26,763 research outputs found
Optimal randomized incremental construction for guaranteed logarithmic planar point location
Given a planar map of segments in which we wish to efficiently locate
points, we present the first randomized incremental construction of the
well-known trapezoidal-map search-structure that only requires expected preprocessing time while deterministically guaranteeing worst-case
linear storage space and worst-case logarithmic query time. This settles a long
standing open problem; the best previously known construction time of such a
structure, which is based on a directed acyclic graph, so-called the history
DAG, and with the above worst-case space and query-time guarantees, was
expected . The result is based on a deeper understanding of the
structure of the history DAG, its depth in relation to the length of its
longest search path, as well as its correspondence to the trapezoidal search
tree. Our results immediately extend to planar maps induced by finite
collections of pairwise interior disjoint well-behaved curves.Comment: The article significantly extends the theoretical aspects of the work
presented in http://arxiv.org/abs/1205.543
Improved Implementation of Point Location in General Two-Dimensional Subdivisions
We present a major revamp of the point-location data structure for general
two-dimensional subdivisions via randomized incremental construction,
implemented in CGAL, the Computational Geometry Algorithms Library. We can now
guarantee that the constructed directed acyclic graph G is of linear size and
provides logarithmic query time. Via the construction of the Voronoi diagram
for a given point set S of size n, this also enables nearest-neighbor queries
in guaranteed O(log n) time. Another major innovation is the support of general
unbounded subdivisions as well as subdivisions of two-dimensional parametric
surfaces such as spheres, tori, cylinders. The implementation is exact,
complete, and general, i.e., it can also handle non-linear subdivisions. Like
the previous version, the data structure supports modifications of the
subdivision, such as insertions and deletions of edges, after the initial
preprocessing. A major challenge is to retain the expected O(n log n)
preprocessing time while providing the above (deterministic) space and
query-time guarantees. We describe an efficient preprocessing algorithm, which
explicitly verifies the length L of the longest query path in O(n log n) time.
However, instead of using L, our implementation is based on the depth D of G.
Although we prove that the worst case ratio of D and L is Theta(n/log n), we
conjecture, based on our experimental results, that this solution achieves
expected O(n log n) preprocessing time.Comment: 21 page
Automatic Markov Chain Monte Carlo Procedures for Sampling from Multivariate Distributions
Generating samples from multivariate distributions efficiently is an important task in Monte Carlo integration and many other stochastic simulation problems. Markov chain Monte Carlo has been shown to be very efficient compared to "conventional methods", especially when many dimensions are involved. In this article we propose a Hit-and-Run sampler in combination with the Ratio-of-Uniforms method. We show that it is well suited for an algorithm to generate points from quite arbitrary distributions, which include all log-concave distributions. The algorithm works automatically in the sense that only the mode (or an approximation of it) and an oracle is required, i.e., a subroutine that returns the value of the density function at any point x. We show that the number of evaluations of the density increases slowly with dimension. (author's abstract)Series: Preprint Series / Department of Applied Statistics and Data Processin
The Unreasonable Success of Local Search: Geometric Optimization
What is the effectiveness of local search algorithms for geometric problems
in the plane? We prove that local search with neighborhoods of magnitude
is an approximation scheme for the following problems in the
Euclidian plane: TSP with random inputs, Steiner tree with random inputs,
facility location (with worst case inputs), and bicriteria -median (also
with worst case inputs). The randomness assumption is necessary for TSP
Cell-probe Lower Bounds for Dynamic Problems via a New Communication Model
In this paper, we develop a new communication model to prove a data structure
lower bound for the dynamic interval union problem. The problem is to maintain
a multiset of intervals over with integer coordinates,
supporting the following operations:
- insert(a, b): add an interval to , provided that
and are integers in ;
- delete(a, b): delete a (previously inserted) interval from
;
- query(): return the total length of the union of all intervals in
.
It is related to the two-dimensional case of Klee's measure problem. We prove
that there is a distribution over sequences of operations with
insertions and deletions, and queries, for which any data
structure with any constant error probability requires time
in expectation. Interestingly, we use the sparse set disjointness protocol of
H\aa{}stad and Wigderson [ToC'07] to speed up a reduction from a new kind of
nondeterministic communication games, for which we prove lower bounds.
For applications, we prove lower bounds for several dynamic graph problems by
reducing them from dynamic interval union
The critical probability for random Voronoi percolation in the plane is 1/2
We study percolation in the following random environment: let be a
Poisson process of constant intensity in the plane, and form the Voronoi
tessellation of the plane with respect to . Colour each Voronoi cell black
with probability , independently of the other cells. We show that the
critical probability is 1/2. More precisely, if then the union of the
black cells contains an infinite component with probability 1, while if
then the distribution of the size of the component of black cells containing a
given point decays exponentially. These results are analogous to Kesten's
results for bond percolation in the square lattice.
The result corresponding to Harris' Theorem for bond percolation in the
square lattice is known: Zvavitch noted that one of the many proofs of this
result can easily be adapted to the random Voronoi setting. For Kesten's
results, none of the existing proofs seems to adapt. The methods used here also
give a new and very simple proof of Kesten's Theorem for the square lattice; we
hope they will be applicable in other contexts as well.Comment: 55 pages, 20 figures; minor changes; to appear in Probability Theory
and Related Field
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