Steiner Minimum Trees in Uniform Orientation Metrics

Brazil, Marcus

doi:10.1007/978-1-4613-0255-1_1

Cited by 8 publications

(5 citation statements)

References 34 publications

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“…It has been conjectured that the bound in Theorem 2.29 can be improved [45,259,261]. There are nevertheless arguments that support the opposite fact, and we conjecture that the bound in Theorem 2.29 is tight.…”

Section: Theorem 229mentioning

confidence: 65%

Optimal Interconnection Trees in the Plane

Brazil

Zachariasen

2015

Algorithms and Combinatorics

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Section: Theorem 229mentioning

confidence: 65%

Optimal Interconnection Trees in the Plane

Brazil

Zachariasen

2015

Algorithms and Combinatorics

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“…Minimum Induced Steiner Subgraph [7] O(m) amortized O(n t−2 +n 2 m) O(n t−2 + n 2 m) The Steiner tree problem is a classical combinatorial optimization problem and has arisen in several areas [3,16,24,25,28]. This problem is proven to be NPhard in Karp's seminal work [21] and has been studied from several perspectives, such as approximation algorithms [4,15], parameterized algorithms [10], and algorithms in practice [1,20].…”

Section: Timementioning

confidence: 99%

Linear-Delay Enumeration for Minimal Steiner Problems

Kobayashi¹,

Kurita²,

Wasa³

2020

Preprint

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Let G = (V, E) be a undirected graph and let W ⊆ V be a set of terminals. A Steiner subgraph of (G, W ) is a subgraph of G that contains all vertices of W and there is a path between every pair of vertices of W in the subgraph. We say that a Steiner subgraph is minimal if it has no proper Steiner subgraph. It is easy to observe that every minimal Steiner subgraph forms a tree, which is called a minimal Steiner tree. We propose a linear delay and polynomial space algorithm for enumerating all minimal Steiner trees of (G, W ), which improves a previously known polynomial delay enumeration algorithm in [Kimelfeld and Sagiv, Inf. Syst., 2008]. Our enumeration algorithm can be extended to other Steiner problems: minimal Steiner forests, minimal terminal Steiner trees, minimal directed Steiner trees. As another variant of the minimal Steiner subgraph problem, we study the problem of enumerating minimal induced Steiner subgraphs. We propose a polynomial delay and exponential space enumeration algorithm of minimal induced Steiner subgraphs for claw-free graphs, whereas the problem on general graphs is shown to be at least as hard as the problem of enumerating minimal transversals in hypergraphs. Contrary to these tractable results, we show that the problem of enumerating minimal group Steiner trees is at least as hard as the minimal transversal enumeration problem on hypergraphs.

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“…We only consider the problem for 4 ≤ λ < ∞ because strong results for the cases where λ = 2, 3 or ∞ are already known [1], [4], [5], [8].…”

Section: Background and Preliminariesmentioning

confidence: 99%

“…(Degree 4 Steiner points only occur in the cases where λ = 4 or 6. The proof of the above statement for λ = 4 is given in [1]; the proof for λ = 6 is similar. )…”

Section: Background and Preliminariesmentioning

confidence: 99%

“…We can use the concept of a 0-shift to resolve the question of precisely when degree 4 Steiner points can occur if λ = 4 or 6. Some properties of degree 4 Steiner points were established in [1] and [2], but the following theorem is a stronger result. Note that a full λ-SMT is said to be fulsome if every λ-SMT on the same set of terminals is full.…”

Section: Length Preserving Shiftsmentioning

confidence: 99%

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Canonical Forms and Algorithms for Steiner Trees in Uniform Orientation Metrics

et al. 2005

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We present some fundamental structural properties for minimum length networks (known as Steiner minimum trees) interconnecting a given set of points in an environment in which edge segments are restricted to λ uniformly oriented directions. We show that the edge segments of any full component of such a tree contain a total of at most four directions if λ is not a multiple of 3, or six directions if λ is a multiple of 3. This result allows us to develop useful canonical forms for these full components.The structural properties of these Steiner minimum trees are then used to resolve an important open problem in the area: does there exist a polynomial time algorithm for constructing a Steiner minimum tree if the topology of the tree is known? We obtain a simple linear time algorithm for constructing a Steiner minimum tree for any given set of points and a given Steiner topology.

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Steiner Minimum Trees in Uniform Orientation Metrics

Cited by 8 publications

References 34 publications

Optimal Interconnection Trees in the Plane

Optimal Interconnection Trees in the Plane

Linear-Delay Enumeration for Minimal Steiner Problems

Canonical Forms and Algorithms for Steiner Trees in Uniform Orientation Metrics

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