24,087 research outputs found
Recommended from our members
Rapid preconditioning of data for accelerating convex hull algorithms
Given a dataset of two-dimensional points in the plane with integer
coordinates, the method proposed reduces a set of n points down to
a set of s points s ≤ n, such that the convex hull on the set of s
points is the same as the convex hull of the original set of n points.
The method is O(n). It helps any convex hull algorithm run faster.
The empirical analysis of a practical case shows a percentage reduction
in points of over 98%, that is reflected as a faster computation with a
speedup factor of at least 4
Simple and Robust Dynamic Two-Dimensional Convex Hull
The convex hull of a data set is the smallest convex set that contains
.
In this work, we present a new data structure for convex hull, that allows
for efficient dynamic updates. In a dynamic convex hull implementation, the
following traits are desirable: (1) algorithms for efficiently answering
queries as to whether a specified point is inside or outside the hull, (2)
adhering to geometric robustness, and (3) algorithmic simplicity.Furthermore, a
specific but well-motivated type of two-dimensional data is rank-based data.
Here, the input is a set of real-valued numbers where for any number its rank is its index in 's sorted order. Each value in can be mapped
to a point to obtain a two-dimensional point set. In this work,
we give an efficient, geometrically robust, dynamic convex hull algorithm, that
facilitates queries to whether a point is internal. Furthermore, our
construction can be used to efficiently update the convex hull of rank-ordered
data, when the real-valued point set is subject to insertions and deletions.
Our improved solution is based on an algorithmic simplification of the
classical convex hull data structure by Overmars and van Leeuwen~[STOC'80],
combined with new algorithmic insights. Our theoretical guarantees on the
update time match those of Overmars and van Leeuwen, namely ,
while we allow a wider range of functionalities (including rank-based data).
Our algorithmic simplification includes simplifying an 11-case check down to a
3-case check that can be written in 20 lines of easily readable C-code. We
extend our solution to provide a trade-off between theoretical guarantees and
the practical performance of our algorithm. We test and compare our solutions
extensively on inputs that were generated randomly or adversarially, including
benchmarking datasets from the literature.Comment: Accepted for ALENEX2
General models in min-max planar location
This paper studies the problem of deciding whether the present iteration point of some algorithm applied to a planar singlefacility min-max location problem, with distances measured by either anl p -norm or a polyhedral gauge, is optimal or not. It turns out that this problem is equivalent to the decision problem of whether 0 belongs to the convex hull of either a finite number of points in the plane or a finite number of differentl q -circles . Although both membership problems are theoretically solvable in polynomial time, the last problem is more difficult to solve in practice than the first one. Moreover, the second problem is solvable only in the weak sense, i.e., up to a predetermined accuracy. Unfortunately, these polynomial-time algorithms are not practical. Although this is a negative result, it is possible to construct an efficient and extremely simple linear-time algorithm to solve the first problem. Moreover, this paper describes an implementable procedure to reduce the second decision problem to the first with any desired precision. Finally, in the last section, some computational results for these algorithms are reported.optimality conditions;continuous location theory;computational geometry;convex hull;Newton-Raphson method
- …