33,690 research outputs found
Orthogonal Range Reporting and Rectangle Stabbing for Fat Rectangles
In this paper we study two geometric data structure problems in the special
case when input objects or queries are fat rectangles. We show that in this
case a significant improvement compared to the general case can be achieved.
We describe data structures that answer two- and three-dimensional orthogonal
range reporting queries in the case when the query range is a \emph{fat}
rectangle. Our two-dimensional data structure uses words and supports
queries in time, where is the number of points in the
data structure, is the size of the universe and is the number of points
in the query range. Our three-dimensional data structure needs
words of space and answers queries in time. We also consider the rectangle stabbing problem on a set of
three-dimensional fat rectangles. Our data structure uses space and
answers stabbing queries in time.Comment: extended version of a WADS'19 pape
Down the Rabbit Hole: Robust Proximity Search and Density Estimation in Sublinear Space
For a set of points in , and parameters and \eps, we present
a data structure that answers (1+\eps,k)-\ANN queries in logarithmic time.
Surprisingly, the space used by the data-structure is \Otilde (n /k); that
is, the space used is sublinear in the input size if is sufficiently large.
Our approach provides a novel way to summarize geometric data, such that
meaningful proximity queries on the data can be carried out using this sketch.
Using this, we provide a sublinear space data-structure that can estimate the
density of a point set under various measures, including:
\begin{inparaenum}[(i)]
\item sum of distances of closest points to the query point, and
\item sum of squared distances of closest points to the query point.
\end{inparaenum}
Our approach generalizes to other distance based estimation of densities of
similar flavor. We also study the problem of approximating some of these
quantities when using sampling. In particular, we show that a sample of size
\Otilde (n /k) is sufficient, in some restricted cases, to estimate the above
quantities. Remarkably, the sample size has only linear dependency on the
dimension
On R-trees with low query complexity
The R-tree is a well-known bounding-volume hierarchy that
is suitable for storing geometric data on secondary memory. Unfortu-
nately, no good analysis of its query time exists. We describe a new algo-
rithm to construct an R-tree for a set of planar objects that has provably
good query complexity for point location queries and range queries with
ranges of small width. For certain important special cases, our bounds
are optimal. We also show how to update the structure dynamically, and
we generalize our results to higher-dimensional spaces
Map matching queries on realistic input graphs under the Fr\'echet distance
Map matching is a common preprocessing step for analysing vehicle
trajectories. In the theory community, the most popular approach for map
matching is to compute a path on the road network that is the most spatially
similar to the trajectory, where spatial similarity is measured using the
Fr\'echet distance. A shortcoming of existing map matching algorithms under the
Fr\'echet distance is that every time a trajectory is matched, the entire road
network needs to be reprocessed from scratch. An open problem is whether one
can preprocess the road network into a data structure, so that map matching
queries can be answered in sublinear time.
In this paper, we investigate map matching queries under the Fr\'echet
distance. We provide a negative result for geometric planar graphs. We show
that, unless SETH fails, there is no data structure that can be constructed in
polynomial time that answers map matching queries in query
time for any , where and are the complexities of the
geometric planar graph and the query trajectory, respectively. We provide a
positive result for realistic input graphs, which we regard as the main result
of this paper. We show that for -packed graphs, one can construct a data
structure of size that can answer -approximate
map matching queries in time, where hides lower-order factors and dependence of .Comment: To appear in SODA 202
On Range Summary Queries
We study the query version of the approximate heavy hitter and quantile problems. In the former problem, the input is a parameter ? and a set P of n points in ?^d where each point is assigned a color from a set C, and the goal is to build a structure such that given any geometric range ?, we can efficiently find a list of approximate heavy hitters in ??P, i.e., colors that appear at least ? |??P| times in ??P, as well as their frequencies with an additive error of ? |??P|. In the latter problem, each point is assigned a weight from a totally ordered universe and the query must output a sequence S of 1+1/? weights such that the i-th weight in S has approximate rank i?|??P|, meaning, rank i?|??P| up to an additive error of ?|??P|. Previously, optimal results were only known in 1D [Wei and Yi, 2011] but a few sub-optimal methods were available in higher dimensions [Peyman Afshani and Zhewei Wei, 2017; Pankaj K. Agarwal et al., 2012].
We study the problems for two important classes of geometric ranges: 3D halfspace and 3D dominance queries. It is known that many other important queries can be reduced to these two, e.g., 1D interval stabbing or interval containment, 2D three-sided queries, 2D circular as well as 2D k-nearest neighbors queries. We consider the real RAM model of computation where integer registers of size w bits, w = ?(log n), are also available. For dominance queries, we show optimal solutions for both heavy hitter and quantile problems: using linear space, we can answer both queries in time O(log n + 1/?). Note that as the output size is 1/?, after investing the initial O(log n) searching time, our structure takes on average O(1) time to find a heavy hitter or a quantile! For more general halfspace heavy hitter queries, the same optimal query time can be achieved by increasing the space by an extra log_w(1/?) (resp. log log_w(1/?)) factor in 3D (resp. 2D). By spending extra log^O(1)(1/?) factors in both time and space, we can also support quantile queries.
We remark that it is hopeless to achieve a similar query bound for dimensions 4 or higher unless significant advances are made in the data structure side of theory of geometric approximations
Multidimensional indexing structure for use with linear optimization queries
Linear optimization queries, which usually arise in various decision support and resource planning applications, are queries that retrieve top N data records (where N is an integer greater than zero) which satisfy a specific optimization criterion. The optimization criterion is to either maximize or minimize a linear equation. The coefficients of the linear equation are given at query time. Methods and apparatus are disclosed for constructing, maintaining and utilizing a multidimensional indexing structure of database records to improve the execution speed of linear optimization queries. Database records with numerical attributes are organized into a number of layers and each layer represents a geometric structure called convex hull. Such linear optimization queries are processed by searching from the outer-most layer of this multi-layer indexing structure inwards. At least one record per layer will satisfy the query criterion and the number of layers needed to be searched depends on the spatial distribution of records, the query-issued linear coefficients, and N, the number of records to be returned. When N is small compared to the total size of the database, answering the query typically requires searching only a small fraction of all relevant records, resulting in a tremendous speedup as compared to linearly scanning the entire dataset
Algorithms for distance problems in planar complexes of global nonpositive curvature
CAT(0) metric spaces and hyperbolic spaces play an important role in
combinatorial and geometric group theory. In this paper, we present efficient
algorithms for distance problems in CAT(0) planar complexes. First of all, we
present an algorithm for answering single-point distance queries in a CAT(0)
planar complex. Namely, we show that for a CAT(0) planar complex K with n
vertices, one can construct in O(n^2 log n) time a data structure D of size
O(n^2) so that, given a point x in K, the shortest path gamma(x,y) between x
and the query point y can be computed in linear time. Our second algorithm
computes the convex hull of a finite set of points in a CAT(0) planar complex.
This algorithm is based on Toussaint's algorithm for computing the convex hull
of a finite set of points in a simple polygon and it constructs the convex hull
of a set of k points in O(n^2 log n + nk log k) time, using a data structure of
size O(n^2 + k)
Orthogonal Point Location and Rectangle Stabbing Queries in 3-d
In this work, we present a collection of new results on two fundamental problems in geometric data structures: orthogonal point location and rectangle stabbing.
- Orthogonal point location. We give the first linear-space data structure that supports 3-d point location queries on n disjoint axis-aligned boxes with optimal O(log n) query time in the (arithmetic) pointer machine model. This improves the previous O(log^{3/2} n) bound of Rahul [SODA 2015]. We similarly obtain the first linear-space data structure in the I/O model with optimal query cost, and also the first linear-space data structure in the word RAM model with sub-logarithmic query time.
- Rectangle stabbing. We give the first linear-space data structure that supports 3-d 4-sided and 5-sided rectangle stabbing queries in optimal O(log_wn+k) time in the word RAM model. We similarly obtain the first optimal data structure for the closely related problem of 2-d top-k rectangle stabbing in the word RAM model, and also improved results for 3-d 6-sided rectangle stabbing.
For point location, our solution is simpler than previous methods, and is based on an interesting variant of the van Emde Boas recursion, applied in a round-robin fashion over the dimensions, combined with bit-packing techniques. For rectangle stabbing, our solution is a variant of Alstrup, Brodal, and Rauhe\u27s grid-based recursive technique (FOCS 2000), combined with a number of new ideas
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