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Balancing degree, diameter and weight in Euclidean spanners
 In Proc. of 18th ESA
, 2010
"... Abstract. In a seminal STOC’95 paper, Arya et al. [4] devised a construction that for any set S of n points in R d and any ɛ>0, provides a(1+ɛ)spanner with diameter O(log n), weight O(log 2 n)w(MST(S)), and constant maximum degree. Another construction of [4] provides a (1 + ɛ)spanner with O(n) ..."
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Abstract. In a seminal STOC’95 paper, Arya et al. [4] devised a construction that for any set S of n points in R d and any ɛ>0, provides a(1+ɛ)spanner with diameter O(log n), weight O(log 2 n)w(MST(S)), and constant maximum degree. Another construction of [4] provides a (1 + ɛ)spanner with O(n) edges and diameter α(n), where α stands for the inverseAckermann function. Das and Narasimhan [12] devised a construction with constant maximum degree and weight O(w(MST(S))), but whose diameter may be arbitrarily large. In another construction by Arya et al. [4] there is diameter O(log n)andweightO(log n)w(MST(S)), but it may have arbitrarily large maximum degree. These constructions fail to address situations in which we are prepared to compromise on one of the parameters, but cannot afford it to be arbitrarily large. In this paper we devise a novel unified construction that trades between maximum degree, diameter and weight gracefully. For a positive integer k, our construction provides a (1+ɛ)spanner with maximum degree O(k), diameter O(logk n + α(k)), weight O(k logk n log n)w(MST(S)), and O(n) edges.Fork = O(1) this gives rise to maximum degree O(1), diameter O(log n) andweightO(log 2 n)w(MST(S)), which is one of the aforementioned results of [4]. For k = n 1/α(n) this gives rise to diameter O(α(n)), weight O(n 1/α(n) (log n)α(n))w(MST(S)) and maximum degree O(n 1/α(n)). In the corresponding result from [4] the spanner has the same number of edges and diameter, but its weight and degree may be arbitrarily large. Our construction also provides a similar tradeoff in the complementary range of parameters, i.e., when the weight should be smaller than log 2 n, but the diameter is allowed to grow beyond log n.
Optimal Euclidean spanners: really short, thin and lanky (Extended Abstract)
 STOC'13
, 2013
"... The degree, the (hop)diameter, and the weight are the most basic and wellstudied parameters of geometric spanners. In a seminal STOC’95 paper, titled“Euclidean spanners: short, thin and lanky”, Arya et al. [2] devised a construction of Euclidean (1 + ɛ)spanners that achieves constant degree, diam ..."
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Cited by 7 (5 self)
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The degree, the (hop)diameter, and the weight are the most basic and wellstudied parameters of geometric spanners. In a seminal STOC’95 paper, titled“Euclidean spanners: short, thin and lanky”, Arya et al. [2] devised a construction of Euclidean (1 + ɛ)spanners that achieves constant degree, diameter O(log n), weight O(log 2 n) · ω(MST), and has running time O(n · log n). This construction applies to npoint constantdimensional 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 one of the most central open problems in the area of Euclidean spanners. Nevertheless, the only progress since 1995 towards its resolution was achieved in the lower bounds front: Any spanner with
New Doubling Spanners: Better and Simpler
, 2013
"... In a seminal STOC’95 paper, Arya et al. conjectured that spanners for lowdimensional Euclidean spaces with constant maximum degree, hopdiameter 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 ques ..."
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In a seminal STOC’95 paper, Arya et al. conjectured that spanners for lowdimensional Euclidean spaces with constant maximum degree, hopdiameter 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’13 (even for doubling metrics). In this work we present a simpler construction of spanners for doubling metrics with the above guarantees. Moreover, our construction extends in a simple and natural way to provide kfault tolerant spanners with maximum degree O(k²), hopdiameter O(log n) and lightness O(k² log n).
From Hierarchical Partitions to Hierarchical Covers: Optimal FaultTolerant Spanners for Doubling Metrics
"... A (1+ǫ)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+ǫ. A natural requirement from a spanner is to be robust against node failures, so that even when some of the nodes in the network f ..."
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A (1+ǫ)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+ǫ. A natural requirement from a spanner is to be robust against node failures, so that even when some of the nodes in the network fail, the remaining part would still provide a (1+ǫ)spanner. The spanner H is called a kfaulttolerant (1+ǫ)spanner, for any 0 ≤ k ≤ n−2, if for any subset F ⊆ X with F  ≤ k, the graph H \F (obtained by removing from H the vertices of F and their incident edges) is a (1 + ǫ)spanner for X \ F. In this paper we devise an optimal construction of faulttolerant spanners for doubling metrics. Specifically, for any npoint doubling metric, any ǫ> 0, and any integer 0 ≤ k ≤ n−2, our construction provides a kfaulttolerant (1+ǫ)spanner with optimal degree O(k) within optimal time O(nlog n+kn). We then strengthen this result to provide nearoptimal (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(MST), while preserving the optimal guarantees on the degree O(k) and the running time O(nlog 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 nettree 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 stateoftheart results in this area.