Sweeping Orders for Simplicial Complex Reconstruction

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.