1,606 research outputs found
Planar Visibility: Testing and Counting
In this paper we consider query versions of visibility testing and visibility
counting. Let be a set of disjoint line segments in and let
be an element of . Visibility testing is to preprocess so that we can
quickly determine if is visible from a query point . Visibility counting
involves preprocessing so that one can quickly estimate the number of
segments in visible from a query point .
We present several data structures for the two query problems. The structures
build upon a result by O'Rourke and Suri (1984) who showed that the subset,
, of that is weakly visible from a segment can be
represented as the union of a set, , of triangles, even though
the complexity of can be . We define a variant of their
covering, give efficient output-sensitive algorithms for computing it, and
prove additional properties needed to obtain approximation bounds. Some of our
bounds rely on a new combinatorial result that relates the number of segments
of visible from a point to the number of triangles in that contain .Comment: 22 page
Memory-Constrained Algorithms for Simple Polygons
A constant-workspace algorithm has read-only access to an input array and may
use only O(1) additional words of bits, where is the size of
the input. We assume that a simple -gon is given by the ordered sequence of
its vertices. We show that we can find a triangulation of a plane straight-line
graph in time. We also consider preprocessing a simple polygon for
shortest path queries when the space constraint is relaxed to allow words
of working space. After a preprocessing of time, we are able to solve
shortest path queries between any two points inside the polygon in
time.Comment: Preprint appeared in EuroCG 201
Data Structures for Halfplane Proximity Queries and Incremental Voronoi Diagrams
We consider preprocessing a set of points in convex position in the
plane into a data structure supporting queries of the following form: given a
point and a directed line in the plane, report the point of that
is farthest from (or, alternatively, nearest to) the point among all points
to the left of line . We present two data structures for this problem.
The first data structure uses space and preprocessing
time, and answers queries in time, for any . The second data structure uses space and
polynomial preprocessing time, and answers queries in time. These
are the first solutions to the problem with query time and
space.
The second data structure uses a new representation of nearest- and
farthest-point Voronoi diagrams of points in convex position. This
representation supports the insertion of new points in clockwise order using
only amortized pointer changes, in addition to -time
point-location queries, even though every such update may make
combinatorial changes to the Voronoi diagram. This data structure is the first
demonstration that deterministically and incrementally constructed Voronoi
diagrams can be maintained in amortized pointer changes per operation
while keeping -time point-location queries.Comment: 17 pages, 6 figures. Various small improvements. To appear in
Algorithmic
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
Entropy, Triangulation, and Point Location in Planar Subdivisions
A data structure is presented for point location in connected planar
subdivisions when the distribution of queries is known in advance. The data
structure has an expected query time that is within a constant factor of
optimal. More specifically, an algorithm is presented that preprocesses a
connected planar subdivision G of size n and a query distribution D to produce
a point location data structure for G. The expected number of point-line
comparisons performed by this data structure, when the queries are distributed
according to D, is H + O(H^{2/3}+1) where H=H(G,D) is a lower bound on the
expected number of point-line comparisons performed by any linear decision tree
for point location in G under the query distribution D. The preprocessing
algorithm runs in O(n log n) time and produces a data structure of size O(n).
These results are obtained by creating a Steiner triangulation of G that has
near-minimum entropy.Comment: 19 pages, 4 figures, lots of formula
GeoLens: enabling interactive visual analytics over large-scale, multidimensional geospatial datasets
2015 Spring.Includes bibliographical references.With the rapid increase of scientific data volumes, interactive tools that enable effective visual representation for scientists are needed. This is critical when scientists are manipulating voluminous datasets and especially when they need to explore datasets interactively to develop their hypotheses. In this paper, we present an interactive visual analytics framework, GeoLens. GeoLens provides fast and expressive interactions with voluminous geospatial datasets. We provide an expressive visual query evaluation scheme to support advanced interactive visual analytics technique, such as brushing and linking. To achieve this, we designed and developed the geohash based image tile generation algorithm that automatically adjusts the range of data to access based on the minimum acceptable size of the image tile. In addition, we have also designed an autonomous histogram generation algorithm that generates histograms of user-defined data subsets that do not have pre-computed data properties. Using our approach, applications can generate histograms of datasets containing millions of data points with sub-second latency. The work builds on our visual query coordinating scheme that evaluates geospatial query and orchestrates data aggregation in a distributed storage environment while preserving data locality and minimizing data movements. This paper includes empirical benchmarks of our framework encompassing a billion-file dataset published by the National Climactic Data Center
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