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Abstract  We present a novel reconstruction algorithm that, given an input point set sampled from an object S, builds a one-parameter family of complexes that approximate S at different scales. At a high level, our method is very similar in spirit to Chew’s surface meshing algorithm, with one notable difference though: the restricted Delaunay triangulation is replaced by the witness complex, which makes our algorithm applicable in any metric space. To prove its correctness on curves and surfaces, we highlight the relationship between the witness complex and the restricted Delaunay triangulation in 2d and in 3d. Specifically, we prove that both complexes are equal in 2d and closely related in 3d, under some mild sampling assumptions. [ABSTRACT FROM AUTHOR]

In real sensor network deployments, spatial distributions of sensors are usually far from being uniform. Such networks often contain regions without enough sensor nodes, which we call holes. In this paper, we show that holes are important topological features that need to be studied. In routing, holes are communication voids that cause greedy forwarding to fail. Holes can also be defined to denote regions of interest, such as the “hot spots” created by traffic congestion or sensor power shortage. In this paper, we define holes to be the regions enclosed by a polygonal cycle which contains all the nodes where local minima can appear. We also propose simple and distributed algorithms, the Tent rule and BoundHole , to identify and build routes around holes. We show that the boundaries of holes marked using BoundHole can be used in many applications such as geographic routing, path migration, information storage mechanisms and identification of regions of interest. [ABSTRACT FROM AUTHOR]