Accurately determining the distribution of rare variants is an important goal of human genetics, but resequencing of a sample large enough for this purpose has been unfeasible until now. Here, we applied Sanger sequencing of genomic PCR amplicons to resequence the diabetes-associated genes KCNJ11 and HHEX in 13,715 people (10,422 European Americans and 3,293 African Americans) and validated amplicons potentially harbouring rare variants using 454 pyrosequencing. We observed far more variation (expected variant-site count ∼578) than would have been predicted on the basis of earlier surveys, which could only capture the distribution of common variants. By comparison with earlier estimates based on common variants, our model shows a clear genetic signal of accelerating population growth, suggesting that humanity harbours a myriad of rare, deleterious variants, and that disease risk and the burden of disease in contemporary populations may be heavily influenced by the distribution of rare variants.

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Nested simplicial meshes generated by the simplicial bisection decomposition proposed by Maubach [Mau95] have been widely used in 2D and 3D as multi-resolution models of terrains and three-dimensional scalar fields, They are an alternative to octree representation since they allow generating crack-free representations of the underlying field. On the other hand, this method generates conforming meshes only when all simplices sharing the bisection edge are subdivided concurrently. Thus, efficient representations have been proposed in 2D and 3D based on a clustering of the simplices sharing a common longest edge in what is called a diamond. These representations exploit the regularity of the vertex distribution and the diamond structure to yield an implicit encoding of the hierarchical and geometric relationships among the triangles and tetrahedra, respectively. Here, we analyze properties of d-dimensional diamonds to better understand the hierarchical and geometric relationships among the simplices generated by Maubach's bisection scheme and derive closed-form equations for the number of vertices, simplices, parents and children of each type of diamond. We exploit these properties to yield an implicit pointerless representation for d-dimensional diamonds and reduce the number of required neighbor-finding accesses from O(d!) to O(d).

Figure 1: Overview our scheme for tetrahedral meshes (illustrated in 2D). (a) We interpret the Morse complex of a simplicial mesh in terms of the primal mesh Σ (solid lines) and its dual Σ d (dashed lines). (b) Encoding the Discrete Morse gradient field entirely with the tetrahedra enables the use of compact topological data structures for morphological extraction. We associate the descending Morse complexes with the cells of Σ (c-d), the ascending Morse complexes with the cells of Σ d (e-f) and the Morse-Smale complex with the dually subdivided tetrahedral mesh Σ S (g), whose hexahedral cells are defined by a tetrahedron and one of its vertices. All relations are encoded strictly in terms of the vertices and tetrahedra of Σ. AbstractWe consider the problem of computing discrete Morse and Morse-Smale complexes on an unstructured tetrahedral mesh discretizing the domain of a 3D scalar field. We use a duality argument to define the cells of the descending Morse complex in terms of the supplied (primal) tetrahedral mesh and those of the ascending complex in terms of its dual mesh. The Morse-Smale complex is then described combinatorially as collections of cells from the intersection of the primal and dual meshes. We introduce a simple compact encoding for discrete vector fields attached to the mesh tetrahedra that is suitable for combination with any topological data structure encoding just the vertices and tetrahedra of the mesh. We demonstrate the effectiveness and scalability of our approach over large unstructured tetrahedral meshes by developing algorithms for computing the discrete gradient field and for extracting the cells of the Morse and Morse-Smale complexes. We compare implementations of our approach on an adjacency-based topological data structure and on the PR-star octree, a compact spatio-topological data structure.

We propose a compact, dimension-independent data structure for manifold, non-manifold and non-regular simplicial complexes, that we call the Generalized Indexed Data structure with Adjacencies (IA * data structure). It encodes only top simplices, i.e., the ones that are not on the boundary of any other simplex, plus a suitable subset of the adjacency relations. We describe the IA * data structure in arbitrary dimensions, and compare the storage requirements of its two-dimensional and three-dimensional instances with both dimension-specific and dimension-independent representations. We show that the IA * data structure is more cost effective than other dimension-independent representations and is even slightly more compact than the existing dimension-specific ones. We present efficient algorithms for navigating a simplicial complex described as an IA * data structure. This shows that the IA * data structure allows retrieving all topological relations of a given simplex by considering only its local neighborhood and thus it is a more efficient alternative to incidence-based representations when information does not need to be encoded for boundary simplices.