IA*: An adjacency-based representation for non-manifold simplicial shapes in arbitrary dimensions

Abstract

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.

Computing discrete Morse complexes from simplicial complexes

“…Here, we focused on data structures with available public-domain implementations. Our experiments clearly show that the Generalized Indexed data structure with Adjacencies (IA * ) [17], a data structure encoding only the vertices and a subset of the simplices of the complex, is the only one that can suitably scale to higher dimensions without being affected by the exponential growth in the number of simplices. We propose a solution to compactly encode a Forman gradient attached to the IA * data structure.…”

“…To be able to extract boundary, coboundary and adjacency relations efficiently, the simplest representation would encode: (i) for each top k-simplex σ, its boundary defined by the references to its k + 1 vertices, and its adjacencies defined by references to the simplices adjacent to σ along a (k − 1)-face; (ii) for each vertex v, its star, defined by the the list of all top simplices incident in v. It can be noticed that storing the entire star of a vertex v is not necessary, since the star can be efficiently reconstructed by navigating the top simplices incident in v through the encoded adjacencies. This constitutes the basis for the Generalized Indexed data structure with Adjacencies (IA * ) [17].…”

Efficient Homology‐Preserving Simplification of High‐Dimensional Simplicial Shapes

“…In this section, we present an algorithm for homology‐preserving edge contraction on the top‐based representation of a simplicial complex. Notice that it can be applied to any top‐based representation, like an adjacency‐based data structure [PBCF93, CDFW11, DFH03, DFMPS04], a hierarchical topological data structure [FWD17] and a corner‐based data structure [RSS01, GR09, GR10, GLLR11]. Given the top‐based representation of a simplicial complex Σ, the objective is to obtain, through a homology‐preserving edge contraction applied to , the top‐based representation of the resulting complex.…”

Efficient Computation of Simplicial Homology through Acyclic Matching

“…The data structure we used for encoding the simplicial complex is an adjacency-based data structure, called the Generalized Indexed data structure with Adjacencies (IA*) [31]. The IA* encodes all vertices and top simplices in a simplicial d-complex Σ.…”