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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 6 (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.
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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Cited by 3 (1 self)
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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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Cited by 1 (1 self)
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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.
VIETORISRIPS COMPLEXES OF PLANAR POINT SETS
"... 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 VietorisRips complex is a combinatorial simplicial complex based on proximity of neighbors that serves as an easilycomputed but highdimensional approximation to the ..."
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Cited by 1 (0 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 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. 1.
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.