Improved computation of plane Steiner Minimal Trees

Cockayne, E. J.; Hewgill, Denton

doi:10.1007/bf01758759

Cited by 28 publications

(29 citation statements)

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“…Exact algorithms have been proposed that determine the minimal Steiner tree for a set of N cities [18]- [25] and [4]. See [26], [27], and [4] for recent surveys. All these exact algorithms have exponential complexity in N , making them usable only for small size problems.…”

Section: The Steiner Problemmentioning

confidence: 99%

“…All these exact algorithms have exponential complexity in N , making them usable only for small size problems. To date, an upper limit is set in [4] where problems of size up to N = 100 are exactly solved.…”

Section: The Steiner Problemmentioning

confidence: 99%

“…Before interaction each one of these two Steiner points possesses three connections, among which is the connection SS which will remain untouched in the interaction process. The triplet of connections for S are with the set of nodes {A 1 , A 2 , S } and for S with the nodes {A 3 , A 4 , S}. If, in its approach, S comes within a distance of T from S , then an interaction will be allowed.…”

Section: Interaction Processmentioning

confidence: 99%

“…As a basis for comparison, we selected -the exact method of [22], which offers results up to N = 15, knowing that for the exact resolutions extended up to N = 100 in [4] quantitative data that would fit into our comparison were not available; -the O(N log N ) heuristic of [31], which represents, among the efficient heuristics, the one with the smallest algorithmic complexity; -two heuristics of [36] and [37], which represent, among the efficient heuristics, the ones that generally yield the shortest suboptimal trees. No exact algorithmic complexities are derived in [36] or [37] for these two methods, but estimations are proposed, O(N 1.317 ) and O(N 2.…”

Section: Evaluation and Comparison Inmentioning

confidence: 99%

“…An interesting characteristic of this heuristic that is shared by very few of the algorithms evoked here is that it can solve Steiner problems in a space of arbitrary dimension, while our approach in its present form, as well as those, for instance, in [22,4,31,36,37], is limited to the plane. The complexity of this heuristic is not explicitly established, but it is certainly above O (N log N ).…”

Section: Evaluation and Comparison Inmentioning

confidence: 99%

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A Dynamic Adaptive Relaxation Scheme Applied to the Euclidean Steiner Minimal Tree Problem

Chapeau‐Blondeau¹,

Janez²,

Ferrier³

1997

SIAM J. Optim.

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Abstract. The Steiner problem is an NP-hard optimization problem which consists of finding the minimal-length tree connecting a set of N points in the Euclidean plane. Exact methods of resolution currently available are exponential in N , making exact minimal trees accessible for only small size problems (up to N ≈ 100). An acceptable suboptimal solution is provided by the minimum spanning tree (MST) which has been shown computable in an O(N log N ) step. We propose here an O(N ) process that is able to relax a given initial Steiner tree into a local minimum of its length. This process, based on a physical analogy, simulates the dynamics of a fluid film which relaxes under surface tension forces and stabilizes in an equilibrium configuration minimizing its total length, through purely local interactions. To improve the solution to the Steiner problem, this O(N ) relaxation scheme is applied to reduce the length of the MST. This results in a heuristic of a very low O(N log N ) complexity for the Steiner problem, whose performance is shown to compare quite favorably with that of the best available heuristics. Large problem sizes up to N = 10000 were successfully tackled. A characterization of the asymptotic behavior of the solution of the Steiner problem shows a stabilization to a nonvanishing positive value of the average length reduction achieved over the MST and predicts an average length for the minimal Steiner tree of about 3% below 0.65N 1/2 for large N .

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Section: The Steiner Problemmentioning

confidence: 99%

Section: The Steiner Problemmentioning

confidence: 99%

Section: Interaction Processmentioning

confidence: 99%

Section: Evaluation and Comparison Inmentioning

confidence: 99%

Section: Evaluation and Comparison Inmentioning

confidence: 99%

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A Dynamic Adaptive Relaxation Scheme Applied to the Euclidean Steiner Minimal Tree Problem

Chapeau‐Blondeau¹,

Janez²,

Ferrier³

1997

SIAM J. Optim.

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Interconnecting networks in the plane: The steiner case

Trietsch

1990

Networks

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It is required to interconnect n existing networks in the plane by new links of minimal total length. We assume that the edges of the networks are straight and that it is possible to make connections along the edges or to the vertices. The use o f Steiner points is also allowed. Though the theoretical number of solutions to the problem is uncountable, we prove that it is enough to consider a finite set of locally optimal solutions with a limit on the number of connections along edges. This makes it possible to solve the problem in finite time by methods similar to those used for the Euclidean Steiner tree problem. The problem can be generalized to include flow‐dependent costs for the various links.

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Probabilistic analysis of an enhanced partitioning algorithm for the steiner tree problem in R^d

Kalpakis

Sherman

1994

Networks

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The Geometric Steiner Minimum Tree problem (GSMT) is to connect at minimum cost n given points (called terminals) in d-dimensional Euclidean space. We generalize a GSMT approximation partitioning algorithm by Komlbs and Shing (KS) and analyze its performance under more relaxed conditions. These generalizations have practical applications for multilayer VLSl routing. Whereas KS assumed d = 2, our algorithm works for any dimension d 2 2. Moreover, whereas their analysis assumed the rectilinear norm and a uniform distribution of input points, our analysis holds for any norm on Rd and whenever the terminals are any independent identically distributed random variables taking on values in any bounded subset of R d . Both algorithms depend on a parameter t through which the user can trade off time for solution quality. We evaluate our algorithm in terms of its performance ratio-the ratio of the cost of the Steiner tree computed by the algorithm divided by the cost of a Steiner minimum tree. Applying a probability theorem on subadditive Euclidean functionals by Steele, we prove the following: Under the aforementioned distribution of inputs, the limit as n + co of the supremum of the performance ratio of our algorithm is 1 + O(t-'/d(d-')), almost surely. This result generalizes the corresponding 1 + O(t-'/') bound proven by KS. Along the way, we prove a useful combinatorial lemma about d-dimensional rectangle slicings. We prove that the worst-case time and space complexity of our algorithm is o(n lg ( n / t ) + TMsT(U, u ) + n T s M T ( t ) / t ) and O(SMsT(u, u ) + S s M T ( t ) ) , respectively, where u I n t 2 d + ' ( n / t )~( t )is the number of vertices in the resulting Steiner tree. Here, T S M T ( f ) and SsMT(t) are the time and space required to solve exactly any GSMT problem of size less than t ; T, (n, rn) and S,T(n, m) are the time and space required to find a minimum spanning tree of a graph with n nodes and m edges; and .(t) is the maximum number of Steiner points for any Steiner minimum tree with t terminals. For example, for R2 and the rectilinear norm, the time is O(n Ig(n/t) + n Ig*n + nTSMT(f)/t) and the space is O(n Ig*n + S,(t)). 0 1994 John Wi/ey& Sons, Inc.

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Improved computation of plane Steiner Minimal Trees

Cited by 28 publications

References 10 publications

A Dynamic Adaptive Relaxation Scheme Applied to the Euclidean Steiner Minimal Tree Problem

A Dynamic Adaptive Relaxation Scheme Applied to the Euclidean Steiner Minimal Tree Problem

Interconnecting networks in the plane: The steiner case

Probabilistic analysis of an enhanced partitioning algorithm for the steiner tree problem in R^d

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Improved computation of plane Steiner Minimal Trees

Cited by 28 publications

References 10 publications

A Dynamic Adaptive Relaxation Scheme Applied to the Euclidean Steiner Minimal Tree Problem

A Dynamic Adaptive Relaxation Scheme Applied to the Euclidean Steiner Minimal Tree Problem

Interconnecting networks in the plane: The steiner case

Probabilistic analysis of an enhanced partitioning algorithm for the steiner tree problem in Rd

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Probabilistic analysis of an enhanced partitioning algorithm for the steiner tree problem in R^d