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39
Resilient Routing for Sensor Networks Using Hyperbolic Embedding of Universal Covering Space
"... Abstract—We study how to characterize the families of paths between any two nodes s, t in a sensor network with holes. Two paths that can be deformed to one another through local changes are called homotopy equivalent. Two paths that pass around holes in different ways have different homotopy types. ..."
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Abstract—We study how to characterize the families of paths between any two nodes s, t in a sensor network with holes. Two paths that can be deformed to one another through local changes are called homotopy equivalent. Two paths that pass around holes in different ways have different homotopy types. With a distributed algorithm we compute an embedding of the network in hyperbolic space by using Ricci flow such that paths of different homotopy types are mapped naturally to paths connecting s with different images of t. Greedy routing to a particular image is guaranteed with success to find a path with a given homotopy type. This leads to simple greedy routing algorithms that are resilient to both local link dynamics and large scale jamming attacks and improve load balancing over previous greedy routing algorithms. I.
Hardness results for homology localization
 In SODA ’10: Proc. 21st Ann. ACMSIAM Sympos. Discrete Algorithms (2010
"... We address the problem of localizing homology classes, namely, finding the cycle representing a given class with the most concise geometric measure. We focus on the volume measure, that is, the 1norm of a cycle. Two main results are presented. First, we prove the problem is NPhard to approximate w ..."
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Cited by 17 (1 self)
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We address the problem of localizing homology classes, namely, finding the cycle representing a given class with the most concise geometric measure. We focus on the volume measure, that is, the 1norm of a cycle. Two main results are presented. First, we prove the problem is NPhard to approximate within any constant factor. Second, we prove that for homology of dimension two or higher, the problem is NPhard to approximate even when the Betti number is O(1). A side effect is the inapproximability of the problem of computing the nonbounding cycle with the smallest volume, and computing cycles representing a homology basis with the minimal total volume. We also discuss other geometric measures (diameter and radius) and show their disadvantages in homology localization. Our work is restricted to homology over the Z2 field. 1
Euclidean versus hyperbolic congestion in idealized versus experimental networks
"... Abstract: This paper proposes a mathematical justification of the phenomenon of extreme congestion at a very limited number of nodes in very large networks. It is argued that this phenomenon occurs as a combination of the negative curvature property of the network together with minimum length routin ..."
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Cited by 12 (3 self)
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Abstract: This paper proposes a mathematical justification of the phenomenon of extreme congestion at a very limited number of nodes in very large networks. It is argued that this phenomenon occurs as a combination of the negative curvature property of the network together with minimum length routing. More specifically, it is shown that, in a large ndimensional hyperbolic ball B of radius R viewed as a roughly similar model of a Gromov hyperbolic network, the proportion of traffic paths transiting through a small ball near the center is Θ(1), whereas, in a Euclidean ball, the same proportion scales as Θ 1
A distributed triangulation algorithm for wireless sensor networks
 on 2d and 3d surface,” in IEEE INFOCOM
, 2011
"... Abstract—Triangulation serves as the basis for many geometrybased algorithms in wireless sensor networks. In this paper we propose a distributed algorithm that produces a triangulation for an arbitrary sensor network, with no constraints on communication model or granularity of the triangulation. We ..."
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Cited by 9 (6 self)
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Abstract—Triangulation serves as the basis for many geometrybased algorithms in wireless sensor networks. In this paper we propose a distributed algorithm that produces a triangulation for an arbitrary sensor network, with no constraints on communication model or granularity of the triangulation. We prove its correctness in 2D, and further extend it to sensor networks deployed on 3D open and closed surfaces. Our simulation results show that the proposed algorithms can tolerate distance measurement errors, and thus work well under practical sensor network settings and effectively promote the performance a range of applications that depend on triangulations. I.
Scalable and Fully Distributed Localization With Mere Connectivity
"... Abstract—This work proposes a novel connectivitybased localization algorithm, well suitable for largescale sensor networks with complex shapes and nonuniform nodal distribution. In contrast to current stateofart connectivitybased localization methods, the proposed algorithm is fully distribute ..."
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Cited by 9 (1 self)
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Abstract—This work proposes a novel connectivitybased localization algorithm, well suitable for largescale sensor networks with complex shapes and nonuniform nodal distribution. In contrast to current stateofart connectivitybased localization methods, the proposed algorithm is fully distributed, where each node only needs the information of its neighbors, without cumbersome partitioning and merging process. The algorithm is highly scalable, with limited error propagation and linear computation and communication cost with respect to the size of the network. Moreover, the algorithm is theoretically guaranteed and numerically stable. Extensive simulations and comparison with other methods under various representative network settings are carried out, showing superior performance of the proposed algorithm. I.
Fast Decentralized Averaging via MultiScale Gossip
"... Abstract. We are interested in the problem of computing the average consensus in a distributed fashion on random geometric graphs. We describe a new algorithm called Multiscale Gossip which employs a hierarchical decomposition of the graph to partition the computation into tractable subproblems. U ..."
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Cited by 8 (2 self)
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Abstract. We are interested in the problem of computing the average consensus in a distributed fashion on random geometric graphs. We describe a new algorithm called Multiscale Gossip which employs a hierarchical decomposition of the graph to partition the computation into tractable subproblems. Using only pairwise messages of fixed size that travel at most O(n 1 3) hops, our algorithm is robust and has communication cost of O(n log log n log ɛ −1) transmissions, which is orderoptimal up to the logarithmic factor in n. Simulated experiments verify the good expected performance on graphs of many thousands of nodes. 1
CABET: ConnectivityBased Boundary Extraction
 of LargeScale 3D Sensor Networks,” Proc. IEEE INFOCOM
, 2011
"... Abstract—Sensor networks are invariably coupled tightly with the geometric environment in which the sensor nodes are deployed. Network boundary is one of the key features that characterize such environments. While significant advances have been made for 2D cases, so far boundary extraction for 3D se ..."
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Cited by 8 (3 self)
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Abstract—Sensor networks are invariably coupled tightly with the geometric environment in which the sensor nodes are deployed. Network boundary is one of the key features that characterize such environments. While significant advances have been made for 2D cases, so far boundary extraction for 3D sensor networks has not been thoroughly studied. We present CABET, a novel ConnectivityBased Boundary Extraction scheme for largescale 3D sensor networks. To the best of our knowledge, CABET is the first 3Dcapable and pure connectivitybased solution for detecting sensor network boundaries. It is fully distributed, and is highly scalable, requiring overall message cost linear with the network size. A highlight of CABET is its nonuniform critical node sampling, called r0sampling, that selects landmarks to form boundary surfaces with bias toward nodes embodying salient topological features. Simulations show that CABET is able to extract a wellconnected boundary in the presence of holes and shape variation, with performance superior to that of some stateoftheart alternatives. In addition, we show how CABET benefits a range of sensor network applications including 3D skeleton extraction, 3D segmentation, and 3D localization. Index Terms—Sensor networks, algorithm/protocol design, 3D boundary Ç 1
Differential Forms for Target Tracking and Aggregate Queries in Distributed Networks
"... Consider mobile targets moving in a plane and their movements being monitored by a network such as a field of sensors. We develop distributed algorithms for innetwork tracking and range queries for aggregated data (for example returning the number of targets within any user given region). Our schem ..."
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Consider mobile targets moving in a plane and their movements being monitored by a network such as a field of sensors. We develop distributed algorithms for innetwork tracking and range queries for aggregated data (for example returning the number of targets within any user given region). Our scheme stores the target detection information locally in the network, and answers a query by examining the perimeter of the given range. The cost of updating data about mobile targets is proportional to the target displacement. The key insight is to maintain in the sensor network a function with respect to the target detection data on the graph edges that is a differential oneform such that the integral of this oneform along any closed curve C gives the integral within the region bounded byC. The differential oneform has great flexibility making it appropriate for tracking mobile targets. The basic range query can be used to find a nearby target or any given identifiable target with cost O(d) where d is the distance to the target in question. Dynamic insertion, deletion, coverage holes and mobility of sensor nodes can be handled with only local operations, making the scheme suitable for a highly dynamic network. It is extremely robust and capable of tolerating errors in sensing and target localization. Due to limited space, we only elaborate the advantages of differential forms in tracking of mobile targets. The same routine can be applied for organizing many other types of informations, for example streaming scalar sensor data (such as temperature data field), to support efficient range queries. We demonstrate through analysis and simulations that this scheme compares favorably with existing schemes that use location services for answering aggregated range queries of target detection data.
CONSEL: Connectivitybased Segmentation in LargeScale 2D/3D Sensor Networks
"... Abstract—A cardinal prerequisite for the system design of a sensor network, is to understand the geometric environment where sensor nodes are deployed. The global topology of a largescale sensor network is often complex and irregular, possibly containing obstacles/holes. A convex network partition, ..."
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Abstract—A cardinal prerequisite for the system design of a sensor network, is to understand the geometric environment where sensor nodes are deployed. The global topology of a largescale sensor network is often complex and irregular, possibly containing obstacles/holes. A convex network partition, socalled segmentation, is to divide a network into convex regions, such that traditional algorithms designed for a simple geometric region can be applied. Existing segmentation algorithms highly depend on concave node detection on the boundary or sink extraction on the medial axis, thus leading to quite sensitive performance to the boundary noise. More severely, since they exploit the network’s 2D geometric properties, either explicitly or implicitly, so far there has been no general 3D segmentation solution. In this paper, we bring a new view to segmentation from a Morse function perspective, bridging the convex regions and the Reeb graph of a network. Accordingly, we propose a novel distributed and scalable algorithm, named CONSEL, for CONnectivitybased SEgmentation in Largescale 2D/3D sensor networks. Specifically, several boundary nodes first perform flooding to construct the Reeb graph. The ordinary nodes then compute mutex pairs locally, thereby generating the coarse segmentation. Next the neighbor regions which are not mutex pair are merged together. Finally, by ignoring mutex pairs which leads to small concavity, we provide the constraints for approximately convex decomposition. CONSEL is more desirable compared with previous studies: (1) it works for both 2D and 3D sensor networks; (2) it only relies on network connectivity information; (3) it guarantees a bound for the regions ’ deviation from convexity. Extensive simulations show that CONSEL works well in the presence of holes and shape variation, always yielding appropriate segmentation results. I.
Geometric Algorithms for Sensor Networks
, 2011
"... Networked embedded sensors provide a unique opportunity for real time, large scale, high resolution environmental monitoring. Such systems are becoming ubiquitous across many activities important to our economy and life, from manufacturing and industrial sensing, to traffic and powergrid management, ..."
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Cited by 5 (0 self)
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Networked embedded sensors provide a unique opportunity for real time, large scale, high resolution environmental monitoring. Such systems are becoming ubiquitous across many activities important to our economy and life, from manufacturing and industrial sensing, to traffic and powergrid management, to wildlife, agriculture and environmental monitoring, to hospital operations and patient observation, all the