A Spatial Alignment Problem

This work concerns the harmonization of geospatial data to improve linkages between place-based characteristics and health outcomes. Such data are typically available as geographic layers, each representing a distinct attribute (e.g., income or distance to a clinic). Since layers are typically constructed independently, their boundaries tend to be spatially incongruent, which can create inconsistencies and introduce bias. This motivates developing algorithmic approaches for aligning such layers while aiming to preserve spatial integrity. This paper formalizes the problem of aligning k collections of m spatial supports over n spatial units in a d-dimensional Euclidean space such that maximum distortion to any collection is minimized. In the above setting, k is the number of layers; n is an indivisible population unit (e.g., census tract); m denotes supports, which are larger regions aggregating a set of contiguous units in order to capture broader regional patterns or enhance statistical stability; and d=2. It is shown that: (1) the one-dimensional case is solvable in time polynomial in k, m, and n; (2) the two-dimensional case is NP-hard for two collections of two supports each; and (3) a heuristic can be provided for aligning a set of collections in the two-dimensional case, which is of practical importance.