5 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
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