Sweeping Orders for Simplicial Complex Reconstruction

TL;DR

This paper introduces a generalized sweeping order for reconstructing simplicial complexes from vertex data, extending previous graph reconstruction methods to higher-dimensional structures with improved efficiency.

Contribution

It generalizes sweep algorithms to simplicial complexes, enabling reconstruction from vertex locations and degree info with a new sweeping order and better runtime performance.

Findings

Successfully reconstructs simplicial complexes from vertex data.

Generalizes existing graph reconstruction algorithms to higher dimensions.

Improves the runtime of key subroutines in the reconstruction process.

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 $\sigma$, all $(i+1)$-dimensional cofaces of $\sigma$ in the halfspace below $\sigma$ have an $i$-dimensional face that appeared earlier in the order ("below" with respect to some direction perpendicular to $\sigma$). 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.