16 research outputs found
The Emergence of Sparse Spanners and Greedy Well-Separated Pair Decomposition
A spanner graph on a set of points in contains a shortest path between
any pair of points with length at most a constant factor of their Euclidean
distance. In this paper we investigate new models and aim to interpret why good
spanners 'emerge' in reality, when they are clearly built in pieces by agents
with their own interests and the construction is not coordinated. Our main
result is to show that if edges are built in an arbitrary order but an edge is
built if and only if its endpoints are not 'close' to the endpoints of an
existing edge, the graph is a (1 + \eps)-spanner with a linear number of
edges, constant average degree, and the total edge length as a small
logarithmic factor of the cost of the minimum spanning tree. As a side product,
we show a simple greedy algorithm for constructing optimal size well-separated
pair decompositions that may be of interest on its own
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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