Voronoi Diagrams for Parallel Halflines and Line Segments in Space

We consider the Euclidean Voronoi diagram for a set of $n$ parallel halflines in 3-space. A relation of this diagram to planar power diagrams is shown, and is used to analyze its geometric and topological properties. Moreover, an easy-to-implement space sweep algorithm is proposed that computes the Voronoi diagram for parallel halflines at logarithmic cost per face. Previously only an approximation algorithm for this problem was known. Our method of construction generalizes to Voronoi diagrams for parallel line segments, and to higher dimensions

Piecewise-Linear Farthest-Site Voronoi Diagrams

Voronoi diagrams induced by distance functions whose unit balls are convex polyhedra are piecewise-linear structures. Nevertheless, analyzing their combinatorial and algorithmic properties in dimensions three and higher is an intriguing problem. The situation turns easier when the farthest-site variants of such Voronoi diagrams are considered, where each site gets assigned the region of all points in space farthest from (rather than closest to) it. We give asymptotically tight upper and lower worst-case bounds on the combinatorial size of farthest-site Voronoi diagrams for convex polyhedral distance functions in general dimensions, and propose an optimal construction algorithm. Our approach is uniform in the sense that (1) it can be extended from point sites to sites that are convex polyhedra, (2) it covers the case where the distance function is additively and/or multiplicatively weighted, and (3) it allows an anisotropic scenario where each site gets allotted its particular convex distance polytope

Unbounded Regions of High-Order Voronoi Diagrams of Lines and Segments in Higher Dimensions

We study the behavior at infinity of the farthest and the higher-order Voronoi diagram of n line segments or lines in a d-dimensional Euclidean space. The unbounded parts of these diagrams can be encoded by a Gaussian map on the sphere of directions S^(d-1). We show that the combinatorial complexity of the Gaussian map for the order-k Voronoi diagram of n line segments or lines is O(min{k,n-k} n^(d-1)), which is tight for n-k = O(1). All the d-dimensional cells of the farthest Voronoi diagram are unbounded, its (d-1)-skeleton is connected, and it does not have tunnels. A d-cell of the Voronoi diagram is called a tunnel if the set of its unbounded directions, represented as points on its Gaussian map, is not connected. In a three-dimensional space, the farthest Voronoi diagram of lines has exactly n^2-n three-dimensional cells, when n >= 2. The Gaussian map of the farthest Voronoi diagram of line segments or lines can be constructed in O(n^(d-1) alpha(n)) time, while if d=3, the time drops to worst-case optimal O(n^2)

Approximate Nearest-Neighbor Search for Line Segments

Approximate nearest-neighbor search is a fundamental algorithmic problem that continues to inspire study due its essential role in numerous contexts. In contrast to most prior work, which has focused on point sets, we consider nearest-neighbor queries against a set of line segments in $\mathbb{R}^d$, for constant dimension $d$. Given a set $S$ of $n$ disjoint line segments in $\mathbb{R}^d$ and an error parameter $\varepsilon > 0$, the objective is to build a data structure such that for any query point $q$, it is possible to return a line segment whose Euclidean distance from $q$ is at most $(1+\varepsilon)$ times the distance from $q$ to its nearest line segment. We present a data structure for this problem with storage $O((n^2/\varepsilon^{d}) \log (\Delta/\varepsilon))$ and query time $O(\log (\max(n,\Delta)/\varepsilon))$, where $\Delta$ is the spread of the set of segments $S$. Our approach is based on a covering of space by anisotropic elements, which align themselves according to the orientations of nearby segments.Comment: 20 pages (including appendix), 5 figure

Computations to Obtain Wider Tunnels in Protein Structures

Finding wide tunnels in protein structures is an important problem in Structural Bioinformatics with applications in various areas such as drug design. Several algorithms have been proposed for finding wide tunnels in a fixed protein conformation. However, to the best of our knowledge, none of the existing work have considered widening the tunnel, i.e., finding a wider tunnel in an alternative conformation of the given structure. In this thesis we initiate this line of research by proposing a tunnel-widening algorithm which aims to make the tunnel wider by a slight local change in the structure of the protein. Given a fixed conformation of a protein with a point located inside it, we first describe an algorithm to identify the widest tunnel from that point to the outside environment of the protein. Then we try to make the tunnel wider by considering various alternative conformations of the protein. We only consider conformations whose energies are not much higher than the energy of the initial conformation. Among these alternative conformations we select the one with the widest tunnel. However, the alternative conformation with the widest tunnel might not be accessible from the initial structure. Thus, in the next step we develop three algorithms for finding a feasible transition pathway from the initial structure to the alternative conformation, i.e., a sequence of intermediate conformations between the initial structure and the alternative conformation such that the energy values of all these intermediate conformations are close to the energy of the initial structure. We evaluate our tunnel-finding and tunnel-widening algorithms on various proteins. Our experiments show that in most cases we can make the tunnel wider in an alternative conformation. However, there are cases in which we find a wider tunnel in an alternative conformation, but the energy value of the alternative conformation is much higher than the energy of the initial structure. We also implemented our three pathway-finding algorithms and tested them on various instances. Our experiments show that although in most cases we can find a feasible transition pathway, there are cases in which the alternative conformation has energy close to the initial structure, but our algorithms cannot find any feasible pathway from the initial structure to the alternative conformation. Furthermore, there is a trade-off between the running time and accuracy of the three pathway-finding algorithms