5 research outputs found
An optimal algorithm for computing angle-constrained spanners
Let S be a set of n points in ℝd. A graph G = (S,E) is called a t-spanner for S, if for any two points p and q in S, the shortest-path distance in G between p and q is at most t|pq|, where |pq| denotes the Euclidean distance between p and q. The graph G is called θ-angle-constrained, if any two distinct edges sharing an endpoint make an angle of at least θ. It is shown that, for any θ with 0 < θ < π/3, a θ-angle-constrained t-spanner can be computed in O(n logn) time, where t depends only on θ
Optimal Euclidean spanners: really short, thin and lanky
In a seminal STOC'95 paper, titled "Euclidean spanners: short, thin and
lanky", Arya et al. devised a construction of Euclidean (1+\eps)-spanners
that achieves constant degree, diameter , and weight , and has running time . This construction
applies to -point constant-dimensional Euclidean spaces. Moreover, Arya et
al. conjectured that the weight bound can be improved by a logarithmic factor,
without increasing the degree and the diameter of the spanner, and within the
same running time.
This conjecture of Arya et al. became a central open problem in the area of
Euclidean spanners.
In this paper we resolve the long-standing conjecture of Arya et al. in the
affirmative. Specifically, we present a construction of spanners with the same
stretch, degree, diameter, and running time, as in Arya et al.'s result, but
with optimal weight .
Moreover, our result is more general in three ways. First, we demonstrate
that the conjecture holds true not only in constant-dimensional Euclidean
spaces, but also in doubling metrics. Second, we provide a general tradeoff
between the three involved parameters, which is tight in the entire range.
Third, we devise a transformation that decreases the lightness of spanners in
general metrics, while keeping all their other parameters in check. Our main
result is obtained as a corollary of this transformation.Comment: A technical report of this paper was available online from April 4,
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Lower Bounds for Computing Geometric Spanners and Approximate Shortest Paths
We consider the problems of constructing geometric spanners, possibly containing Steiner points, for sets of points in the d-dimensional space IR d , and constructing spanners and approximate shortest paths among a collection of polygonal obstacles in the plane. The complexities of these problems are shown to be \Omega\Gamma n log n) in the algebraic computation tree model. Since O(n log n)-time algorithms are known for solving these problems, our lower bounds are tight up to a constant factor. 1 Introduction Geometric spanners are data structures that approximate the complete graph on a set of points in the d-dimensional space IR d , in the sense that the shortest path (based on such a spanner) between any pair of given points is not more than a factor of t longer than the distance between the points in IR d . Let ø be a fixed constant such that 1 ø 1. Throughout this paper, we measure distances between points in the d-dimensional space IR d with the L ø -metric, where d ..