Obtaining subtrees from graphs: an ant colony approach

Given an edge-weighted graph and a set of known seed vertices of interest, a network scientist often desires to understand the graph relationships to explain connections between the seed vertices. If the size of the seed set is 2, shortest path calculations are an attractive computational kernel to explore the connections between the two vertices. When the seed set is 3 or larger (say up to 1,000s) Steiner minimal tree – min-weight acyclic connected subgraph (of the input graph) that contains all the seed vertices – is an attractive generalization of shortest weighted paths. In general, computing a Steiner minimal tree is NP-hard, but decades ago several polynomial-time algorithms were designed and proven to yield Steiner trees whose total weight is bounded within 2 times the minimal Steiner tree. Despite its rich theoretical literature, works related to parallel Steiner minimal tree computation and their scalable implementations are rather scarce. In this paper, we present a parallel 2-approximation Steiner minimal tree algorithm (with theoretical guarantees) and its MPI-based distributed implementation. In place of distance computation between all pairs of seed vertices, an expensive phase in many approximation algorithms, the solution we employ, exploits Voronoi cell computation. Also, this approach has higher parallel efficiency than others that involve minimum spanning tree computation on the entire graph. Furthermore, our distributed design exploits asynchronous processing and a message prioritization scheme to accelerate convergence of distance computation, employs techniques to avoid inefficient distributed spanning tree computation on the entire graph, and harnesses a combination of vertex and edge centric processing to offer fast time-to-solution. We demonstrate scalability and performance of our solution using real-world graphs with up to 128 billion edges and 512 compute nodes (8K processes), show the ability to find Steiner trees with up to 10K seed vertices in under one minute, and present in-depth analyses that highlight the benefits of our design choices. Using four real-world graphs and three seed sets for each, we compare our solution with the state-of-the-art exact Steiner minimal tree solver, SCIP-Jack, and two sequential algorithms with the same approximation bound as our algorithm. Our distributed solution comfortably outperforms these related works on graphs with 10s million edges and offers decent strong scaling – up to 90% efficient. We empirically show that, on average, the total distance (sum of edge weights) of the Steiner tree identified by our solution is 1.0527 times greater than the Steiner minimal tree (i.e., the optimal solution) – well within the theoretical bound of less than equal to 2.

The following problems are worth studying: Open problems on the Steiner ratio, such as Chung-Gilbert’s conjecture, Graham-Hwang’s conjecture, and Cielick’s conjecture, etc.. Find better approximation for network Steiner trees and establish an explicit lower bound for the approximation performance ratio of network Steiner trees. Close the gap between the lower bound and the upper bound for the approximation performance ratio of Steiner minimum arborescence. Find more efficient approximation algorithms for on-line and dynamic Steiner minimum trees and various Steiner tree packing problems. Find good approximation algorithms for multi-weighted Steiner trees and multiphase Steiner trees and study close relationship between multi-weighted Steiner trees, multiphase Steiner trees, and phylogenetic trees. Make clear whether there exists a polynomial-time approximation scheme for class Steiner tree in the special case with the real world background and highway interconnection problem. Close the gap between the lower bound and the upper bound for the approximation performance ratio of bottleneck Steiner tree in the Euclidean plane and make clear whether there exists a polynomial-time approximation scheme for the Steiner tree with minimum number of Steiner points and bounded edge-length. Implement efficient approximation algorithms to meet the requests from industries.

In this letter, we address the problem of minimizing the energy consumption of disseminating a library of files to a set of Internet-of-Things (IoT) devices using an unmanned aerial vehicle (UAV). A framework is provided, in which a subset of devices are selected to receive data from a UAV and then forward the required data to other devices. Furthermore, optimal energy-efficient path selection is considered in order to realize efficient data dissemination. Specifically, an optimization problem is formulated to minimize the energy expenditure of the IoT devices and UAV while the latter tours to disseminate the required files to the former. An ant colony optimization (ACO) algorithm is developed to solve the optimization problem. Simulation results show that the proposed framework is more energy-efficient compared to a baseline approach, where the UAV hovers above each device to deliver the data. Results also illustrate that the proposed ACO algorithm provides performance close to the optimal solution, which is obtained through exhaustive search.

The Steiner tree problem is one of the fundamental optimization problems in distributed graph algorithms. Recently Saikia and Karmakar [27] proposed a deterministic distributed algorithm for the Steiner tree problem that constructs a Steiner tree in \(O(S + \sqrt{n} \log ^* n)\) rounds whose cost is optimal upto a factor of \(2(1 - 1/\ell )\), where n and S are the number of nodes and shortest path diameter [17] respectively of the given input graph and \(\ell \) is the number of terminal leaf nodes in the optimal Steiner tree. The message complexity of the algorithm is \(O(\varDelta (n - t) S + n^{3/2})\), which is equivalent to \(O(mS + n^{3/2})\), where \(\varDelta \) is the maximum degree of a vertex in the graph, t is the number of terminal nodes (we assume that \(t < n\)), and m is the number of edges in the given input graph. This algorithm has a better round complexity than the previous best algorithm for Steiner tree construction due to Lenzen and Patt-Shamir [21]. In this paper we present a deterministic distributed algorithm which constructs a Steiner tree in \(\tilde{O}(S + \sqrt{n})\) rounds and \(\tilde{O}(mS)\) messages and still achieves an approximation factor of \(2(1 - 1/\ell )\). Note here that \(\tilde{O}(\cdot )\) notation hides polylogarithmic factors in n. This algorithm improves the message complexity of Saikia and Karmakar’s algorithm by dropping the additive term of \(O(n^{3/2})\) at the expense of a logarithmic multiplicative factor in the round complexity. Furthermore, we show that for sufficiently small values of the shortest path diameter \((S=O(\log n))\), a \(2(1 - 1/\ell )\)-approximate Steiner tree can be computed in \(\tilde{O}(\sqrt{n})\) rounds and \(\tilde{O}(m)\) messages and these complexities almost coincide with the results of some of the singularly-optimal minimum spanning tree (MST) algorithms proposed in [9, 12, 23].

Efficient data collection in wireless sensor networks (SNs) plays a key role in power conservation. It has spurred a number of research projects focusing on effective algorithms that reduce power consumption with effective in-network aggregation techniques. Up to now, most approaches are based on the assumption that data collection involves all nodes of a network. There is a large number of queries that in fact select only a subset of the nodes in a SN. Thus, we concentrate on selective queries, i.e., queries that request data from a subset of a SN. The task of optimal data collection in such queries is an instance of the NP-hard minimal Steiner tree problem. We argue that selective queries are an important class of queries that can benefit from algorithms that are tailored for partial node participation of a SN. We present an algorithm, called Pocket Driven Trajectories (PDT), that optimizes the data collection paths by approximating the global minimal Steiner tree using solely local spatial knowledge. We identify a number of spatial factors that play an important role for efficient data collection, such as the distribution of participating nodes over the network, the location and dispersion of the data clusters, the location of the sink issuing a query, as well as the location and size of communication holes. In a series of experiments, we compare performance of well-known algorithms for aggregate query processing against the PDT algorithm in partial node participation scenarios. To measure the efficiency of all algorithms, we also compute a near-optimal solution, the globally approximated minimal Steiner tree. We outline future research directions for selective queries with varying node participation levels, in particular scenarios in which node participation is the result of changing physical phenomena as well as reconfigurations of the SN itself.KeywordsSensor NetworkSensor NodeWireless Sensor NetworkSteiner TreeSensor ReadingThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Boolean satisfiability (SAT) is a well-studied N P-complete problem for formulating and solving other combinatorial problems like planning and scheduling. The Steiner tree problem (STP) asks to find a minimal cost tree in a graph that spans a set of nodes. STP has been shown to be N P-hard. In this paper, we propose to solve the STP by formulating it as a variation of SAT, and to solve it using a heuristic search method guided by the backbone of the problem. The algorithm is tested on a well known set of benchmark instances. Experimental results demonstrate the applicability of the proposed approach, and show that substantial quality improvement can be obtained compared to other heuristic methods.

In this paper, an algorithm based ant colony approach (ACA) is designed to find the optimal configuration for redundancy apportionment problem (RAP) of series-parallel multi-state system (MSS). In order to achieve maximum system reliability under cost and performance constraints, the algorithm takes advantage of ACA’s combinatorial optimization, uses heuristic information about reliability to arrange components from available choice. Different from traditional nominal performance and reliability system, universal generating function (UGF) is introduced and used to estimate the multi-state system reliability. Finally the experiment has shown that the presented algorithm can provide optimal and near optimal solutions, and have efficient and convenient calculation performance.

Ant colony optimization (ACO) is a well-known heuristic intelligent method which is used to solve combinatorial optimization problem and can obtain high-quality results in a reasonable execution time. Steiner Tree Problem (STP) is a classic NPcomplete problem. The computational complexity of the exact algorithm increases exponentially with the increment of the size of the problem. In this paper, we propose an ant colony optimization algorithm with data parallelism to solve the STP. The simulation results show that the algorithm has good performance in result quality.

In this paper, we study the interconnect layout optimization problem under a higher order resistance-inductance-capacitance model to optimize not only delay, but also waveform for interconnects with nonmonotone signal response in the context of multichip-module global routing. We propose a unified approach that considers topology optimization and waveform optimization simultaneously. Using a new incremental moment-computation algorithm, we interleave topology construction with moment computation to facilitate accurate delay calculation and evaluation of waveform quality. Our algorithm considers a large class of routing topologies, ranging from shortest path Steiner trees to bounded-radius Steiner trees and Steiner routings. We construct a set of required arrival-time Steiner (RATS) trees, providing smooth tradeoffs among signal delay, waveform, and routing area. When combined with the MINOTAUR MCM global router (Cong and Madden, 1998), (Madden, 1998) that we have developed, the RATS-tree solutions prove to be effective in reducing overall routing congestion.

Finding a shortest network interconnecting a given set of points in a metric space is called the Steiner minimum tree problem. The Steiner ratio is the largest lower bound for the ratio between lengths of a Steiner minimum tree and a minimum spanning tree for the same set of points. In this paper, we show that in a metric space, if the Steiner ratio is less than one and finding a Steiner minimum tree for a set of size bounded by a fixed number can be performed in polynomial time, then there exists a polynomialtime heuristic for the Steiner minimum tree problem with performance ratio bigger than the Steiner ratio. It follows that in the Euclidean plane, there exists a polynomial-time heuristic for Steiner minimum trees with performance ratio bigger than $${\textstyle{1 \over 2}}\sqrt 3 $$ . This solves a long-standing open problem.

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.

This paper describes and uses an ant colony meta-heuristic optimization method to solve the redundancy optimization problem. This problem is known as total investment-cost minimization of series-parallel power system configuration. Redundant components are included to achieve a desired level of availability. System availability is represented by a multi-state availability function. The power systems components are characterized by their performance (capacity), availability and cost. These components are chosen among a list of products available on the market. The proposed meta-heuristic seeks to the best minimal cost power system configuration with desired availability. To estimate the series-parallel power system availability, a fast method based on universal moment generating function (UMGF) is suggested. The ant colony approach is used as an optimization technique. An example of electrical power system is presented.

Particle swarm optimizer, ant colony strategy and harmony search scheme hybridized for optimization of truss structures

A Steiner Minimal Tree (SMT) for a given set P of points is a shortest network interconnecting the points of P whose vertex set may include some additional points in order to get the minimum possible total length in a metric space. When no additional points are allowed the minimum interconnection network is the well-known minimum spanning tree (MST) of P. The Steiner ratio is the greatest lower bound of the ratio of the length of an SMT over that of an MST of P. In this paper we study the Steiner minimal tree problem in which all the edges of SMT have fixed orientations. We call it the SMT problem in the λ-geometry plane, where λ is the number of possible orientations.Here is the summary of our results. 1. We show that the Steiner ratio for ¦P¦≥3 is √3/2 cos(π/2λ), for λ=6m+3 and integer m≥0, and is √3/2, for λ=6k and integer k≥1, disproving a a conjecture of Du et al.[3] that the ratio is √3/2 iff the unit disk in normed planes is an ellipse. 2. We derive the Steiner ratios for ¦P¦ ≤ 4 for all possible λ's and show that for ¦P¦≥3 there exists an SMT whose Steiner points lie in a multi-level Hanan-grid, generalizing a result that holds for rectilinear case, i.e., λ=2. These results show that the Steiner ratio is not a monotonically increasing function of λ, as believed by many researchers. We conjecture that the Steiner ratios obtained above (¦P¦≤4) are actually true for all ¦P¦≥3.KeywordsMinimum Span TreeSteiner TreeRegular PointSteiner PointNormed PlaneThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Steiner tree problem (STP) in graphs aims to find a tree of minimum weight in the graph that connects a given set of vertices. It is a classic NP-hard combinatorial optimization problem and has many applications in real world. Many approximate algorithms have been developed for STP, but they suffer from high computational complexity and weak worst-case solution guarantees, respectively. Heuristic algorithms are also developed. But each of them requires application domain knowledge to design and is only suitable for specific scenarios. In this paper, we design a novel framework Cherrypick based on novel graph embedding and deep reinforcement learning to tackle the STP. Given an STP instance, Cherrypick uses this embedding to encode its path-related information and sends the encoded graph to a deep reinforcement learning component based on deep Q network (DQN) to find solutions. We implement the Cherrypick and demonstrate its efficacy and efficiency with extensive experiments using real-world and synthetic datasets.