IMF: Iterative Max-Flow for Node Localizability Detection in Barycentric Linear Localization
IIMF: An Iterative Max-Flow Method forLocalizability Detection
Haodi Ping †† School of Information, Renmin University of China, Beijing, P.R.China, 100872Email: [email protected]
Abstract —Determining whether nodes can be localized, calledlocalizability detection, is a basic work in network localization.We present a novel approach for localizability detection given aset of nodes and their connectivity relationships. The key ideais transforming the problem of detecting three disjoint pathsinto a max-flow problem, which can be efficiently solved byexisting algorithms. Furthermore, we prove the correctness ofour algorithm.
Index Terms —Localizability Detection; Max-Flow
I. I
NTRODUCTION
An essential problem in network localization algorithms isto detect whether a network or a node in the network is localiz-able [1]. Based on the underlying objective function, networklocalization algorithms can be classified into two categories:1) most of the existing literature solve the localization problemusing a least-square optimization model [2]–[8], which wecall
Non-Linear Localization (NLL) methods; 2) a recentbarycentric coordinate based linear localization frameworktransforms the localization problem to solving linear euqations[9]–[13], which we call
Barycentric coordinate-based linearlocalization (BLL) .In NLL, the network localizability problem is closely relatedto the graph rigidity problem [14], [15].
Lemma 1 (NLL Localizability Condition [14]) . A network G is localizable in 2D, if and only if it contains at least 3 anchorsand it is global rigid. In BLL, the network localizability is based on a generatedgraph G A associated with the barycentric representation, in-stead of G . Definition 1 (Generated Graph, G A ) . Given G = ( V , E ) , theassociated generated graph G A = ( V , E A ) is defined as agraph having the same node set V . An edge ( i, j ) ∈ E A if j is a neighbor of i in G and j can form at least one trianglewith another two neighbors of i . Then, the network localizability condition is proposed:
Lemma 2 (BLL Localizability Condition [12]) . Using BLLalgorithms, nodes in a generic graph G are localizable in 2Dif every node can find at least three node-disjoint paths toanchors through only edges in G A . However, given a network G , there is no algorithm thatcan tell which of the nodes are localizable from Lemma 2.The key issue in Lemma 2 is verifying the condition of three node disjoint paths (3DP). Thus, we first prove that3DP is equivalent to having a flow of more than 3 in anappropriately-constructed graph. Then, a centralized iterativemax-flow based algorithm to detect nodes satisfying the BLLlocalizability condition is designed.II. M AX -F LOW AND D ISJOINT P ATH
The key constraint in Lemma 2 is node disjoint paths,first we prove that finding disjoint paths in G = ( V , E ) canbe reduced to a max-flow problem using an appropriately-constructed graph G (cid:48) = ( V (cid:48) , E (cid:48) ) . Each agent v i ∈ V is firstlydivided into two copies { v ini , v outi } and then added to V (cid:48) ,while anchors of V and an additional virtual target node Ω are added directly. For each agent v i ∈ V , ( v ini , v outi ) ∈ E (cid:48) .For each neighbor v j of an agent v i , ( v outi , v j ) ∈ E (cid:48) if v j isan anchor, ( v outi , v inj ) ∈ E (cid:48) , otherwise; For each anchor node i ∈ V (cid:48) , ( i, Ω) ∈ E (cid:48) . Finally, each arc of E (cid:48) is assigned with acapacity of 1 in the capacity matrix C . Lemma 3.
Consider a network G and its corresponding G (cid:48) ,the following two are equivalent. • The count of disjoint paths from i to anchors in G . • The max flow from i out to Ω in G (cid:48) constrained by C .Proof: First we prove that every path to anchors in G isincluded in G (cid:48) . Suppose an arbitrary edge ( v i , v j ) on any path P = v · · · v k in G . If v i is an agent and v j is an anchor,then the edge is maintained by v ini v outi v j . If both v i and v j are agents, the edge becomes v ini v outi v inj v outj . Since we focuson paths to anchors, the case that v i is an anchor and v j isan agent is not considered. Thus, any path P in G can betransformed to a new path in G (cid:48) .Suppose any source node v outi , it is straightforward that anypath from v outi to Ω must go through one edge of the minimumcut. And each node v j is passed by at most one path since thecapacity of ( v inj , v outj ) is one. Since the maximum flow equalsthe minimum edge cut, the count of disjoint paths from i toanchors in G equals the max flow from v outi to Ω in G (cid:48) .A casse study is shown in Fig. 1. A network G of 5 nodesis given in Fig. 1(a). Its corresponding G (cid:48) is in Fig. 1(b). Theconstructed capacity matrix C is shown in Fig. 1(c). a r X i v : . [ c s . N I] F e b
54 2 3 (a) G Ω in in out out in in out out (b) G (cid:48) A g e n t s A n c h o r s Ω in out in out in out in out Ω (c) C Fig. 1. A demonstration of the construction of G (cid:48) . Algorithm 1 Iterative Max-Flow (IMF) BLL-LocalizabilityDetection
Input: G = ( V , E ) . Output: V ∗ : localizable nodes of G .3: ι : localizability of each node of V .4: ι i ← , for all i ∈ A ; ι i ← , otherwise.5: run Graph Construction .6: S (cid:48) ← S .7: ι (cid:48) ← ι .8: set the i − th and i th rows and columns of C to 0 for all i ∈ S \ S (cid:48) .9: for each i ∈ S (cid:48) do if MAX FLOW ( C , i, k ) ≥ then ι i ← .11: else ι i ← , S (cid:48) ← S (cid:48) \ { i } .12: end if end for if ι (cid:48) = ι then return ι else go back to 7.16: end if V ∗ ← nodes with ι = 1 . Graph Construction C ← n + m +1 , n + m +1 .2: for each v i ∈ S do C i − , i ← . // The index i − is for i in , while i is for i out . N ∗ i ← Calculate Barycentric Neighbors ( N i ) .5: for each j ∈ N ∗ i do if v j is an anchor then C i,j ← .7: else C i, j − ← .8: end if end for end for k ← n + m + 1 . // The index n + m + 1 is for Ω . for each i ∈ A do C i,k ← .13: end for III. I
TERATIVE M AX -F LOW B ASED L OCALIZABILITY V ERIFICATION
A. Algorithm
Lemma 3 provides an efficient approach to count the numberof disjoint paths. To find the subgraph with most verticesthat satisfies Lemma 2, we iteratively remove nodes nothaving three disjoint paths and the corresponding capacitiesof these nodes in each round until no more nodes can beremoved. Finally, every node of the remaining graph has threedisjoint edges to anchors so that the constraint in Lemma2 is met. The iterative procedure is detailed in Algorithm1. The operation
Calculate Barycentric Neighbors returns N ∗ i , which is a subset of neighbors N i and each of them canform a triangle with additional d − nodes of N ∗ i . The function MAX FLOW is implemented with Goldberg’s push-relabelmaximum-flow minimum-cut algorithm [16].
B. Validity of the IMF algorithm
Theorem 1 (The IMF algorithm can detect all nodes satisfyingthe BLL localizability condition) . For a graph G , G [ V ∗ ] selected by Algorithm 1 is the subgraph with most verticesthat can satisfy Lemam 2.Proof: Index the nodes removed by Algorithm 1 as
V \ V ∗ = { s · · · s c } . Let V (cid:48) be an arbitrary feasible solutionof Lemam 2 and G [ V (cid:48) ] be the induced subgraph of V (cid:48) . s is removed because s has not three disjoint paths to anchorsin G . There must be that s cannot find three disjoint pathsin G [ V (cid:48) ] since G [ V (cid:48) ] is a subgraph of G . i.e., s will not beincluded by any other feasible solution. Similarly, every nodeof V \ V ∗ is not included by a feasible solution. Additionally,every node of V ∗ satisfies Lemam 2, otherwise it will beremoved. Thus, Algorithm 1 selects the most vertices that cansatisfy Lemam 2. C. Analysis1) Time Complexity:
The complexity of the Goldberg’spush-relabel maximum-flow algorithm is O ( kn m ) , where n is the number of nodes and m is the number of edges. k is theiteration count, in the worst case, only one node is removed.Thus, the time complexity of IMF is O ( n m ) at worst.
2) Extension to NLL Localizability Detection:
Lemma 1shows that IMF selects all BLL localizable nodes. If werun IMF on the original graph G rather than G A , the outputsubgraph is also global rigid, i.e., they are NLL localizable. Theorem 2 (Global Rigidity Condition [14]) . A graph G isglobal rigid in 2D if and only if G is redundant rigid and3-connected. Theorem 3.
The subgraph returned by IMF is NLL localizableif the graph construction procedure is based on N .Proof: In the subgraph returned by IMF, each node hasthree node disjoint paths to anchors. Thus, there is no binaryvertex cut. The subgraph is certainly redundant rigid and 3-connected. Thus, the NLL localizability condition is met.
3) Extension to Arbitrary Dimensions:
IMF works in anydimensions, denoted by d . A network in (cid:60) d needs at least d + 1 anchors to be localized. If we change the threshold inLine 10 of IMF from 3 to d + 1 and run IMF in 3D networks,then each node has d + 1 node disjoint paths to anchors inthe output subgraph. However, in 3D space, whether havingone generically globally rigid formation is sufficient to implythat the graph is generically globally rigid is an open problem[14], [17]. At least, IMF provides an approach to detect awell-structured subgraph.IV. R ELATED W ORK
A. Centralized Localization Detection Algorithms
Determining whether an entire network can be decom-posed to detection of redundant rigidity and detection of 3-connectivity. Determining 3-connectivity are detailed in [18],[19]. Determining redundant rigidity in (cid:60) has been derived[20] based on variants of Laman’s theorem [21]. . Distributed Localization Detection Algorithms The most representative distributed localizability detectionmethod is trilateration protocol (TP) [14]. The main idea isthat a node is detected as localizable if it finds three or morelocalizable neighbors. TP is simple and efficient, but TP missestheoretically localizable nodes that are on a geographicalgap or a geographical border . Moreover, none node can belocalized if the anchors do not have common neighbors.To identify nodes on a geographical gap or a border, Yanget al. [22] proposed a fully distributed approach, called wheelextension (WE). A structure of the wheel graph is definedand proved to be global rigid. Then each node tries to find awheel graph in neighbors. If three or more nodes of a wheelare localizable, all other nodes of the wheel are localizable.WE inherits the efficiency of TP, but it may fail if there are notat least three anchors coexist in one wheel at the beginning,then no node can be localized.To supplement TP and WE, Wu et al. [23] defines theconcept of the branch, and proves that a branch includingthree anchors is localizable. A triangle extension (TE) methodis proposed to construct branches. The branch constructionstarts from a K (a complete graph of two nodes) of anchorsand extends the branch by extension operations. Once theextension reaches the third anchor or a localizable node, allnodes on the branch are localizable. However, TE may failwhen no node can find a K of anchors in its neighbors, suchthat no node can be localized.R EFERENCES[1] Z. Yang and Y. Liu, “Understanding node localizability of wireless adhoc and sensor networks,”
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