On a Registration-Based Approach to Sensor Network Localization

1 On a Regstraton-Based Approach to Sensor Network Localzaton R. Sanyal, M. Jaswal, and K. N. Chaudhury arxv:177.2866v2 [math.oc] 9 Nov 217 Abstract We consder a regstraton-based approach for localzng sensor networks from range measurements. Ths s based on the assumpton that one can fnd overlappng clques spannng the network. That s, for each sensor, one can dentfy geometrc neghbors for whch all nter-sensor ranges are known. Such clques can be effcently localzed usng multdmensonal scalng. However, snce each clque s localzed n some local coordnate system, we are requred to regster them n a global coordnate system. In other words, our approach s based on transformng the localzaton problem nto a problem of regstraton. In ths context, the man contrbutons are as follows. Frst, we descrbe an effcent method for parttonng the network nto overlappng clques. Second, we study the problem of regsterng the localzed clques, and formulate a necessary rgdty condton for unquely recoverng the global sensor coordnates. In partcular, we present a method for effcently testng rgdty, and a proposal for augmentng the parttoned network to enforce rgdty. A recently proposed semdefnte relaxaton of global regstraton s used for regsterng the clques. We present smulaton results on random and structured sensor networks to demonstrate that the proposed method compares favourably wth state-of-the-art methods n terms of run-tme, accuracy, and scalablty. Index Terms Sensor networks, localzaton, scalablty, rgdty, clque, multdmensonal scalng, semdefnte programmng. I. INTRODUCTION Recent developments n wreless communcaton and mcro-electro-mechancs have prolferated the deployment of wreless sensor networks (WSN) [1]. A typcal WSN may consst of few tens to thousands of nodes. Each node s a low-power devce equpped wth transducers, power supply, memory, processor, rado transmtter, and actuators. A global postonng system (GPS) s often nstalled on some of the nodes. Such nodes are referred to as anchor nodes. However, only a small fracton of the nodes are equpped wth GPS to mnmze weght and power consumpton. In ths paper, we wll use sensor to specfcally refer to a node that does not have a GPS, whle the term node wll be used for both sensors and anchors. WSNs are mostly deployed n remote locatons, and nodes have lmted memory capacty, so wreless transmtters are used to transfer the sensor data to base statons. Due to power constrants, two nodes can communcate f and only f the nter-node dstance s wthn some rado range, whch we wll denote by r [1]. We would lke to note that although GPS modules are gettng cheaper, deployng them n large scale would stll be costly. Moreover, GPS comes wth ts own lmtatons [1]. To calculate the poston of a sensor usng GPS alone, at least four lne-of-sghts (wth satelltes) are requred. Ths mght not be vable n case of bad weather. Furthermore, for underwater surveys and mnng applcatons, t s not even feasble to have lne-of-sghts. In fact, n applcatons where the poston nformaton s crucal, localzaton algorthms can be used to back up GPS postonng. To meanngfully nterpret the sensor data, one requres the locatons of the sensors. A central problem n ths regard s to estmate the sensor locatons from the nter-sensor dstances and the anchor locatons. Ths problem s referred to as sensor network localzaton (SNL) [2], [3]. To set up the mathematcal The authors were supported by a Startup Grant from IISc Bangalore and an EMR Grant SERB/F/647/216-217 from Department of Scence and Technology, Government of Inda. The second author was supported by a NBHM Postdoctoral Fellowshp from Department of Atomc Energy, Government of Inda. Address: Department of Electrcal Engneerng, Indan Insttute of Scence, Inda. Correspondence: {rajatsanyal,monka,kunal}@ee.sc.ernet.n.

2 descrpton of SNL, we ntroduce some notatons that wll be follow throughout the paper. Assume that we have a total of N sensors and K anchors. We label the sensors usng S = {1,..., N}, the anchors usng A = {N + 1,..., N + K}, and N = S A denotes the nodes n general. Let X s = { x : S} and X a = {ā k : k A} (1) denote the sensor and anchor locatons. We assume x and ā k to be n R d, where d s typcally 2 or 3 [1], [2]. The dstance between two nodes and j (that are wthn the rado range r) can be calculated usng dfferent technques, such as the receved sgnal strength or the tme of arrval [3]. A measurement graph G s used to encode the dstance nformaton [4], [5]. Partcularly, G = (V, E), where V(G) = N, and (, j) E(G) f and only f the dstance between the -th and the j-th node s known. The problem s to compute the unknown sensor locatons X s from the measured dstances and the anchor locatons X a. We make the standard assumpton that the anchor locatons are nose-free [4], [6]. A. Optmzaton Algorthms The decson verson of the SNL problem s known to be computatonally ntractable [7]. The presence of nose makes the problem even more challengng n practce. Nonetheless, several methods have been proposed that can compute approxmate solutons. A survey of the lterature on SNL s beyond the scope of ths paper. Instead, we wll focus on some of the recent optmzaton methods that are related to the present work. We refer the nterested reader to [3] for a survey of algorthms that are not based on optmzaton. The smplest optmzaton framework for SNL s that of stran mnmzaton [8]. In ths approach, the sensor locatons x 1,..., x N are obtaned by mnmzng the stran functon ( x x j 2 d 2 2+ ( j) x a k 2 d 2 ) 2 k. (2) (,j) E (,k) E In (2), the ndces, j are reserved for S, and the ndex k for A. Unfortunately, t s dffcult to compute the global mnmum of (2) snce t s non-convex n the varables [2]. In ths regard, several approxmaton algorthms based on convex programmng have been proposed, whch can provably compute the global mnmum under certan condtons. Based on the computng paradgm, one can broadly classfy these as centralzed and dstrbuted algorthms. Centralzed algorthms employ a server to store the transmtted range measurements, based on whch the sensor locatons are computed. It was observed n [9] that the dstance bounds n SNL can be posed as semdefnte constrants. Later, n the semnal paper [8], the authors showed how (2) can be approxmated usng a convex semdefnte program (SDP). The man advantage of posng SNL as a convex program s that we can fnd the global mnmzer of the problem ndependent of the ntalzaton. The flp sde, however, s that standard SDP solvers (e.g., SeDuM [1]) are memory and computaton ntensve, and hence cannot be scaled to large-szed problems. For example, the SDP-based algorthm n [8] can scale only up to a few hundred nodes [11]. To mprove the scalablty, a further edge-based relaxaton of [8] was proposed n [11]. Whle the relaxaton can ndeed scale up to 8 nodes, ts performance s nevertheless nferor to that of the orgnal SDP for medum-szed problems. On the other hand, dstrbuted algorthms dvde the processng over the nodes. As a result, they exhbt better scalablty compared to centralzed methods. The man drawback s that they suffer from error propagaton [3]. Moreover, dstrbuted methods such as the ones n [4], [6], [12], [13] can operate only n the presence of anchors (whch mght not be avalable, e.g., n ndoor WSN). The dstrbuted algorthm n [12] that can handle mllon sensors wthout any sgnfcant communcaton overhead. However, the localzaton accuracy of ths method s condtoned on a good ntalzaton. More recently, dstrbuted methods based on convex programmng have been proposed n [4], [6], [13]. In partcular, a dstrbuted algorthm based on the alternatng drecton method of multplers was proposed n [6]. However, as reported n [4], the approach s computatonally demandng snce each node s requred to solve an SDP per teraton, and also the communcaton overhead s sgnfcant. A dstrbuted algorthm that s

3 cheaper and requres a smaller communcaton overhead was later proposed n [4]. One shortcomng of the latter method s that t requres the sensors to be n the convex hull of the anchors, whch s dffcult to guarantee n practce. The present work was motvated by a class of centralzed algorthms that use dvde-and-conquer approaches to mprove scalablty [5], [14], [15], [16]. The general mechansm s to partton G nto overlappng subgraphs, localze each subgraph usng the nduced dstances, and fnally regster the subgraphs. The dea s to construct subgraphs that are denser than the orgnal graph. Moreover, the smaller graphs can be effcently localzed. The algorthms essentally dffer on how each subproblem s solved. For example, a Cuthll-McKee-type permutaton s used n [15] to partton G. In [5], [16], G s parttoned usng neghborhood subgraphs. To mprove the localzaton, rgd subgraphs [17] are extracted from each neghborhood subgraph n [5]. Recently, recursve spectral clusterng was used n [14]. We note that the subgraphs obtaned usng the graph parttonng n [14], [15] are not guaranteed to be rgd [18], [19], and hence can result n poor localzaton. In fact, a few poorly localzed subgraphs can adversely affect the overall regstraton. The algorthms n [15], [16] regster the subgraphs n a sequental fashon, and nevtably suffer from error propagaton. Recently, a least-square method was proposed n [2] that can regster the subgraphs n a globally-consstent manner. In partcular, t was demonstrated n [2] that global regstraton can successfully operate n adversaral stuatons where sequental methods fal. In ths regard, we note that a lateraton condton was ntroduced n [2] that can guarantee exact recovery n the nose-free settng. However, there s no known effcent algorthm for testng lateraton. B. Contrbutons We propose a dvde-and-conquer algorthm buldng on the deas n [14], [2], [21]. In partcular, we address the followng ssues that emerged from ths lne of work: testng and ensurng that each subgraph s rgd, formulatng a testable condton for recoverng the sensor coordnates, and developng a scalable algorthm for regsterng the localzed subgraphs. In ths context, the contrbutons are as follows: () To bypass the rgdty ssue assocated wth the localzaton of each subgraph, we propose to use clques. Clques are trvally rgd and can be effcently localzed usng multdmensonal scalng [22]. However, gven that fndng clques n a large graph s challengng, we frst partton G nto neghborhood graphs [5], [16]. We then extract a clque from each neghborhood graph usng the algorthm from [23]. Fnally, we augment the vertces of a clque to expand t nto a maxmal clque. We expermentally demonstrate that the complete process s fast for both random and structured geometrc graphs. () We study the problem of regsterng a system of localzed clques. In partcular, we establsh a rgdty condton that s necessary for recoverng the orgnal sensor coordnates. The proposed condton can be effcently tested smply by computng the maxmum flow between the vertces of an approprate graph. We present supportng examples to conjecture that the proposed rgdty condton s also suffcent for exact recovery. Moreover, we demonstrate usng numercal examples that the regstraton performance can be mproved n the nosy settng by enforcng the rgdty condton. We note that a regstraton-based approach for anchorless SNL was earler proposed n [21] that uses clques and cmds. In the present work, we focus on anchored SNL (though the method can also be used for anchorless SNL). Moreover, we consder a dfferent clque exploraton process. Importantly, we nvestgate the rgdty problem assocated wth regstraton whch was not dscussed n [21]. C. Organzaton The rest of the paper s organzed as follows. In Secton II, we propose a rgdty crtera for the regstraton problem, and explan how ths can be tested effcently. The proposed graph parttonng s descrbed n Secton III keepng the regstraton problem n mnd. Classcal multdmensonal scalng s revewed n Secton IV, whch s used to localze the clques. The regstraton algorthm s descrbed n Secton V. Expermental results and comparsons are provded n Secton VI.

4 Fg. 1. A confguraton of two patches, two sensors and three anchors (left) and ts correspondence graph (rght). Crcles, damonds, and squares are used to represent the sensors, anchors, and patches; note that C s the anchor patch. It s clear that ths confguraton s rgd n two dmensons (see text for precse defntons). We have marked the edges of the three dsjont paths between patch vertces A and B usng dfferent colors (rght). See text for comments. II. THE RIGIDITY PROBLEM We frst study the fundamental problem of rgdty whose resoluton wll be useful durng the graph parttonng phase n Secton III. Ths problem s also relevant for other dvde-and-conquer approaches [5], [14], [16], [2], where a system of pont sets are requred to be regstered. More precsely, consder the sensors X s and the anchors X a n (1), and subsets C 1,..., C M N. Followng [5], [14], we wll refer to each C as a patch. Moreover, we create an addtonal patch C M+1 consstng solely of the anchors A. Assume that the ponts n each patch have been derved from the respectve ponts n X s X a va a rgd transform. Let x k = R (x k, ) = O x k, + t (k C \A), (3) and ā l = R (ā l ) (l C A), (4) where x k, s the coordnate of the k-th pont n the -th patch, and R = (O, t ) s the rgd transform assocated wth the -th patch, where the orthogonal matrx O represents rotaton (or reflecton) and t s the translaton component. We wll refer to the (x k, ) s as the patch coordnates. The patches and the patch coordnates together form a confguraton. The regstraton problem s one of determnng the unknown X s from the gven confguraton. Problem II.1 (Regstraton). Fnd x 1,..., x N and rgd transforms Q 1,..., Q M such that, for 1 M, where k C \ A and l C A. x k = Q (x k, ) and ā l = Q (ā l ), In the noseless settng, the soluton (ponts and transforms) sought above exsts trvally, namely, the ground truth x k = x k and Q = R. The rgdty problem s to determne whether the soluton s unque (upto a global rgd transformaton).

5 Fg. 2. Example of a confguraton wth four patches and four sensors (left). Holdng patches A and B fxed, we can reflect patches C and D along the red dotted lne. Thus, the confguraton s not rgd. Notce that there are just two V 1(Γ)-dsjont paths between patch vertces A and C; we have marked the edges of these paths wth sold lnes (rght). See text for comments. Problem II.2 (Unqueness). Determne whether Problem VIII.4 have a unque soluton up to a rgd transform. That s, f x 1,..., x N s a soluton of Problem VIII.4, then s t necessary that for some rgd transform R, x k = R( x k ) and ā l = R(ā l ), where k S and l A? A set of ponts n R d s sad to be non-degenerate f ther affne span s R d. Clearly, the cardnalty of such ponts must be d + 1 or more. For example, three ponts are non-degenerate n two-dmensons f and only f they are not collnear. We note that the transforms Q 1,..., Q M n Problem VIII.4 are latent varables and do not appear n the Problem VIII.5. Defnton II.3 (Rgdty). A confguraton s sad to be rgd n R d (or smply rgd) f the soluton s unque n the sense of Problem VIII.5; otherwse, the confguraton s sad to be flexble. To provde geometrc nsghts to the rgdty problem, we consder smple nstances of rgd and flexble confguratons n Fgures 1 and 2. In partcular, we wsh to hghlght the mportance of overlaps among patches n determnng rgdty. In Fgure 1, patches A and B share two sensors. The patches can be reflected along the lne jonng sensors 1 and 2, but due to the presence of anchors n A and B, reflecton s ruled out. The confguraton s thus rgd n the sense of Defnton II.3. On the other hand, the confguraton n Fgure 2 s flexble snce patches C and D can be reflected along the red dotted lne. The above concepts and defntons were motvated by the rgdty aspects of the SNL problem [2], and more generally, the dstance-geometry problem [18], [19]. Here, the problem s to determne f the avalable dstance measurements unquely defne the sensor locatons (modulo a rgd transform whch leaves the dstances unchanged). A fundamental result n ths regard s that, f the orgnal sensor locatons are generc [19], then the unqueness problem can be completely resolved usng just the measurement graph [17], [18], [19]. Our present objectve s to come up wth smlar results for Problem VIII.5. At ths pont, we wsh to emphasze that rgdty theory s solely concerned wth exact measurements [19], [24]. The pont s that the combnatoral structure of the problem should, n prncple, be able to guarantee exact recovery of the ground truth when the measurements are perfect. The desgn of an algorthm that can provably recover the ground truth s however a completely dfferent topc. We wll present some representatve examples n Secton VI, whch suggest that rgdty can also help mprove the algorthmc performance n the nosy settng. We wll assume the followng n the rest of the dscusson.

6 Assumpton II.4 (Non-degeneracy). There are at least d + 1 non-degenerate ponts n each patch. Under the above assumpton, we provde a necessary condton for rgdty. Before dong so, we note that a lateraton crtera was earler proposed n [2] that can guarantee rgdty. However, t s not known f there exsts an effcent test for lateraton. Moreover, a path confguraton can be rgd wthout beng laterated, that s, lateraton s not necessary for rgdty. Ths fact s demonstrated wth an example n Fgure 3. Ths motvated us to look for a crtera that s both necessary and suffcent for rgdty. We propose a necessary condton for rgdty that can be tested effcently. We present some examples where the condton s also suffcent, and conjecture that ths s true n general. Before statng the result, we set up a specal bpartte graph that captures the overlap-pattern among patches. Recall that a graph s sad to be bpartte f the vertex set V can be dvded nto dsjont subsets V 1 and V 2 such that there are no edges between the vertces of a gven V. Defnton II.5 (Correspondence graph). We defne the bpartte correspondence graph to be Γ = (V 1, V 2, E), where V 1 (Γ) are the nodes, V 2 (Γ) are the patches, and (k, ) E(Γ) f and only f k C. The correspondence graph for the confguraton n Fgure 1 s shown on the rght. Fnally, we ntroduce a specal noton of connectvty. Recall that a path s an ordered sequence of vertces v 1, v 2,..., v n such that (v t, v t+1 ) s an edge for 1 t n 1. The path s sad to be between vertces α and β (or the path connects α and β) f v 1 = α and v n = β. Two paths n a graph are sad to be vertex-dsjont over a set Θ f they do not share a common vertex from Θ. A set of paths are sad to be Θ-dsjont f any two paths are vertex-dsjont over Θ. Defnton II.6 (Quas connected). The correspondence graph Γ s sad to be quas k-connected f any two vertces n V 2 (Γ) have k or more V 1 (Γ)-dsjont paths between them. Moreover, there exst two vertces n V 2 (Γ) that are connected by exactly k paths that are V 1 (Γ)-dsjont. For latter reference, we record the followng characterzaton of quas k-connectvty. The equvalence can be derved by adaptng the proof of Menger s theorem [25, Theorem 3.3.1]. Proposton II.7. The followng are equvalent. (a) The correspondence graph Γ s quas k-connected. (b) E(Γ) can be dvded nto two dsjont subsets E 1 and E 2 such that the edges from E 1 and that from E 2 are () ncdent on at least k common vertces from V 1 (Γ), and () not ncdent on any common vertex from V 2 (Γ). In Fgure 2, notce that there are 3 paths between any par of vertces n V 2 (Γ), but at most two paths are V 1 (Γ)-dsjont. In ths case, the confguraton s not rgd. In fact, we have the followng result (cf. supplementary materal for the proof). Theorem II.8 (Necessary condton). Under Assumpton VIII.1, f a confguraton s rgd n R d, then ts correspondence graph must be quas (d+1)-connected. Moreover, we see from the example n Fgure 2 that, f Γ fals to be quas (d + 1)-connected, then the confguraton s not rgd. We are yet to fnd a counter-example where the confguraton s flexble yet Γ s quas (d + 1)-connected. Based on emprcal evdences, we make the followng conjecture. Conjecture II.9. Suppose Assumpton VIII.1 holds, and that any k d + 1 ponts n X s X a are non-degenerate. Then a confguraton s rgd n R d f and only f ts correspondence graph s quas (d+1)-connected. The second assumpton appears somewhat strngent at frst sght. The relevance of ths assumpton s somewhat clear from the example n Fgure 1. Namely, f sensors 1, 2 and anchor 5 are concurrent, then one can reflect patch B about the lne jonng these ponts. We note that the use of some form of non-degeneracy assumpton s standard n rgdty theory [19]. Based on Defntons II.5 and II.6, t s not dffcult to establsh a relaton between quas connectvty and the maxmum flow between the vertces of V 2 (Γ) [26]. In ths context, recall that a vertex s sad to have capacty κ f the ncomng and outgong flows for the vertex are at most κ [26].

7 Fg. 3. Example of a rgd confguraton (left) n two dmensons [2]. The confguraton s not laterated, but the correspondence graph (rght) s quas 3-connected. The edges of the three dsjont paths between patch vertces A and B are marked n sold. Proposton II.1 (Connectvty usng flow). Assume that each vertex n V 1 (Γ) s assgned unt capacty whle computng the flow. Then the maxmum flow between the vertces of V 2 (Γ) s at least k f and only f Γ s quas k-connected. The key pont s that one can effcently check f, under the assumpton that the vertces n V 1 (Γ) have unt capacty, the maxmum flow between the vertces of V 2 (Γ) s at least k. Ths can be done usng the Ford-Fulkerson algorthm [26]. Note that we do not need to check the maxmum flow for all pars of vertces n V 2 (Γ). We can smply fx a vertex and check the maxmum flow between ths vertex and the remanng vertces. III. PARTITIONING We now descrbe a heurstc for parttonng G nto overlappng patches such that the correspondng Γ s quas (d + 1)-connected. In ths relaton, we note that dvde-and-conquer approaches have been proposed n [14], [5], [15], [16], where G s parttoned nto overlappng patches. The dffculty wth the approaches n [14], [15] s that the patches and the resultng patch confguraton are not guaranteed to be rgd. We propose to bypass the former rgdty ssue by usng clques, that s, complete subgraphs of the measurement graph. In other words, each patch s a clque n our approach. Ths s precsely why we choose to denote the patches as C n Secton II. Clques are trvally rgd [18], [19], and can be localzed usng multdmensonal scalng [22]. In partcular, for each V(G), we extract a maxmal clque contanng. The system of clques forms a clque-cover. We recall that a clque s sad to be maxmal f t s not contaned n a strctly larger clque. By targetng maxmal clques, we wsh to mnmze the number of clques that are requred to be regstered n the fnal phase. Let G denote the neghborhood graph of some V(G). Namely, G s the subgraph of G nduced by and ts one-hop neghbors. For each vertex, we want to fnd a maxmal clque C V(G) contanng. In ths regard, we note that t suffces to restrct the search to G. Proposton III.1. Let C be a maxmal clque. Then C f and only f C V(G ).

8 Proof. Let C be a maxmal clque contanng. Then, for any j C, we have (, j) E(G). Hence, j V(G ). In the other drecton, suppose that C V(G ), but does not belong to C. Then, by appendng vertex to C, we obtan a clque that strctly contans C, whch contradcts the maxmalty of C. Unfortunately, fndng clques s generally ntractable [23]. Based on Proposton III.1, we frst extract a clque from a gven subgraph G usng the algorthm n [23]. In ths work, the combnatoral problem of fndng maxmal clques s relaxed nto a contnuous optmzaton problem. The statonary ponts of the latter are computed usng projected gradent descent. The key result of the paper s that one can provably locate a clque by runnng the gradent-descent for suffcent number of teratons and roundng the output [23, Theorem 7, Corollary 3]. The authors emprcally notced that the clque retuned by the algorthm s often maxmal. Snce the subgraphs G are typcally small for practcal values of r, we found the algorthm n [23] to be qute effcent for our purpose. The clque located wthn a gven G usng the above clque-fndng algorthm may not contan vertex. In ths case, we can n fact obtan a larger clque smply by appendng vertex to the found clque. Generally, snce the subgraphs are small, one can effcently test for maxmalty, and keep appendng nodes untl the maxmal clque s found. In practce, we notced that the clques returned by the algorthm n [23] are often maxmal or near-maxmal. As a result, the combned process of appendng vertces and testng for maxmalty s qute fast. We note that an extracted clque can belong to two or more subgraphs. We dscard the redundant clques durng the clque-fndng process. At the end, suppose that we have located, say, m maxmal clques, C 1,..., C m, that cover the vertces of G. We next test the rgdty of the patch confguraton. To do so, we append to the exstng clques an addtonal clque C m+1 contanng the anchors, and test f Γ s quas (d + 1)-connected. If so, we set M = m, and proceed to the localzaton phase n Secton IV. If Γ fals the test, we proceed as follows. Followng Proposton II.1, we know that there exst s, t V 2 (Γ) for whch the maxmum s-t flow s k d, where recall that the vertces n V 1 (Γ) are assgned unt capacty. In fact, the Ford-Fulkerson algorthm returns the value k and the correspondng mnmum cut C = (S, T ), where s S and t T [26]. Let I S and I T be the ndces of the clques n S and T, that s, I S = S V 2 (Γ) and I T = T V 2 (Γ). Proposton III.2. Let A = α IS C α and B = β IT C β. Then A B = k. Proof. From the max-flow-mn-cut theorem [26], A B k. If A B s less than k, then there would be a vertex n V 1 (Γ) wth more than one edge n the cut set [26]. However, snce each vertex n V 1 (Γ) has unt capacty, ths s not possble. We wsh to ncrease the maxmum flow by extractng a clque and appendng t to the exstng confguraton. In partcular, the appended clque must contan some A \ B and j B \ A. Our task s to fnd a maxmal clque contanng and j. Defne G j be the common subgraph of G and G j, that s, G j s the subgraph of G nduced by the vertces V(G ) V(G j ). Smlar to Proposton III.1, we note the followng. Proposton III.3. Let C be a maxmal clque. Then, j C f and only f C V(G j ). After appendng C to the exstng clques, we obtan a new confguraton. Accordngly, we update V 2 (Γ) and E(Γ), and ncrease m by 1. In partcular, we reorder the ndces of the clques so that C m+1 contnues to be the anchor clque. For the updated Γ, we recompute I S or I T, and note that m belongs to ether of these. As a result, we conclude the followng. Proposton III.4. For the updated Γ, A B > k. Moreover, f the maxmum flow s unquely acheved for the vertces dentfed by the Ford-Fulkerson algorthm, then appendng C actually ncreases the maxmum flow. We contnue ths process, n whch we alternately augment the confguraton and test rgdty, untl we attan the maxmum flow of d + 1. It s possble that the process s prematurely termnated f we are unable to fnd a clque of sze at least d + 1. Ths typcally happens n adversaral settngs where r s small, makng G extremely sparse. At the

9 end of the process, assume that we have M + 1 clques, C 1..., C M+1, where C M+1 s the anchor clque. For the smulatons n Secton VI, we found that M s typcally about 3% of the total number of nodes. IV. LOCALIZATION Havng parttoned the measurement graph nto a system of overlappng clques, we now localze them n parallel. Snce all the nter-node dstances are avalable n a clque, we can effcently localze a clque usng multdmensonal scalng [22]. However, there are two types of clques, namely, clques wthout anchors and those wth anchors. For the former, we can drectly use classcal multdmensonal scalng (cmds). In partcular, suppose that the clque has n sensors, and the dstances are {d j : 1, j n}. Consder the n n matrces D and B gven by { d 2 j f j, D j = otherwse, and B = 1 2 ( I n 1 ( n uu ) D I n 1 n uu ), where u s the all-ones vector of length n, and I n s the n n dentty matrx. Snce B s symmetrc, t has real egenvalues and a full set of orthonormal egenvectors. Let λ 1 λ n be the sorted egenvalues, and q 1,..., q n be the correspondng egenvectors. Theorem IV.1 (Multdmensonal Scalng, [22]). Suppose that the avalable dstances are exact, that s, d j = x x j for some x 1,..., x n. Then B and rank(b) d. The sensor locatons can be taken to be x = ( λ1 q 1 (),..., λ d q d () ) (1 n). (5) If the dstances are nosy, B can have negatve egenvalues and ts rank can be greater than d. In ths case, t s customary to use the postve egenvalues and the correspondng egenvectors n (5). The resultng nter-sensor dstances are an approxmaton to the avalable dstances, where the approxmaton error s determned by the rank of B, and the number and magntude of the negatve egenvalues [22]. Perturbaton analyss of cmds s a well-researched topc and the method s known to be stable under deformatons [27]. If a clque has one or more anchors, we have to take nto consderaton the stpulated anchor locatons. More precsely, we have a constraned problem, where we need to reconstruct the sensor locatons keepng the anchor varables fxed. In such scenaros, we can use cmds followed by an algnment. Assume that, of the n nodes, the frst k are anchors and the remanng are sensors. We frst localze the n nodes usng cmds, regardless of the anchor locatons. Then we algn the reconstructed anchors wth the orgnal anchors va a rgd transformaton. In partcular, f there s just one anchor, and f the reconstructed and orgnal locatons are x and ā, then we translate the reconstructed nodes by ā x. If there are more than one anchor, then we perform an optmal algnment usng least-square fttng: mn O O(d),t R d k Ox + t ā 2, (6) =1 where x and ā are the reconstructed and the orgnal anchor locatons. As s well-known, the mnmum of (6) has a smple closed-form soluton [28]. In partcular, the optmal transform s gven by O = VU and t = µ O ν, where µ = 1 k x and ν = 1 k ā, k k =1 and C = UΣV s the SVD of C = k =1 (x µ)(ā ν). We apply the transform x O x + t on the reconstructed sensors, and place the anchors n ther stpulated locatons. Fnally, we refne the localzaton by mnmzng the stress functon usng a gradent-based method [8]. The refnement s partcularly effectve when the dstances are nosy. =1

1 V. REGISTRATION As a fnal step, we need to regster the localzed clques n a global coordnate system. Whle the least-square formulaton of the regstraton problem has a closed-form soluton for two clques [28], the problem s generally ntractable when there are three or more clques [2]. Recently, t was demonstrated n [2] that the least-square optmzaton can be approxmated usng a semdefnte program (SDP). Later, a scalable ADMM-based solver for ths SDP was proposed n [21]. For completeness, we revew the SDP relaxaton and the ADMM solver. In the absence of nose, the relaton between the local and global coordnates are gven by (12) and (13). Snce these are not expected to hold exactly n the presence of nose, the authors n [14] proposed to mnmze the least-square objectve where M ( =1 k C \A α 2 k, + λ l C A α k, = x k O x k, t and β l, = O M+1 ā l O ā l t β 2 l, ), (7) are the regstraton errors for sensors and anchors. The scale λ > s used to combne the gross errors. The varables are the sensor coordnates x k and the rgd transformatons (O, t ). The dummy varable O M+1 s ntroduced to make the objectve homogenous [14]. In terms of the matrx varables we can express (7) as where Z = [x 1 x N t 1... t M ] and O = [O 1 O M+1 ], =1 ( [Z ] [ J B Trace ] [ ]) Z O, (8) B D M [ J = e k, e k, + λ B = D = M+1 =1 k C \A [ k C \A δ N+M N+ l C A ( ) δ M+1 I d x k, e k, + λ ] (f I d ) ā l δ N+M, l C A M+1 =1 [ k C \A ( δ M+1 N+ I d ) x k, x k, + λ ] (f I d ) ā l ā l (f I d ). l C A Here, s the Kronecker product, δ L e k, = δ N+M k O ] δ N+M N+, ( ) δ M+1 I d s the all-zero vector of length L wth unty at the -th poston, δ N+M N+ and f = δ M+1 M+1 δm+1. The mnmum of (8) over Z s attaned when Z = OBJ 1. On substtutng Z n (8), we get the followng problem: mn Trace (CG) G (9) s.t. [G] = I d ( = 1,..., M +1), rank (G) = d. where C = D BJ 1 B, G = O O, and the d d matrx [G] denotes the -th dagonal block of G. It was observed n [2] that the objectve and the constrants n (9) are convex, except for the rank condton. By droppng the rank constrant, the authors arrved at the followng SDP relaxaton: mn Trace (CG) s.t. [G] = I d ( = 1,..., M +1). (1) G

11 The global mnmum of (1) can be computed for small or even medum-szed problems usng an nterorpont solver [1]. However, such solvers are both memory and computaton ntensve. In partcular, the cost of approxmatng the global mnmum of (1) wthn a gven accuracy s O((Md) 4.5 ) [2]. Snce M s of the order O( N ) n our case, the sze of the SDP varables can be few hundreds or thousands. Interorpont solvers run out of memory for such large problems. To acheve scalablty, an teratve solver based on ADMM was proposed n [21]. The ADMM updates are summarzed n Algorthm 1, where S + s the set of symmetrc postve semdefnte matrces of sze L = (M +1)d, Ω s the set of symmetrc matrces of sze L whose d d dagonal blocks are dentty, and Π S (A) denotes the projecton of A onto a convex set S. Notce that the only non-trval computaton s determnng Π S+ (A). Ths s obtaned by computng the egendecomposton of A and settng the negatve egenvalues to zero. The projecton Π Ω (A) amounts to settng the d d dagonal blocks of A to I d, whle keepng the non-dagonal blocks unchanged. We ntalze H usng the spectral algorthm n [2]. The Lagrange multpler Λ s ntally set to be the zero matrx. We use a condton from [29] to termnate the teratons. Algorthm 1: ADMM Solver. Input: C, ρ > Output: G. 1 Intalze H and Λ. 2 whle stoppng crtera s not met 3 G Π S+ [ H ρ 1 (C Λ) ]. 4 H Π Ω [ G ρ 1 Λ ]. 5 Λ Λ + ρ(h G). 6 end We can establsh the convergence of Algorthm 1 usng the analyss n [29]. In partcular, we have the followng result. Theorem V.1. Startng wth H and Λ, let (G k, H k, Λ k ) k 1 be the varables generated by Algorthm 1. Then Objectve convergence: If F s the optmum of (1), then Asymptotc feasblty: For 1 M, That s, (G k ) approaches the feasble set n (1). lm k Trace(CGk ) = F. lm k [Gk ] = I d. In fact, snce updates 3 and 4 n Algorthm 1 are convex projectons, we can establsh the Theorem V.1 usng elementary results from convex analyss. Ths and other techncal results wll be reported separately [3]. Havng approxmated the optmal G usng Algorthm 1, we compute O = [O 1 O M+1 ] usng the roundng n [2]. The frst N columns of Z = OBJ 1 are taken to be the estmated sensor locatons. As a fnal step, we refne the locatons usng stress mnmzaton [8] and denote the result as x 1,..., x N. VI. NUMERICAL EXPERIMENTS In ths secton, we conduct numercal smulatons to demonstrate the performance of the proposed method. In partcular, we llustrate the mpact of rgdty on the performance of the regstraton algorthm, and study the tmng of dfferent phases of the proposed method and the scalng of the localzaton error wth the nose level. We then compare wth some of the state-of-the-art algorthms [6], [8], [11], [4] n terms of accuracy, run-tme, and scalablty. The comparsons are performed on planar networks, namely, the random geometrc graph (RGG) [8], [11], [14], and the structured PACM logo [5]. The dameter

12.4.2.2.4.4.2.2.4.4.2.2.4 (a) ANE = 1.9e-1..5.4.2.2.4 (b) ANE = 7.1e-12..4.2.2.4.4.2.2.4.4.2.2.4 (c) ANE = 8.6e-2..4.2.2.4 (d) ANE = 6.1e-3. Fg. 4. Illustraton of the mpact of rgdty on the regstraton accuracy. The parameters for the RGG are N = 5, K = 1, and r = 7. The top and bottom rows correspond to the nose levels η = and η =.1. For a fxed η, the fgure on the left corresponds to the stuaton where Γ fals to be quas 3-connected, whle that on the rght corresponds to the stuaton where Γ s modfed to ensure quas 3-connectvty (usng the procedure descrbed n the text). Green crcles ( ) denote orgnal sensor locatons, red stars ( ) denote estmated locatons, and blue damonds ( ) denote anchor locatons.

13.45.4.35 r = 2 r = 5 r = 8 ANE.3.25.2.15.1.5.2.4.6.8 Fg. 5. ANE versus nose level η for a RGG consstng of 5 sensors and 5 anchors. We consder the rado ranges r = 2, 5, and 8. η (maxmum dstance between any two ponts) of the logo s 16.92. We consder the followng nose model that was used n [8], [11], [14]: and d j = 1 + ɛ j x x j d k = 1 + ɛ k x ā k (, j S), ( S, k A), where ɛ j and ɛ k are..d. Gaussans wth mean zero and standard devaton η. As mentoned earler, we enforce symmetry by replacng d j and d j wth ther average [31]. For a quanttatve comparson of the localzaton accuraces, we use the average normalzed error (ANE) [5] gven by { N } 1/2 =1 ANE = x x 2 N =1 x, x c 2 where x c s the centrod of the orgnal sensor locatons. Of course, we assume that the reconstructon has been optmally algned wth the ground truth before computng the ANE [28]. We also present vsual comparson of the localzatons obtaned usng dfferent methods. For all experments, we have used λ = 1 n (7) and ρ =.1 for the augmented Lagrangan. TABLE I RUN-TIMES (IN SECONDS) OF DIFFERENT PHASES OF THE ALGORITHM PARTITIONING (t 1), LOCALIZATION (t 2) AND REGISTRATION (t 3). N K r η t 1 t 2 t 3 1 1.4.79.4.2.73.6.92 5 5 8 3.9.7.21 3.9 12.8 8 8 4 7.2.8 7.5.2 7.9 1 1 2 9.2 1.7 8.1.2 1.5

14 TABLE II COMPARISON OF THE RUN-TIME OF THE PROPOSED METHOD WITH THAT OF SNLSDP [8], ESDP [11] AND THE LOCALIZATION ACCURACY OF THE PROPOSED METHOD WITH E-ML [6], SNLSDP [8], ESDP [11] AND SNLDR [4], FOR RANDOM GEOMETRIC GRAPHS ON THE UNIT-SQUARE. THE RUN-TIME AND ANE WERE AVERAGED OVER 1 REALIZATIONS OF THE RANDOM GRAPH AND THE MEASURED DISTANCES. WE HAVE USED TO MARK THOSE INSTANCES WHERE THE ALGORITHM TOOK INDEFINITE TIME TO SOLVE THE PROBLEM. THE INSTANCES WHERE THE INTERIOR POINT SOLVER RAN OUT OF MEMORY ARE MARKED WITH. Tme Accuracy (ANE) N K r η Proposed ESDP [11] SNLSDP [8] Proposed ESDP [11] SNLDR [4] SNLSDP [8] E-ML [6] 1 5 1.25 sec.2sec sec 3.9e-16 7.9e-8 1.3e-4 1.3e-9 1.4e-3.2sec sec.4sec 9.6e-2 9.5e-2 2e-1 9.6e-2 1.3e-1 2 6.88 sec.6sec.5sec 1.3e-15 1.1e-8 1e-4 1.1e-8 2.5e-3.6sec.6sec 1sec 6.4e-2 8.8e-2 1.6e-1 6.4e-2 9.2e-2 4 8.63.5sec 3.3sec.6sec 2.3e-15 4.1e-8 1e-2 1.2e-9 2.3e-3 1.2sec 1.2sec.8sec 4e-2 7.3e-2 1.5e-1 4e-2 9.6e-2 2 24.28 2sec 3sec 19sec 4e-14 3.1e-7 1.4e-2 2.5e-9 4sec 7sec 4sec 1.7e-2 3.1e-2 1.1e-1 1.7e-2 5 54 8 5sec 1.5mn 5.8mn 4.7e-14 1.5e-7 1.6e-2 8.8e-7 1mn 25sec 7.6mn 1e-2 2e-2 7.2e-2 1e-2 1 14 2 1sec 2.6mn 26mn 1.3e-13 9.9e-7 5.9e-3 3.6e-2 5.4mn 1.2mn 26mn 7e-3 1.3e-2 4.6e-2 4.1e-2 4 44.6 3.2mn 32.8mn 5.6e-13 2.1e-7.5 4.2mn 32.8mn 1.7e-3 3.2e-3 6 64.5 8.5mn 1hr 7.6e-13 2.3e-3.5 6.8mn 42.2mn 1.3e-3 3.4e-3 8 84.4 15.3mn 1.4hr 2e-12 1.1e-5.1 2mn 1.4hr 2.5e-4 4.3e-4 A. Performance Analyss To assess the performance of our method, we present smulaton results on RGGs over the unt square [8], [11], [14], [5]. To generate a RGG, we unformly sample N ponts on the unt square [.5,.5] 2 and fx them to be the sensors. We addtonally pck K ponts at random from the square (dstnct from the sensors) and fx them to be the anchors. We assume that the dstance between two sensors, or between a sensor and an anchor, s known f t s at most r. Experment 1: To understand the mportance of rgdty, a network consstng of 5 sensors and 1 anchors s consdered where r s taken to be 7. We generate several nstances of random graphs wth the above parameters untl we have an nstance where the correspondng Γ fals to be quas 3-connected. We run our algorthm at nose levels η = and on these nstances and record the localzaton results. We then augment the exstng clque system to ensure that Γ s quas 3-connected. Ths s done usng the heurstcs proposed n Secton III. We agan run our algorthm at nose levels η = and and record the results. A partcular nstance s reported n Fgure 4. We notce that the proposed algorthm performs poorly f Γ s not quas 3-connected. In partcular, notce that the regstraton mechansm fals n specfc regons of the network. Ths can be attrbuted to the fold-over phenomena assocated wth patches that are loosely connected to the rest of the patch system [16]. However, when Γ s forced to be quas 3-connected, we notce that the regstraton output mproves sgnfcantly for both the noseless and nosy cases. In fact, we acheve almost machne-level precson for the noseless case. Experment 2: In Table I, we report the run-tmes of the three phases of the algorthm for networks of dfferent szes (we round K to 1% of N n each case). The experments were performed usng MATLAB 8.2 on a 4-core workstaton wth 3.4 GHz processor and 32 GB memory. Notce that the tmng of the localzaton and the parttonng phase ncreases almost lnearly wth the number of sensors. Interestngly, the tmng does not vary much wth the nose level for a fxed N. However, the tmng of the fnal regstraton phase appears to depend heavly on the nose level. An explanaton for ths s that we use the soluton of the spectral relaxaton of (1) as an ntalzaton for the ADMM solver [2]. If the spectral relaxaton turns out to be a good approxmaton of the optmal soluton, then the ADMM solver converges n few teratons. Else, a large number of ADMM teratons are requred. Experment 3: As a fnal analyss, we study the scalng of ANE wth the nose level, when r = 2, 5, and 8.

15 A fxed network consstng of 5 sensors and 5 anchors was used. For a partcular η and r, we averaged the ANEs obtaned over 1 realzatons of the random graph and the measured dstances. The results from a typcal experment are plotted n Fgure 5. We notce that the ANE ncreases almost lnearly wth η when r = 2 and 5. However, when r = 2, the ANE tends to ncrease abruptly at large nose levels. The reason for ths s that the regstraton process can fal when r s low and η s large (low sgnal-to-nose rato scenaro). B. Comparson We now compare the proposed method wth the followng optmzaton-based methods: Edge-based Maxmum Lkelhood (E-ML) relaxaton [6], SNL usng SDP (SNLSDP) [8], Edge-based SDP (ESDP) relaxaton [11], SNL usng Dsk Relaxaton (SNLDR) [4]. E-ML uses dstrbuted optmzaton to mnmze a surrogate of the ML estmator for the dstance measurements n SNL. SNLDR uses dstrbuted optmzaton for a novel convex relaxaton of the SNL problem. On the other hand, SNLSDP s a centralzed algorthm whch s based on an SDP-based relaxaton of the SNL problem. ESDP s a further relaxaton of SNLSDP that can handle large networks. Experment 4: The proposed method s compared wth E-ML, SNLSDP, ESDP and SNLDR on random geometrc graphs. The results are reported n Table II. For a far comparson wth SNLDR, we addtonally placed an anchor at each of the four corners of the unt square to ensure that the sensors are n the convex hull of the anchors [4]. The localzaton obtaned usng our method s comparable wth that obtaned from SNLSDP for small networks (N 5). The performance of SNLSDP starts degradng when N > 5, and t cannot handle large networks (N > 1). On the other hand, the proposed method s able to mantan ts performance across networks of all szes. Notce that, though ESDP can scale up to networks of sze 8, ts performance falls off abruptly when N > 4. The proposed method s generally faster than ESDP and SNLSDP. TABLE III VISUAL COMPARISON OF THE PROPOSED ALGORITHM WITH SNLSDP [8], ESDP [11] AND SNLDR [4] FOR A RANDOM GEOMETRIC GRAPH ON THE UNIT SQUARE CONSISTING OF 1 SENSORS AND 14 ANCHORS. THE RADIO-RANGE USED IS r =.4. SEE FIGURE 4 FOR A DESCRIPTION OF THE SYMBOLS,, AND. THE (ANE, RUN-TIME) ARE REPORTED IN THE CAPTION. η Proposed SNLSDP [8] ESDP [11] SNLDR [4] (7.1e-14, 1sec) (1.5e-8, 6sec) (1.5e-7, 1sec) (1.3e-2, ).5.5.5.5 - -.5.5.5 -.5.5.5.5.5.5.5 -.5 - -.5 (2.3e-2, 3sec) (2.4e-2, 6sec) (4.4e-2, 3sec) (1.2e-1, ).5.5.5.5 - -.5.5.5 -.5.5.5.5.5.5.5 -.5 - -.5 Experment 5: We provde some vsual comparson n Fgures III and IV for RGGs and the PACM logo [5]. The latter conssts of 425 ponts sampled from the logo. We randomly set 43 ponts as anchors. Notce

16 TABLE IV COMPARISON OF LOCALIZATION RESULTS FOR THE PACM LOGO [5]. THE PARAMETERS ARE N = 382, K = 43, AND r = 1.9. BLUE CIRCLES ( ) DENOTE ORIGINAL (COLUMN 1) AND RECONSTRUCTED (COLUMNS 2-5) SENSOR LOCATIONS, AND RED DIAMONDS ( ) DENOTE ANCHOR LOCATIONS. η Orgnal Proposed SNLSDP [8] ESDP [11] SNLDR [4] (2.5e-13, 3sec) (6.3e-2, 2mn) (3.7e-2, 12sec) (2.9e-2, ).5 (5.6e-2, 5sec) (5e-2, 2mn) (1e-1, 14sec) (1.5e-1, ) that the reconstructon from the proposed method s vsbly superor to the competng methods n ether case, whch s also reflected by the ANE. The accuracy s compettve wth SNLSDP, but consstently better than the other methods. In partcular, notce the poor localzatons obtaned usng SNLDR when η =.5. Addtonal comparsons wth [4], [8], [11], and [12] are provded n the supplementary materal. The MATLAB code of our algorthm s publcly avalable [32]. VII. CONCLUSION We demonstrated that by transformng the localzaton problem nto a regstraton problem, one can acheve scalablty wthout compromsng the localzaton accuracy. In partcular, the convex relaxaton of the regstraton problem appears to be better behaved n terms of scalablty and approxmaton qualty compared to the convex relaxatons of the localzaton problem. For example, the proposed algorthm can localze a network of 8 nodes n 15 mnutes wth almost machne-precson accuracy of 1e-12. In contrast, the convex relaxaton n [8] cannot be scaled beyond 1 nodes. An excepton n ths regard s ESDP, whch can be scaled to networks wth thousands of nodes. However, ts localzaton accuracy starts fallng off wth the ncrease n network sze. A key contrbuton of the paper s that we formulated and analysed the rgdty problem assocated wth mult-patch regstraton. An open queston that emerged from ths analyss s whether quas-connectvty s suffcent for the patch confguraton to be rgd. Another relevant queston that remans unaddressed s the mpact of rgdty on the performance of the regstraton algorthm, both n terms of tghtness and stablty. These wll be nvestgated n future work. A. Proof of Theorem II.8 VIII. SUPPLEMENTARY In ths secton, we prove Theorem II.8. Frst, we recall a basc assumpton that was made n ths regard. Assumpton VIII.1 (Non-degeneracy). There are at least d + 1 non-degenerate ponts n each patch. We now restate Theorem II.8. Theorem VIII.2 (Necessary condton). Under Assumpton VIII.1, f a confguraton s rgd n R d, then ts correspondence graph must be quas (d+1)-connected. To prove Theorem VIII.2, we wll need the followng proposton:

17 Proposton VIII.3. The followng are equvalent. (a) The correspondence graph Γ s quas k-connected. (b) E(Γ) can be dvded nto two dsjont subsets E 1 and E 2 such that the edges from E 1 and that from E 2 are () ncdent on at least k common vertces from V 1 (Γ), and () not ncdent on any common vertex from V 2 (Γ). For completeness, we recall Problems II.1 and II.2 from the man manuscrpt. Problem VIII.4 (Regstraton). Fnd x 1,..., x N and rgd transforms Q 1,..., Q M such that, for 1 M, where k C \ A and l C A. x k = Q (x k, ) and ā l = Q (ā l ), (11) Problem VIII.5 (Unqueness). Determne whether Problem VIII.4 have a unque soluton up to a rgd transform. That s, f x 1,..., x N s a soluton of Problem VIII.4, then s t necessary that for some rgd transform R, where k S and l A? x k = R( x k ) and ā l = R(ā l ), Moreover, we assume that the ponts n each patch have been derved from the respectve ponts n X s X a va a rgd transform. Let and x k = R (x k, ) = O x k, + t (k C \A), (12) ā l = R (ā l ) (l C A), (13) We consder a dfferent regstraton problem where the patch coordnates are replaced by the orgnal coordnates. Problem VIII.6 (Regstraton). Fnd x 1,..., x N and rgd transforms T 1,..., T M such that, for 1 M, where k C \ A and l C A. x k = T ( x k ) and ā l = T (ā l ), (14) A trval soluton s x k = x k and T = (I d, ). As wth Problem VIII.5, we can ask whether ths s the only soluton. It turns out that the questons are related. Proposton VIII.7 (Equvalence). Problem VIII.4 has an unque soluton f and only f Problem VIII.6 has an unque soluton. Proof. Combnng (12), (13) and (14), we can wrte x k = (T R )(x k, ) and ā l = (T R )(ā l ). (15) Comparng (15) wth (11), we have Q = T R. It follows that the T s are unque f and only f the Q s are unque. Moreover, the unqueness of the x k s follows from the unqueness of the transforms and relatons (11) and (14). We also make the observaton concernng Problem VIII.6 that T 1,..., T M satsfyng (14) are unque,.e., T = R(I d, ) for some rgd transform R, f and only f the correspondng x 1,..., x N are related to x 1,..., x N va a rgd transform. If T = R(I d, ), then t follows from (14) that x k = R( x k ). Conversely, f x k = R( x k ) for some rgd transform R, then the correspondng T should necessarly be of the form T = R(I d, ). Indeed, f some T R(I d, ), then we can construct a soluton that s not related to x 1,..., x N va a rgd transform, and ths would lead to a contradcton. To complete the proof of Theorem VIII.2, t remans to show that, f the soluton of Problem VIII.6 s unque, then Γ must be quas (d + 1)-connected. We wll prove ths by contradcton. As a frst step, we note that Γ s at least quas-1 connected.

18 Proposton VIII.8. If Problem VIII.6 has a unque soluton, then Γ must be quas-k connected for some k 1. Proof. Indeed, suppose that there exst non-empty subsets S and T of V 2 (Γ) such that there s no path between any S and j T. Defne A = α S C α and B = β T Clearly, A B must be empty. Else, we can fnd a path between some S and j T, whch would volate our assumpton. However, on settng T = (I d, ) for S, and T j = ( I d, ) for j T, we obtan a soluton to Problem VIII.6 whch s dfferent from the trval soluton. Hence, our assumpton about the exstence of S and T must be wrong. In fact, we can make the stronger clam that Γ s quas k-connected, where k d + 1. To establsh the clam, we show that the rgdty assumpton s volated f k d. Frst, we ntroduce few notatons about paths. Suppose that there are one or more paths between two vertces of V 2 (Γ). We denote the j-th vertex on the -th path usng σ j. In partcular, σ1 and σ p are the ntal and fnal vertces, where p s the number of vertces on the path. Snce Γ s bpartte, p must be odd, and { σ j V 1 (Γ) for j = 2, 4,..., p 1, V 2 (Γ) for j = 1, 3,..., p. For 1 j (p 1)/2, consder the vertces σ 2j, σ 2j 1, and σ 2j+1. The frst vertex represents a node, whle the latter two represent patches. Moreover, the node belongs to both the patches. Therefore, for 1 k and 1 j (p 1)/2, O σ 2j 1 x σ 2j + t σ 2j 1 = O σ 2j+1 x σ 2j C β. + t σ 2j+1. (16) To arrve at a contradcton, we show that f k d, then there exsts at least some T = (O, t ), 1 M, dfferent 1 from (I d, ) for whch the system of equatons n (14) hold. To do so, we dvde V 2 (Γ) nto two dsjont sets. Note that, from Proposton VIII.3, we can dentfy dsjont subsets E 1, E 2 E(Γ) such that the edges from E 1 and that from E 2 are not ncdent on any common vertex of V 2 (Γ). In partcular, defne S V 2 (Γ) to be the vertces on whch the edges of E 1 are ncdent. Smlarly, let T V 2 (Γ) be the vertces on whch the edges of E 2 are ncdent. Then, S and T are non-empty and dsjont. Wthout loss of generalty, we assume that the vertex correspondng to the anchor patch belongs to S. Snce Γ s quas k-connected, we can fnd a dstnct vertex t T whch s connected wth the anchor patch vertex by paths σ 1,..., σ k that are V 1 (Γ)-dsjont. Note that, snce the anchor patch s fxed, O M+1 = I d and t M+1 =. Therefore, we set { (O, t), f T, T = (O, t ) = (I d, ) f S, and show that f k d, then we can fnd (O, t) (I d, ) such that (14) holds. Note that Proposton VIII.3 also tells us that the edges from E 1 and that from E 2 are ncdent on exactly k common vertces from V 1 (Γ); we denoted these vertces usng Ω. It s also be reasoned that each path contans exactly one vertex from Ω. Assume that σ 2q Ω be the vertex on the path σ. Therefore, x σ 2q = O x σ 2q + t. (17) If k = 1, then O = I d and t = 2 x 2q σ satsfy (17), and hence the equatons n (14). On the other hand, f 2 k d, then we have k equatons smlar to (17), one for each path. We elmnate t by subtractng the equatons correspondng to 2 k from the equaton correspondng to = 1. Ths gves us O( x σ 2q x 2q σ 1 ) = x 2q 1 σ x 2q σ 1 1 1 Wthout loss of generalty, we omt the global rgd transform R. (2 k).