Abstract The minimal convex hulls of disks problem is to find such arrangements of circular disks in the plane that minimize the length of the convex hull boundary. The mixed-integer non-linear programming model, named [17], works only for small to moderate-sized problems. Here we propose a polylithic framework of the problem for big problem instances by combining the following algorithms and models: (i) A fast disk-packing algorithm based on Voronoi diagrams, non-linear programming (NLP) models for packing disks, and an NLP model for minimizing the discretized perimeter of convex hull; (ii) A fast convex-hull algorithm to compute the convex hulls of disk arrangements and their perimeter lengths; (iii) A mixed-integer NLP model taking the output of as its input. We present complete analytic solutions for small problems up to four disks and a semi-analytic mixed-integer linear programming model which yields exact solutions for strip packing problems with up to one thousand congruent disks. It turns out that the proposed polylithic approach works fine for large problem instances containing up to 1,000 disks. Monolithic and polylithic solutions using usually outperform other approaches. The polylithic approach yields better solutions than the results in [17] and provides a benchmark suite for further research.

Voronoi diagrams are powerful for understanding spatial properties. However, few reports have been made for moving generators despite their important applications. We present a topology-oriented event-increment (TOI-E) algorithm for constructing a Voronoi diagram of moving circular disks in the plane over the time horizon$[0, t^{\infty })$. The proposed TOI-E algorithm computes the event history of the Voronoi diagram over the entire time horizon in$O(k_F \log n + k_C n \log n)$time with$O(n \log n)$preprocessing time and$O(n + k_F + k_C)$memory for$n$disk generators,$k_F$edge flips, and$k_C$disk collisions during the time horizon. Given an event history, the Voronoi diagram of an arbitrary moment$t^{\ast}

Geometric properties are critical for the function of molecules consisting of atoms which are usually modeled as a set of spheres in 3D. We propose "Molecular Geometry" which is a theoretical framework of computational understanding of the geometry of molecules in the claim that most, hopefully all, molecular structure problems can be effectively and efficiently facilitated by the "geometrization" of the problem at hand. We also report BULL!, the molecular geometry engine based on the Voronoi diagram of spheres, the quasi-triangulation, and the beta-complex. Being a program implemented in C++, application programmers can simply call API-functions of BULL! to create application programs correctly, efficiently, and conveniently. The BULL! engine, which will be freely available from the Voronoi Diagram Research Center at Hanyang University, is designed so that application programs are completely independent of future modifications and improvements.