem 3), we saythat graph G has the Laman property in R3 if and only ifit has 3n � 6 edges and no subgraph of k vertices has morethan 3k � 6 edges. However, such conditions are necessarybut not sufficient for a graph to be rigid in R3. A famouscounterexample is reported in ‘double bananas’, wherethe two bananas can rotate relative to one another alongtheir implied hinge [36]. There has been a lot of effort try-ing to examine appropriate refinements that would yield acombinatorial characterization of generic rigidity in R3.

While Laman’s condition does not characterize genericrigidity in 3-space, the other characterizations of planarrigidity are not known to fail. It may also be possible toaugment Laman’s condition. The obvious question, whichremains unanswered, is whether every 3-connected graphsatisfying Laman’s condition is rigid.

V2

V4 V5

V6 V7

V3

V1

Fig. 5. Let X1 = {V1, V2, V3}, X2 = {V2, V4, V6}, X3 = {V3, V5, V7}, X4 = {V6, V7},and v = {X1, X2, X3, X4}. Then v is a cover of G. Furthermore,

PXev

(2|X| � 3) = 10 < 11 = 2|V| � 3, so G is not rigid by Theorem 5.

3.2. Algorithms for rigidity verification

We have seen that there are a number of character-izations of Laman graphs (generically rigid graphs) andmany of them lead to an associated algorithm for verifi-cation. After all, what we ultimately concern is the posi-tions of nodes in the WSN that can be found throughalgorithm performance. Except for those examinedabove, we compile a list of them here for comparison,in the form of a theorem.

Theorem 6. For a graph G = (V, E) with |V| P 2 the followingare equivalent:

(1) G is generically rigid in R2;(2) there exists a Henneberg 2-sequence for G;(3) for every pair of vertices i, j in V, the multigraph

Gij = (V, E [ ij) obtained by adding an edge betweenvertices i, j is the edge-disjoint union of two spanningtrees;

(4) for every edge ij e E, the multigraph Gij obtained bydoubling edge ij is the union of two edge-disjoint span-ning trees;

(5) G has a proper 3T2 partition;(6) G admits a red–black hierarchy (RBH).

Characterizations (3) and (4) are given in terms of cos-panning trees. Thanks to the existence of polynomial timealgorithms for decomposing a graph into two spanningtrees [48,52], this approach leads to the best known algo-rithm for the Laman decision problem, which runs in timeOðn

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinlog n

pÞ.

Character (5), based on Crapo’s notion of a proper 3T2decomposition [49], was from Tay [53] without usingLaman’s Theorem. Crapo’s algorithm always produces a3T2 decomposition such that one of the trees is spanning.However there are many 3T2 decompositions where allthree trees have approximately the same size. It is anopen question if Crapo’s algorithm can be altered to yieldsuch a balanced decomposition, which would improve therunning time.

Characterization (6) is due to Bereg [54]. The RBH for agraph G is a hierarchical decomposition of the graph intotrees which is a certificate for generic rigidity in the plane.Moreover, an RBH can be constructed in O(n2) time and ahierarchy can be verified to be RBH in O(n) time. Havingan RBH of a Laman graph G enables us to compute itsHenneberg 2-sequence in O(n2) time. Recently Daescu,etc. extended these results and obtained an algorithm forverifying Laman graphs in O(Tst(n) + nlogn) time, whereTst(n) is the best time to extract two edge disjoint spanningtrees from G or decide that no such trees exist [55]. More-over, they speeded up the construction of an RBH toO(nlgn).

3.3. Conditions for global rigidity

Most of the known results on globally rigid graphs areconcerned with the two-dimensional case. Only some ofthem can extend to higher dimensions, leading to partial re-sults on the unique localizability of three-dimensional net-

a

b

ce

fd

a

b

c

d

e f

Fig. 6. Two realizations of a rigid triconnected graph in the plane.


works. The main tool of working with higher dimensionalglobal rigidity is the stress matrix, which plays a similar rolefor global rigidity as the rigidity matrix does for rigidity.

Definition 13. Stress matrix: Let G = (V, E) be a graph and(G, p) be a realization of G in Rd. For each i 2 V, let E(i) be theset of edges of G which are incident to i. An equilibriumstress for (G, p) is a map x: E ? R such that, for each i e V,P

e=ijeE(i)x(e)(p(i) � p(j)) = 0. We associate a symmetric|V| � |V| stress matrix X with each equilibrium stress xfor (G, p) as follows. For each distinct i, j e V, the entry inrow j and column i of X is �x(e) if e = ij e E and zerootherwise. The diagonal entries of X are then chosen sothat its row and column sums are equal to zero.

So a stress matrix of a framework is a rearrangement ofa stress vector into a matrix form, with suitably chosendiagonal entries. It can be proved that the rank of X is atmost |V| � d � 1 and that having a stress matrix with thismaximum possible rank is a sufficient condition for theglobal rigidity of a generic framework [56].

Theorem 7. Let G = (V, E) be a graph. Then G is globally rigidin Rd if and only if either |V| 6 d + 1 and G is complete, or|V| P d + 2 and the associated stress matrix X of an equilib-rium stresss x has rank |V| � (d + 1).

We can find that Theorem 7 states much the same wayas Theorem 2 does in Section 3.1. It implies that globalrigidity in Rd is a generic property. The sufficient conditionfor unique realizability expressed by Theorem 7 is not ofmuch practical use for us unless we can readily computethe rank of a stress matrix.

As is already clear, a necessary condition for uniquerealization of generic framework is rigidity. Hendrickson[44] pointed out that d + 1-connectivity of G is another nec-essary condition for a d-dimensional framework (G, p) to beglobally rigid.

Definition 14. k-connectivity: A graph G is k-connected ifit remains connected upon removal of any set of <kvertices, together with the removal of the edges incidenton them. Or equivalently, between any two vertices of thegraph, there must exist at least k paths which have no edgeor vertex in common.

Rigidity and d + 1-connectivity are necessary but notsufficient to ensure that a graph has a unique realization.A two-dimensional example of a rigid, triconnectedgraph with two satisfying realizations is given in Fig. 6.We need the graph to be generically redundantly rigidto ensure generic global rigidity.

Definition 15. Redundantly rigid: G = (V, E) is redundantlyrigid if G � e is rigid for all e e E, i.e. the removal of a singleedge e from the rigid graph G does not destroy rigidity.

If G is not redundantly rigid and G has more thand + 1 vertices, which is quite common in WSNs, thenalmost all realizations of G are not unique. In R2, wehave Theorem 8.

Theorem 8. [35]. A graph with n P 4 vertices is genericallyglobally rigid if and only if it is 3-connected and redundantlyrigid.

In R3, it is necessary that a graph be 4-connected andredundantly rigid to be globally rigid. However, these con-ditions are known to be insufficient [35]. No necessary andsufficient conditions for generic global rigidity are known,and it is not clear that such conditions have to exist, in con-trast to the two-dimensional case. That is, there may beexample of three-dimensional graphs for which specifica-tion of a set of lengths confined to certain intervals for eachlength always guarantees global rigidity, while specifica-tion of the lengths for the same sensor pairs but confinedto other intervals for each length results in lack of globalrigidity.

It is a natural thought that densely connected WSN maylead to a unique realization. It has been proved that 6-con-nected graphs are redundantly rigid. Combining this withTheorem 7, we show that sufficiently high connectedgraphs are globally rigid.

Theorem 9. [57]. Let G be 6-connected. Then G is globallyrigid.

An infinite family of 5-connected non-rigid graphs givenin [51] show that the hypothesis on vertex connectivity inTheorem 9 cannot be reduced from six to five. On the otherhand, Jackson et al. showed in [58] that the connectivityhypothesis can be replaced by a slightly weaker hypothesisof ‘essentially 6-connected graph’ or even weaker but alsosufficient connectivity condition to guarantee global rigid-ity. Recently Jackson et al. provided another sufficient con-nectivity condition for global rigidity [59].

3.4. Inductive construction

One of the most useful methodologies for devising adistributed WSN localization algorithm is through sequen-tial extension. It is therefore quite helpful to derive induc-tive constructions for generically globally rigid graphs (i.e.,solvable) associated with WSNs. That is, given a graph G,we try to create, hopefully in polynomial time, a globallyrigid realization ( G, p) in Rd if such a realization exits.Henneberg construction [46] becomes our important toolthroughout this subsection.

Definition 16. Let G = (V, E) be a graph. The Henneberg-I(H1) step (or vertex addition) applied to G, inserts one newvertex that gets connected to 2 existing ones. TheHenneberg-II (H2) step (or edge split) applied to G, replaces


an edge by a new vertex that gets connected to itsendpoints and to one more arbitrary vertex. For anexample of a H1 and a H2 step. See Fig. 7.

Henneberg 2-sequences are a systematic way of generat-ing minimally rigid graphs in R2 based on the H1 and H2

operations. The formal definition follows.

Definition 17. A Henneberg 2-sequence for a graph G is asequence of graphs G1, . . . , Gn with the following proper-ties : (1) G1 = K3; (2) Gn = G; and (3) Gi+1 is obtained from Gi,through a H1 or a H2 step, 8i 2 f2; . . . ;n� 1g.

See Fig. 8 for an example of a Henneberg 2-sequence forthe K3,3 graph. It is important to notice that the Henneberg2-sequence for a graph G may not be unique, i.e., a graph Gcan have many Henneberg 2-sequences. The other opera-tion is edge addition by adding a new edge connecting somepair of non-neighbor nodes [56].

The following Theorem by Tay and Whiteley [46] fullyjustifies our interest in Henneberg 2-sequence.

Theorem 10. A graph is minimally rigid in R2 if and only if ithas a Henneberg 2-sequence.

Naturally, if H is a minimally rigid graph and G is ob-tained from H by a Henneberg 2-sequence. Then G is min-imally rigid. The analogue for global rigidity is in Theorem11. Again, in R3, Theorem 10 becomes a sufficient but notnecessary condition.

Theorem 11. [60]. Let H be a globally rigid graph with atleast four vertices and let G be obtained from H by Henneberg-II. Then G is globally rigid.

A slightly weaker result was previously obtained byConnelly [35], who showed that if G can be obtained fromK4 by Henneberg-II then G is globally rigid. Combined withanother key ingredient of inductive construction due toJackson and Jordán [57] we have Theorem 12.

Theorem 12. [56]. A graph with at least four vertices isglobally rigid if and only if it can be obtained from K4 byHenneberg-II and edge additions.

(a) (b) (c)Fig. 7. (a) Graph before Henneberg step; (b/c) a representative case of H1

step and H2 step. The newly added vertexes and the new edges aredepicted in grey.

H1 H1

Fig. 8. A Henneberg 2-sequence for the K3,3 graph. At each step,

Another interesting result on the construction of glob-ally rigid realizations is by trilateration. When a two-dimensional graph has more than 3n � 6 edges and satisfiesa trilaterative ordering [31], it is globally rigid. A construc-tion process is also given in [31]. Note that as more verticesand edges are added, most of the graph becomes quicklywell connected. Thus if it is sufficient to located just a highpercentage of the nodes, the connectivity requirementsmay be much less harsh than the result in Theorem 8.

4. Bearing constraint based localizability analysis

Since rigidity of framework with distance information iswell understood in two-dimension and partially under-stood in three-dimension, pure distance information isused extensively in studying WSN localization problem.Alternatively, the distance between each pair of nodescan be held fixed with constraints prescribing directionsand bearings between nodes along with fewer distances.However, in the most general form, there is no completetheory for frameworks based solely on bearings. In thissection we present the conditions for unique WSN localiza-tion by using rigidity and global rigidity of graphs that rep-resent the network topologies, in which bearingconstraints on directed links are satisfied among nodes.Note that direction constraints, determined by the direc-tion of line joining two nodes, can be easily converted tobearing constraints, and vice versa.

There are two important differences between the caseswhen all constraints are lengths. The direction problem canbe solved for all d, whereas the length problem has beensolved only when d = 1, 2. Furthermore, there is no needto assume that the framework is generic to solve the direc-tion problem. We first look at the simpler case when allconstraints are bearing constraints.

Given a graph G, we can interpret the edges as line seg-ments in the plane whose direction is to be fixed and therebyobtain the concept of parallel drawings, which is equivalentto the linearized problem obtained from interpreting theedges of G as constraints. Two frameworks F p and F q onthe same graph are parallel drawings if all correspondingedges are parallel to each other. See Fig. 9. Since a bearingconstraint can be written as a parallel drawing constraint[61], we are interested in a parallel framework F q in whichqi � qj is parallel to pi � pj for all ði; jÞ 2 B, where B is the setof maintenance links with bearing constraints.

A parallel rigid motion is a trajectory along which frame-works contained in this trajectory are translations or dila-tions of each other. All other motions are non-trivial.

H2

the added vertex and the new edges are depicted in grey.

(a) (b)Fig. 10. (a) Length edge; (b) bearing edge.


Definition 18. Parallel rigidity [62]: If parallel rigidmotions are the only possible trajectories the frameworkis called parallel rigid, otherwise parallel flexible. SeeFig. 9e.

Two parallel rigid frameworks can be obtained fromeach other by congruence and scaling transformations.They are also called similar, or direction congruent or glob-ally direction rigid. We say that (G, p) and (G, q) are directioncongruent if there exists a scalar k and a vector t such thatq(i) = kp(i) + t for all i e V.

Lengths and directions play symmetric roles in thetheory, leading to a basic duality between these two con-straints. Because of this geometric switching, the generictype of rigidity is defined in the same manner as in the caseof distances. The translation of Laman’s conditions for gen-eric direction frameworks can be given as follows.

Theorem 13. [62]. A graph G ¼ ðV ;BÞ, where B denotes theset of links with bearing measurements, is generically parallelrigid in 2-space if and only if there is a subset B0#B satisfyingthe following two conditions: (1) jB0j ¼ 2jV j � 3, (2) For allB00#B0;B00–0; jB00j 6 2jVðB00Þj � 3, where jVðB00Þj is the num-ber of vertices that are end-vertices of the edges in B00.

This characterization of parallel rigid graphs can be gen-eralized for arbitrary dimensions.

Theorem 14. [63]. A graph G = (V, E) is generically parallelrigid in Rd, d P 2, if and only if it contains a set of edges E0# Ewith (d � 1)|E0| = d|V| � (d + 1) such that for all subsetsE00# E, (d � 1) |E00| 6 d|V(E00)| � (d + 1)

The fact that the direction constraint is a linear con-straint enabled Whiteley to characterize parallel rigid d-dimensional frameworks in terms of the rank of their direc-tion rigidity matrix. He then used this result to obtain acombinatorial characterization for the generic case [64].The following result can be derived from this characteriza-tion of d-dimensional parallel rigid frameworks.

Theorem 15. Let (G, p) be a d-dimensional generic frame-work. Then (G, p) is parallel rigid if and only if

PXe{(d|X| �

d � 1) P d|V| � (d + 1) for all covers v of G.Note that the characterization of two-dimensional par-

allel rigid frameworks given by Theorem 15 is identical tothe characterization of two-dimensional rigid genericframeworks given in Theorem 5.

We may suppose from Section 3 that the conditions forglobal rigidity are much stronger than those for parallelrigidity. But the key constraints here are linear equations.

(a) (b) (c)Fig. 9. (a–d) are all parallel drawings. In particular, (b and c) are dilations of thflexible since (d) shows a non-trivial parallel drawing of (a). While the framewo

If we satisfy 2n � 3 bearing constraints of a parallel rigidframework and add one distance constraint, we will have aglobally rigid framework. For networks with pure bearingconstraints between nodes, if there exist two non-similarparallel frameworks with points p and q, then both frame-works are not rigid. On the other hand, if the framework isparallel rigid, at least one distance constraint is necessaryto rule out scaling. Only translations of the framework willbe the trivial frameworks. In practical terms, one anchornode with distance measurement, possibly with world coor-dinates to pin down the whole network to exclude transla-tions, is needed to exclude scaling. In fact, parallel rigidityup to translation and dilation is equivalent to global rigidity.

5. Mixed constraints based localizability analysis

In this section we consider WSNs in which both dis-tance information and bearing information are known forsome parts of vertices. We now mix these two types ofconstraints into a single system. The unique localizationproblem for such networks seems to be also NP-hard sowe still restrict our attention to the two-dimensional case.

A mixed graph is a graph together with a bipartitionB [ L of its edge set. We refer to edges in B as bearing edgesand edges in L as length edges. To distinguish the two kindsof constraints in figure of designs, we will follow the con-vention of indicating a length constraint between twopoints as an ordinary edge, and a bearing constraint be-tween two points as an edge with two arrowheads alongits interior, see Fig. 10.

A mixed framework (G, p) is a mixed graph G = (V; B, L)together with a map p: V ? R2. Related terminologies canbe defined analogously as those of pure frameworks [65].Two mixed frameworks (G, p) and (G, q) are equivalent ifp(u) � p(v) is a scalar multiple of q(u) � q(v) for alluv 2 B and ||p(u) � p(v)|| = ||q(u) � q(v)|| for all uv 2 L.The mixed frameworks ( G, p) and ( G, q) are congruent ifthere exists a vector t e R2 and k e {�1, 1} such that

(d) (e)e framework in (a). It is obvious that framework (a) is rigid but parallelrk in (e) is parallel rigid.


q(v) = kp(v) + t for all v e V . This is equivalent to saying that(G, q) can be obtained from (G, p) by a rotation of 0� or 180�and a translation. The mixed framework (G, p) is globallyrigid if every framework which is equivalent to (G, p) iscongruent to (G, p). It is rigid if there exists an e > 0 suchthat every framework (G, q) which is equivalent to (G, p)and satisfies ||p(v) � q(v)||<e for all v e V, is congruent to (G, p).

Servatius and Whiteley [66] developed a rigidity theoryfor mixed framework analogous to that for ‘distance con-strained’ frameworks. One may construct a jB [ Lj � 2jV jmixed rigidity matrix for a mixed framework ( G, p) anduse its rows to define the mixed rigidity matroid of (G, p).A generic mixed framework is rigid if and only if its rigiditymatrix has rank 2|V| � 2.

A framework using distance-bearing constraints can beunique up to translation. This will require 2|V| � 2 inde-pendent constraints. For example, it is easy to see that:(i) an independent set of 2|V| � 3 distances plus any singlebearing is an appropriate set of 2|V| � 2 constraints, see forexample Fig. 11a; (ii) an independent set of 2|V| � 3 bear-ings plus any single distance is an appropriate set of2|V| � 2 constraints, see for example Fig. 11b. It is clearfrom Fig. 11c that 2|V| � 2 mixed constraints is only a nec-essary condition for a framework to be rigid.

For a set X of vertices in a mixed graph G, we use B (X)and L(X) to denote the set of bearing edges, resp. lengthedges in G[X]. The following theorem presents an analogueto Laman’s conditions for mixed graphs.

Theorem 16. Let G ¼ ðV ;B; LÞ be a mixed graph withjBj þ jLj ¼ 2jV j � 2. Then G is rigid if and only if for allX # V with |X| P 2, (1) i(X) 6 2|X| � 2 when BðXÞ–£–LðXÞ,and (2) i(X) 6 2|X| � 3 when BðXÞ ¼£ or LðXÞ ¼£.

There exist efficient algorithms to check whether amixed graph G ¼ ðV ;B; LÞ satisfies condition (1) and (2).See [66] and the references therein.

As for parallel rigidity determination, Theorem 16 alsoholds. We rewrite it in a slightly modified form below.

Theorem 17. Let G ¼ ðV ;B; LÞ be a mixed graph, then G isgenerically parallel rigid in 2-space if and only if the followingconditions hold: (i) jBj þ jLj ¼ 2jV j � 2; (ii) for all subsets V0 ofvertices: jL0j þ jB0j 6 2jV 0j � 2; (iii) for all subsets V0 of at leasttwo vertices: jB0j 6 2jV 0j � 3 and (iv) for all subsets V0 of atleast two vertices: |L0| 6 2|V0| � 3.

The problem of characterizing when a generic mixedframework (G, p) is globally rigid is still an open problem.Recently, Jackson et al. give some partial results that may

(a) (b) (c)Fig. 11. (a) and (b) are rigid mixed constraints based graphs, while (c) isflexible.

serve as a building block to a complete characterization[65]. They proved a necessary condition for global rigidity,which is analogous to the ‘3-connectedness condition’ ofTheorem 8. They also described some sufficient conditionsfor global mixed-rigidity by using Henneberg construction.

Theorem 18. [67]. Let H be a globally rigid mixed graph withat least three vertices and let G be obtained from H by a1-extension on an edge uw. If H � uw is rigid, then G isglobally rigid.

In mixed graphs, a special kind of 0-extension also pre-serves global rigidity.

Theorem 19. [67] Let G and H be mixed graphs with |V(H)|P 2. Suppose that G can be obtained from H by a -extensionwhich adds a vertex v incident to two bearing edges. Then G isglobally rigid if and only if H is globally rigid.

We close this section by using Theorems 18 and 19 toshow that a special family of d-dimensional generic mixedframeworks is globally rigid.

Theorem 20. [65]. Let G be a mixed graph in which everypair of adjacent vertices is connected by both a length and abearing edge, and (G, p) be a generic realization of G in Rd.Then (G, p) is globally rigid if and only if G is 2-connected.

6. Conclusion

Despite a significant number of approaches developedfor WSN localization, there are still many unsolved prob-lems in this area. The challenges to be addressed are bothin analytical characterization of the sensor networks anddevelopment of efficient localization algorithms for variousclasses of sensor networks under a variety of conditions.

In this paper we have drawn on powerful results fromrigidity theory, distance geometry, geometric constraintsin CAD, and combinatorial theory. These fields have beenactive for decades and made plentiful fruit. By selectingwhat we consider to be most efficient from these tech-niques (terminologies, concepts, theorems, algorithms,etc.), we cover in an in-depth way the main theoreticalinvestigations of WSN unique localization problem. By effi-ciency we mean there may exist efficient algorithms that,given an instance of the WSN localization problem in R2

and R3, verifies the uniqueness or declare that the given in-stance is infeasible. We show that both distance constraintbased localizability and bearing constraint based localiz-ability are well understood in R2. Only partial analogous re-sults are available in R3.

Other interesting ideas in literature, though not studiedsystematically in this work, have also been exploited in re-cent years. For instance, node localizability problem is dis-tinct from network localizability problem discussed above.It was observed that even in networks with non-globallyrigid graphs, there may exist uniquely localizable nodes[60,68]. A theoretical investigation of this phenomenon ofglobally linked nodes was later given by Jackson et al.[69], together with a sufficient generic condition. Our pre-liminary idea is to make use of hierarchical structures [70]in analyzing these partially localizable networks. Another


concept put forward recently is universal rigidity, which iseven more restrictive than global rigidity [71]. As real pro-ject works in either two-dimension or three-dimensionthat is easy to distinguish, the new concept’s effectivenessneeds further study.

It is evident that the aforementioned heuristics for theunique localization problem depend heavily upon thosemathematic theories. Some of the important open issuesare already shown in the former sections. What we aremostly concern is to achieve more satisfying fruits in R3.It is also highly desirable that the verification algorithmscan directly bring forth efficient realization of the given in-stance of a WSN.

Acknowledgements

This research was partially supported by the NaturalScience Foundation of China (60874009 and 10971120),and the Natural Science Foundation of Shandong Province(ZR2010FM010 and ZR2010FM047).

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Yuan Zhang received the B.Sc. degree inElectronics Engineering from Nankai Univer-
sity, and the M.Sc. degree in CommunicationSystems from Shandong University, China, in1996 and 2003 respectively. From October2003 to October 2004, he was a research fel-low at Kyung Hee University, Korea. He iscurrently working as a Senior Lecturer atUniversity of Jinan, China, and is also a PhDcandidate at Shandong University, China. Hisresearch interests are in wireless networksand mobile communications, especially
focusing on wireless sensor networks and ubiquitous computing. As thecorresponding author he has published more than 10 technical papers inrelated fields.

rks 10 (2012) 623–634

Shutang Liu was born in 1960. He receivedhis Ph.D. in Control Theory and Control Engi-neering from South China University ofTechnology in 2002. He is currently a profes-sor at the School of the Control Science andEngineering, Shandong University, China. Hisresearch interests include bifurcations andchaotic theory of nonlinear dynamical sys-tems and its application, qualitative control ofcomplex system, and algorithm design forwireless sensor networks. His research hasbeen supported by the China National Science

Foundation. He has published more than 80 papers in prestigious journalsand conference proceedings.

Zhao Xiuyang was born in 1974 in ShandongProvince of China. He received his B.Sc. degreein material science and engineering from theShandong University of Technology of Chinain 1998, and Master and Ph.D. degree incomputational material science from theShandong University of China in 2001 and2006. During 2008–2010, he had worked asthe postdoctor in School of computer science,Shandong University. He is currently a asso-ciate professor of the School of InformationScience and Engineering of University of Jinan.

His main research interests include computer graphic, machine learning,and skeletal tissue engineering.

Zhongtian Jia received his B.Sc. degree inApplied Mathematics from Qingdao Univer-sity, China, in June 1996 and his M.Sc. degreein Computer Science from Shandong Univer-sity, China, in June 2005. He is currentlyworking towards his Ph.D. in Cryptography atBeijing University of Posts and telecommuni-cations, China. His current research interestincludes wireless network security andapplied cryptography.

theoretic analysis of unique localization for wireless sensor networks

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