Minimum Convex Partitions and Maximum Empty Polytopes

Let $S$ be a set of $n$ points in $\mathbb{R}^d$. A Steiner convex partition is a tiling of ${\rm conv}(S)$ with empty convex bodies. For every integer $d$, we show that $S$ admits a Steiner convex partition with at most $\lceil (n-1)/d\rceil$ tiles. This bound is the best possible for points in general position in the plane, and it is best possible apart from constant factors in every fixed dimension $d\geq 3$. We also give the first constant-factor approximation algorithm for computing a minimum Steiner convex partition of a planar point set in general position. Establishing a tight lower bound for the maximum volume of a tile in a Steiner convex partition of any $n$ points in the unit cube is equivalent to a famous problem of Danzer and Rogers. It is conjectured that the volume of the largest tile is $\omega(1/n)$. Here we give a $(1-\varepsilon)$-approximation algorithm for computing the maximum volume of an empty convex body amidst $n$ given points in the $d$-dimensional unit box $[0,1]^d$.Comment: 16 pages, 4 figures; revised write-up with some running times improve

Optimal Point Placement for Mesh Smoothing

We study the problem of moving a vertex in an unstructured mesh of triangular, quadrilateral, or tetrahedral elements to optimize the shapes of adjacent elements. We show that many such problems can be solved in linear time using generalized linear programming. We also give efficient algorithms for some mesh smoothing problems that do not fit into the generalized linear programming paradigm.Comment: 12 pages, 3 figures. A preliminary version of this paper was presented at the 8th ACM/SIAM Symp. on Discrete Algorithms (SODA '97). This is the final version, and will appear in a special issue of J. Algorithms for papers from SODA '9

Pattern Matching for sets of segments

In this paper we present algorithms for a number of problems in geometric pattern matching where the input consist of a collections of segments in the plane. Our work consists of two main parts. In the first, we address problems and measures that relate to collections of orthogonal line segments in the plane. Such collections arise naturally from problems in mapping buildings and robot exploration. We propose a new measure of segment similarity called a \emph{coverage measure}, and present efficient algorithms for maximising this measure between sets of axis-parallel segments under translations. Our algorithms run in time O(n^3\polylog n) in the general case, and run in time O(n^2\polylog n) for the case when all segments are horizontal. In addition, we show that when restricted to translations that are only vertical, the Hausdorff distance between two sets of horizontal segments can be computed in time roughly O(n^{3/2}{\sl polylog}n). These algorithms form significant improvements over the general algorithm of Chew et al. that takes time $O(n^4 \log^2 n)$. In the second part of this paper we address the problem of matching polygonal chains. We study the well known \Frd, and present the first algorithm for computing the \Frd under general translations. Our methods also yield algorithms for computing a generalization of the \Fr distance, and we also present a simple approximation algorithm for the \Frd that runs in time O(n^2\polylog n).Comment: To appear in the 12 ACM Symposium on Discrete Algorithms, Jan 200

Applied Similarity Problems Using Frechet Distance

In the first part of this thesis, we consider an instance of Frechet distance problem in which the speed of traversal along each segment of the curves is restricted to be within a specfied range. This setting is more realistic than the classical Frechet distance setting, specially in GIS applications. We also study this problem in the setting where the polygonal curves are inside a simple polygon. In the second part of this thesis, we present a data structure, called the free-space map, that enables us to solve several variants of the Frechet distance problem efficiently. Our data structure encapsulates all the information available in the free-space diagram, yet it is capable of answering more general type of queries efficiently. Given that the free-space map has the same size and construction time as the standard free-space diagram, it can be viewed as a powerful alternative to it. As part of the results in Part II of the thesis, we exploit the free-space map to improve the long-standing bound for computing the partial Frechet distance and obtain improved algorithms for computing the Frechet distance between two closed curves, and the so-called minimum/maximum walk problem. We also improve the map matching algorithm for the case when the map is a directed acyclic graph. As the last part of this thesis, given a point set S and a polygonal curve P in R^d, we study the problem of finding a polygonal curve Q through S, which has a minimum Frechet distance to P. Furthermore, if the problem requires that curve Q visits every point in S, we show it is NP-complete.Comment: arXiv admin note: text overlap with arXiv:1003.0460 by other author