8396933
Hierarchical morse complexes for piecewise linear 2-manifolds
We present algorithms for constructing a hierarchy of increasingly coa rse Morse complexes that decompose a piecewise linear 2-manifold. While Morse complexes are defined only in the smooth category, we extend the construction to the piecewise linear category by ensuring structural integrity and simulating differentiability. We then simplify Morse complexes by cancelling pairs of critical points in order of increasing persistence.
44420378
Triangulating Topological Spaces
Given a subspace ${\Bbb X} \subseteq {\Bbb R}^d$ and a finite set S⊆ℝd, we introduce the Delaunay complex, ${\cal D}_{\Bbb X}$, restricted by ${\Bbb X}$. Its simplices are spanned by subsets T⊆S for which the common intersection of Voronoi cells meets ${\Bbb X}$ in a non-empty set. By the nerve theorem, $\bigcup {\cal D}_{\Bbb X}$ and ${\Bbb X}$ are homotopy equivalent if all such sets are contractible. This paper proves a sufficient condition for $\bigcup {\cal D}_{\Bbb X}$ and ${\Bbb X}$ be homeomorphic.
7437367
Vines and vineyards by updating persistence in linear time
Persistent homology is the mathematical core of recent work on shape, including reconstruction, recognition, and matching. Its pertinent information is encapsulated by a pairing of the critical values of a function, visualized by points forming a diagram in the plane. The original algorithm in [10] computes the pairs from an ordering of the simplices in a triangulation and takes worst-case time cubic in the number of simplices. The main result of this paper is an algorithm that maintains the pairing in worst-case linear time per transposition in the ordering. A side-effect of the algorithm's analysis is an elementary proof of the stability of persistence diagrams [7] in the special case of piecewise-linear functions. We use the algorithm to compute 1-parameter families of diagrams which we apply to the study of protein folding trajectories.
13608907
180 Wrapped Tubes
The 180 models collected in this paper are produced by sampling and wrapping point sets on tubes. The surfaces are represented as triangulated 2-manifolds and available as stl- files from the author's web site atwww.cs.duke.edu/ edels. Each tube is obtained by thickening a circle or a smooth torus knot, and for some we use the degrees of freedom in the thickening process to encode meaningful information, such as curvature or torsion.
30538700
Computing a Ham-Sandwich Cut in Two Dimensions
Let B be a set of n"b black points and W a set of n"w, white points in the Euclidean plane. A line h is said to bisect B (or W) if, at most, half of the points of B (or W) lie on any one side of h. A line that bisects both B and W is called a ham-sandwich cut of B and W. We give an algorithm that computes a ham-sandwich cut of B and W in 0((n"h+n"w) log (min {n"b, n"w}+ 1)) time. The algorithm is considerably simpler than the previous most efficient one which takes 0((n"b + n"w) log (n"b + n"w)) time.
31366541
An optimal algorithm for constructing the weighted voronoi diagram in the plane
Let S denote a set of n points in the plane such that each point p has assigned a positive weight w(p) which expresses its capability to influence its neighbourhood. In this sense, the weighted distance of an arbitrary point x from p is given by de(x,p)/w(p) where de denotes the Euclidean distance function. The weighted Voronoi diagram for S is a subdivision of the plane such that each point p in S is associated with a region consisting of all points x in the plane for which p is a weighted nearest point of S. 
 
An algorithm which constructs the weighted Voronoi diagram for S in O(n2) time is outlined in this paper. The method is optimal as the diagram can consist of Θ(n2) faces, edges and vertices.
34884676
Cutting dense point sets in half
A halving hyperplane of a set S of n points in ℝd contains d affinely independent points of S so that equally many of the points off the hyperplane lie in each of the two halfspaces. We prove bounds on the number of halving hyperplanes under the condition that the ratio of largest over smallest distance between any two points is at most δn(1/d), δ some constant. Such a set S is called dense.In d = 2 dimensions the number of halving lines for a dense set can be as much as Ω (n log n), and it cannot exceed O (n5/4/log*n). The upper bound improves over the current best bound of O(n3/2/log*n) which holds more generally without any density assumption. In d = 3 dimensions we show that O(n7/3) is an upper bound on the number of halving planes for a dense set. The proof is based on a metric argument that can be extended to d ≥ 4 dimensions, where it leads to O(nd-2/d) as an upper bound for the number of halving hyperplanes.