Sweeping Orders for Simplicial Complex Reconstruction

Simplicial complexes arising from real-world settings may not be directly observable. Hence, for an unknown simplicial complex in Euclidean space, we want to efficiently reconstruct it by querying local structure. In particular, we are interested in queries for the indegree of a simplex $\sigma$ in some direction: the number of cofacets of $\sigma$ contained in some halfspace "below" $\sigma$. Fasy et al. proposed a method that, given the vertex set of a simplicial complex, uses indegree queries to reconstruct the set of edges. In particular, they use a sweep algorithm through the vertex set, identifying edges adjacent to and above each vertex in the sweeping order. The algorithm relies on a natural but crucial property of the sweeping order: at a given vertex $v$, all edges adjacent to $v$ contained in the halfspace below $v$ have another endpoint that appeared earlier in the order. The edge reconstruction algorithm does not immediately extend to higher-dimensional simplex reconstruction. In particular, it is not possible to sweep through a set of $i$-simplices in a fixed direction and maintain that all $(i+1)$-cofacets of a given simplex $\sigma$ that come below $\sigma$ are known. We circumvent this by defining a sweeping order on a set of $i$-simplices, that additionally pairs each $i$-simplex $\sigma$ with a direction perpendicular to $\sigma$. Analogous to Fasy et al., our order has the crucial property that, at any $i$-simplex $\sigma$ paired with direction $s$, each $(i+1)$-dimensional coface of $\sigma$ that lies in the halfspace below $\sigma$ with respect to the direction $s$ has an $i$-dimensional face that appeared earlier in the order. We show how to compute such an order and use it to extend the edge reconstruction algorithm of Fasy et al. to simplicial complex reconstruction. Our algorithm can reconstruct arbitrary embedded simplicial complexes.