Improved Grid Map Layout by Point Set Matching

Abstract

Associating the regions of a geographic subdivision with the cells of a grid is a basic operation that is used in various types of maps, like spatially ordered treemaps and Origin-Destination maps (OD maps). In these cases the regular shapes of the grid cells allow easy representation of extra information about the regions. The main challenge is to find an association that allows a user to find a region in the grid quickly. We call the representation of a set of regions as a grid a grid map.We introduce a new approach to solve the association problem for grid maps by formulating it as a point set matching problem: Given two sets [Formula: see text] (the centroids of the regions) and [Formula: see text] (the grid centres) of [Formula: see text] points in the plane, compute an optimal one-to-one matching between [Formula: see text] and [Formula: see text]. We identify three optimisation criteria that are important for grid map layout: maximise the number of adjacencies in the grid that are also adjacencies of the regions, minimise the sum of the distances between matched points, and maximise the number of pairs of points in [Formula: see text] for which the matching preserves the directional relation (SW, NW, etc.). We consider matchings that minimise the [Formula: see text]-distance (Manhattan-distance), the ranked [Formula: see text]-distance, and the [Formula: see text]-distance, since one can expect that minimising distances implicitly helps to fulfill the other criteria.We present algorithms to compute such matchings and perform an experimental comparison that also includes a previous method to compute a grid map. The experiments show that our more global, matching-based algorithm outperforms previous, more local approaches with respect to all three optimisation criteria.