Research Article Efficient Deterministic Anchor Deployment for …downloads.hindawi.com/journals/ijdsn/2013/429065.pdf · 2015-11-21 · Nevertheless, numerous disadvantages of the

Hindawi Publishing CorporationInternational Journal of Distributed Sensor NetworksVolume 2013 Article ID 429065 13 pageshttpdxdoiorg1011552013429065

Research ArticleEfficient Deterministic Anchor Deployment forSensor Network Positioning

Yongle Chen12 Ci Chen3 Hongsong Zhu1 and Limin Sun1

1 State Key Laboratory of Information Security Institute of Information Engineering Chinese Academy of SciencesBeijing 100093 China

2University of the Chinese Academy of Sciences Beijing 100190 China3The School of Software and Microelectronics Peking University Beijing 102600 China

Correspondence should be addressed to Limin Sun sunliminiieaccn

Received 10 January 2013 Accepted 24 February 2013

Academic Editor Jianwei Niu

Copyright copy 2013 Yongle Chen et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Sensor network positioning systems have been extensively studied in recent years Most of the systems share a common assumptionthat some known-position anchor nodes have existed However a more fundamental question is always being overlooked that ishow to acquire the anchorrsquos position In general GPS-based measures and the artificial calibration are two dominant methods toacquire anchor positions Due to the high energy cost and failures in occlusion regions of the GPSmodules the artificial calibrationmethod is adopted extensively Nevertheless numerous disadvantages of the artificial calibration such as the expensive labor costand error-prone features also make it hard to be an efficient solution for the anchor positioning For this reason we designan efficient mapping algorithm between anchors and their positions (MD-SKM) to avoid the complicated artificial calibrationAdditionally we propose a best feature matching (BFM) method to further relax the restriction of MDS-KM where three or morecalibrated anchors are neededWe evaluate ourMDS-KMalgorithmunder various topologies and connectivity settings Experimentresults show that at a slightly higher connectivity level our algorithm can achieve the exactly correct matching between anchorsand their positions without any calibrated anchors

1 Introduction

Sensor networks have been extensively studied due to theirsalient advantages of monitoring and controlling relatedapplications such as the battlefield monitoring medicalsurveillance and structure monitoring Furthermore inthese applications a basic service requirement implied isto determine the exact location of the happening eventthrough pre-deployed sensors in order that the operatorsare prone to execute appropriate control actions in responseto the event In this sense the positioning technology ofthe sensor network is getting more and more attentionsCurrently most developed positioning systems depend onfourmetrics including TOA AOA RSS and the connectivityof the signals In addition these systems also have a commonassumption that is some known-position nodes exist in thesensor network which are also named as anchors Basedon the known-position anchors they are able to localize

other unknown-position nodes However how to acquire theanchorrsquos position is still an unsolved question

At present two methods are mainly used for acquiringanchorsrsquo positions One is leveraging GPS modules andthe other is called the artificial calibration [1] In practicethe placement ways to construct the sensor network decidewhich method to be adopted to acquire anchorsrsquo positionsThe placement ways can be summarized in two types thestochastic way and the deterministic way In terms of thestochastic way the most typical example is the battlefieldmonitoring In the battlefield scenario military sensors arerandomly scattered from the air where the anchorsrsquo positionsare stochastic In this case to acquire anchorsrsquo positions wehave to rely on the GPS modules attached with sensors TheGPS modules are usually limited by a series of disadvantagessuch as high energy costs and failures in occlusion regionswhich make this method not applicable for the low-powersensor networks Different from the stochastic way in some

2 International Journal of Distributed Sensor Networks

scenarios the anchorsrsquo positions are deterministic accordingto predesigned placement blueprint such as that in medicalsurveillance and structure monitoring In these scenariosthe correspondence is recorded between anchorsrsquo physicalpositions in the blueprint and anchorsrsquo IDs such that eachanchor unambiguously knows its own position We namedthis method the artificial calibration Also many works [2 3]have pointed out that optimizing the anchor placement is ableto accelerate the convergence of the positioning algorithmand improve the positioning accuracy Nevertheless this kindof methods always suffers from the complicated and error-prone mapping between physical locations and the node IDswhich is even more severe in a large sensor network

In this paper we design an efficient MDS-KM matchingalgorithm to avoid the artificial calibration cost in deter-ministic anchor placement To the best of our knowledgewe are the first to consider solving the artificial calibrationproblem of anchors placement Given sufficient calibratedanchors (C-anchors for short) we first design a distributedMDS-MAP(A)method to construct an absolute radiomap byusing estimated distance or hop distance between anchorsIn the absolute radiomap each anchor has an absolutecoordinate of its position corresponding to the physicalposition in the blueprint In order to map the radiomapwith the blueprint we use the kNN method to select the119896-nearest physical positions in the blueprint away from theanchor absolute positions in the radiomap and then build acomplete bipartite graph Based on the bipartite graph weadopt the Kuhn-Munkres (KM) algorithm to get a maximumweighted matching Accordingly we achieve the correspon-dence between anchor nodes in the radiomap and physicalpositions in the blueprint Meanwhile in order to relax ourMDS-KM method for the cases without calibrated anchorswe design a best feature matching (BFM) method to activelymap parts of anchors in the radiomap to positions in theblueprint Our method will greatly improve the efficiency ofanchor placement through avoiding the artificial calibrationThe experiment in Section 6 shows that the mapping fromthe radiomap to the blueprint is exactly correct when theconnectivity level of network is not excessively low

The remainder of the paper is organized as follows Therelated work is shown in Section 2We formulate the problemin Section 3 while leaving the details of our algorithm designfor Section 4 The further improved strategy is presentedin Section 5 Then we show the experiment and simulationresults of our MDS-KM algorithm in Section 6 Finally weconclude the paper in Section 7

2 Related Work

Many works have pointed out that anchor placement wayswill help to improve positioning performance The pioneeranchor placement ways are mainly based on the empiricalevidence in positioning system For example Shang et al [2]randomly place anchors in their experiment and find that aselection of collinear anchors in one test is rather unluckyRecently Akl et al [3] study the anchor placement for passivepositioning and they find that the optimal placement is that

no three anchor nodes are collinear at the center of networkThe authors of [4] point out that the optimal placement ofanchors should be around the corners of the network andalso find that the more nonlinearity results in the betterpositioning performance

Doherty et al [5] place the anchors at the corners of thenetwork to acquire a better positioning results However thealgorithmhas a constraint requirements that all the unknownnodes should be placed within the convex hull of the anchorswhich reduces the algorithm generality Ash and Moses [6]analyse and prove that the anchors on the corners of networkwill help to improve positioning result when the network isa rectangle Hara and Fukumura [7] also propose an anchorplacement algorithm applied to the rectangle network andthat they point out the anchors must be placed in thecenters of subrectangle regions divided from the rectanglenetwork

Some anchor placements focus on the effect of theenvironment For example the authors of [8] conduct someexperiments where anchors are placed either on the ceilingor the floor The study find that anchors or on the floor arebetter for monitoring moving people in the room Althoughmany anchor placement works are developed they only focuson how to improve the positioning performance based onanchor positions and ignore how to acquire the positions ofanchorsThis paper analyses the artificial calibration problemto acquire the anchor positions after deterministic placementIn order to efficiently acquire the anchor positions we intro-duce MDS method to construct a radiomap correspondingto the blueprint Then anchors physical positions can be self-calibrated by mapping the radiomap to the blueprint

MDS method is a series of analysis techniques used fordisplaying the data proximity as a geometrical picture [9] Atpresent there are many variants of MDS positioning algo-rithm including classical metric MDS-MAP(C) distributedMDS-MAP(P) local MDS and weighted dwMDS(G) Cen-tralized MDS-MAP(C) [10] algorithm is the earliest usage ofMDS techniques in sensor network positioning Since MDS-MAP(C) uses the shortest hop distance as the estimate of thetrue Euclidean distance it is not good for irregular networkA distributed MDS method MDS-MAP(P) [11] is proposedto be applied to different network topologies MDS-MAP(P)first constructs a 2-hop local map by executing MDS-MAP(C) method for nearby nodes then merges each localmap into a global map based on the common nodes LocalMDS [12] is another distributed variant of MDS-MAP(C)improved for irregular topologiesThe difference fromMDS-MAP(P) is that the nearby nodes of constructing local maponly include 1-hop neighbors and the weights are restricted to0 or 1 Meanwhile a least square optimizationmethod is usedfor refining the local mapsThe dwMDS(G) [13] is a weighteddistributedMDSmethod in which aweighted (Gauss kernel)cost function is adopted for adaptively emphasizing the mostaccurate range measurements Besides dwMDS(G) designsa neighbor selection method to avoid the biasing effects ofnoisy range measurements neighbors

In this paper we design a distributed MDS-KM methodto increase the efficiency of anchor placement At first wedesign an MDS-MAP(A) method focusing on the anchor

International Journal of Distributed Sensor Networks 3

11

12 9

8

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710

13

14

minus75 minus73minus56

minus57

minus59

minus67minus60

minus68 minus69

minus60

minus79 minus56

minus74minus76minus56

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minus63

minus53

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minus77

1

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55

1

2

3

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Figure 1 The blueprint and the corresponding radiomap

Radiomap (RM)

Blueprint (BP)

Absolute RM by MDS-MAP

RM-to-BPKM matching

Figure 2 The MDS-KM algorithm framework

positioning to construct a radiomap with absolute coordi-nates which is not subject to the irregular anchors dis-tribution Afterwards we use the KM algorithm to obtainthe maximal weighted matching of complete bipartite graphconstructed by the radiomap and the blueprint Besides ourMDS-KM method can also avoid the error-prone mappingduring the artificial calibration

3 Problem Specification

The optimized placement of anchors has a very importantimpact for the positioning performance For improvingthe positioning accuracy a predefined blueprint is usuallyconstructed to guide the anchor placement before deployingthe positioning system That is called deterministic anchorplacement For example the left graph in Figure 1 is ablueprint where the black cycles are the positions to placeanchors During the placement anchor node ID will be one-to-one mapped to the anchor position marked on blueprintwhich is called artificial calibration This process will con-sume a higher labor cost and lead to error-prone mappingIn order to solve this problem we build a radiomap usingthe connectivity in large sparse network or signal strengthbetween anchors in small dense network and then adaptivelymap the radiomap to the blueprint with little or no artificialcalibration As shown in the right graph of Figure 1 thevertices in the radiomap represent anchor nodes and the edgeweights represent the signal strength in the small networkIn sparse network many anchor nodes may have only fewneighbor anchors or even none Here we make the shortesthop distance fromone anchor to another anchor as theweightin the radiomap

Accordingly the problem to be solved becomes the exactmapping from the radiomap to the blueprint Intuitively theradiomap has similar characteristics with the blueprint Theradiomap-to-blueprint mapping should be graph isomor-phism (GI) problem [14] But in the small network it does not

11988911

11988912

11988922

11988923

11988934

11988943

11988933

11988944

119881 119881998400

1998400

2998400

3998400

4998400

1

2

3

4

Figure 3 A complete bipartite graph

Radiomap (RM)

Blueprint (BP)

Relative RM byMDS

Advanced graph(AG)

Absolute RM byMDS-MAP

RM-to-BPKM matching

Figure 4 The improved MDS-KM algorithm framework

strictly belong to graph isomorphismproblem Supposing theradiomap and the blueprint are isomorphic each vertex andedge in both graphs must have a corresponding bijectionAs the physical distance increases in the blueprint theRSS in the radiomap damps and even disappears but thephysical distance can still be measured Thus the blueprint isa complete graph while the radiomap is a subgraph of theblueprint Even though we limited the maximum measuredistance in the blueprint the edges in the blueprint maystill not have a corresponding bijection to the edges in theradiomap due to the effect of the surrounding noise Theedges in the radiomap only have a corresponding bijectionwith the subset of the blueprint This is a typical subgraphisomorphism problem [15]

However subgraph isomorphism is an NP-completeproblem [16] Furthermore the distances between verticesin the blueprint do not exactly reflect the RSS values inthe radiomap subjected to the surrounding noise Thereforethe existing heuristic subgraph isomorphism algorithm isnot suitable for the radiomap-to-blueprint mapping In thispaper we design an MDS-KM matching algorithm to solvethis mapping problem in the small network or the sparsenetwork We introduce the multidimensional scaling (MDS)method in the anchor placement which is well suited to com-pute a relative coordinates map in a low-dimensional spaceby one matrix representing distance information betweennodes Based onMDSmethod and sufficient known-positioncalibrated anchors (3 or more) we design a distributedMDS-MAP(A) method to construct the radiomap with absolutecoordinates Then the Euclidean distances of vertices in the

4 International Journal of Distributed Sensor Networks

5

1

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1014

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4

minus75 minus73

minus56

minus60

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minus69

minus67

minus60

minus79

minus56

minus74

minus76

minus53

minus68

minus87

minus69minus55

minus58

minus76

minus56

minus74

Figure 5 The advanced graph in the small network

AnchorsConcrete column

Wooden wallCorridor

1 2

3

4 5 6

121110

13 14 15 16 17 18

242322212019

8 97

(a)

AnchorsConcrete column

Wooden wallCorridor

1 2

3

4 5 6

121110987

16

20191817

151413

(b)

Figure 6 Two experiments in uniform and n-sharp distribution

radiomap and the blueprint are computed as the weights toconstruct a weighted bipartite graph where one part of thebipartite graph includes all the vertices in the radiomap andthe other part of the bipartite graph includes all the verticesin the blueprint Afterwards we adopt the classical Kuhn-Munkres (KM) method [17] to carry out a maximum weightmatching of the bipartite and then get a one-to-one mappingbetween anchor node IDs in the radiomap and positions inthe blueprint

4 Radiomap-to-Blueprint Mapping

41 Algorithm Overview As mentioned above the matchingbetween the radiomap and the blueprint is our primaryobjective The MDS-KM matching process is illustrated inFigure 2 In general the MDS method utilizes the physicaldistance between anchors to construct a relative coordinate

radiomap But the edge weights in the radiomap of the smallnetwork represent the RSS values We need to transformRSS value to the estimated distance according to the signalpropagation model Then we use the MDS method to get aradiomap with relative coordinates Having sufficient anchornode positions (3 for 2D networks and 4 for 3D networks)we can map the relative coordinates of anchors to absolutecoordinates through a linear transformation [10]ThenweuseKM algorithm to compute the optimal complete matchingbetween the blueprint and the radiomap with absolute coor-dinates Since the KM algorithm is applied to the weightedbipartite graph matching we need to construct a bipartitegraph utilizing the radiomap and the blueprint Thus wedesign an error-torrent kNN vertex selection method tobuild a bipartite graph Finally we achieve the mappingfrom the radiomap to the blueprint through computing themaximum weighted matching of the bipartite In Section 5

International Journal of Distributed Sensor Networks 5

0 2 4 6 8 100

2

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(a)

0 2 4 6 8 100

2

4

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(b)

Figure 7 Two simulations in grid and random distribution

0 10 20 300

5

10

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20

25

(a)

0 10 20 300

5

10

15

20

25

(b)

Figure 8 The average errors in both experiment scenarios

we further design a best feature matching (BFM) methodto relax the restriction of MDS-KM where three or morecalibrated anchors are needed

42 Absolute Radiomap Construction

421 Collecting Distance Information In order to constructa radiomap we need to compute the estimated Euclideandistance based on the RSS or hop distance between anchorsThe RSS or hop distance of each pair of anchors should beobtained at first Intuitively flooding is a better selectionIn the small dense network each anchor node broadcaststhe beacon packet periodically and keeps on receiving thebeacon packets from other anchors then computes the RSSsof these beacon packets After awhile each anchorwill recordan RSS sequence from other anchor nodes within its 1-hopcommunication range Finally each anchor sends its node

ID and RSS sequences to the backend positioning server forconstructing the radiomap In order to avoid the sendingcollision we will make the broadcast cycle of each anchordifferent in our experiment

Additionally in large sparse network many anchorsmay not be within the communication range of any otheranchors These anchors are isolated We will use the shortesthop distances as the estimated distance There are someintermediate unknown-position nodes scattered within theanchorsThe shortest hop distance is defined as theminimumhop count between anchors multiplied by the average signalhop distance In this process each anchor will broadcastits beacon packet periodically Each intermediate unknown-position node records the minimal hop value and adds itselfto the value and then forwards the hop count continuallywith initial anchor ID until the beacon packet arrives to anew anchor or achieves our hop limit In order to reduce the

6 International Journal of Distributed Sensor Networks

communication cost we set a hop upper limit (eg 10) toconstruct local map Each anchor records all the minimumhop counts from nearby anchors and sends them and theirnode IDs to the positioning server

422 Estimated Distance In the large sparse network wecan compute the Euclidean distance between the calibratedanchors According to the minimum hop counts betweenthem we further compute the average single hop distanceAccordingly we can compute the hop distance between eachpairwise anchors as the estimated distance In the small densenetworkweneed to use signal propagationmodel to computethe estimated distance based on the RSS value According towhether the travel distance is short or large the propagationmodels can be classified into large scale and small scale[18] In general the small-scale model needs to characterizethe rapid fluctuations of RSS over short travel distance Ithas a better accuracy than large-scale model but it is verydifficult to determine the model parameters In this paperwe concentrate on the generality of the designed algorithmand do not consider a specific scenario Hence we select agood compromise between simplicity and accuracy which iscalled the wall attenuation factor propagation model (WAF)[19] This model provides flexibility when applied to indoorscenario while considering outdoor large-scale fading Thismodel is described as

119901 (119889) [119889119887119898] = 119901 (1198890) [119889119887119898] minus 10120572 log(119889

1198890

) minus 120575

120575 = 119899119908 timesWAF 119899119908 lt 119862

119862 timesWAF 119899119908 ge 119862

(1)

where 119889 is the transmitter-receiver distance 119875(1198890) is the

signal power at some reference distance 1198890 120572 indicates the

rate at which the signal fades 119862 is the maximum number ofobstacles up to which the attenuation factor makes a differ-ence 119899119908 is the number of obstacles between the transmitterand the receiver and WAF is the obstacle attenuation factorIn general the values of 120572 and WAF depend on the specificpropagation environment and should be derived empiricallyGiven the RSS value we can further compute the estimateddistance 119889

119890as follows

119889119890= 1198890times 10(119901(1198890)[119889119887119898]minus119901(119889)[119889119887119898]minus120575)(10120572) (2)

Additionally there are some optimization methods totune parameters of propagation model so that the RSSmeasurements can characterize the accurate distances [20ndash22] In our algorithm the MDS method can tolerate errorgracefully due to the overdetermined nature of the solution[9] Hence we do not need exactly RSS values depending onoptimizing the propagation model

423 Constructing Absolute Radiomap In this part we willuse the MDS method to construct the absolute radiomap Atpresent many types of MDS techniques have been developed[9] In our algorithm we design a distributed MDS-MAP(A)algorithm focusing on the anchor placement The MDS-MAP(A) algorithm consists of four main steps as follows

First we use the above estimated distance to construct the1-hop proximity matrix 119875 for each anchor where the 1-hopneighbors of anchors in large network will be the anchorsin the range of hop upper limit We denote the proximitymeasure between anchor 119894 and 119895 as 119901

119894119895 Then assuming

an m-dimensional space given the anchor 119894 coordinates119883119894= (1199091198941 1199091198942 119909

119894119898) and the anchor 119895 coordinates 119883

119895=

(1199091198951 1199091198952 119909

119895119898) the practical Euclidean distance between

anchor 119894 and 119895 is denoted by 119889119894119895which will construct a

Euclidean distances matrix119863 as

119889119894119895= radic

119898

sum

119896=1

(119909119894119896minus 119909119895119896)2

(3)

In theory the matrix 119875 should be equal to the matrix 119863 Butthe estimated distance with errors makes them unequal Inthis case the MDS method can ensure 119875 is approximate to119863as far as possible

Second we run the MDS algorithm for each distancematrix 119875 to get a local map with relative coordinates Inclassical metric MDS the proximity matrix 119875 can be trans-formed to a double centered matrix 119861 which is symmetricand positive semidefinite matrix as

119861 = minus1

2(1199012

119894119895minus1

119899

119899

sum

119895=1

1199012

119894119895minus1

119899

119899

sum

119894=1

1199012

119894119895+1

1198992

119899

sum

119894=1

119899

sum

119895=1

1199012

119894119895) (4)

When we shift 119875 to the center 119861 can also be expressed asfollows

119861 = 119883119883119879=

119898

sum

119896=1

119909119894119896119909119895119896 (5)

We perform the singular value decomposition (SVD) on 119861 toget 119861 = 119881119860119881

119879 which has complexity of O(1198963) where 119896 isthe number of anchors in the local mapThus the complexityof computing 119899 local maps is O(1198963119899) where 119899 is the numberof anchors in the radiomap The coordinate matrix is 119883 =

119881119860(12) where 119860 = diag(119897

1 1198972 119897119899) is the eigenvalue

diagonal matrix in descending order 119881 = [1198811 1198812 119881

119899] is

the eigenvector corresponding to the eigenvalue We selectthe first 119898 eigenvectors to construct a coordinate matrix inlower dimension This is the best low-rank approximationbetween matrix 119875 and119863 in the least-squares sense

Third we merge all local maps to the whole relativeradiomap Each local map is a group of 1-hop neighborsWe randomly select a local map as the base map and thensequentially merge the neighbor local map according to thecommon nodes Eventually the base map grows to cover thewhole radiomap As known from [11] the complexity of thisstep is the same as step 2

Finally given sufficient calibrated anchors we map therelative coordinates to the absolute coordinates of anchorsthrough a liner transformation [10] which include scal-ing reflection and rotation The radiomap with absolutepositions can be achieved eventually For 119903 anchors thecomplexity of this step is O(1199033 + 119899)

International Journal of Distributed Sensor Networks 7

43 Radiomap-to-BlueprintMatching Since the surroundingnoise and irregular topology affect the precision of estimateddistance and lead to the inaccuracy absolute coordinates ofanchors in the radiomap the absolute coordinates in theradiomap are not completely consistent with the coordinatesof anchor physical positions in the blueprint Hence theabove two groups of coordinates cannot be correspond-ing completely We only search for the most approximatematching of two coordinates Therefore the objective of theradiomap-to-blueprint matching turns into minimizing thesum of corresponding Euclidean distances between the phys-ical positions in the blueprint and the absolute coordinatepositions in the radiomap We present a k-nearest neighbor(kNN) method to find the best approximate positions intwo graphs The k-nearest neighbor is a simple classificationmethod in the data mining field This algorithm can selectthe 119896-nearest ones through evaluating Euclidean distancebetween positions For each anchor in the radiomap weutilize the kNNmethod to find the 119896-nearest positions in theblueprint away from itThenwe can build aweighted bipartitegraph whose weights on edges are the Euclidean distancesAn example with 119896 = 2 is shown in Figure 3 Additionallythe value of parameter 119896 is task specific In our algorithm weselect the minimal 119896 to guarantee that all the positions in theblueprint will be selected into 1198811015840 when all anchors 119881 in theradiomap have been carried out in the kNN operation Thusthe bipartite graph has a complete matching where everyvertex of the graph is exactly incident to only one edge

Accordingly the radiomap-to-blueprint matching prob-lem will be transformed into a minimum weighted matchingproblem in a weighted bipartite graph where the sum ofthe weight of all the edges in the bipartite matching isminimal Such a matching is also known as the optimalassignment problem It can be solved by Kuhn-Munkres(KM) algorithm in polynomial time However the KMalgorithm just applies to solving the maximum weightedmatching problem We need to pick the minus of the weightsin the bipartite so that the minimum weighted matchingproblem is further transformed into a maximum weightedmatching problemTheKMalgorithmwill use vertex labelingmethod to transform the maximum weighted matching intocomplete matching in unweighted bipartite graph and thenuse the classical Hungarian algorithm to solve the maximummatching problem of unweighted bipartite graph

Algorithm 1 is a simplified KM algorithm procedureWe first initialize a feasible vertex labeling Normally eachvertex in one side of the bipartite graph is labeled withthe maximum weight of its incident edges connected to thevertices in the other side and each vertex in the other sideis labeled zero (line 2ndash6) The bipartite graph will becomean unweighted bipartite graph Then we seek a maximummatching usingHungarian algorithm and decide whether themaximum matching is a complete matching or not (line 7-8) If the maximum matching is a complete matching wesave the matching and return Otherwise we need to relabelthe vertices following the KM algorithm rules and literatelycarry out the Hungarian algorithm (line 12-13) Finally wecan achieve a complete matching and get the mappingrelationships between the radiomap and the blueprint

5 Without Calibrated Anchors

In this section we try to relax our MDS-KM algorithm to beapplied to the situation without any artificial calibrations Wedesign a best feature matching (BFM) method to actively getparts of mapping from anchors in the radiomap to positionsin the blueprint without any artificial calibration In order todistinguish the feature of vertices in the radiomap and theblueprint we bring in the vertex weighted sequence as thefeature metric where the edge weight is RSS value or hopcount Then some vertices with best unique feature in theradiomap can be selected and their corresponding verticesare found in the blueprint by our BFMmethod However theedge weight in the blueprint is physical distance The vertexweighted sequences in the radiomap are not comparableto those in the blueprint because of the different types ofthe edge weight Hence we transform the blueprint to anadvanced graph (AG) whose vertex features are the RSSsequences in the small network and hop count sequences inthe large network The new matching process of MDS-KMalgorithm is also changed to Figure 4 The advanced graphis used to seek the parts of anchors with a unique featureinstead of the calibrated anchors to construct the absoluteradiomap

51 Blueprint to the Advanced Graph In the small networkthe distances between vertices in the blueprint are not exactlyreflecting the RSSs in the radiomap due to the surroundingobstacles and noise We first use the signal propagationmodel mentioned in the above subsection to transform thedistances between vertices in the blueprint into the RSSvalues which is constructed in an advanced graph denotedby 119866119860= (119881119860 119864119860) These RSS values represent the weights of

the edges in the advanced graph and the number of verticesand edges in the advanced graph is the same as that of theblueprint Since any two vertices in the blueprint have oneedge the advanced graph is also a complete graph Figure 5is an example of the advanced graph from the blueprint inFigure 1 In the large network we compute the minimal hopcounts between pairwise anchors in the blueprint after settingthe communication range of node and then construct anadvanced graph whose edge weights represent minimal hopcounts Similarly the advanced graph in the large network isalso a complete graph

52 Best Feature Matching Before executing the MDS-MAP(A) method the radiomap 119866

119877= (119881

119877 119864119877) has the

vertex set 119881119877and edge set 119881

119877 The edge weight represents the

RSS or hop count We first make the vertices distinguishabledepending on their invariants which are the fixed propertiesof vertices during matching A simple invariant is the vertexdegree However in a graph the vertex degree is not uniqueThere is likely to be many vertices having the same degreeTherefore we bring the weights into the vertex invariantsfor example 119868(V

119894119882) = (V

119894 1199081 1199082 119908

119889) by following

the arrangement 1199081gt 1199082gt sdot sdot sdot gt 119908

119889 and 119889 is the

degree of the vertex V119894 Similarly we can formulate the

corresponding vertex invariants of the advanced graph For

8 International Journal of Distributed Sensor Networks

2 4 6 8 10 12 14 16

55

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Erro

r (m

)

Connectivity

3 C-anchors5 C-anchors7 C-anchors

(a)

3 C-anchors5 C-anchors7 C-anchors

2 4 6 8 10 12 14Connectivity

55

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)(b)

Figure 9 The error analysis in both experiment scenarios

5

4

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03 6 9 12 15

The v

alue

of p

aram

eter119870

Connectivity

Uniform topology119899-sharp topology

Figure 10 The selection of the K using for constructing bipartitegraph

example 1198681015840(V1015840119894 1199081015840) = (V1015840

119894 1199081015840

1 1199081015840

2 119908

1015840

(119899minus1))11990810158401gt 1199081015840

2gt sdot sdot sdot gt

1199081015840

(119899minus1) n is the number of all vertices Each vertex degree is

119899 minus 1 since the advanced graph is a complete graphWe will select the vertices invariants in the radiomap

which are the most easy to distinguish We noted thatthe degrees of many vertices in the radiomap are dif-ferent so that the number of weights in some vertexinvariants is inconsistent This brings inconvenience toour feature comparison Therefore we need to normal-ize the vertex invariants of the radiomap We first com-pute the maximal degree of all vertices Max(d) in theradiomap then extend the vertex invariant 119868(V

119894119882) from

(V119894 1199081 119908

119889) to (V

119894 1199081 119908

119889 119908(119889+1)

119908Max(119889)) where

minus60

minus65

minus70

minus75

minus80

minus85

minus900 100 200 300 400 500

Number of packets

Rece

ived

sign

al st

reng

th

Without wallWooden wallConcrete column

Figure 11 The obstacle affecting

119889 le Max(119889) 119908(119889+1)

= 119908(119889+2)

= 119908Max(119889) = 119908min 119908minis the minimum RSS value measured from anchor device inthe small network or hop count of zero in the larger networkWe can compute the Euclidean distance 119889

119877119860between vertices

invariants in two graphs as follows

119889119877119860= radic

Max(119889)sum

119894=1

(119908119894minus 1199081015840119894)2 (6)

We still adopt the k-nearest neighbor (119896 = 2) method tofind the two minimum 119889

119877119860between vertices in the radiomap

and vertices in the advanced graph For each vertex in the

International Journal of Distributed Sensor Networks 9

20

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Connectivity

The n

umbe

r of c

orre

ct m

atch

ing

Uniform topology119899-sharp topology

Figure 12 The results of BFMmethod in the small network

radiomap the absolute value of the difference of the twominimum Euclidean distances can be computed and sortedin descending order The bigger the absolute value of thedifference themore unique the vertex features So the verticesin front of the order are the most possible unique and distin-guishable ones They can actively catch their correspondingminimum Euclidean distance vertices in the advanced graphTo some extent this method is subject to the symmetryof anchors in the blueprint But we can artificially designthe blueprint keeping asymmetric Meanwhile the irregularenvironment also affects the symmetry of the blueprintTherefore in practice the weights of anchors in the blueprintare hardly perfectly symmetric

6 Implantation and Experiment

61 Experiment Design In our experiment we will runMDS-KM algorithm on a variety of anchor topologies inthe small and large networks In the 30m times 25m roomthe anchors are installed on the ceiling or concrete columns(1) Figure 6(a) is the placement blueprint where there are24 positions to place anchor nodes Concrete columns andwoodenwalls in the room are the principal obstacles affectingcommunication quality between anchors (2) We simplifythe topology of Figure 6(a) into an n-sharp topologies of 20positions as shown in Figure 6(b)

In the large network we simulate the anchors in theMATLAB placed with grid distribution and random distri-bution respectively as shown in Figures 7(a) and 7(b) Anumber of 100 nodes are placed uniformly and randomly in a10 119903times10 119903multihopnetwork where 85 nodes are intermediateunknown-position nodes denoted by the circle and 15 nodesare anchors denoted by the stars (lowast) For the purpose offacilitating the comparison of positioning error we select thesimilar anchor positions in both topologies to construct theradiomap

It should be noted that the complicated office room ismore sensitive to the noise than outdoors Meanwhile themost indoor positioning systems are usually deployed deter-ministically according to the placement blueprint Thereforewe choose the indoor environment as the case of the smallnetwork which is more powerful to verify the MDS-KMperformance

62 The Small Network During the radiomap constructionwe set each anchor ID numbermultiplied by 100millisecondsas its broadcast cycle to avoid the sending collision Afterrunning 2 minutes we compute the average RSS valuesbetween anchors We use our MDS-MAP(A) method in thetopologies (1) and (2) for constructing the absolute radiomapbased on 3 random calibrated anchors denoted by the stars(lowast) as shown in Figure 8 The circles represent the estimatedabsolute positions and the solid lines represent the errorsbetween the estimated positions and the true positions Thelonger the solid line the larger the positioning error Thetransmitting power of TelosB in TinyOS system is classifiedinto 1 to 31 levelsWith the level rising the transmitting powerbecomes higher We set the highest level of transmittingpower in this group of experiments The results show that wehave the average estimation errors of 305mand 325m in twotopologies

Figure 9 shows the average performance of MDS-MAP(A) positioning affected by connectivity and numbersof calibrated anchors Figures 9(a) and 9(b) show the resultsof MDS-MAP(A) positioning of two topologies respectivelyWe set the transmitting power levels as 11 17 21 26 and31 respectively in our experiments Three five and sevencalibrated anchors are used Then we get the connectivitylevels of 26 43 68 106 and 152 in the uniform topologyand 21 36 58 96 and 132 in the n-sharp topologyWith the lowering of the connectivity level the positioningperformance declines significantly When the connectivitylevel is less than 3 the average error will be achieved toaround 55m Besides the positioning error becomes lightlylower with the increasing of C-anchors Meanwhile thedifferent numbers of calibrated anchors also have very closepositioning errors Therefore a certain range of a number ofvariations of calibrated anchors has no significant influenceon positioning performance

We obtain a radiomap with absolute coordinates afterMDS-MAP(A) operation Before running the KMmatchingwe need to set the parameter 119896 for constructing a bipar-tite graph In our experiment we show the minimal 119896 toproducing a complete bipartite graph in Figure 10 With theconnectivity level rising the value of 119896 reduces graduallyWhen the connectivity level is 152 in uniform topologyand 96 and 132 in n-sharp topology the value of 119896 is 1That means that the bipartite graph is already a one-to-onemapping complete graph Then we can obtain the optimalmatching between the blueprint and the radiomap withoutthe KM method Meanwhile we find that this mapping isalso exactly correct Under other connectivity levels we mustuse the KM method to find the optimal matching We findthat the rate of correct matching between anchors in the

10 International Journal of Distributed Sensor Networks

0 2 4 6 8 100

2

4

6

8

10

(a)

0 2 4 6 8 100

2

4

6

8

10

(b)

Figure 13 The average errors in both simulation scenarios

15

1

05

05 10 15 20

Connectivity

Erro

r (119903)

3 C-anchors5 C-anchors7 C-anchors

(a)

5 10 15Connectivity

1

2

3

0

Erro

r (119903)

3 C-anchors5 C-anchors7 C-anchors

(b)

Figure 14 The error analysis in both simulation scenarios

radiomap and positions in the blueprint can achieve 100when connectivity level is over 3 Only when the connectivitylevel is less than 3 there are two anchor nodes with errormapping in both topologies where the node IDs are 3 and10 respectively This is because both nodes are close to eachotherThe positioning error from theMDS-MAP(A) methodwill make their positions confused so that the maximumweighted matching of the KM method is not exactly themapping from the radiomap to the blueprint Meanwhile wealso observe that themore calibrated anchors cannot help theaccuracy of the KM matching unless the anchors with errormatching are calibrated anchors

In order to validate the performance of our BFM algo-rithm we need to exactly transform the physical distance ofthe blueprint into RSS value of the advanced graph At first

we make a measurement test for determining the parametersWAF and 120572 in (1) During our experiment we test twotypes of obstacle materials 40 cm width wooden wall and60 cm times 60 cm width concrete column Two TelosB nodeslie in two sides of obstacle and 2m away from the obstacleOne node broadcasts beacon packet every 10 seconds whileanother node receives the packet and computes the RSS valueWe spend 80 minutes to get the results shown in Figure 11We find that the wooden wall and concrete column canapproximately reduce RSS 5 db and 10 db respectively Basedon the measurement we further compute the fading factor 120572in our environment which is approximate to 3 Then we usethe experimental values to construct the advanced graph

Figure 12 is the number of correct matching anchors withconnectivity increasing during the BFMprocessThe number

International Journal of Distributed Sensor Networks 11

7

6

5

4

3

2

13 6 9 12 15 18

Connectivity

The v

alue

of p

aram

eter119870

Grid topologyRandom topology

(a)

12

9

6

3

The n

umbe

r of c

orre

ct m

atch

ing

Grid topologyRandom topology

3 6 9 12 15 18Connectivity

(b)

Figure 15 The results of BFMmethod in the large network

1 119866(119883 119884119882)lowast119866 is a bipartite Graph119882 is the Weightlowast2 for all (119909 isin 119883 and 119910 isin 119884) do3 lowastInitialize all vertices labelinglowast4 119897(119909) =Max119908(119909 119910) 119910 isin 1198845 119897(119910) = 0

6 end for7119872 =Hungarian(119866(119883 119884 119897))8 if (119872 is complete matching of 119866) then9 = save(119872)10 return 11 else12 relabeling(119897)lowastas KM ruleslowast13 goto 714 end if

Algorithm 1 The Kuhn-Munkres Algorithm

in the uniform case is lightly more than that in the randomcase which is mainly due to more quantity of anchors inuniform topology Meanwhile we find that there are three ormore anchors at least with correct matching even when theconnectivity is lower than 3 in two topologies Therefore wecan run our MDS-KM method in all the above experimentswithout any calibrated anchors which further reduces thelabor cost But unfortunately our BFM method cannot helpto solve the error mapping of the MDS-KM method underthe lower connectivity

63 The Large Network We run MDS-MAP(A) methodfor the grid and random topologies of the large networkto construct the absolute radiomap based on 3 randomcalibrated anchors as shown in Figure 13The circles representunknown-position intermediate nodes The stars representthe anchor nodes and the solid lines represent the errors

between the estimated positions and the true positions Inthe 10 119903 times 10 119903 area we set the communication range as 15 rand 2 r respectively in the grid and random topologies Theaverage connectivity levels of both topologies are 67 and 63respectively Although both connectivity levels are similarthe positioning errors have a big difference After runningthe MDS-MAP(A) method for the radiomap we have thecorresponding average estimation errors of 087 r and 135 rin both topologies This is because the connectivity level ofnodes in the random case is uneven so that its estimatederror of hop distance is significantly bigger than that in thegrid case Therefore the corresponding absolute radiomapin the random case has also a bigger average estimationerror

Additionally we compare the performance of the MDS-MAP(A) method in different connectivity levels and cali-brated anchors In both topologies we select 3 5 and 7 cali-brated anchors randomly to construct the absolute radiomap

12 International Journal of Distributed Sensor Networks

during every trail In the grid topology the radio ranges arefrom 1 119903 to 2 119903 with an increment of 025 119903 which result inthe connectivity of 39 55 67 122 and 182 respectively asshown in Figure 14(a) We find that the higher connectivitylevel will bring about a better positioning result and themorecalibrated anchors also improve the positioning performanceWhen connectivity level is lower than 67 especially theaverage estimated error will increase significantly In therandom topology the radio ranges are from 1 119903 to 3 119903 with anincrement of 05 119903 which lead to average connectivity of 4551 63 102 and 156 respectively as shown in Figure 14(b)This design is to compare the performance of the MDS-MAP(A) algorithm under the similar connectivity levels ofboth topologiesWe can see that the positioning performancein the random topology has a significant reduction than thatin the grid topology The maximum average estimated erroris even twice that in the grid topologyThat is mainly becausethe estimated hop distance in the random topology is ratherinaccurate

Figure 15(a) is the 119896-value selection of both topologiesWe can find that the 119896 in the random topology has a highervalue than that in the grid topology This is because thehigher errors of the estimated hop distance in the randomtopology produce the bigger position errors of the absoluteradiomap Thus the anchors in the radiomap cannot exactlycorrespond with the positions in the blueprint In order toget a complete bipartite graph 119896-value must be increasedAfterwards we find that the KM method can reach a 100rate of correct matching except that there are 3 and 2 error-matching anchors respectively under the connectivity of 45and 51 in the random topology It is further suggested that theMDS-KM algorithm is well suited to the higher connectivitynetwork

Figure 15(b) reflects the BFM method performance inboth topologies of the large network In the random topologythe BFMmethod can obtain a better feature matching resultThis is because many vertices in the grid topology havethe same hop count sequences subjected to the symmetryof anchor distribution Therefore the vertices invariants inthe grid topology are hard to be distinguished while in therandom topology there aremore distinguishable verticeswithunique invariants But in both topologies we can also findthat there are more than three anchors with correct featurematching In other words the MDS-KM method can runsuccessfully in two simulation scenarios of the large networkwithout any calibrated anchors

7 Conclusion

In this paper we consider the anchor self-positioning prob-lem in detail During the deterministic anchor placement wedesign an efficient mapping algorithm between anchors andpositions (MDS-KM) to avoid the expensive labor cost anderror-prone features of artificial calibration Additionally wepropose a best feature matching (BFM) method to obtainsome mappings between anchors and positions in advanceso that any calibrated anchors are not needed Experimentalresults show that the MDS-KM algorithm can achieve the

100 correct matching between anchors and positions undera higher connectivity level Meanwhile in our experimentsand simulations the BFM method can obtain sufficientknown-position anchors to support the successful running ofthe MDS-KMmethod

Acknowledgments

This work is supported by the General Program of NationalNatural Science Foundation of China (NSFC) under Grantno 61073180 and the National Key Basic Research Programof China (973) under Grant no 2011CB302902

References

[1] H S AbdelSalam and S Olariu ldquoTowards enhanced RSSI-Based distance measurements and localization in WSNsrdquo inProceedings of the IEEE INFOCOM Workshops 2009 pp 1ndash2April 2009

[2] Y Shang W Ruml Y Zhang and M Fromherz ldquoLocalizationfrom connectivity in sensor networksrdquo IEEE Transactions onParallel and Distributed Systems vol 15 no 11 pp 961ndash9742004

[3] R Akl K Pasupathy and M Haidar ldquoAnchor nodes placementfor effective passive localizationrdquo in Proceedings of the Inter-national Conference on Selected Topics in Mobile and WirelessNetworking (iCOST rsquo11) pp 127ndash132 October 2011

[4] T Kunz and B Tatham ldquoLocalization in wireless sensor net-works and anchor placementrdquo Journal of Sensor and ActuatorNetworks vol 1 no 1 pp 36ndash58 2012

[5] L Doherty K S J Pister and L El Ghaoui ldquoConvex positionestimation in wireless sensor networksrdquo in Proceedings ofthe 20th Annual Joint Conference of the IEEE Computer andCommunications Societies (INFOCOM rsquo01) vol 3 pp 1655ndash1663 April 2001

[6] J N Ash and R L Moses ldquoOn optimal anchor node placementin sensor localization by optimization of subspace principalanglesrdquo in Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing (ICASSP rsquo08) pp 2289ndash2292 April 2008

[7] S Hara and T Fukumura ldquoDetermination of the placementof anchor nodes satisfying a required localization accuracyrdquo inProceedings of the IEEE International Symposium on WirelessCommunication Systems (ISWCS rsquo08) pp 128ndash132 October2008

[8] R Zemek M Takashima S Hara et al ldquoAn effect of anchornodes placement on a target location estimation performancerdquoin Proceedings of the IEEE Region 10 Conference (TENCON rsquo06)pp 1ndash4 November 2006

[9] I Borg and P Groenen ldquoModern multidimensional scalingtheory and applicationsrdquo Journal of Educational Measurementvol 40 no 3 pp 277ndash280 2003

[10] Y Shang W Ruml Y Zhang and M P J Fromherz ldquoLocal-ization from mere connectivityrdquo in Proceedings of the 4th ACMInternational Symposium on Mobile Ad Hoc Networking andComputing (MobiHoc rsquo03) pp 201ndash212 ACM New York NYUSA June 2003

[11] Y Shang and W Ruml ldquoImproved MDS-based localizationrdquoin Proceedings of the 23th Annual Joint Conference of the IEEEComputer andCommunications Societies (IEEE INFOCOM rsquo04)vol 4 pp 2640ndash2651 March 2004

International Journal of Distributed Sensor Networks 13

[12] X Ji and H Zha ldquoSensor positioning in wireless ad-hoc sensornetworks using multidimensional scalingrdquo in Proceedings ofthe 23th Annual Joint Conference of the IEEE Computer andCommunications Societies (IEEE INFOCOM rsquo04) vol 4 pp2652ndash2661 March 2004

[13] J A Costa N Patwari and A O Hero ldquoDistributed weighted-multidimensional scaling for node localization in sensor net-worksrdquo ACM Transactions on Sensor Networks vol 2 no 1 pp39ndash64 2006

[14] D C Schmidt and L E Druffel ldquoA fast backtracking algorithmto test directed graphs for isomorphism using distance matri-cesrdquo Journal of the Association for Computing Machinery vol23 no 3 pp 433ndash445 1976

[15] Sansone and M Vento ldquoSubgraph transformations for theinexact matching of attributed relational graphsrdquo Computingvol 12 pp 43ndash52 1998

[16] S A Cook ldquoThe complexity of theorem-proving proceduresrdquoin Proceedings of the 3rd annual ACM symposium on Theory ofcomputing (STOC rsquo71) pp 151ndash158 ACM New York NY USA1971

[17] J Munkres ldquoAlgorithms for the assignment and transportationproblemsrdquo Journal of the Society for Industrial and AppliedMathematics vol 5 no 1 pp 32ndash38 1957

[18] T S RappaportWireless Communications Principles and Prac-tice IEEE Press Piscataway NJ USA 1st edition 1996

[19] P Bahl and V Padmanabhan ldquoRadar an in-building rf-baseduser location and tracking systemrdquo in Proceedings of the 9thAnnual Joint Conference of the IEEEComputer and Communica-tions Societies (IEEE INFOCOM rsquo00) vol 2 pp 775ndash7784 2000

[20] K Benkic M Malajner P Planinsic and Z Cucej ldquoUsing RSSIvalue for distance estimation in wireless sensor networks basedon ZigBeerdquo in Proceedings of the 15th International Conferenceon Systems Signals and Image Processing (IWSSIP rsquo08) pp 303ndash306 June 2008

[21] P Barsocchi S Lenzi S Chessa and G Giunta ldquoVirtual cali-bration for RSSI-based indoor localization with IEEE 802154rdquoin Proceedings of the IEEE International Conference on Commu-nications (ICC rsquo09) pp 1ndash5 June 2009

[22] K Srinivasan and P Levis ldquoRSSI is under appreciatedrdquo inProceedings of the 3rd Workshop on Embedded NetworkedSensors (EmNets rsquo06) 2006

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