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Distributed coverage verification in sensor networks without location information
 IEEE TRANSACTIONS ON AUTOMATIC CONTROL
, 2008
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Rips complexes of planar point sets
, 2007
"... ABSTRACT. Fix a finite set of points in Euclidean nspace E n, thought of as a pointcloud sampling of a certain domain D ⊂ E n. The Rips complex is a combinatorial simplicial complex based on proximity of neighbors that serves as an easilycomputed but highdimensional approximation to the homotopy ..."
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Cited by 8 (2 self)
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ABSTRACT. Fix a finite set of points in Euclidean nspace E n, thought of as a pointcloud sampling of a certain domain D ⊂ E n. The Rips complex is a combinatorial simplicial complex based on proximity of neighbors that serves as an easilycomputed but highdimensional approximation to the homotopy type of D. There is a natural “shadow ” projection map from the Rips complex to E n that has as its image a more accurate ndimensional approximation to the homotopy type of D. We demonstrate that this projection map is 1connected for the planar case n = 2. That is, for planar domains, the Rips complex accurately captures connectivity and fundamental group data. This implies that the fundamental group of a Rips complex for a planar point set is a free group. We show that, in contrast, introducing even a small amount of uncertainty in proximity detection leads to ‘quasi’Rips complexes with nearly arbitrary fundamental groups. This topological noise can be mitigated by examining a pair of quasiRips complexes and using ideas from persistent topology. Finally, we show that the projection map does not preserve higherorder topological data for planar sets, nor does it preserve fundamental group data for point sets in dimension larger than three. 1.
VIETORISRIPS COMPLEXES OF PLANAR POINT SETS
"... Fix a finite set of points in Euclidean nspace E n, thought of as a pointcloud sampling of a certain domain D ⊂ E n. The VietorisRips complex is a combinatorial simplicial complex based on proximity of neighbors that serves as an easilycomputed but highdimensional approximation to the homotopy ..."
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Fix a finite set of points in Euclidean nspace E n, thought of as a pointcloud sampling of a certain domain D ⊂ E n. The VietorisRips complex is a combinatorial simplicial complex based on proximity of neighbors that serves as an easilycomputed but highdimensional approximation to the homotopy type of D. There is a natural “shadow” projection map from the VietorisRips complex to E n that has as its image a more accurate ndimensional approximation to the homotopy type of D. We demonstrate that this projection map is 1connected for the planar case n = 2. That is, for planar domains, the VietorisRips complex accurately captures connectivity and fundamental group data. This implies that the fundamental group of a VietorisRips complex for a planar point set is a free group. We show that, in contrast, introducing even a small amount of uncertainty in proximity detection leads to ‘quasi’VietorisRips complexes with nearly arbitrary fundamental groups. This topological noise can be mitigated by examining a pair of quasiVietorisRips complexes and using ideas from persistent topology. Finally, we show that the projection map does not preserve higherorder topological data for planar sets, nor does it preserve fundamental group data for point sets in dimension larger than three.
Testing contractibility in planar Rips complexes
 In Proc. Symp. on Comp. Geom. (SoCG) 2008
"... The (Vietoris)Rips complex of a discrete pointset P is an abstract simplicial complex in which a subset of P defines a simplex if and only if the diameter of that subset is at most 1. We describe an efficient algorithm to determine whether a given cycle in a planar Rips complex is contractible. Ou ..."
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The (Vietoris)Rips complex of a discrete pointset P is an abstract simplicial complex in which a subset of P defines a simplex if and only if the diameter of that subset is at most 1. We describe an efficient algorithm to determine whether a given cycle in a planar Rips complex is contractible. Our algorithm requires O(m log n) time to preprocess a set of n points in the plane in which m pairs have distance at most 1; after preprocessing, deciding whether a cycle of k Rips edges is contractible requires O(k) time. We also describe an algorithm to compute the shortest noncontractible cycle in a planar Rips complex in O(n 2 log n + mn) time.
Run to Potential: Sweep Coverage in Wireless Sensor Networks
"... Abstract—Wireless sensor networks have become a promising technology in monitoring physical world. In many applications with wireless sensor networks, it is essential to understand how well an interested area is monitored (covered) by sensors. The traditional way of evaluating sensor coverage requir ..."
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Abstract—Wireless sensor networks have become a promising technology in monitoring physical world. In many applications with wireless sensor networks, it is essential to understand how well an interested area is monitored (covered) by sensors. The traditional way of evaluating sensor coverage requires that every point in the field should be monitored and the sensor network should be connected to transmit messages to a processing center (sink). Such a requirement is too strong to be financially practical in many scenarios. In this study, we address another type of coverage problem, sweep coverage, when we utilize mobile nodes as supplementary in a sparse and probably disconnected sensor network. Different from previous coverage problem, we focus on retrieving data from dynamic Points of Interest (POIs), where a sensor network does not necessarily have fixed data rendezvous points as POIs. Instead, any sensor node within the network could become a POI. We first analyze the relationship among information access delay, information access probability, and the number of required mobile nodes. We then design a distributed algorithm based on a virtual 3D map of local gradient information to guide the movement of mobile nodes to achieve sweep coverage on dynamic POIs. Using the analytical results as the guideline for setting the system parameters, we examine the performance of our algorithm compared with existing approaches. I.
CONTROL OF MULTIAGENT NETWORKS: FROM NETWORK DESIGN TO DECENTRALIZED COORDINATION Approved by:
, 2012
"... To my parents Jenn and Yu who never gave up, my sunshine Cheryl, my dog Michael, and my friends. Thank you for your motivation, patience, and understanding. iii PREFACE This dissertation represents a culmination of my research in the GRITS (Georgia Robotics and Intelligent Systems) Lab at Georgia Te ..."
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To my parents Jenn and Yu who never gave up, my sunshine Cheryl, my dog Michael, and my friends. Thank you for your motivation, patience, and understanding. iii PREFACE This dissertation represents a culmination of my research in the GRITS (Georgia Robotics and Intelligent Systems) Lab at Georgia Tech under the supervision of Dr. Magnus Egerstedt from Fall 2008 to Spring 2012. In particular, it presents a suite of tools that I have developed which support the various stages of multiagent design: ranging from initial network design, to local execution using decentralized coordination strategies. Together, the tools support a multiagent system design methodology that is showcased through examples in three application domains: air traffic merging and spacing under the FAA’s NextGen program, collaborative multiUAV convoy protection in dynamic environments, and an educational tools for robotics. It is my firm belief that as autonomous and unmanned systems become more affordable, com
EXTREMAL BETTI NUMBERS OF RIPS COMPLEXES
, 910
"... Abstract. Upper bounds on the topological Betti numbers of VietorisRips complexes are established, and examples of such complexes with high Betti numbers are given. 1. ..."
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Abstract. Upper bounds on the topological Betti numbers of VietorisRips complexes are established, and examples of such complexes with high Betti numbers are given. 1.
Complex Networks: New Models and Distributed Algorithms
"... Over the past few years, a consensus has emerged among scientists and engineers that netcentric technology can provide unprecedented levels of performance, robustness, and efficiency. Success stories such as the Internet, distributed sensor networks, and multiagent networks of mobile robots are on ..."
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Over the past few years, a consensus has emerged among scientists and engineers that netcentric technology can provide unprecedented levels of performance, robustness, and efficiency. Success stories such as the Internet, distributed sensor networks, and multiagent networks of mobile robots are only a few examples that support this view. The important role played by complex networks has been widely observed in various physical, natural, and social systems. Given the complexity of many of these systems, it is important to understand the fundamental rules that govern them and introduce appropriate models that capture such principles, while abstracting away the redundant details. The main goal of this thesis is to contribute to the emerging field of "network science' ' in two ways. The first part of the thesis focuses on the question of information aggregation over complex networks. The problem under study is the asymptotic behavior of agents in a network when they are willing to share information with their neighbors. We start by focusing on conditions under which all agents in the network will asymptotically agree on some quantity of interest, what is known as the consensus problem. We present conditions that guarantee asymptotic agreement when interagent communication links change randomly over time. We then propose a distributed (nonBayesian) algorithm that enables agents to not only agree, but also learn the true