Light graphs with small routing cost

Wu, Bang Ye; Chao, Kun-Mao; Tang, Chuan Yi

doi:10.1002/net.10019

Cited by 29 publications

(22 citation statements)

References 20 publications

(37 reference statements)

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“…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: For Every Vertex V ∈ V D T (Rt V) ≤ α · D G (Rt V) and mentioning

confidence: 99%

Steiner Shallow-Light Trees Are Exponentially Lighter than Spanning Ones

Elkin¹,

Solomon²

2015

SIAM J. Comput.

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Abstract-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: For Every Vertex V ∈ V D T (Rt V) ≤ α · D G (Rt V) and mentioning

confidence: 99%

Steiner Shallow-Light Trees Are Exponentially Lighter than Spanning Ones

Elkin¹,

Solomon²

2015

SIAM J. Comput.

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“…SLTs were found useful for numerous data gathering and dissemination tasks in overlay networks [10,44,34], in the message-passing model of distributed computing [5,6], and in wireless and sensor networks [45,9,18,36,35,43]. Moreover, SLTs find applications in routing [3,42,28,46] and in network and VLSI-circuit design [15,16,17,41]. In addition, SLTs are embedded within various related structures, such as light approximate routing trees [46], shallow-low-light trees [19,20], light spanners [6,40], and others [41,36,35].…”

Section: Journal Of Computational Geometry Jocgorgmentioning

confidence: 99%

“…Moreover, SLTs find applications in routing [3,42,28,46] and in network and VLSI-circuit design [15,16,17,41]. In addition, SLTs are embedded within various related structures, such as light approximate routing trees [46], shallow-low-light trees [19,20], light spanners [6,40], and others [41,36,35].…”

Section: Journal Of Computational Geometry Jocgorgmentioning

confidence: 99%

Euclidean Steiner Shallow-Light Trees

Solomon

2014

Proceedings of the Thirtieth Annual Symposium on Computational Geometry

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Abstract. A spanning tree that simultaneously approximates a shortest-path tree and a minimum spanning tree is called a shallow-light tree (shortly, SLT). More specifically, an (α, β)-SLT of a weighted undirected graph G = (V, E, w) with respect to a designated vertex rt ∈ V is a spanning tree of G with: (1) root-stretch α -it preserves all distances between rt and the other vertices up to a factor of α, and (2) lightness β -it has weight at most β times the weight of a minimum spanning tree M ST (G) of G. In this paper we show that Steiner points lead to a quadratic improvement in Euclidean SLTs, by presenting a construction of Euclidean Steiner (1 + , O( 1 ))-SLTs in arbitrary 2-dimensional Euclidean spaces. The lightness bound β = O( 1 ) of our construction is optimal up to a constant. The runtime of our construction, and thus the number of Steiner points used, are bounded by O(n).

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“…The total distance has been investigated by several authors (e.g. [13]) and also under different names, such as transmission [15], total routing cost [18], and Wiener index [5,17], with the latter being the oldest and most common. The average distance has also been investigated under the name mean distance [7].…”

Section: Introductionmentioning

confidence: 99%

On thek-ary hypercube tree and its average distance

Alsuwaiyel

2011

International Journal of Computer Mathematics

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To cite this article: M. H. Alsuwaiyel (2011) On the k-ary hypercube tree and its average distanceThe d-dimensional k-ary hypercube tree T k (d) is a generalization of the hypercube tree, also known in the literature as the spanning binomial tree. We present some of its structural properties and investigate in detail its average distance. For instance, it is shown that the binary hypercube tree has the anomaly of having two nodes in its centre as opposed to having one in hypercube trees of arity k > 2. However, in all dimensions, the centre, centroid and median coincide. We show that its total distance is σ (T k whose limiting value is 2. This answers a generalization of a conjecture of Dobrynin et al. [Wiener index of trees: Theory and applications, Acta Appl. Math. 66 (2001), pp. 211-249] for the binary hypercube by the affirmative.

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Light graphs with small routing cost

Cited by 29 publications

References 20 publications

Steiner Shallow-Light Trees Are Exponentially Lighter than Spanning Ones

Steiner Shallow-Light Trees Are Exponentially Lighter than Spanning Ones

Euclidean Steiner Shallow-Light Trees

On thek-ary hypercube tree and its average distance

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