Search results for "Steiner tree problem"
showing 10 items of 16 documents
On the Low-Dimensional Steiner Minimum Tree Problem in Hamming Metric
2011
It is known that the d-dimensional Steiner Minimum Tree Problem in Hamming metric is NP-complete if d is considered to be a part of the input. On the other hand, it was an open question whether the problem is also NP-complete in fixed dimensions. In this paper we answer this question by showing that the problem is NP-complete for any dimension strictly greater than 2. We also show that the Steiner ratio is 2 - 2/d for d ≥ 2. Using this result, we tailor the analysis of the so-called k-LCA approximation algorithm and show improved approximation guarantees for the special cases d = 3 and d = 4.
An efficient distributed algorithm for generating and updating multicast trees
2006
As group applications are becoming widespread, efficient network utilization becomes a growing concern. Multicast transmission represents a necessary lower network service for the wide diffusion of new multimedia network applications. Multicast transmission may use network resources more efficiently than multiple point-to-point messages; however, creating optimal multicast trees (Steiner Tree Problem in networks) is prohibitively expensive. This paper proposes a distributed algorithm for the heuristic solution of the Steiner Tree Problem, allowing the construction of effective distribution trees using a coordination protocol among the network nodes. Furthermore, we propose a novel distribut…
Computing Euclidean Steiner trees over segments
2020
In the classical Euclidean Steiner minimum tree (SMT) problem, we are given a set of points in the Euclidean plane and we are supposed to find the minimum length tree that connects all these points, allowing the addition of arbitrary additional points. We investigate the variant of the problem where the input is a set of line segments. We allow these segments to have length 0, i.e., they are points and hence we generalize the classical problem. Furthermore, they are allowed to intersect such that we can model polygonal input. As in the GeoSteiner approach of Juhl et al. (Math Program Comput 10(2):487–532, 2018) for the classical case, we use a two-phase approach where we construct a superse…
On the low-dimensional Steiner minimum tree problem in Hamming metric
2013
While it is known that the d-dimensional Steiner minimum tree problem in Hamming metric is NP-complete if d is part of the input, it is an open question whether this also holds for fixed dimensions. In this paper, this question is answered by showing that the Steiner minimum tree problem in Hamming metric is already NP-complete in 3 dimensions. Furthermore, we show that, the minimum spanning tree gives a 2-2d approximation on the Steiner minimum tree for d>=2. Using this result, we analyse the so-called k-LCA and A"k approximation algorithms and show improved approximation guarantees for low dimensions.
Using a TSP heuristic for routing order pickers in warehouses
2010
In this paper, we deal with the sequencing and routing problem of order pickers in conventional multi-parallel-aisle warehouse systems. For this NP-hard Steiner travelling salesman problem (TSP), exact algorithms only exist for warehouses with at most three cross aisles, while for other warehouse types literature provides a selection of dedicated construction heuristics. We evaluate to what extent reformulating and solving the problem as a classical TSP leads to performance improvements compared to existing dedicated heuristics. We report average savings in route distance of up to 47% when using the LKH (Lin-Kernighan-Helsgaun) TSP heuristic. Additionally, we examine if combining problem-sp…
The Random Neural Network Model for the On-line Multicast Problem
2005
In this paper we propose the adoption of the Random Neural Network Model for the solution of the dynamic version of the Steiner Tree Problem in Networks (SPN). The Random Neural Network (RNN) is adopted as a heuristic capable of improving solutions achieved by previously proposed dynamic algorithms. We adapt the RNN model in order to map the network characteristics during a multicast transmission. The proposed methodology is validated by means of extensive experiments.
An Efficient Distributed Algorithm for Generating Multicast Distribution Trees
2005
Multicast transmission may use network resources more efficiently than multiple point-to-point messages; however, creating optimal multicast trees (Steiner Tree Problem in Networks) is prohibitively expensive. For this reason, heuristic methods are generally employed. Conventional centralized Steiner heuristics provide effective solutions, but they are unpractical for large networks, since they require complete knowledge of the network topology. This paper proposes a distributed algorithm for the heuristic solution of the Steiner Tree Problem. The algorithm allows the construction of effective distribution trees using a coordination protocol among the network nodes. The algorithm has been i…
A Grid Enabled Parallel Hybrid Genetic Algorithm for SPN
2004
This paper presents a combination of a parallel Genetic Algorithm (GA) and a local search methodology for the Steiner Problem in Networks (SPN). Several previous papers have proposed the adoption of GAs and others metaheuristics to solve the SPN demonstrating the validity of their approaches. This work differs from them for two main reasons: the dimension and the features of the networks adopted in the experiments and the aim from which it has been originated. The reason that aimed this work was namely to assess deterministic and computationally inexpensive algorithms which can be used in practical engineering applications, such as the multicast transmission in the Internet. The large dimen…
New Algorithms for Computing Phylogenetic Biodiversity
2014
A common problem that appears in many case studies in ecology is the following: given a rooted phylogenetic tree \(\mathcal{T}\) and a subset R of its leaf nodes, we want to compute the distance between the elements in R. A very popular distance measure that can be used for this reason is the Phylogenetic Diversity (PD), which is defined as the cost of the minimum weight Steiner tree in \(\mathcal{T}\) that spans the nodes in R. To analyse the value of the PD for a given set R it is important also to calculate the variance of this measure. However, the best algorithm known so far for computing the variance of the PD is inefficient; for any input tree \(\mathcal{T}\) that consists of n nodes…
Recent mathematical approaches to reconstruct phylogenies: A chemosystematist's and botanist's view
1989
Some basic problems of mathematical phylogenetics are discussed. While algorithms regularly depend on the principle of parsimony, some features of phylogenesis interfere with that principle. Nonrandomness of the distribution of mutations as well as the inconstancy of the molecular clock in time and within a given sequence can bias the calculated relationships of closely related taxa. True comparability of sequences is difficult to establish, since this requires defining of homology of positions and of functions of amino acids as well. Parallelism and convergence can give rise to errors in establishing homology. Furthermore, they are difficult to be integrated into a consistent mathematical …