Stable algorithms and kinetic mesh refinement

In many applications there has been an increasing interest in computing certain properties of a changing set of input objects where the changes may be of dynamic nature in terms of insertions and deletions of an input object, or of kinetic nature in terms of continuous motion of these objects. For solving problems that involve dynamic or kinetic modifications to the input, one needs to first solve the static version of the same problem where no modifications are allowed, and then develop efficient update algorithms for handling various changes to the input. For developing dynamic and kinetic update algorithms, Acar et al. recently proposed a framework called self-adjusting computation. Given a dynamic or kinetic problem, the principal algorithmic technique of the self-adjusting computation framework, called change propagation, uses a static solution to the problem to automatically generate a dynamic or a kinetic update algorithm. The efficiency of their update algorithm directly depends on the stability of the static algorithm: a static algorithm is stable if its executions with similar inputs produce outputs and intermediate data that are different only by a small fraction. Under this framework, designing an efficient update algorithm can therefore be reduced to designing a stable static algorithm. Motivated by the self-adjusting computation framework, we follow a stable design approach in this thesis. We first design static algorithms that are stable, and then present update algorithms that are in the form of change propagation and guarantee efficient responses to dynamic and kinetic changes. We apply this approach for solving several open problems in computational geometry. First, we propose a robust motion simulator and experimentally evaluate its effectiveness on kinetically maintaining convex hulls in three dimensions. Then, we consider the mesh refinement problem and provide update algorithms that dynamically and kinetically maintain quality meshes. Mesh refinement is an essential step in many applications in scientific computing, graphics, etc. The idea behind mesh refinement is to break up a physical domain into well-shaped discrete elements, e.g., almost equilateral triangles in two dimensions, so that certain functions defined on the domain may be computed approximately by considering these discrete elements. The refinement process is carried out by inserting additional Steiner points into the given point set, taking care to insert a small number of them. This problem has been studied extensively in the static setting with several recent results achieving fast runtimes. In the dynamic and kinetic settings, however, there has been relatively little progress. In this thesis, we propose efficient solutions in both settings: in the dynamic setting, we design a dynamic algorithm for the closely related problem of well-spaced point sets in arbitrary dimensions; in the kinetic setting, we propose the first kinetic data structure for maintaining quality meshes of continuously moving points on a plane. The results we present in this thesis demonstrate that the stable design approach not only provides an alternative perspective in designing dynamic and kinetic algorithms, but also transfers the inherent complexity of the update algorithms to the stable design and analysis of a static algorithm. This in turn strengthens the connection between static algorithms and dynamic and kinetic algorithms assisting us to solve several open problems.