Sublinear convex-hull reconstruction from general supersequences

Determine whether, given a sequence P of n points in the Euclidean plane that is guaranteed to contain a subsequence Q whose points are in counterclockwise order and whose convex hull equals the convex hull of P, one can compute the convex hull of P in o(n log n) time without assuming any smoothness properties of the sequence.

Background

The paper introduces supersequences as an auxiliary structure for preprocessing uncertain geometric data where the output (e.g., convex hull) is an ordered subsequence. Their reconstruction algorithms currently require the supersequence to be smooth, a property that constrains how far out of order elements can be and provides packing bounds, enabling linear-time reconstruction.

To further decouple preprocessing from reconstruction, the authors formalize the general reconstruction task without smoothness: given a sequence that contains the convex hull as a subsequence in the correct cyclic order, can the convex hull be computed faster than the standard O(n log n) time? This open problem is central because a positive resolution would allow reconstruction from arbitrary supersequences, strengthening the preprocessing framework. The authors also note the problem remains open even when all points are unique.