Sweeping Orders for Simplicial Complex Reconstruction

TL;DR

Reconstruct a simplicial complex $K$ embedded in ${\mathbb{R}}^d$ from only its vertex set $K_0$ and indegree queries $Indeg(\sigma,s)$. The method introduces sweeping orders $\mathrm{SO}_i$ for each skeleton $K_i$, pairs each $i$-simplex with a direction perpendicular to it, and uses candidate-ordering circles $\gamma_\sigma$ to radially order cofacets and reconstruct $K_{i+1}$ from $K_i$ iteratively until $K$ is recovered. The main contributions are (i) a higher-dimensional generalization of graphsearch reconstruction via indegree queries, (ii) a radial subroutine FindUnfound with improved runtime over prior methods, and (iii) a complete reconstruction framework with proven correctness and indegree-query bounds, plus connections to faithful discretizations through verbose persistence diagrams. This approach enables faithful directional discretizations for shape reconstruction from vertex data, with potential applications in computational geometry and directional topological transforms.

Abstract

Standard sweep algorithms require an order of discrete points in Euclidean space, and rely on the property that, at a given point, all points in the halfspace below come earlier in this order. We are motivated by the problem of reconstructing a graph in $\mathbb{R}^d$ from vertex locations and degree information, which was addressed using standard sweep algorithms by Fasy et al. We generalize this to the reconstruction of general simplicial complexes. As our main ingredient, we introduce a generalized \emph{sweeping order} on $i$-simplices, maintaining the property that, at a given $i$-simplex $σ$, all $(i+1)$-dimensional cofaces of $σ$ in the halfspace below $σ$ have an $i$-dimensional face that appeared earlier in the order ("below" with respect to some direction perpendicular to $σ$). We then go on to incorporate computing such sweeping orders to reconstruct an unknown simplicial complex $K$, starting with only its vertex locations, i.e., its $0$-skeleton. Specifically, once we have found the $i$-skeleton of $K$, we compute a sweeping order for the $i$-simplices, and use it to reconstruct the $(i+1)$-skeleton of $K$ by querying the \emph{indegree}, or the number of $(i+1)$-simplices incident to and below a given $i$-simplex. In addition to generalizing the algorithm by Fasy et al. to simplicial complexes, we improve upon the running time of their central subroutine of radially finding edges above a vertex.

Figures (10)

• Figure 1: Here, every edge $e$ has other edges in the halfspaces on either side of $e$. This general higher-dimensional phenomenon contrasts the special zero-dimensional case; in every direction, we can find at least one vertex with no other vertices below it. However, for each dimension of simplex, there do exist simplices with no cofacets in a halfspace below it, which we will see is true in general.
• Figure 2: Examples of complexes in ${\mathbb {R}}\xspace^2$ that are (a) both locally injective and embedded, (b) locally injective but not embedded, and (c) neither locally injective nor embedded.
• Figure 3: (a)-(b) $\gamma(\alpha)$ is the $\gamma$-normal of $p$. (c) since $\sigma$ has codimension one with the ambient space, any point above (or below) $\sigma$ has a $\gamma$-normal of $\gamma(0)$ (or $\gamma(\pi)$, respectively).
• Figure 4: For some $\sigma_j=\rho\cup\{v_j\}$, we consider a cofacet $\tau=\rho\cup\{v_j,v_h\}$ (unshaded) for which the vertex $v_h$ lies below $\sigma_j$ with respect to the direction $\gamma_\rho(\alpha_{v_j})$. The simplex $\sigma_h=\rho\cup\{v_h\}$ is a cofacet of $\rho$ that also has $\tau$ as a cofacet. The simplex $\rho$ has a perpendicular circle of directions $\gamma_\rho$.
• Figure 5: Suppose that (a) is the one-skeleton of an otherwise unknown simplicial complex $K$. If $K$ is known to be locally injective, (b) depicts the set of candidate triangles. If $K$ is known to be embedded, we instead get the set of candidate triangles shown in (c).

Theorems & Definitions (39)

• Definition 1: Simplex
• Definition 2: Simplicial Complex
• Definition 3: Embedded and locally injective
• Definition 4: $\gamma$-Normal of $p$
• Definition 5: Sweeping Order
• Definition 6: Maximally Perpendicular Circle
• Remark 7
• Lemma 8: Directions are Perpendicular to their Paired Simplices
• proof
• Lemma 9: Halfspace Property