For many geometric graph problems for points in the unit square, such as minimum spanning trees, matching, and traveling salesmen, the sum of edge lengths is O(sqrt n) and the sum of dth powers of edge lengths is O(log n). We provide a "gap theorem" showing that if these bounds do not hold for a class of graphs, both sums will instead be Omega(n). For traveling salesmen the O(log n) bound is tight but for some other graphs the sum of dth powers of edge lengths is O(1).
(BibTeX -- Citations -- Preprint of SCG version -- CiteSeer)
Beta-skeletons are geometric graphs used among other purposes for empirical network analysis and minimum weight triangulation. A fractal construction shows that, for any beta>0, the beta-skeleton of a point set can have arbitrarily large dilation: Omega(nc), where c is a constant depending on beta and going to zero in the limit as beta goes to zero. In particular this applies to the Gabriel graph. We also show upper bounds on dilation of the same form.
(BibTeX -- Citations -- ACM DL)
Surveys results in geometric network design theory, including algorithms for constructing minimum spanning trees and low-dilation graphs.
(BibTeX -- Citations -- CiteSeer)
We show how to find shortest paths along the segments of an arrangement of n vertical and horizontal line segments in the plane, in time O(n3/2).
(BibTeX -- Citations -- CiteSeer -- ACM DL)
We consider the problem of "connect the dots": if we have an unknown smooth curve from which sample points have been selected, we would like to find a curve through the sample points that approximates the unknown curve. We show that if the local sample density is sufficiently high, a simple algorithm suffices: form the Delaunay triangulation of the sample points together with their Voronoi vertices, and keep only those Delaunay edges connecting original sample points.
(Tech. report version -- BibTeX -- Citations -- CiteSeer -- ACM DL)
We show how to find shortest paths between two points on the lines of an arrangement of n lines with k distinct orientations, in time O(n + k2).
(BibTeX -- SODA paper -- Citations -- CiteSeer)
We show how to test the dilation of a star, embedded in a Euclidean space of bounded dimension, in time O(n log n), and how to find the star center that has the minimum dilation for a given set of leaf points in randomized expected time O(n log n). For two-dimensional points, we can find the minimum dilation center, constrained to be one of the input points, in time O(n 2α(n) log2n). The unconstrained center placement algorithm involves quasiconvex programming, and is used as a subroutine in the constrained center placement algorithm.
(SoCG05 talk slides -- Citations -- BibTeX)
The problem is to quickly find, in a set of sites with weights, the site maximizing the product of its weight with its distance from the query point. Our solution combines known results on core-sets with a reduction from the weighted to the unweighted problem that works in any metric space. This leads to fast approximation algorithms for the constrained minimum dilation star problem in any fixed dimension.
We show that the triangulations of a finite point set form a flip graph that can be embedded isometrically into a hypercube, if and only if the point set has no empty convex pentagon. Point sets of this type include convex subsets of lattices, points on two lines, and several other infinite families. As a consequence, flip distance in such point sets can be computed efficiently.
Geometry -- Publications -- David Eppstein -- Theory Group -- Inf. & Comp. Sci. -- UC Irvine
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