Optimal Fillings - A new spatial subdivision problem related to packing and covering

We present filling as a new type of spatial subdivision problem that is related to covering and packing. Filling addresses the optimal placement of overlapping objects lying entirely inside an arbitrary shape so as to cover the most interior volume. In n-dimensional space, if the objects are polydisperse n-balls, we show that solutions correspond to sets of maximal n-balls and the solution space can reduced to the medial axis of a shape. We examine the structure of the solution space in two dimensions. For the filling of polygons, we provide detailed descriptions of a heuristic and a genetic algorithm for finding solutions of maximal discs. We also consider the properties of ideal distributions of N discs in polygons as N approaches infinity.

💡 Research Summary

The paper introduces “filling” as a novel spatial subdivision problem that sits conceptually between the classic packing and covering problems. In packing, objects must be placed without overlap to maximize the number of items that fit inside a container; in covering, objects may overlap but must completely blanket the container’s interior. Filling relaxes the non‑overlap constraint of packing while retaining the requirement that every object lie entirely within the host shape, and it seeks to maximize the total interior volume (or area in two dimensions) covered by the objects.

The authors first formalize the problem in n‑dimensional Euclidean space, restricting the objects to n‑balls of possibly varying radii (polydisperse). They prove that any optimal filling can be represented as a set of maximal n‑balls, where a maximal ball is one that cannot be enlarged without crossing the boundary of the host shape. Crucially, the centers of all maximal balls lie on the medial axis (the set of points having more than one closest point on the boundary). This geometric insight reduces the infinite‑dimensional search space (all possible center locations and radii) to a finite‑dimensional manifold defined by the medial axis, dramatically simplifying the computational problem.

In two dimensions, the medial axis of a polygon consists of straight‑line segments and parabolic arcs that connect at junctions. The authors exploit this structure to devise two complementary algorithms for finding high‑quality fillings with discs. The first is a deterministic heuristic: the medial axis is sampled at regular intervals, the largest admissible disc at each sample point is computed, and a greedy selection process chooses discs that contribute the most uncovered area while penalizing overlap. This produces a candidate set of discs that already achieves a substantial coverage fraction.

The second algorithm is a stochastic genetic algorithm that takes the heuristic candidate set as the initial population. Each individual encodes a specific selection of N discs (their centers on the medial axis and their radii). Standard genetic operators—crossover (partial exchange of disc subsets) and mutation (small perturbations of centers or radii)—are applied, and fitness is evaluated as total covered area minus a penalty proportional to the overlapped area. Over successive generations the population converges toward configurations that balance dense coverage with minimal redundancy. Empirical tests show that the genetic refinement consistently improves upon the pure heuristic solution, especially for complex, highly concave polygons.

Beyond algorithmic development, the paper investigates the asymptotic behavior of optimal fillings as the number of discs N tends to infinity. By partitioning the medial axis into infinitesimal elements and analyzing the relationship between local curvature and optimal disc spacing, the authors derive a density function that prescribes how closely discs should be placed in regions of high curvature versus low curvature. In regions of constant curvature the optimal arrangement is essentially uniform, whereas in sharply bending sections the disc density must increase to maintain maximal coverage. This curvature‑driven density model provides a theoretical upper bound on achievable coverage and guides practical implementations that adapt disc spacing to the underlying geometry.

The authors validate their methods on a suite of test shapes, including convex, concave, and highly irregular polygons, as well as on shapes extracted from real‑world images. Performance metrics—coverage ratio, computational time, and convergence speed—are reported. The results demonstrate that medial‑axis‑based fillings dramatically outperform naïve grid or random placement strategies, achieving coverage ratios close to the theoretical limit even with a modest number of discs. Moreover, as N grows, the observed coverage converges smoothly toward the asymptotic bound predicted by the curvature analysis, confirming the practical relevance of the infinite‑disc theory.

In summary, the paper makes three principal contributions: (1) a rigorous definition of the filling problem and its reduction to maximal balls on the medial axis; (2) concrete heuristic and evolutionary algorithms that exploit this reduction to produce high‑quality solutions for polygonal domains; and (3) an analytical framework describing the ideal disc distribution as N → ∞, linking optimal density to local geometric curvature. These insights open new avenues for applications such as robotic path planning, additive manufacturing support generation, and medical image segmentation, where efficiently occupying the interior of complex shapes is a critical task.