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Algorithms for continuous optimization problems have a rich history of design and innovation over the past several decades, in which mathematical analysis of their convergence and complexity properties plays a central role. Besides their…
Starting with the irreducible triangulations of a fixed surface and splitting vertices, all the triangulations of the surface up to a given number of vertices can be generated. The irreducible triangulations have previously been determined…
The rapid advances in 3D scanning and acquisition techniques have given rise to the explosive increase of volumetric digital models in recent years. This dissertation systematically trailblazes a novel volumetric modeling framework to…
Normed spaces appear to have very little going for them: aside from the hackneyed linear structure, you get a norm whose only virtue, aside from separating points, is the Triangle Inequality. What could you possibly prove with that? As it…
Let M be the interior of a compact 3-manifold with non-empty boundary, and T be an ideal (topological) triangulation of M. This paper describes necessary and sufficient conditions for the existence of angle structures, semi-angle structures…
To solve the combinatorial optimization problems especially the minimal Vertex-cover problem with high efficiency, is a significant task in theoretical computer science and many other subjects. Aiming at detecting the solution space of…
We investigate the complexity of finding an embedded non-orientable surface of Euler genus $g$ in a triangulated $3$-manifold. This problem occurs both as a natural question in low-dimensional topology, and as a first non-trivial instance…
We describe an efficient algorithm to compute a pseudotriangulation of a finite planar family of pairwise disjoint convex bodies presented by its chirotope. The design of the algorithm relies on a deepening of the theory of visibility…
Typical performance of approximation algorithms is studied for randomized minimum vertex cover problems. A wide class of random graph ensembles characterized by an arbitrary degree distribution is discussed with some theoretical frameworks.…
Machine Learning models incorporating multiple layered learning networks have been seen to provide effective models for various classification problems. The resulting optimization problem to solve for the optimal vector minimizing the…
Homology features of spaces which appear in applications, for instance 3D meshes, are among the most important topological properties of these objects. Given a non-trivial cycle in a homology class, we consider the problem of computing a…
Delaunay flip is an elegant, simple tool to convert a triangulation of a point set to its Delaunay triangulation. The technique has been researched extensively for full dimensional triangulations of point sets. However, an important case of…
A theory of transversely oriented spun-normal immersed surfaces in ideally triangulated 3--manifolds is developed in this paper, including linear functionals determining the boundary curves, Euler characteristic and homology class of these…
In terms of the number of triangles, it is known that there are more than exponentially many triangulations of surfaces, but only exponentially many triangulations of surfaces with bounded genus. In this paper we provide a first geometric…
Many problems in computational geometry are not stated in graph-theoretic terms, but can be solved efficiently by constructing an auxiliary graph and performing a graph-theoretic algorithm on it. Often, the efficiency of the algorithm…
With the rapid development of high-resolution 3D vision applications, the traditional way of manipulating surface detail requires considerable memory and computing time. To address these problems, we introduce an efficient surface detail…
We describe a new algorithm to compute the geometric intersection number between two curves, given as edge vectors on an ideal triangulation. Most importantly, this algorithm runs in polynomial time in the bit-size of the two edge vectors.…
We study the problem of estimating the number of triangles in a graph stream. No streaming algorithm can get sublinear space on all graphs, so methods in this area bound the space in terms of parameters of the input graph such as the…
We show that in any triangulated 3-manifold, every index n topologically minimal surface can be transformed to a surface which has local indices (as computed in each tetrahedron) that sum to at most n. This generalizes classical theorems of…
A common representation of a three dimensional object in computer applications, such as graphics and design, is in the form of a triangular mesh. In many instances, individual or groups of triangles in such representation need to satisfy…