1,176 research outputs found
Approximate Minimum Diameter
We study the minimum diameter problem for a set of inexact points. By
inexact, we mean that the precise location of the points is not known. Instead,
the location of each point is restricted to a contineus region (\impre model)
or a finite set of points (\indec model). Given a set of inexact points in
one of \impre or \indec models, we wish to provide a lower-bound on the
diameter of the real points.
In the first part of the paper, we focus on \indec model. We present an
time
approximation algorithm of factor for finding minimum diameter
of a set of points in dimensions. This improves the previously proposed
algorithms for this problem substantially.
Next, we consider the problem in \impre model. In -dimensional space, we
propose a polynomial time -approximation algorithm. In addition, for
, we define the notion of -separability and use our algorithm for
\indec model to obtain -approximation algorithm for a set of
-separable regions in time
Preprocessing Imprecise Points for Delaunay Triangulation: Simplified and Extended
Suppose we want to compute the Delaunay triangulation of a set P whose points are restricted to a collection R of input regions known in advance. Building on recent work by Löffler and Snoeyink, we show how to leverage our knowledge of R for faster Delaunay computation. Our approach needs no fancy machinery and optimally handles a wide variety of inputs, e.g., overlapping disks of different sizes and fat regions. Keywords: Delaunay triangulation - Data imprecision - Quadtree
Triangulating the Square and Squaring the Triangle: Quadtrees and Delaunay Triangulations are Equivalent
We show that Delaunay triangulations and compressed quadtrees are equivalent
structures. More precisely, we give two algorithms: the first computes a
compressed quadtree for a planar point set, given the Delaunay triangulation;
the second finds the Delaunay triangulation, given a compressed quadtree. Both
algorithms run in deterministic linear time on a pointer machine. Our work
builds on and extends previous results by Krznaric and Levcopolous and Buchin
and Mulzer. Our main tool for the second algorithm is the well-separated pair
decomposition(WSPD), a structure that has been used previously to find
Euclidean minimum spanning trees in higher dimensions (Eppstein). We show that
knowing the WSPD (and a quadtree) suffices to compute a planar Euclidean
minimum spanning tree (EMST) in linear time. With the EMST at hand, we can find
the Delaunay triangulation in linear time.
As a corollary, we obtain deterministic versions of many previous algorithms
related to Delaunay triangulations, such as splitting planar Delaunay
triangulations, preprocessing imprecise points for faster Delaunay computation,
and transdichotomous Delaunay triangulations.Comment: 37 pages, 13 figures, full version of a paper that appeared in SODA
201
Online Searching with an Autonomous Robot
We discuss online strategies for visibility-based searching for an object
hidden behind a corner, using Kurt3D, a real autonomous mobile robot. This task
is closely related to a number of well-studied problems. Our robot uses a
three-dimensional laser scanner in a stop, scan, plan, go fashion for building
a virtual three-dimensional environment. Besides planning trajectories and
avoiding obstacles, Kurt3D is capable of identifying objects like a chair. We
derive a practically useful and asymptotically optimal strategy that guarantees
a competitive ratio of 2, which differs remarkably from the well-studied
scenario without the need of stopping for surveying the environment. Our
strategy is used by Kurt3D, documented in a separate video.Comment: 16 pages, 8 figures, 12 photographs, 1 table, Latex, submitted for
publicatio
Distant Representatives for Rectangles in the Plane
The input to the distant representatives problem is a set of n objects in the plane and the goal is to find a representative point from each object while maximizing the distance between the closest pair of points. When the objects are axis-aligned rectangles, we give polynomial time constant-factor approximation algorithms for the L?, L?, and L_? distance measures. We also prove lower bounds on the approximation factors that can be achieved in polynomial time (unless P = NP)
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