243 research outputs found
New bounds on the average distance from the Fermat-Weber center of a planar convex body
The Fermat-Weber center of a planar body is a point in the plane from
which the average distance to the points in is minimal. We first show that
for any convex body in the plane, the average distance from the
Fermat-Weber center of to the points of is larger than , where is the diameter of . This proves a conjecture
of Carmi, Har-Peled and Katz. From the other direction, we prove that the same
average distance is at most . The new bound substantially improves the previous bound of
due to
Abu-Affash and Katz, and brings us closer to the conjectured value of . We also confirm the upper bound conjecture for centrally
symmetric planar convex bodies.Comment: 13 pages, 2 figures. An earlier version (now obsolete): A. Dumitrescu
and Cs. D. T\'oth: New bounds on the average distance from the Fermat-Weber
center of a planar convex body, in Proceedings of the 20th International
Symposium on Algorithms and Computation (ISAAC 2009), 2009, LNCS 5878,
Springer, pp. 132-14
Semidefinite Representation of the -Ellipse
The -ellipse is the plane algebraic curve consisting of all points whose
sum of distances from given points is a fixed number. The polynomial
equation defining the -ellipse has degree if is odd and degree
if is even. We express this polynomial equation as
the determinant of a symmetric matrix of linear polynomials. Our representation
extends to weighted -ellipses and -ellipsoids in arbitrary dimensions,
and it leads to new geometric applications of semidefinite programming.Comment: 16 pages, 5 figure
Solving a "Hard" Problem to Approximate an "Easy" One: Heuristics for Maximum Matchings and Maximum Traveling Salesman Problems
We consider geometric instances of the Maximum Weighted Matching Problem
(MWMP) and the Maximum Traveling Salesman Problem (MTSP) with up to 3,000,000
vertices. Making use of a geometric duality relationship between MWMP, MTSP,
and the Fermat-Weber-Problem (FWP), we develop a heuristic approach that yields
in near-linear time solutions as well as upper bounds. Using various
computational tools, we get solutions within considerably less than 1% of the
optimum.
An interesting feature of our approach is that, even though an FWP is hard to
compute in theory and Edmonds' algorithm for maximum weighted matching yields a
polynomial solution for the MWMP, the practical behavior is just the opposite,
and we can solve the FWP with high accuracy in order to find a good heuristic
solution for the MWMP.Comment: 20 pages, 14 figures, Latex, to appear in Journal of Experimental
Algorithms, 200
Approximating Median Points in a Convex Polygon
We develop two simple and efficient approximation algorithms for the
continuous -medians problems, where we seek to find the optimal location of
facilities among a continuum of client points in a convex polygon with
vertices in a way that the total (average) Euclidean distance between
clients and their nearest facility is minimized. Both algorithms run in
time. Our algorithms produce solutions within a
factor of 2.002 of optimality. In addition, our simulation results applied to
the convex hulls of the State of Massachusetts and the Town of Brookline, MA
show that our algorithms generally perform within a range of 5\% to 22\% of
optimality in practice
On the Fermat-Weber Point of a Polygonal Chain and Its Generalizations
In this paper, we study the properties of the Fermat-Weber point for a set of fixed points, whose arrangement coincides with the vertices of a regular polygonal chain. A k-chain of a regular n-gon is the segment of the boundary of the regular n-gon formed by a set of k (≤ n) consecutive vertices of the regular n-gon. We show that for every odd positive integer k, there exists an integer N(k), such that the Fermat-Weber point of a set of k fixed points lying on the vertices a k-chain of a n-gon coincides with a vertex of the chain whenever n ≥ N(k). We also show that ⌈πm(m + 1) - π2/4⌉ ≤ N(k) ≤ ⌊πm(m + 1) + 1⌋, where k (= 2m + 1) is any odd positive integer. We then extend this result to a more general family of point set, and give an O(hk log k) time algorithm for determining whether a given set of k points, having h points on the convex hull, belongs to such a family
Certified Approximation Algorithms for the Fermat Point and n-Ellipses
Given a set A of n points in ?^d with weight function w: A??_{> 0}, the Fermat distance function is ?(x): = ?_{a?A}w(a)?x-a?. A classic problem in facility location dating back to 1643, is to find the Fermat point x*, the point that minimizes the function ?. We consider the problem of computing a point x?* that is an ?-approximation of x* in the sense that ?x?*-x*? ?(x*) and d = 2. Finally, all our planar (d = 2) algorithms are implemented in order to experimentally evaluate them, using both synthetic as well as real world datasets. These experiments show the practicality of our techniques
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