Euclidean Voronoi labelling on the multidimensional grid

Abstract Given a set of k points (sites) S ⊂ R d , and a finite d - dimensional grid G n d =( I , n ) d , we describe an efficient algorithm computing the mapping D : G n d → { l …, k } such that D ( p ) is the index of the site closest to p under the Euclidean norm. This mapping is called the Voronoi labelling of the grid. The algorithm traverses the points of G on a “locality-preserving” space-filling curve, exploiting the correlation between the value of D on adjacent grid points to reduce computation time. The advantage of this algorithm over existing ones is that it works in any dimension, and is general-purpose, in the sense that it is easily modified to accomodate many variants of the problem, such as non-integral sites, and computation on a subset of G . The runtimes of our algorithm in two dimensions compare with those of the existing algorithms, and are better than the few existing algorithms operating in higher dimensions.