993 research outputs found
The Skip Quadtree: A Simple Dynamic Data Structure for Multidimensional Data
We present a new multi-dimensional data structure, which we call the skip
quadtree (for point data in R^2) or the skip octree (for point data in R^d,
with constant d>2). Our data structure combines the best features of two
well-known data structures, in that it has the well-defined "box"-shaped
regions of region quadtrees and the logarithmic-height search and update
hierarchical structure of skip lists. Indeed, the bottom level of our structure
is exactly a region quadtree (or octree for higher dimensional data). We
describe efficient algorithms for inserting and deleting points in a skip
quadtree, as well as fast methods for performing point location and approximate
range queries.Comment: 12 pages, 3 figures. A preliminary version of this paper appeared in
the 21st ACM Symp. Comp. Geom., Pisa, 2005, pp. 296-30
Computational Geometry Column 42
A compendium of thirty previously published open problems in computational
geometry is presented.Comment: 7 pages; 72 reference
On the complexity of range searching among curves
Modern tracking technology has made the collection of large numbers of
densely sampled trajectories of moving objects widely available. We consider a
fundamental problem encountered when analysing such data: Given polygonal
curves in , preprocess into a data structure that answers
queries with a query curve and radius for the curves of that
have \Frechet distance at most to .
We initiate a comprehensive analysis of the space/query-time trade-off for
this data structuring problem. Our lower bounds imply that any data structure
in the pointer model model that achieves query time, where is
the output size, has to use roughly space in
the worst case, even if queries are mere points (for the discrete \Frechet
distance) or line segments (for the continuous \Frechet distance). More
importantly, we show that more complex queries and input curves lead to
additional logarithmic factors in the lower bound. Roughly speaking, the number
of logarithmic factors added is linear in the number of edges added to the
query and input curve complexity. This means that the space/query time
trade-off worsens by an exponential factor of input and query complexity. This
behaviour addresses an open question in the range searching literature: whether
it is possible to avoid the additional logarithmic factors in the space and
query time of a multilevel partition tree. We answer this question negatively.
On the positive side, we show we can build data structures for the \Frechet
distance by using semialgebraic range searching. Our solution for the discrete
\Frechet distance is in line with the lower bound, as the number of levels in
the data structure is , where denotes the maximal number of vertices
of a curve. For the continuous \Frechet distance, the number of levels
increases to
Data Structure Lower Bounds for Document Indexing Problems
We study data structure problems related to document indexing and pattern
matching queries and our main contribution is to show that the pointer machine
model of computation can be extremely useful in proving high and unconditional
lower bounds that cannot be obtained in any other known model of computation
with the current techniques. Often our lower bounds match the known space-query
time trade-off curve and in fact for all the problems considered, there is a
very good and reasonable match between the our lower bounds and the known upper
bounds, at least for some choice of input parameters. The problems that we
consider are set intersection queries (both the reporting variant and the
semi-group counting variant), indexing a set of documents for two-pattern
queries, or forbidden- pattern queries, or queries with wild-cards, and
indexing an input set of gapped-patterns (or two-patterns) to find those
matching a document given at the query time.Comment: Full version of the conference version that appeared at ICALP 2016,
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