Approximations and Lower Bounds for the Length of Minimal Euclidean Steiner Trees

Rubinstein, J. Hyam; Weng, J. F.; Wormald, Nicholas C.

doi:10.1007/s10898-005-4207-8

Cited by 3 publications

(18 citation statements)

References 8 publications

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“…for small ε. Theorem 6.5 sharpens Lemma 2.2 of [29] in the range ε ∈ (π/3, 2π/3) by giving a lower bound for F d for all d 2 of the form n α(ε) where α(ε) is an explicit function of ε. In particular, it will follow that if ε > 105.6 .…”

Section: Results For Large εmentioning

confidence: 91%

“…Note that a 0-approximate Steiner tree is the same as a locally minimum Steiner tree. (In [29] the distinction was made between a pseudo-Steiner point of an ε-approximate Steiner tree and a Steiner point of a locally minimum Steiner tree. For the sake of simplicity we make no such distinction and use the term Steiner point for both.)…”

Section: Formulation Of the Problem Conjectures And Previous Resultsmentioning

confidence: 99%

“…The second conjecture is weaker than the first, but it seems difficult to deduce an upper bound for F d that cannot already be deduced for F d . Rubinstein, Weng and Wormald [29] showed that for ε < 1/n 2 , F d (ε, n) C d (ε log n + ε 2 n 3 ). They also consider larger values of ε.…”

Section: Formulation Of the Problem Conjectures And Previous Resultsmentioning

confidence: 99%

“…In [29] an example is given that shows that Conjecture 3.1 is sharp for each d 3. Our second main result, Theorem 4.3, is a lower bound for F 2 that shows that Conjecture 3.1 is already sharp in the plane for sufficiently small ε.…”

Section: Rangementioning

confidence: 99%

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Approximate Euclidean Steiner Trees

Ras

Swanepoel

Thomas

2016

J Optim Theory Appl

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An approximate Steiner tree is a Steiner tree on a given set of terminals in Euclidean space such that the angles at the Steiner points are within a specified error from 120 • . This notion arises in numerical approximations of minimum Steiner trees. We investigate the worst-case relative error of the length of an approximate Steiner tree compared to the shortest tree with the same topology. It has been conjectured that this relative error is at most linear in the maximum error at the angles, independent of the number of terminals. We verify this conjecture for the two-dimensional case as long as the maximum angle error is sufficiently small in terms of the number of terminals. In the two-dimensional case we derive a lower bound for the relative error in length. This bound is linear in terms of the maximum angle error when the angle error is sufficiently small in terms of the number of terminals. We find improved estimates of the relative error in length for larger values of the maximum angle error and calculate exact values in the plane for three and four terminals.Communicated by

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Section: Results For Large εmentioning

confidence: 91%

Section: Formulation Of the Problem Conjectures And Previous Resultsmentioning

confidence: 99%

Section: Formulation Of the Problem Conjectures And Previous Resultsmentioning

confidence: 99%

Section: Rangementioning

confidence: 99%

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Approximate Euclidean Steiner Trees

Ras

Swanepoel

Thomas

2016

J Optim Theory Appl

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“…But this result is useful in helping determine lower bounds for a full Steiner tree. The above theorem and its applications will be discussed in more detail in [6].…”

Section: Lemma 32 If T Is a Full Smt On Points A; B; C Then The Linmentioning

confidence: 99%

Upper and Lower Bounds for the Lengths of Steiner Trees in 3-Space

2004

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We present a method of determining upper and lower bounds for the length of a Steiner minimal tree in 3-space whose topology is a given full Steiner topology, or a degenerate form of that full Steiner topology. The bounds are tight, in the sense that they are exactly satisfied for some configurations. This represents the first nontrivial lower bound to appear in the literature. The bounds are developed by first studying properties of Simpson lines in both two and three dimensional space, and then introducing a class of easily constructed trees, called midpoint trees, which provide the upper and lower bounds. These bounds can be constructed in quadratic time. Finally, we discuss strategies for improving the lower bound.

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Reorganizing topologies of Steiner trees to accelerate their eliminations

Grodet

Tsuchiya

2019

Discrete Math. Algorithm. Appl.

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We describe a technique to reorganize topologies of Steiner trees by exchanging neighbors of adjacent Steiner points. We explain how to use the systematic way of building trees, and therefore topologies, to find the correct topology after nodes have been exchanged. Topology reorganizations can be inserted into the enumeration scheme commonly used by exact algorithms for the Euclidean Steiner tree problem in d-space, providing a method of improvement different than the usual approaches. As an example, we show how topology reorganizations can be used to dynamically change the exploration of the usual branch-and-bound tree when two Steiner points collide during the optimization process. We also turn our attention to the erroneous use of a pre-optimization lower bound in the original algorithm and give an example to confirm its usage is incorrect. In order to provide numerical results on correct solutions, we use planar equilateral points to quickly compute this lower bound, even in dimensions higher than two. Finally, we describe planar twin trees, identical trees yielded by different topologies, whose generalization to higher dimensions could open a new way of building Steiner trees.

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Approximations and Lower Bounds for the Length of Minimal Euclidean Steiner Trees

Cited by 3 publications

References 8 publications

Approximate Euclidean Steiner Trees

Approximate Euclidean Steiner Trees

Upper and Lower Bounds for the Lengths of Steiner Trees in 3-Space

Reorganizing topologies of Steiner trees to accelerate their eliminations

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