Euclidean Steiner Minimal Tree Research Articles

Optimization of Pipeline Network Layout for Multiple Heat Sources Distributed Energy Systems Considering Reliability Evaluation

Due to the target of carbon neutrality, energy saving has become more important than ever. At the same time, the widespread use of distributed energy systems and the regional utilization of industrial waste heat leads to the existence of multiple heat sources in an area. Therefore, how to design an economical and reliable pipeline network to meet energy-saving demand under multiple heat source conditions becomes a problem. In this work, an optimization method is established to determine the optimal pipeline network topology with minimum total annual cost. In this optimization method, Star tree algorithm, Kruskal algorithm and GeoSteiner algorithm are combined with a linear programming model to establish a distributed energy pipeline network for multiple heat sources. The model incorporates Euclidean Steiner Minimum Tree and Rectilinear Steiner Minimum Tree in the consideration of the topology optimization of Distributed Energy System pipeline networks. Four pipeline network topologies, STAR, Minimum Spanning Tree, Euclidean Steiner Minimum Tree and Rectilinear Steiner Minimum Tree, are evaluated in this paper from economic and reliability perspectives. A case extracted from a real industrial park where steam is the medium is used to prove the validity of the model. The optimization results show that a Euclidean Steiner Minimum Tree pipeline network has a lower total annual cost than three other types of pipeline network and ranks second in reliability. Considering the comprehensive economy and reliability, ESMT is the optimal pipeline network type of distributed energy system with steam as the medium.

Computing Euclidean Steiner trees over segments

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 superset of so-called full components of an SMT in the first phase. We prove a structural theorem for these full components, which allows us to use almost the same GeoSteiner algorithm as in the classical SMT problem. The second phase, the selection of a minimal cost subset of constructed full components, is exactly the same as in GeoSteiner approach. Finally, we report some experimental results that show that our approach is more efficient than the approximate solution that is obtained by sampling the segments.

Exact and heuristic algorithms for Space Information Flow.

Space Information Flow (SIF) is a new promising research area that studies network coding in geometric space, such as Euclidean space. The design of algorithms that compute the optimal SIF solutions remains one of the key open problems in SIF. This work proposes the first exact SIF algorithm and a heuristic SIF algorithm that compute min-cost multicast network coding for N (N ≥ 3) given terminal nodes in 2-D Euclidean space. Furthermore, we find that the Butterfly network in Euclidean space is the second example besides the Pentagram network where SIF is strictly better than Euclidean Steiner minimal tree. The exact algorithm design is based on two key techniques: Delaunay triangulation and linear programming. Delaunay triangulation technique helps to find practically good candidate relay nodes, after which a min-cost multicast linear programming model is solved over the terminal nodes and the candidate relay nodes, to compute the optimal multicast network topology, including the optimal relay nodes selected by linear programming from all the candidate relay nodes and the flow rates on the connection links. The heuristic algorithm design is also based on Delaunay triangulation and linear programming techniques. The exact algorithm can achieve the optimal SIF solution with an exponential computational complexity, while the heuristic algorithm can achieve the sub-optimal SIF solution with a polynomial computational complexity. We prove the correctness of the exact SIF algorithm. The simulation results show the effectiveness of the heuristic SIF algorithm.

Connectivity Restoration in Wireless Sensor Networks via Space Network Coding

The problem of finding the number and optimal positions of relay nodes for restoring the network connectivity in partitioned Wireless Sensor Networks (WSNs) is Non-deterministic Polynomial-time hard (NP-hard) and thus heuristic methods are preferred to solve it. This paper proposes a novel polynomial time heuristic algorithm, namely, Relay Placement using Space Network Coding (RPSNC), to solve this problem, where Space Network Coding, also called Space Information Flow (SIF), is a new research paradigm that studies network coding in Euclidean space, in which extra relay nodes can be introduced to reduce the cost of communication. Unlike contemporary schemes that are often based on Minimum Spanning Tree (MST), Euclidean Steiner Minimal Tree (ESMT) or a combination of MST with ESMT, RPSNC is a new min-cost multicast space network coding approach that combines Delaunay triangulation and non-uniform partitioning techniques for generating a number of candidate relay nodes, and then linear programming is applied for choosing the optimal relay nodes and computing their connection links with terminals. Subsequently, an equilibrium method is used to refine the locations of the optimal relay nodes, by moving them to balanced positions. RPSNC can adapt to any density distribution of relay nodes and terminals, as well as any density distribution of terminals. The performance and complexity of RPSNC are analyzed and its performance is validated through simulation experiments.

들로네 삼각망과 최소신장트리를 결합한 효율적인 유클리드 스타이너 최소트리 생성

스타이너 트리의 생성은 NP-Complete 영역에 속하므로, 이것을 위한 휴리스틱들은, 다수의 입력 노드에 대해서 많은 시간과 계산을 요구한다. 본 논문에서는 많은 입력노드에 대해, 들로네 삼각망과 Prim의 최소신장트리를 결합한 효과적인 유클리드 스타이너 최소트리 구성방법을 제안한다. 이 방법은 Prim의 최소신장트리와 최소신장트리기반 스타이너 트리와 각각 비교 분석되었다. 제안된 방법은 30,000개의 입력노드에 대해 최소신장트리에 비해 연결 길이는 2.1% 감소, 실행시간은 138.2% 증가하였고, 최소신장트리기반 스타이너최소트리에 비해 실행시간 18.9% 감소, 연결 길이 0.013% 감소의 실험결과를 보였다. 따라서 본 연구의 제안방법은 실행시간이 주요 요인이 되지 않는 환경에서 연결 길이를 단축해야 할 응용에 잘 적용될 수 있을 것이다. As Steiner minimum tree building belongs to NP-Complete problem domain, heuristics for the problem ask for immense amount execution time and computations in numerous inputs. In this paper, we propose an efficient mechanism of euclidean Steiner minimum tree construction for numerous inputs using combination of Delaunay triangulation and Prim's minimum spanning tree algorithm. Trees built by proposed mechanism are compared respectively with the Prim's minimum spanning tree and minimums spanning tree based Steiner minimum tree. For 30,000 input nodes, Steiner minimum tree by proposed mechanism shows about 2.1% tree length less and 138.2% execution time more than minimum spanning tree, and does about 0.013% tree length less and 18.9% execution time less than minimum spanning tree based Steiner minimum tree in experimental results. Therefore the proposed mechanism can work moderately well to many useful applications where execution time is not critical but reduction of tree length is a key factor.

The Process of Ant Algorithm for Solving the Degree-Constrained Euclidean Steiner Minimal Tree

The degree-constrained Euclidean Steiner minimal tree problem was discussed based on the Euclidean Steiner minimal tree with each original point being added with a degree constraint. The property of the problem was analyzed and the implementation process of solving the problem by using the ant algorithm was presented.The algorithms is coded in Delphi and run on the Windows XP environment.

Euclidean Steiner trees optimal with respect to swapping 4-point subtrees

The Steiner tree problem in Euclidean space \(E^3\) asks for a minimum length network \(T\), called a Euclidean Steiner Minimum Tree (ESMT), spanning a given set of points. This problem is NP-hard and the hardness is inherently due to the number of feasible topologies (underlying graph structure of \(T\)) which increases exponentially as the number of given points increases. Planarity is a very strong condition that gives a big difference between the ESMT problem in the Euclidean plane \(E^2\) and in Euclidean \(d\)-space \(E^d (d\ge 3)\): the ESMT problem in the plane is practically solvable whereas the ESMT problem in \(d\)-space is really intractable. The simplest tree rearrangement technique is to repeatedly replace a subtree spanning four nodes in \(T\) with another subtree spanning the same four nodes. This technique is referred to as the Swapping 4-point Topology/ Tree technique in this paper. An indicator (or quasi-indicator) of \(T\) plays a similar role in the optimization of the length \(L(T)\) of \(T\) in the discrete topology space (the underlying graph structure of \(T\)) to the derivative of a differentiable function which indicates a fastest direction of descent. \(T\) will be called S4pT-optimal if it is optimal with respect to swapping 4-point subtrees. In this paper we first make a complete analysis of 4-point trees in Euclidean space exploring all possible types of 4-point trees and their properties. We review some known indicators of 4-point ESMTs in \(E^2\), and give some simple geometric proofs of these indicators. Then, we translate these indicators to \(E^3\), producing eight quasi-indicators in \(E^3\) using computational experiments, the best quasi-indicator \(\rho _\mathrm{osr}\) is sifted with an effectiveness for randomly generated 4-point sets as high as 98.62 %. Finally we show how \(\rho _\mathrm{osr}\) is used to find an S4pT-optimal ESMT on 14 probability vectors in \(4d\)-space with a detailed example.

Multi-terminal pipe routing by Steiner minimal tree and particle swarm optimisation

Computer-aided design of pipe routing is of fundamental importance for complex equipments' developments. In this article, non-rectilinear branch pipe routing with multiple terminals that can be formulated as a Euclidean Steiner Minimal Tree with Obstacles (ESMTO) problem is studied in the context of an aeroengine-integrated design engineering. Unlike the traditional methods that connect pipe terminals sequentially, this article presents a new branch pipe routing algorithm based on the Steiner tree theory. The article begins with a new algorithm for solving the ESMTO problem by using particle swarm optimisation (PSO), and then extends the method to the surface cases by using geodesics to meet the requirements of routing non-rectilinear pipes on the surfaces of aeroengines. Subsequently, the adaptive region strategy and the basic visibility graph method are adopted to increase the computation efficiency. Numeral computations show that the proposed routing algorithm can find satisfactory routing layouts while running in polynomial time.

Solving the euclidean steiner minimum tree using cellular stochastic diffusion search algorithm

The Euclidean Steiner minimum tree problem is a classical NP-hard combinatorial optimization problem. Because of the intrinsic characteristic of the hard computability, this problem cannot be solved accurately by efficient algorithms up to now. Due to the extensive applications in real world, it is quite important to find some heuristics for it. The stochastic diffusion search algorithm is a newly population-based algorithm whose operating mechanism is quite different from ordinary intelligent algorithms, so this algorithm has its own advantage in solving some optimization problems. This paper has carefully studied the stochastic diffusion search algorithm and designed a cellular automata stochastic diffusion search algorithm for the Euclidean Steiner minimum tree problem which has low time complexity. Practical results show that the proposed algorithm can find approving results in short time even for the large scale size, while exact algorithms need to cost several hours.

Traffic-Aware Relay Node Deployment: Maximizing Lifetime for Data Collection Wireless Sensor Networks

Wireless sensor networks have been widely used for ambient data collection in diverse environments. While in many such networks the nodes are randomly deployed in massive quantity, there is a broad range of applications advocating manual deployment. A typical example is structure health monitoring, where the sensors have to be placed at critical locations to fulfill civil engineering requirements. The raw data collected by the sensors can then be forwarded to a remote base station (the sink) through a series of relay nodes. In the wireless communication context, the operation time of a battery-limited relay node depends on its traffic volume and communication range. Hence, although not bounded by the civil-engineering-like requirements, the locations of the relay nodes have to be carefully planned to achieve the maximum network lifetime. The deployment has to not only ensure connectivity between the data sources and the sink, but also accommodate the heterogeneous traffic flows from different sources and the dominating many-to-one traffic pattern. Inspired by the uniqueness of such application scenarios, in this paper, we present an in-depth study on the traffic-aware relay node deployment problem. We develop optimal solutions for the simple case of one source node, both with single and multiple traffic flows. We show however that the general form of the deployment problem is difficult, and the existing only connectivity-guaranteed solutions cannot be directly applied here. We then transform our problem into a generalized version of the Euclidean Steiner Minimum Tree problem (ESMT). Nevertheless, we face further challenges as its solution is in continuous space and may yield fractional numbers of relay nodes, where simple rounding of the solution can lead to poor performance. We thus develop algorithms for discrete relay node assignment, together with local adjustments that yield high-quality practical solutions. Our solution has been evaluated through both numerical analysis and ns-2 simulations and compared with state-of-the-art approaches. The results show that for all test cases where the continuous space optimal solution can be computed within acceptable time frames, the network lifetime achieved by our solution is very close to the upper bound of the optimal solution (the difference is less than 13.5 percent). Moreover, it achieves up to 6-14 times improvement over the existing traffic-oblivious strategies.

Solving the Degree-Constrained Euclidean Steiner Minimal Tree Problem

The degree-constrained Euclidean Steiner minimal tree problem was discussed based on the Euclidean Steiner minimal tree with each original point being added with a degree constraint. The property of the problem was analyzed and the implementation process of solving the problem by using the simulated annealing algorithm and the ant algorithm was presented. Both algorithms are coded in Delphi and run on the Windows XP environment. Series of numerical examples were tested and the efficiency of these algorithms was validated.

Algorithms for degree-constrained Euclidean Steiner minimal tree

A new problem of degree-constrained Euclidean Steiner minimal tree is discussed, which is quite useful in several fields. Although it is slightly different from the traditional degree-constrained minimal spanning tree, it is also NP-hard. Two intelligent algorithms are proposed in an attempt to solve this difficult problem. Series of numerical examples are tested, which demonstrate that the algorithms also work well in practice.

A Sausage Heuristic for Steiner Minimal Trees in Three-Dimensional Euclidean Space

Given a set V of size N≥4 vertices in a metric space, how can one interconnect them with the possible use of a set S of size M vertices not in the set V, but in the same metric space, so that the cumulative cost of the inter-connections between all the vertices is a minimum? When one uses the Euclidean metric to compute these inter-connections, this is referred to as the Euclidean Steiner Minimal Tree Problem. This is an NP-hard problem. The Steiner Ratio ρ of a vertex set is the length of this Steiner Minimal Tree (SMT), divided by the length of the Minimum Spanning Tree (MST), and is a popular and tractable measure of solution quality. The ℛ-Sausage heuristic described in this paper employs a decomposition technique to explore the point set. The fixed vertices of the set are connected to a set of centroid vertices of Delaunay tetrahedrons. The path topology is preserved as far as possible, together with a cycle prevention rule, where junctions, and deviations from the ℛ-Sausage structure occur. Furthermore, repeated sweeps, with different root vertices are accommodated. The computational complexity of the heuristic is shown to be O(N2). Experimental results with thousands of vertices are presented. Comparisons with an exponential running time Branch and Bound algorithm are also shown.

Optimal Steiner hull algorithm

Given a set Z of n points in the plane, we consider the problem of finding the Steiner hull for Z which is a non-trivial polygon containing every Euclidean Steiner minimal tree for Z. We give an optimal Θ( nlog n) time and Θ( n) space algorithm exploiting a Delaunay triangulation of Z. If the Delaunay triangulation is given, the algorithm requires linear time and space. Furthermore, we argue that the uniqueness argument for the O( n 3) time Steiner hull algorithm given in [4] is incorrect, and we give a correct uniqueness proof.

Grade of Service Steiner Minimum Trees in the Euclidean Plane

Abstract. Let P = {p1, p2, \ldots, pn } be a set of n {\lilsf terminal points} in the Euclidean plane, where point pi has a {\lilsf service request of grade} g(pi) ∈ {1, 2, \ldots, n} . Let 0 < c(1) < c(2) < ⋅s < c(n) be n real numbers. The {\lilsf Grade of Service Steiner Minimum Tree (GOSST)} problem asks for a minimum cost network interconnecting point set P and some {\lilsf Steiner points} with a service request of grade 0 such that (1) between each pair of terminal points pi and pj there is a path whose minimum grade of service is at least as large as \min(g(pi), g(pj)) ; and (2) the cost of the network is minimum among all interconnecting networks satisfying (1), where the cost of an edge with service of grade g is the product of the Euclidean length of the edge with c(g) . The GOSST problem is a generalization of the Euclidean Steiner minimum tree problem where all terminal points have the same grade of service request. When there are only two (three, respectively) different grades of service request by the terminal points, we present a polynomial time approximation algorithm with performance ratio \frac 4 3 ρ (((5+4\sqrt 2 )/7)ρ , respectively), where ρ is the performance ratio achieved by an approximation algorithm for the Euclidean Steiner minimum tree problem. For the general case, we prove that there exists a GOSST that is the minimum cost network under a full Steiner topology or its degeneracies. A powerful interior-point algorithm is used to find a (1+e) -approximation to the minimum cost network under a given topology or its degeneracies in O(n1.5(log n + log (1/e))) time. We also prove a lower bound theorem which enables effective pruning in a branch-and-bound method that partially enumerates the full Steiner topologies in search for a GOSST. We then present a k -optimal heuristic algorithm to compute good solutions when the problem size is too large for the branch-and-bound algorithm. Preliminary computational results are presented.

Euclidean Steiner minimal trees, minimum energy configurations, and the embedding problem of weighted graphs in E3

We have found that a triple helix configuration of points in E 3 yields the best value of the Steiner ratio for the Euclidean Steiner Minimal Tree (ESMT) problem. In this paper we explore the properties, configurations, and implications of this topology which yields this best Steiner ratio and its relationship to the Euclidean Graph embedding problem (EGEP) for weighted graphs in E 3. The unique equivalence between these problems is also explored in their application for identification and modelling of minimum energy configurations (MECs) such as the biochemical protein structures of Collagen.

The role of Steiner hulls in the solution to Steiner tree problems

ASteiner tree problem on the plane is that of finding a minimum lengthSteiner tree connecting a given setK ofterminals and lying within a given regionR of the Euclidean plane; it includes as special cases the Euclidean Steiner minimal tree problem (ESMT), the rectilinear Steiner tree problem (RST), and the Steiner tree problem on graphs (STG). ASteiner hull forK inR generically refers to any subregion ofR known to contain a Steiner tree. This paper gives a survey of the role of Steiner hulls in the Steiner tree problem. The significance of Steiner hulls in the efficient solution of Steiner tree problems is outlined, and then a compendium is given of the known Steiner hull constructions for ESMT, RST, and STG problems.

Some upper bounds for minimal trees

This paper introduces two upper bounds for the length of two kinds of minimal trees. The first upper bound, σ(n) < π ([((n − 1) π) 1 2 ] + 1 2 )+ 1 2 , is related to the Euclidean Steiner Minimal Tree obtained from any set of n points enclosed i nside a circumference of radius 1. The second upper bound, Φ(n)⩽2(n − 1) k , is related to the Rectilinear Minimal Tree obtained from any set of n = [(k(k+2)+1) 2] points ( k=1, 2, 3,…),, enclosed in a square with sides equal to 1. Moreover, we also present the only case where the well-known bound σ(n) = 1 + n is attained for the length of the Rectilinear Steiner Minimal Tree for any set of n = t 2 points ,( t=2, 3,…), enclosed in a square with sides equal to 1.

An Improved Algorithm for Steiner Trees

Two basic results for the Euclidean Steiner minimal tree problem are shown: (1) all admissible partitions of the set of nodes (which the tree should span) correspond to degenerate full Steiner topologies, and can be systematically derived from these topologies if degeneracy occurs; (2) for any given full Steiner tree topology there exists a lower bound on the value of the corresponding Steiner tree (this bound holds as an equality if the tree is full). Taken together these two results suggest a relatively efficient (although still exponential) branch and bound algorithm for the Steiner minimal tree construction. The algorithm uses the lower bound to discard some topologies after a brief check, and does away with the need to check all the possible partitions of the node set.

A decomposition theorem on Euclidean Steiner minimal trees

The Euclidean Steiner minimal tree problem is known to be an NP-complete problem and current alogorithms cannot solve problems with more than 30 points. Thus decomposition theorems can be very helpful in extending the boundary of workable problems. There have been only two known decomposition theorems in the literature. This paper provides a 50% increase in the reservoir of decomposition theorems.