The Emergence of Sparse Spanners and Greedy Well-Separated Pair Decomposition

A spanner graph on a set of points in $R^d$ 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

New Doubling Spanners: Better and Simpler

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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 $O(\log n)$, and weight $O(\log^2 n) \cdot \omega(MST)$, and has running time $O(n \cdot \log n)$. This construction applies to $n$-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 $O(\log n) \cdot \omega(MST)$. 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, 201

ε\varepsilon-Coresets for Clustering (with Outliers) in Doubling Metrics

We study the problem of constructing $\varepsilon$-coresets for the $(k, z)$-clustering problem in a doubling metric $M(X, d)$. An $\varepsilon$-coreset is a weighted subset $S\subseteq X$ with weight function $w : S \rightarrow \mathbb{R}_{\geq 0}$, such that for any $k$-subset $C \in [X]^k$, it holds that $\sum_{x \in S}{w(x) \cdot d^z(x, C)} \in (1 \pm \varepsilon) \cdot \sum_{x \in X}{d^z(x, C)}$. We present an efficient algorithm that constructs an $\varepsilon$-coreset for the $(k, z)$-clustering problem in $M(X, d)$, where the size of the coreset only depends on the parameters $k, z, \varepsilon$ and the doubling dimension $\mathsf{ddim}(M)$. To the best of our knowledge, this is the first efficient $\varepsilon$-coreset construction of size independent of $|X|$ for general clustering problems in doubling metrics. To this end, we establish the first relation between the doubling dimension of $M(X, d)$ and the shattering dimension (or VC-dimension) of the range space induced by the distance $d$. Such a relation was not known before, since one can easily construct instances in which neither one can be bounded by (some function of) the other. Surprisingly, we show that if we allow a small $(1\pm\epsilon)$-distortion of the distance function $d$, and consider the notion of $\tau$-error probabilistic shattering dimension, we can prove an upper bound of $O( \mathsf{ddim}(M)\cdot \log(1/\varepsilon) +\log\log{\frac{1}{\tau}} )$ for the probabilistic shattering dimension for even weighted doubling metrics. We believe this new relation is of independent interest and may find other applications. We also study the robust coresets and centroid sets in doubling metrics. Our robust coreset construction leads to new results in clustering and property testing, and the centroid sets can be used to accelerate the local search algorithms for clustering problems.Comment: Appeared in FOCS 2018, this is the full versio