17,997 research outputs found
Computing the Largest Empty Rectangle
We consider the following problem: Given a rectangle containing N points, find the largest area subrectangle with sides parallel to those of the original rectangle which contains none of the given points. If the rectangle is a piece of fabric or sheet metal and the points are flaws, this problem is finding the largest-area rectangular piece which can be salvaged. A previously known result [13] takes worst-case and expected time. This paper presents an time, space algorithm to solve this problem. It uses a divide-and-conquer approach similar to the ones used by Bentley [1] and introduces a new notion of Voronoi diagram along with a method for efficient computation of certain functions over paths of a tree
Minimum Convex Partitions and Maximum Empty Polytopes
Let be a set of points in . A Steiner convex partition
is a tiling of with empty convex bodies. For every integer ,
we show that admits a Steiner convex partition with at most 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 . 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 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 .
Here we give a -approximation algorithm for computing the
maximum volume of an empty convex body amidst given points in the
-dimensional unit box .Comment: 16 pages, 4 figures; revised write-up with some running times
improve
Querying for the Largest Empty Geometric Object in a Desired Location
We study new types of geometric query problems defined as follows: given a
geometric set , preprocess it such that given a query point , the
location of the largest circle that does not contain any member of , but
contains can be reported efficiently. The geometric sets we consider for
are boundaries of convex and simple polygons, and point sets. While we
primarily focus on circles as the desired shape, we also briefly discuss empty
rectangles in the context of point sets.Comment: This version is a significant update of our earlier submission
arXiv:1004.0558v1. Apart from new variants studied in Sections 3 and 4, the
results have been improved in Section 5.Please note that the change in title
and abstract indicate that we have expanded the scope of the problems we
stud
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