175 research outputs found
Partitioning Regular Polygons into Circular Pieces I: Convex Partitions
We explore an instance of the question of partitioning a polygon into pieces,
each of which is as ``circular'' as possible, in the sense of having an aspect
ratio close to 1. The aspect ratio of a polygon is the ratio of the diameters
of the smallest circumscribing circle to the largest inscribed disk. The
problem is rich even for partitioning regular polygons into convex pieces, the
focus of this paper. We show that the optimal (most circular) partition for an
equilateral triangle has an infinite number of pieces, with the lower bound
approachable to any accuracy desired by a particular finite partition. For
pentagons and all regular k-gons, k > 5, the unpartitioned polygon is already
optimal. The square presents an interesting intermediate case. Here the
one-piece partition is not optimal, but nor is the trivial lower bound
approachable. We narrow the optimal ratio to an aspect-ratio gap of 0.01082
with several somewhat intricate partitions.Comment: 21 pages, 25 figure
Fat Polygonal Partitions with Applications to Visualization and Embeddings
Let be a rooted and weighted tree, where the weight of any node
is equal to the sum of the weights of its children. The popular Treemap
algorithm visualizes such a tree as a hierarchical partition of a square into
rectangles, where the area of the rectangle corresponding to any node in
is equal to the weight of that node. The aspect ratio of the
rectangles in such a rectangular partition necessarily depends on the weights
and can become arbitrarily high.
We introduce a new hierarchical partition scheme, called a polygonal
partition, which uses convex polygons rather than just rectangles. We present
two methods for constructing polygonal partitions, both having guarantees on
the worst-case aspect ratio of the constructed polygons; in particular, both
methods guarantee a bound on the aspect ratio that is independent of the
weights of the nodes.
We also consider rectangular partitions with slack, where the areas of the
rectangles may differ slightly from the weights of the corresponding nodes. We
show that this makes it possible to obtain partitions with constant aspect
ratio. This result generalizes to hyper-rectangular partitions in
. We use these partitions with slack for embedding ultrametrics
into -dimensional Euclidean space: we give a -approximation algorithm for embedding -point ultrametrics
into with minimum distortion, where denotes the spread
of the metric, i.e., the ratio between the largest and the smallest distance
between two points. The previously best-known approximation ratio for this
problem was polynomial in . This is the first algorithm for embedding a
non-trivial family of weighted-graph metrics into a space of constant dimension
that achieves polylogarithmic approximation ratio.Comment: 26 page
Binary Space Partitions for Fat Rectangles
This is the published version. Copyright © 2000 Society for Industrial and Applied Mathematic
Constructing Binary Space Partitions for Orthogonal Rectangles in Practice
The original publication is available at www.springerlink.comIn this paper, we develop a simple technique for constructing
a I3inary Space Partition (nSP) for a set of orthogonal rectangles in IR3.
OUf algorithm has the novel feature that it tunes its performance to the
geometric properties of the rectangles, e.g., their aspect ratios.
"Fe have implemented our algorithm and tested its performance on real
data scti). V\.Tc have also systematically compared the performance of our
algorithm with that of other techniques presented in the literature. Our
studies show that our algorithm constructs nsps of near-linear size and
small height in practice, has fast running times, and answers queries
efficiently. It is a method of choice for constructing BSPs for orthogonal
rectangles
Decomposing and packing polygons / Dania el-Khechen.
In this thesis, we study three different problems in the field of computational geometry: the partitioning of a simple polygon into two congruent components, the partitioning of squares and rectangles into equal area components while minimizing the perimeter of the cuts, and the packing of the maximum number of squares in an orthogonal polygon. To solve the first problem, we present three polynomial time algorithms which given a simple polygon P partitions it, if possible, into two congruent and possibly nonsimple components P 1 and P 2 : an O ( n 2 log n ) time algorithm for properly congruent components and an O ( n 3 ) time algorithm for mirror congruent components. In our analysis of the second problem, we experimentally find new bounds on the optimal partitions of squares and rectangles into equal area components. The visualization of the best determined solutions allows us to conjecture some characteristics of a class of optimal solutions. Finally, for the third problem, we present three linear time algorithms for packing the maximum number of unit squares in three subclasses of orthogonal polygons: the staircase polygons, the pyramids and Manhattan skyline polygons. We also study a special case of the problem where the given orthogonal polygon has vertices with integer coordinates and the squares to pack are (2 {604} 2) squares. We model the latter problem with a binary integer program and we develop a system that produces and visualizes optimal solutions. The observation of such solutions aided us in proving some characteristics of a class of optimal solutions
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
A Framework for Index Bulk Loading and Dynamization
In this paper we investigate automated methods for externalizing
internal memory data structures. We consider a class of balanced trees that we
call weight-balanced partitioning trees (or wp-trees) for indexing a set of points
in Rd. Well-known examples of wp-trees include fed-trees, BBD-trees, pseudo
quad trees, and BAR trees. These trees are defined with fixed degree and are
thus suited for internal memory implementations. Given an efficient wp-tree
construction algorithm, we present a general framework for automatically obtaining
a new dynamic external data structure. Using this framework together
with a new general construction (bulk loading) technique of independent interest,
we obtain data structures with guaranteed good update performance in
terms of I /O transfers. Our approach gives considerably improved construction
and update I/O bounds of e.g. fed-trees and BBD-trees
- …