Suneeta Ramaswami

Efficient Approximation Algorithms for Tiling and Packing Problems with Rectangles

We provide improved approximation algorithms for several rectangle tiling and packing problems (RTILE, DRTILE, and d-RPACK) studied in the literature. Most of our algorithms are highly efficient since their running times are near-linear in the sparse input size rather than in the domain size. In addition, we improve the best known approximation ratios.

Converting triangulations to quadrangulations

Abstract We study the problem of converting triangulated domains to quadrangulations, under a variety of constraints. We obtain a variety of characterizations for when a triangulation (of some structure such as a polygon, set of points, line segments or planar subdivision) admits a quadrangulation without the use of Steiner points, or with a bounded number of Steiner points. We also investigate the effect of demanding that the Steiner points be added in the interior or exterior of a triangulated simple polygon and propose efficient algorithms for accomplishing these tasks. For example, we give a linear-time method that quadrangulates a triangulated simple polygon with the minimum number of outer Steiner points required for that triangulation. We show that this minimum can be at most ⌊ n 3 ⌋ , and that there exist polygons that require this many such Steiner points. We also show that a triangulated simple n-gon may be quadrangulated with at most ⌊ n 4 ⌋ Steiner points inside the polygon and at most one outside. This algorithm also allows us to obtain, in linear time, quadrangulations from general triangulated domains (such as triangulations of polygons with holes, a set of points or line segments) with a bounded number of Steiner points.

Efficient computation of location depth contours by methods of computational geometry

The concept of location depth was introduced as a way to extend the univariate notion of ranking to a bivariate configuration of data points. It has been used successfully for robust estimation, hypothesis testing, and graphical display. The depth contours form a collection of nested polygons, and the center of the deepest contour is called the Tukey median. The only available implemented algorithms for the depth contours and the Tukey median are slow, which limits their usefulness. In this paper we describe an optimal algorithm which computes all bivariate depth contours in O(n2) time and space, using topological sweep of the dual arrangement of lines. Once these contours are known, the location depth of any point can be computed in O(log2n) time with no additional preprocessing or in O(log n) time after O(n2) preprocessing. We provide fast implementations of these algorithms to allow their use in everyday statistical practice.

Computing Constrained Minimum-Width Annuli of Point Sets

We study the problem of determining whether a manufactured disc of certain radius r is within tolerance. More precisely, we present algorithms that, given a set of n probe points on the surface of the manufactured object, compute the thinnest annulus whose outer (or inner, or median) radius is r and that contains all the probe points. Our algorithms run in O(n log n) time.

Computing constrained minimum-width annuli of point sets

We study the problem of determining whether a manufactured disk of certain radius r is within tolerance. More precisely, we present algorithms that, given a set of n probe points on the surface of the manufactured object, compute the thinnest annulus whose outer (or inner, or median) radius is r and that contains all the probe points. Our algorithms run in O(nlogn) time.

Quadrilateral meshes with bounded minimum angle

We present an algorithm that constructs a strictly convex quadrilateral mesh for a simple polygonal region in which no newly created angle is smaller than D(18.43) (=arctan(1/3)). This is the first known result on quadrilateral mesh generation with a provable guarantee on the minimum angle.

An O(n log n)-time algorithm for the restriction scaffold assignment problem.

The restriction scaffold assignment problem takes as input two finite point sets S and T (with S containing more points than T ) and establishes a correspondence between points in S and points in T , such that each point in S maps to exactly one point in T and each point in T maps to at least one point in S. An algorithm is presented that finds a minimum-cost solution for this problem in O(n log n) time, provided that the points in S and T are restricted to lie on a line and the cost function delta is the L(1) metric. This algorithm runs in linear time, if S and T are presorted. This improves the previously best-known O(n (2))-time algorithm for this problem.

Optimal parallel randomized algorithms for the Voronoi diagram of line segments in the plane and related problems

In this paper, we present an optimal parallel randomized algorithm for the Voronoi diagram of a set of <italic>n</italic> non-intersecting (except possibly at endpoints) line segments in the plane. Our algorithm runs in <italic>O</italic>(log<italic>n</italic>) time with very high probability and uses <italic>O</italic>(<italic>n</italic>) processors on a CRCW PRAM. This algorithm is optimal in terms of <italic>P.T</italic> bounds since the sequential time bound for this problem is &OHgr;(<italic>n</italic>log<italic>n</italic>). Our algorithm improves by an <italic>O</italic>(log<italic>n</italic>) factor the previously best known deterministic parallel algorithm which runs in <italic>O</italic>(log<supscrpt>2</supscrpt><italic>n</italic>) time using <italic>O</italic>(<italic>n</italic>) processors. We obtain this result by using random sampling at “two stages” of our algorithm and using efficient randomized search techniques. This technique gives a direct optimal algorithm for the Voronoi diagram of points as well (all other optimal parallel algorithms for this problem use reduction from the 3-d convex hull construction).

Efficient Many-To-Many Point Matching in One Dimension

Let S and T be two sets of points with total cardinality n. The minimum-cost many-to-many matching problem matches each point in S to at least one point in T and each point in T to at least one point in S, such that sum of the matching costs is minimized. Here we examine the special case where both S and T lie on the line and the cost of matching s ∈S to t ∈T is equal to the distance between s and t. In this context, we provide an algorithm that determines a minimum-cost many-to-many matching in O(n log n) time, improving the previous best time complexity of O(n2) for the same problem.

CONSTRAINED QUADRILATERAL MESHES OF BOUNDED SIZE

We introduce a new algorithm to convert triangular meshes of polygonal regions, with or without holes, into strictly convex quadrilateral meshes of small bounded size. Our algorithm includes all vertices of the triangular mesh in the quadrilateral mesh, but may add extra vertices (called Steiner points). We show that if the input triangular mesh has t triangles, our algorithm produces a mesh with at most quadrilaterals by adding at most t+2 Steiner points, one of which may be placed outside the triangular mesh domain. We also describe an extension of our algorithm to convert constrained triangular meshes into constrained quadrilateral ones. We show that if the input constrained triangular mesh has t triangles and its dual graph has h connected components, the resulting constrained quadrilateral mesh has at most quadrilaterals and at most t+3h Steiner points, one of which may be placed outside the triangular mesh domain. Examples of meshes generated by our algorithm, and an evaluation of the quality of these meshes with respect to a quadrilateral shape quality criterion are presented as well.