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

Chan, T.-H. Hubert; Li, Mingfei; Li, Ning

doi:10.1007/s00453-013-9779-y

Cited by 5 publications

(9 citation statements)

References 15 publications

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“…All previous constructions of FT spanners for doubling metrics [14,15,41,16] are quite simple, but they are inherently suboptimal. Our construction is more involved, but it produces much better spanners.…”

Section: Our and Previousmentioning

confidence: 99%

“…Spanners for doubling metrics were also subject of intensive research [26,13,11,31,38,27,28,39,14,25,15,41,16]. In many of these works the objective is to devise spanners that achieve one parameter or more among small number of edges, degree, diameter, lightness, and runtime.…”

Section: Question 4 Is There An Algorithm That Constructs a K-mentioning

confidence: 99%

“…O(kn) unspecified unspecified unspecified unspecified doubling [14] O(km) unspecified O(α(m, n)) unspecified unspecified doubling [14] O(k 2 n) O(k 2 ) unspecified unspecified unspecified doubling [15] O In ICALP'12, [14] devised a construction of k-FT spanners for doubling metrics with O(kn) edges, and also another construction with degree O(k 2 ). The other papers in this context are follow-ups [15,41,16] to [25] (they were mentioned in Section 1.1, but they apply to arbitrary doubling metrics), which provide in O(n log n + k 2 n) time, a k-FT spanner with degree O(k 2 ), diameter O(log n) and lightness O(k 2 log n).…”

Section: # Edgesmentioning

confidence: 99%

“…(In the net-tree spanner of [26,13], the radius of i-level nodes is 2 i ; we use radius 5 i as in [38,28] for efficiency reasons; see Section 2 for more details.) To obtain a k-FT spanner, [14,15] use k + 1 net-trees rather than one, and in each of these trees, each tree node x is associated with a single point from D(x); the FT spanner is obtained as a union of k + 1 net-tree spanners, each corresponding to a different net-tree. Another idea that is used in [41,16] is to use a single nettree, but to associate each tree node x with a set S(x) of (up to) k + 1 points from D(x) rather than a single point; the FT spanner is obtained by replacing each edge (x, y) of the basic net-tree spanner by a bipartite clique between the corresponding sets S(x) and S(y).…”

Section: Our and Previousmentioning

confidence: 99%

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From hierarchical partitions to hierarchical covers

Solomon

2014

Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing

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A (1 + ϵ)-spanner for a doubling metric (X, δ) is a subgraph H of the complete graph corresponding to (X, δ), which preserves all pairwise distances to within a factor of 1 + ϵ. A natural requirement from a spanner, which is essential for many applications (mainly in distributed systems or wireless networks), is to be robust against vertex and edge failuresso that even when some vertices and edges in the network fail, we still have a (1 + ϵ)-spanner for what remains. TheIn this paper we devise an optimal construction of faulttolerant spanners for doubling metrics: For any n-point doubling metric, any ϵ > 0, and any integer 1 ≤ k ≤ n − 2, our construction provides a k-fault-tolerant (1 + ϵ)-spanner with optimal degree O(k) within optimal time O(n log n + kn).We then strengthen this result to provide near-optimal (up to a factor of log k) guarantees on the diameter and weight of our spanners, namely, diameter O(log n) and weight O(k 2 + k log n) · ω(M ST ), while preserving the optimal guarantees on the degree O(k) and the runtime O(n log n + kn).Our result settles several fundamental open questions in this area, culminating a long line of research that started with the STOC'95 paper of Arya et al. and the STOC'98 paper of Levcopoulos et al.On the way to this result we develop a new technique for constructing spanners in doubling metrics. In particular, our spanner construction is based on a novel hierarchical cover of the metric, whereas most previous constructions of spanners for doubling and Euclidean metrics (such as the net-tree spanner) are based on hierarchical partitions of the metric. We demonstrate the power of hierarchical covers in the context of geometric spanners by improving the stateof-the-art results in this area.

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Section: Our and Previousmentioning

confidence: 99%

Section: Question 4 Is There An Algorithm That Constructs a K-mentioning

confidence: 99%

Section: # Edgesmentioning

confidence: 99%

Section: Our and Previousmentioning

confidence: 99%

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From hierarchical partitions to hierarchical covers

Solomon

2014

Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing

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“…However, in many net-based spanner constructions, each chain of lonely nodes (i.e., a chain of nodes each of which has only one child) will be contracted ; since there can be cross edges incident on each node in a long lonely chain, the degree of the contracted node can be large. The idea of constant degree single-sink spanners (used in [2,5,6]) can be applied to resolve this issue. However, a simpler method is parent replacement, used by Gottlieb and Roditty [21] to build a routing tree (RouTree) and reroute spanner paths, thus pruning unnecessary cross edges.…”

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confidence: 99%

New Doubling Spanners: Better and Simpler

Chan¹,

Li²,

Li³

et al. 2015

SIAM J. Comput.

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In a seminal STOC 1995 paper, Arya et al. conjectured that spanners for lowdimensional Euclidean spaces with constant maximum degree, hop-diameter O(log n), and lightness O(log n) (i.e., weight O(log n) · w(MST)) can be constructed in O(n log n) time. This conjecture, which became a central open question in this area, was resolved in the affirmative by Elkin and Solomon in STOC 2013. In fact, Elkin and Solomon proved that the conjecture of Arya et al. holds even in doubling metrics. However, Elkin and Solomon's spanner construction is complicated. In this work we present a significantly simpler construction of spanners for doubling metrics with the same guarantees as above. Our construction is based on the basic net-tree spanner framework. However, by employing well-known properties of the net-tree spanner in conjunction with numerous new ideas, we managed to get significantly stronger results. First and foremost, our construction extends in a simple and natural way to provide k-fault tolerant spanners with maximum degree O(k 2 ), hopdiameter O(log n), and lightness O(k 2 log n). This is the first construction of fault-tolerant spanners (even for Euclidean metrics) that achieves good bounds (polylogarithmic in n and polynomial in k) on all the involved parameters simultaneously. Second, we show that the lightness bound of our construction can be improved to O(k 2 ) (with high probability), for random points in [0, 1] D , where 2 ≤ D = O(1). Introduction.An n-point metric space (X, d) can be represented by a complete weighted graph G = (X, E), where the weight w(e) of an edge e = (u, v) is given by d (u, v). A t-spanner of X is a weighted subgraph H = (X, E ) of G (where E ⊆ E has the same weights) that preserves all pairwise distances to within a factor of t, i.e., u, v) is the distance between u and v in H. The parameter t is called the stretch of the spanner H. A path between u and v in H with weight at most t · d(u, v) is called a t-spanner path.In this paper we focus on the regime of stretch t = 1 + for an arbitrarily small 0 < < 1 2 . In general, there are metric spaces (such as the one corresponding to uniformly weighted complete graph), where the only possible (1 + )-spanner is the complete graph. A special class of metric spaces, which has been subject to intensive research in the last decade, is the class of doubling metrics. The doubling dimension of a metric space (X, d), denoted by dim(X) (or dim when the context is clear), is the smallest value ρ such that every ball in X can be covered by 2 ρ balls of half the radius [3,11,22]. A metric space is called doubling if its doubling dimension is bounded by some constant. (We will sometimes disregard dependencies on and dim to avoid

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Epsilon-Coresets for Clustering (with Outliers) in Doubling Metrics

Huang

Jiang

et al. 2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

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We study the problem of constructing ε-coresets for the (k, z)-clustering problem in a doubling metric M (X, d). An ε-coreset is a weighted subset S ⊆ X with weight function w : S → R ≥0 , such that for any k-subset C ∈ [X] k , it holds thatWe present an efficient algorithm that constructs an ε-coreset for the (k, z)-clustering problem in M (X, d), where the size of the coreset only depends on the parameters k, z, and the doubling dimension ddim(M ). To the best of our knowledge, this is the first efficient -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 is 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 ± )-distortion of the distance function d (the distorted distance is called the smoothed distance function), the shattering dimension can be upper bounded by O( −O(ddim(M )) ). For the purpose of coreset construction, the above bound does not suffice as it only works for unweighted spaces. Therefore, we introduce the notion of τ -error probabilistic shattering dimension, and prove a (drastically better) upper bound of O(ddim(M )·log(1/ε)+log log 1 τ ) for the probabilistic shattering dimension for weighted doubling metrics. As it turns out, an upper bound for the probabilistic shattering dimension is enough for constructing a small coreset. We believe the new relation between doubling and shattering dimensions is of independent interest and may find other applications.Furthermore, we study robust coresets for (k, z)-clustering with outliers in a doubling metric. We show an improved connection between α-approximation and robust coresets. This also leads to improvement upon the previous best known bound of the size of robust coreset for Euclidean space [Feldman and Langberg, STOC 11]. The new bound entails a few new results in clustering and property testing.As another application, we show constant-sized (ε, k, z)-centroid sets in doubling metrics can be constructed by extending our coreset construction. Prior to our result, constant-sized centroid sets for general clustering problems were only known for Euclidean spaces. We can apply our centroid set to accelerate the local search algorithm (studied in [Friggstad et al., FOCS 2016]) for the (k, z)-clustering problem in doubling metrics.

show abstract

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

Cited by 5 publications

References 15 publications

From hierarchical partitions to hierarchical covers

From hierarchical partitions to hierarchical covers

New Doubling Spanners: Better and Simpler

Epsilon-Coresets for Clustering (with Outliers) in Doubling Metrics

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