A Genetic Algorithm Approach to Euclidean Steiner Tree Problem
Pages 71 - 75
Abstract
We propose a genetic algorithm-based method for solving the Euclidean Steiner Tree Problem and show its effectiveness through experiments using some benchmark problems. Given a finite number of terminal points, the proposed method first generates many non-terminal points by using a simple rule, and initializes a set of chromosomes, each of which corresponds to a part of the non-terminal points generated. It then runs a genetic algorithm to find a chromosome with the highest fitness value, where the fitness value of a chromosome is determined based on the total edge length of the minimum spanning tree obtained from all terminal points and the non-terminal points corresponding to the chromosome. It finally optimizes the locations of the non-terminal points in the minimum spanning tree.
References
[1]
John E Beasley. 1990. OR-Library: distributing test problems by electronic mail. Journal of the Operational Research Society 41, 11 (1990), 1069–1072. https://people.brunel.ac.uk/%7Emastjjb/jeb/info.html.
[2]
Amir Beck and Shoham Sabach. 2015. Weiszfeld’s method: Old and new results. Journal of Optimization Theory and Applications 164 (2015), 1–40.
[3]
Marcus Brazil, Ronald L Graham, Doreen A Thomas, and Martin Zachariasen. 2014. On the history of the Euclidean Steiner tree problem. Archive for History of Exact Sciences 68 (2014), 327–354.
[4]
Guan-Qiang Dong, Zong-Xiao Yang, Lei Song, Kun Ye, and Gen-Sheng Li. 2015. The global shortest path visualization approach with obstructions. Journal of Robotics and Mechatronics 27, 5 (2015), 579–585.
[5]
Derek R Dreyer and Michael L Overton. 1998. Two heuristics for the Euclidean Steiner tree problem. Journal of Global Optimization 13 (1998), 95–106.
[6]
Marcia Fampa and Kurt M Anstreicher. 2008. An improved algorithm for computing Steiner minimal trees in Euclidean d-space. Discrete Optimization 5, 2 (2008), 530–540.
[7]
Michael R Garey, Ronald L Graham, and David S Johnson. 1977. The complexity of computing Steiner minimal trees. SIAM J. Appl. Math. 32, 4 (1977), 835–859.
[8]
Edgar N Gilbert and Henry O Pollak. 1968. Steiner minimal trees. SIAM J. Appl. Math. 16, 1 (1968), 1–29.
[9]
David E. Goldberg. 1989. Genetic Algorithms in Search, Optimization and Machine Learning (1st ed.). Addison-Wesley, Reading, MA, USA.
[10]
John H Holland. 1992. Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence. MIT press.
[11]
Sheryl Hsu, Fidel I Schaposnik Massolo, and Laura P Schaposnik. 2022. A Physarum-inspired approach to the Euclidean Steiner tree problem. Scientific Reports 12, 1 (2022), 14536:1–13.
[12]
Daniel Juhl, David M Warme, Pawel Winter, and Martin Zachariasen. 2018. The GeoSteiner software package for computing Steiner trees in the plane: an updated computational study. Mathematical Programming Computation 10 (2018), 487–532.
[13]
RC Prim. 1957. Shortest connection matrix network and some generalisations. Bell System Technical Journal 36, 6 (1957), 1389–1401.
[14]
J MacGregor Smith, DT Lee, and Judith S Liebman. 1981. An O(nlog n) heuristic for Steiner minimal tree problems on the Euclidean metric. Networks 11, 1 (1981), 23–39.
[15]
Jon W Van Laarhoven and Kurt M Anstreicher. 2013. Geometric conditions for Euclidean Steiner trees in \(\mathbb {R}^d\). Computational Geometry 46, 5 (2013), 520–531.
[16]
Jon W Van Laarhoven and Jeffrey W Ohlmann. 2011. A randomized Delaunay triangulation heuristic for the Euclidean Steiner tree problem in \(\mathbb {R}^d\). Journal of Heuristics 17 (2011), 353–372.
[17]
Endre Weiszfeld. 1937. Sur le point pour lequel la somme des distances de n points donnés est minimum. Tohoku Mathematical Journal, First Series 43 (1937), 355–386.
[18]
Endre Weiszfeld and Frank Plastria. 2009. On the point for which the sum of the distances to n given points is minimum. Annals of Operations Research 167 (2009), 7–41.
[19]
Pawel Winter. 1985. An algorithm for the Steiner problem in the Euclidean plane. Networks 15, 3 (1985), 323–345.
[20]
Pawel Winter and Martin Zachariasen. 1997. Euclidean Steiner minimum trees: An improved exact algorithm. Networks: An International Journal 30, 3 (1997), 149–166.
[21]
Zong-Xiao Yang, Xiao-Yao Jia, Jie-Yu Hao, Yan-Ping Gao, et al. 2013. Geometry-experiment algorithm for Steiner minimal tree problem. Journal of Applied Mathematics 2013 (2013), 1–10.
Index Terms
- A Genetic Algorithm Approach to Euclidean Steiner Tree Problem
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Published In
September 2024
93 pages
ISBN:9798400717031
DOI:10.1145/3702468
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Published: 16 December 2024
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- Japan Society for the Promotion of Science KAKENHI
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ICRSA 2024
ICRSA 2024: 2024 7th International Conference on Robot Systems and Applications
September 12 - 14, 2024
Bangkok, Thailand
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