9,205 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
Partitioning Graph Drawings and Triangulated Simple Polygons into Greedily Routable Regions
A greedily routable region (GRR) is a closed subset of , in
which each destination point can be reached from each starting point by
choosing the direction with maximum reduction of the distance to the
destination in each point of the path.
Recently, Tan and Kermarrec proposed a geographic routing protocol for dense
wireless sensor networks based on decomposing the network area into a small
number of interior-disjoint GRRs. They showed that minimum decomposition is
NP-hard for polygons with holes.
We consider minimum GRR decomposition for plane straight-line drawings of
graphs. Here, GRRs coincide with self-approaching drawings of trees, a drawing
style which has become a popular research topic in graph drawing. We show that
minimum decomposition is still NP-hard for graphs with cycles, but can be
solved optimally for trees in polynomial time. Additionally, we give a
2-approximation for simple polygons, if a given triangulation has to be
respected.Comment: full version of a paper appearing in ISAAC 201
Packing Plane Perfect Matchings into a Point Set
Given a set of points in the plane, where is even, we consider
the following question: How many plane perfect matchings can be packed into
? We prove that at least plane perfect matchings
can be packed into any point set . For some special configurations of point
sets, we give the exact answer. We also consider some extensions of this
problem
CiNCT: Compression and retrieval for massive vehicular trajectories via relative movement labeling
In this paper, we present a compressed data structure for moving object
trajectories in a road network, which are represented as sequences of road
edges. Unlike existing compression methods for trajectories in a network, our
method supports pattern matching and decompression from an arbitrary position
while retaining a high compressibility with theoretical guarantees.
Specifically, our method is based on FM-index, a fast and compact data
structure for pattern matching. To enhance the compression, we incorporate the
sparsity of road networks into the data structure. In particular, we present
the novel concepts of relative movement labeling and PseudoRank, each
contributing to significant reductions in data size and query processing time.
Our theoretical analysis and experimental studies reveal the advantages of our
proposed method as compared to existing trajectory compression methods and
FM-index variants
Algorithms for Subpath Convex Hull Queries and Ray-Shooting Among Segments
In this paper, we first consider the subpath convex hull query problem: Given a simple path ? of n vertices, preprocess it so that the convex hull of any query subpath of ? can be quickly obtained. Previously, Guibas, Hershberger, and Snoeyink [SODA 90\u27] proposed a data structure of O(n) space and O(log n log log n) query time; reducing the query time to O(log n) increases the space to O(nlog log n). We present an improved result that uses O(n) space while achieving O(log n) query time. Like the previous work, our query algorithm returns a compact interval tree representing the convex hull so that standard binary-search-based queries on the hull can be performed in O(log n) time each. Our new result leads to improvements for several other problems.
In particular, with the help of the above result, we present new algorithms for the ray-shooting problem among segments. Given a set of n (possibly intersecting) line segments in the plane, preprocess it so that the first segment hit by a query ray can be quickly found. We give a data structure of O(n log n) space that can answer each query in (?n log n) time. If the segments are nonintersecting or if the segments are lines, then the space can be reduced to O(n). All these are classical problems that have been studied extensively. Previously data structures of O?(?n) query time were known in early 1990s; nearly no progress has been made for over two decades. For all problems, our results provide improvements by reducing the space of the data structures by at least a logarithmic factor while the preprocessing and query times are the same as before or even better
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