Sparse Fault-Tolerant Spanners for Doubling Metrics with Bounded Hop-Diameter or Degree

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Incubators vs Zombies: Fault-Tolerant, Short, Thin and Lanky Spanners for Doubling Metrics

Recently Elkin and Solomon gave a construction of spanners for doubling metrics that has constant maximum degree, hop-diameter O(log n) and lightness O(log n) (i.e., weight O(log n)w(MST). This resolves a long standing conjecture proposed by Arya et al. in a seminal STOC 1995 paper. However, Elkin and Solomon's spanner construction is extremely complicated; we offer a simple alternative construction that is very intuitive and is based on the standard technique of net tree with cross edges. Indeed, our approach can be readily applied to our previous construction of k-fault tolerant spanners (ICALP 2012) to achieve k-fault tolerance, maximum degree O(k^2), hop-diameter O(log n) and lightness O(k^3 log n)

Optimal Vertex Fault Tolerant Spanners (for fixed stretch)

A $k$-spanner of a graph $G$ is a sparse subgraph $H$ whose shortest path distances match those of $G$ up to a multiplicative error $k$. In this paper we study spanners that are resistant to faults. A subgraph $H \subseteq G$ is an $f$ vertex fault tolerant (VFT) $k$-spanner if $H \setminus F$ is a $k$-spanner of $G \setminus F$ for any small set $F$ of $f$ vertices that might "fail." One of the main questions in the area is: what is the minimum size of an $f$ fault tolerant $k$-spanner that holds for all $n$ node graphs (as a function of $f$, $k$ and $n$)? This question was first studied in the context of geometric graphs [Levcopoulos et al. STOC '98, Czumaj and Zhao SoCG '03] and has more recently been considered in general undirected graphs [Chechik et al. STOC '09, Dinitz and Krauthgamer PODC '11]. In this paper, we settle the question of the optimal size of a VFT spanner, in the setting where the stretch factor $k$ is fixed. Specifically, we prove that every (undirected, possibly weighted) $n$-node graph $G$ has a $(2k-1)$-spanner resilient to $f$ vertex faults with $O_k(f^{1 - 1/k} n^{1 + 1/k})$ edges, and this is fully optimal (unless the famous Erdos Girth Conjecture is false). Our lower bound even generalizes to imply that no data structure capable of approximating $dist_{G \setminus F}(s, t)$ similarly can beat the space usage of our spanner in the worst case. We also consider the edge fault tolerant (EFT) model, defined analogously with edge failures rather than vertex failures. We show that the same spanner upper bound applies in this setting. Our data structure lower bound extends to the case $k=2$ (and hence we close the EFT problem for $3$-approximations), but it falls to $\Omega(f^{1/2 - 1/(2k)} \cdot n^{1 + 1/k})$ for $k \ge 3$. We leave it as an open problem to close this gap.Comment: To appear in SODA 201

Sparse fault-tolerant spanners for doubling metrics with bounded hop-diameter or degree

We study fault-tolerant spanners in doubling metrics. A subgraph H for a metric space X is called a k-vertex-fault-tolerant t-spanner ((k, t)-VFTS or simply k-VFTS), if for any subset S â\u8a\u86 X with |S | â\u89¤ k, it holds that dH\S(x, y) â\u89¤ t · d(x, y), for any pair of x, y â\u88\u88 X \ S. For any doubling metric, we give a basic construction of k-VFTS with stretch arbitrarily close to 1 that has optimal O(kn) edges. In addition, we also consider bounded hop-diameter, which is studied in the context of fault-tolerance for the first time even for Euclidean spanners. We provide a construction of k-VFTS with bounded hop-diameter: for m â\u89¥ 2n, we can reduce the hop-diameter of the above k-VFTS to O(α(m, n)) by adding O(km) edges, where α is a functional inverse of the Ackermannâ\u80\u99s function. Finally, we construct a fault-tolerant single-sink spanner with bounded maximum degree, and use it to reduce the maximum degree of our basic k-VFTS. As a result, we get a k-VFTS with O(k 2 n) edges and maximum degree O(k²)

Sparse Fault-Tolerant Spanners for Doubling Metrics with Bounded Hop-Diameter or Degree

Session A6Lecture Notes in Computer Science, Vol. 7391 entitled: Automata, Languages, and Programming: 39th international colloquium, ICALP 2012, Warwick, UK, 9-13 July 2012: ProceedingsWe study fault-tolerant spanners in doubling metrics. A subgraph H for a metric space X is called a k-vertex-fault-tolerant t-spanner ((k,t)-VFTS or simply k-VFTS), if for any subset S ⊆ X with |S| ≤ k, it holds that d H ∖ S (x, y) ≤ t ·d(x, y), for any pair of x, y ∈ X ∖ S. For any doubling metric, we give a basic construction of k-VFTS with stretch arbitrarily close to 1 that has optimal O(kn) edges. In addition, we also consider bounded hop-diameter, which is studied in the context of fault-tolerance for the first time even for Euclidean spanners. We provide a construction of k-VFTS with bounded hop-diameter: for m ≥ 2n, we can reduce the hop-diameter of the above k-VFTS to O(α(m, n)) by adding O(km) edges, where α is a functional inverse of the Ackermann’s function. Finally, we construct a fault-tolerant single-sink spanner with bounded maximum degree, and use it to reduce the maximum degree of our basic k-VFTS. As a result, we get a k-VFTS with O(k 2 n) edges and maximum degree O(k 2)

New Doubling Spanners: Better and Simpler

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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