Parallel Minimum Spanning Tree Heuristic for the steiner problem in graphs

Akbari, Hoda; Iranmanesh, Z.; Ghodsi, Mohammad

doi:10.1109/icpads.2007.4447726

Cited by 4 publications

(3 citation statements)

References 17 publications

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“…We are not aware of any parallel, distributed solution that can compete with the scalability demonstrated in this paper. Several papers discuss parallel, distributed Steiner tree computation from the theoretical perspective only [41], [47]- [49]. PARSTEINER94 [50], algorithmically similar to GeoSteiner, is an early parallel exact solver.…”

Section: Related Workmentioning

confidence: 99%

Towards Distributed 2-Approximation Steiner Minimal Trees in Billion-edge Graphs

Reza¹,

Sanders²,

Pearce³

2022

Preprint

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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 2approximation 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.

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Section: Related Workmentioning

confidence: 99%

Towards Distributed 2-Approximation Steiner Minimal Trees in Billion-edge Graphs

Reza¹,

Sanders²,

Pearce³

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Numerous other generalizations with their applications are described in [40][41][42][43][44][45][46]; however, we omit them from the examination, since they are quite specific and mostly devised to suit a very limited area of applications. They are of little interest to us since almost none of them received a parallel or distributed implementation.…”

Section: -The Obstacle Avoiding Steiner Tree Problemmentioning

confidence: 99%

“…The algorithm of Akbari et al [40] is based on the DNH. They selected a powerful scalable network architecture called the three-dimensional mesh of trees [101] in order for the algorithm to achieve an execution time of O(log 2 n) when run on O(n 3 ) processors.…”

Section: Parallel Mst Heuristic By Akbari Et Al (Stpg)mentioning

confidence: 99%

A Survey of Parallel and Distributed Algorithms for the Steiner Tree Problem

Bezenšek

Robič

2013

Int J Parallel Prog

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Given a set of input points, the Steiner Tree Problem (STP) is to find a minimum-length tree that connects the input points, where it is possible to add new points to minimize the length of the tree. Solving the STP is of great importance since it is one of the fundamental problems in network design, very large scale integration routing, multicast routing, wire length estimation, computational biology, and many other areas. However, the STP is NP-hard, which shatters any hopes of finding a polynomial-time algorithm to solve the problem exactly. This is why the majority of research has looked at finding efficient heuristic algorithms. Additionally, many authors focused their work on utilizing the ever-increasing computational power and developed many parallel and distributed methods for solving the problem. In this way we are able to obtain better results in less time than ever before. Here, we present a survey of the parallel and distributed methods for solving the STP and discuss some of their applications.

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Distributed approximate minimal Steiner trees with millions of seed vertices on billion-edge graphs

Reza

Steil

Sanders

et al. 2023

Journal of Parallel and Distributed Computing

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Parallel Minimum Spanning Tree Heuristic for the steiner problem in graphs

Abstract: Given an undirected graph with weights associated with its edges, the Steiner tree problem consists of finding a minimum weight subtree spanning a given subset of (terminal)

Cited by 4 publications

References 17 publications

Towards Distributed 2-Approximation Steiner Minimal Trees in Billion-edge Graphs

Towards Distributed 2-Approximation Steiner Minimal Trees in Billion-edge Graphs

A Survey of Parallel and Distributed Algorithms for the Steiner Tree Problem

Distributed approximate minimal Steiner trees with millions of seed vertices on billion-edge graphs

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