Narrow-Shallow-Low-Light Trees with and without Steiner Points

Elkin, Michael; Solomon, Shay

doi:10.1007/978-3-642-04128-0_19

Cited by 4 publications

(11 citation statements)

References 17 publications

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“…A similar situation occurs in the context of low-light trees, which combine small lightness with small depth [22], [24]. It is known that Steiner points do not help in this context either, i.e., that any Steiner tree T can be converted into a spanning tree with the the same (up to constant factors) lightness and depth as those of T [24].…”

Section: Steiner Points In Other Metric Structuresmentioning

confidence: 84%

“…SLTs were also used for constructing spanners [6], [40]. Closely related tree structures, such as light approximate routing trees and shallow-low-light trees were investigated in [48], [22], [24]. Moreover, some of the constructions in [48], [22], [24] are based on constructions of SLTs.…”

Section: Resultsmentioning

confidence: 99%

“…It is known that Steiner points do not help in this context either, i.e., that any Steiner tree T can be converted into a spanning tree with the the same (up to constant factors) lightness and depth as those of T [24].…”

Section: Steiner Points In Other Metric Structuresmentioning

confidence: 97%

“…To summarize, Steiner points were studied in many different settings in the context of probabilistic tree embeddings, low-light trees, graph spanners and distance preservers [3], [35], [28], [13], [24]. In all these settings they are known not to help much in improving inherent tradeoffs between the involved parameters.…”

Section: Steiner Points In Other Metric Structuresmentioning

confidence: 99%

“…Shallow-low-light trees were studied in [22], [24]. In addition to small root-distortion and lightness, these trees have small depth.…”

Section: Related Workmentioning

confidence: 99%

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Steiner Shallow-Light Trees are Exponentially Lighter than Spanning Ones

Elkin

Solomon

2011

2011 IEEE 52nd Annual Symposium on Foundations of Computer Science

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For a pair of parameters α, β ≥ 1, a spanning tree T of a weighted undirected n-vertex graph G = (V, E, w) is called an (α, β)-shallow-light tree (shortly, (α, β)-SLT) of G with respect to a designated vertex rt ∈ V if (1) it approximates all distances from rt to the other vertices up to a factor of α, and (2) its weight is at most β times the weight of the minimum spanning tree MST (G) of G. The parameter α (respectively, β) is called the root-distortion (resp., lightness) of the tree T . Shallowlight trees (SLTs) constitute a fundamental graph structure, with numerous theoretical and practical applications. In particular, they were used for constructing spanners, in network design, for VLSIcircuit design, for various data gathering and dissemination tasks in wireless and sensor networks, in overlay networks, and in the message-passing model of distributed computing.Tight tradeoffs between the parameters of SLTs were established by Awerbuch et al.[5], [6] and Khuller et al. [33]. They showed that for any > 0 there always exist (1 + , O( 1 ))-SLTs, and that the upper bound β = O( 1 ) on the lightness of SLTs cannot be improved. In this paper we show that using Steiner points one can build SLTs with logarithmic lightness, i.e., β = O(log 1 ). This establishes an exponential separation between spanning SLTs and Steiner ones.One particularly remarkable point on our tradeoff curve is = 0. In this regime our construction provides a shortest-path tree with weight at most O(log n) · w(MST (G)). Moreover, we prove matching lower bounds that show that all our results are tight up to constant factors.Finally, on our way to these results we settle (up to constant factors) a number of open questions that were raised by Khuller et al. [33] in SODA'93.

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Section: Steiner Points In Other Metric Structuresmentioning

confidence: 84%

Section: Resultsmentioning

confidence: 99%

Section: Steiner Points In Other Metric Structuresmentioning

confidence: 97%

Section: Steiner Points In Other Metric Structuresmentioning

confidence: 99%

“…Shallow-low-light trees were studied in [22], [24]. In addition to small root-distortion and lightness, these trees have small depth.…”

Section: Related Workmentioning

confidence: 99%

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Steiner Shallow-Light Trees are Exponentially Lighter than Spanning Ones

Elkin

Solomon

2011

2011 IEEE 52nd Annual Symposium on Foundations of Computer Science

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Fault-Tolerant Spanners for Doubling Metrics: Better and Simpler

Solomon

2012

Preprint

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In STOC'95 Arya et al. [2] conjectured that for any constant dimensional n-point Euclidean space, a (1 + ǫ)-spanner with constant degree, diameter O(log n) and weight O(log n)•ω(M ST ) can be built in O(n•log n) time. Recently Elkin and Solomon [10] (technical report, April 2012) proved this conjecture of Arya et al. in the affirmative. In fact, the proof of [10] is more general in two ways. First, it applies to arbitrary doubling metrics. Second, it provides a complete tradeoff between the three involved parameters that is tight (up to constant factors) in the entire range. Subsequently, Chan et al. [5] (technical report, July 2012]) provided another proof for Arya et al.'s conjecture, which is simpler than the proof of Elkin and Solomon [10]. Moreover, Chan et al. [5] also showed that one can build a fault-tolerant (FT) spanner with similar properties. Specifically, they showed that there exists a k-FT (1 + ǫ)-spanner with degree O(k 2 ), diameter O(log n) and weight O(k 3 • log n) • ω(M ST ). The running time of the construction of [5] was not analyzed.In this work we improve the results of Chan et al.[5], using a simpler proof. Specifically, we present a simple proof which shows that a k-FTSimilarly to the constructions of [10] and [5], our construction applies to arbitrary doubling metrics. However, in contrast to the construction of Elkin and Solomon [10], our construction fails to provide a complete (and tight) tradeoff between the three involved parameters. The construction of Chan et al. [5] has this drawback too.For random point sets in R d , we "shave" a factor of log n from the weight bound. Specifically, in this case our construction provides within the same time O(n • (log n + k 2 )), a k-FT (1 + ǫ)-spanner with degree O(k 2 ), diameter O(log n) and weight that is with high probability O(k 2 ) • ω(M ST ).

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Computing a Minimum-Cost $k$-hop Steiner Tree in Tree-Like Metrics

Böhm,

Hoeksma,

Megow

et al. 2020

Preprint

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We consider the problem of computing a Steiner tree of minimum cost under a k-hop constraint which requires the depth of the tree to be at most k. Our main result is an exact algorithm for metrics induced by graphs of bounded treewidth that runs in time n O(k) . For the special case of a path, we give a simple algorithm that solves the problem in polynomial time, even if k is part of the input. The main result can be used to obtain, in quasi-polynomial time, a near-optimal solution that violates the k-hop constraint by at most one hop for more general metrics induced by graphs of bounded highway dimension.

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Narrow-Shallow-Low-Light Trees with and without Steiner Points

Cited by 4 publications

References 17 publications

Steiner Shallow-Light Trees are Exponentially Lighter than Spanning Ones

Steiner Shallow-Light Trees are Exponentially Lighter than Spanning Ones

Fault-Tolerant Spanners for Doubling Metrics: Better and Simpler

Computing a Minimum-Cost $k$-hop Steiner Tree in Tree-Like Metrics

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