Abstract
Minimal-length Steiner trees in the two-dimensional Euclidean domain are of special interest to enable the efficient coordination of multi-agent and interconnected systems. We propose an approach to compute obstacle-avoiding Steiner trees by using the hybrid between hierarchical optimization of shortest routes through sequential quadratic programming over constrained two-dimensional convex domains, and the gradient-free stochastic optimization algorithms with a convex search space. Our computational experiments involving 3,000 minimal tree planning scenarios in maps with convex and non-convex obstacles show the feasibility and the efficiency of our approach. Also, our comparative study involving relevant classes of gradient-free and nature inspired heuristics has shed light on the robustness of the selective pressure and exploitation abilities of the Dividing Rectangles (DIRECT), the Rank-based Differential Evolution (RBDE) and the Differential Evolution with Successful Parent Selection (DESPS). Our approach offers the cornerstone mechanisms to further advance towards developing efficient network optimization algorithms with flexible and scalable representations.
Keywords
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Olfati-Saber, R.: Flocking for multi-agent dynamic systems: algorithms and theory. IEEE Trans. Autom. Control 51, 401–420 (2006)
Li, A., Wang, L., Pierpaoli, P., Egerstedt, M.: Formally correct composition of coordinated behaviors using control barrier certificates. In: 2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 3723–3729 (2018)
Luo, W., Khatib, S.S., Nagavalli, S., Chakraborty, N., Sycara, K.: Distributed knowledge leader selection for multi-robot environmental sampling under bandwidth constraints. In: 2016 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 5751–5757 (2016)
de Fermat, P.: Method for determining maxima and minima and tangents to curved lines. Oeuvres 1, 135 (1643)
Vojtěch, J., Kössler, M.: On minimal graphs containing n given points. Časopis pro pěstování matematiky a fysiky 63, 223–235 (1934). (in Czech). Zbl 0009.13106
Robbins, H., Courant, R.: What is Mathematics? Oxford University Press, New York (1941)
Winter, P., MacGregor Smith, J.: Steiner minimal trees for three points with one convex polygonal obstacle. Ann. Oper. Res. 33, 577–599 (1991). https://doi.org/10.1007/BF02067243
Winter, P.: Euclidean Steiner minimal trees with obstacles and Steiner visibility graphs. Discret. Appl. Math. 47, 187–206 (1993)
Zachariasen, M., Winter, P.: Obstacle-avoiding Euclidean Steiner trees in the plane: an exact algorithm. In: Goodrich, M.T., McGeoch, C.C. (eds.) ALENEX 1999. LNCS, vol. 1619, pp. 286–299. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-48518-X_17
Weng, J.F., MacGregor Smith, J.: Steiner minimal trees with one polygonal obstacle. Algorithmica 29, 638–648 (2001). https://doi.org/10.1007/s00453-001-0002-1
Winter, P., Zachariasen, M., Nielsen, J.: Short trees in polygons. Discret. Appl. Math. 118, 55–72 (2002)
Müller-Hannemann, M., Tazari, S.: A near linear time approximation scheme for Steiner tree among obstacles in the plane. Comput. Geom. Theory Appl. 43, 395–409 (2010)
Borradaile, G., Kenyon-Mathieu, C., Klein, P.: A polynomial-time approximation scheme for Steiner tree in planar graphs. In: ACM-SIAM Symposium on Discrete Algorithms, pp. 1285–1294 (2007)
Cohen, N., Nutov, Z.: Approximating Steiner trees and forests with minimum number of Steiner points. J. Comput. Syst. Sci. 98, 53–64 (2018)
Chen, B., Chen, H., Wu, C.: Obstacle-avoiding connectivity restoration based on quadrilateral Steiner tree in disjoint wireless sensor networks. IEEE Access 7, 124116–124127 (2019)
Caleffi, M., Akyildiz, I.F., Paura, L.: On the solution of the Steiner tree np-hard problem via Physarum bionetwork. IEEE/ACM Trans. Network. 23, 1092–1106 (2015)
Sun, Y., Halgamuge, S.: Fast algorithms inspired by Physarum polycephalum for node weighted Steiner tree problem with multiple terminals. In: IEEE Congress on Evolutionary Computation, pp. 3254–3260 (2016)
Camacho-Vallejo, J.F., Garcia-Reyes, C.: Co-evolutionary algorithms to solve hierarchized Steiner tree problems in telecommunication networks. Appl. Soft Comput. 84, 105718 (2019)
Parque, V., Miyashita, T.: Obstacle-avoiding Euclidean Steiner trees by n-star bundles. In: IEEE 30th International Conference on Tools with Artificial Intelligence, pp. 315–319 (2018)
Chuong, T.V., Nam, H.H.: A variable neighborhood search algorithm for solving the Steiner minimal tree problem. In: Cong Vinh, P., Alagar, V. (eds.) ICCASA/ICTCC -2018. LNICST, vol. 266, pp. 218–225. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-06152-4_19
Lai, X., Zhou, Y., Xia, X., Zhang, Q.: Performance analysis of evolutionary algorithms for Steiner tree problems. Evol. Comput. 25, 707–723 (2017)
Chen, X., Liu, G., Xiong, N., Su, Y., Chen, G.: A survey of swarm intelligence techniques in VLSI routing problems. IEEE Access 8, 26266–26292 (2020)
Tan, W.C., Chen, I., Pantazis, D., Pan, S.J.: Transfer learning with PipNet: for automated visual analysis of piping design. In: 2018 IEEE 14th International Conference on Automation Science and Engineering (CASE), pp. 1296–1301 (2018)
Liu, Q., Wang, C.: Multi-terminal pipe routing by Steiner minimal tree and particle swarm optimisation. Enterp. Inf. Syst. 6, 315–327 (2012)
Liu, G., Guo, W., Niu, Y., Chen, G., Huang, X.: A PSO-based timing-driven octilinear Steiner tree algorithm for VLSI routing considering bend reduction. Soft. Comput. 19, 1153–1169 (2015)
Huang, X., Liu, G., Guo, W., Niu, Y., Chen, G.: Obstacle-avoiding algorithm in X-architecture based on discrete particle swarm optimization for VLSI design. ACM Trans. Des. Autom. Electron. Syst. 20, 1–28 (2015)
Sui, H., Niu, W.: Branch-pipe-routing approach for ships using improved genetic algorithm. Front. Mech. Eng. 11, 316–323 (2016). https://doi.org/10.1007/s11465-016-0384-z
Niu, W., Sui, H., Niu, Y., Cai, K., Gao, W.: Ship pipe routing design using NSGA-II and coevolutionary algorithm. Math. Probl. Eng. 2016, 1–21 (2016)
Liu, L., Liu, Q.: Multi-objective routing of multi-terminal rectilinear pipe in 3D space by MOEA/D and RSMT. In: 2018 3rd International Conference on Advanced Robotics and Mechatronics (ICARM), pp. 462–467 (2018)
Jiang, W.Y., Lin, Y., Chen, M., Yu, Y.Y.: A co-evolutionary improved multi-ant colony optimization for ship multiple and branch pipe route design. Ocean Eng. 102, 63–70 (2015)
Ztopuoianu, A.C., et al.: Multi-objective optimal design of obstacle-avoiding two-dimensional Steiner trees with application to ascent assembly engineering. J. Mech. Des. 140, 061401-1–061401-11 (2018)
Wu, H., Xu, S., Zhuang, Z., Liu, G.: X-architecture Steiner minimal tree construction based on discrete differential evolution. In: Liu, Y., Wang, L., Zhao, L., Yu, Z. (eds.) ICNC-FSKD 2019. AISC, vol. 1074, pp. 433–442. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-32456-8_47
Byrd, R., Gilbert, J., Nocedal, J.: A trust region method based on interior point techniques for nonlinear programming. Math. Program. 89(1), 149–185 (2000). https://doi.org/10.1007/PL00011391
Parque, V., Miyashita, T.: Bundling n-Stars in polygonal maps. In: 29th IEEE International Conference on Tools with Artificial Intelligence, ICTAI 2017, Boston, MA, USA, 6–8 November 2017, pp. 358–365 (2017)
Zăvoianu, A.-C., et al.: On the optimization of 2D path network layouts in engineering designs via evolutionary computation techniques. In: Andrés-Pérez, E., González, L.M., Periaux, J., Gauger, N., Quagliarella, D., Giannakoglou, K. (eds.) Evolutionary and Deterministic Methods for Design Optimization and Control With Applications to Industrial and Societal Problems. CMAS, vol. 49, pp. 307–322. Springer, Cham (2019). https://doi.org/10.1007/978-3-319-89890-2_20
Storn, R., Price, K.: Differential evolution - a simple and efficient heuristic for global optimization over continuous spaces. J. Glob. Optim. 11, 341–359 (1997). https://doi.org/10.1023/A:1008202821328
Das, S., Abraham, A., Chakraborty, U.K., Konar, A.: Differential evolution using a neighborhood-based mutation operator. IEEE Trans. Evol. Comput. 13, 526–553 (2009)
Sutton, A.M., Lunacek, M., Whitley, L.D.: Differential evolution and non-separability: using selective pressure to focus search. In: The Genetic and Evolutionary Computation Conference (GECCO), pp. 1428–1435 (2007)
Guo, S., Yang, C., Hsu, P., Tsai, J.: Improving differential evolution with a successful-parent-selecting framework. IEEE Trans. Evol. Comput. 19(5), 717–730 (2015)
Qu, B., Liang, J., Suganthan, P.: Niching particle swarm optimization with local search for multi-modal optimization. Inf. Sci. 197, 131–143 (2012)
Jones, D.R.: Direct global optimization algorithm. In: Floudas, C.A., Pardalos, P.M. (eds.) Encyclopedia of Optimization, pp. 431–440. Springer, Boston (2001). https://doi.org/10.1007/0-306-48332-7_93
Parque, V., Miura, S., Miyashita, T.: Computing path bundles in bipartite networks. In: Proceedings of the 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications, pp. 422–427 (2017)
Parque, V., Miyashita, T.: Numerical representation of modular graphs. In: IEEE 42nd Annual Computer Software and Applications Conference, pp. 819–820 (2018)
Parque, V., Miyashita, T.: On the numerical representation of labeled graphs with self-loops. In: 29th IEEE International Conference on Tools with Artificial Intelligence, pp. 342–349 (2017)
Parque, V., Miyashita, T.: On succinct representation of directed graphs. In: IEEE International Conference on Big Data and Smart Computing, pp. 199–205 (2017)
Parque, V., Miyashita, T.: On graph representation with smallest numerical encoding. In: IEEE 42nd Annual Computer Software and Applications Conference, pp. 817–818 (2018)
Parque, V., Suzaki, W., Miura, S., Torisaka, A., Miyashita, T., Natori, M.: Packaging of thick membranes using a multi-spiral folding approach: flat and curved surfaces. Adv. Space Res. (2020, in press). https://doi.org/10.1016/j.asr.2020.09.040
Acknowledgment
This research was supported by JSPS KAKENHI Grant Number 20K11998.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Parque, V. (2021). On Hybrid Heuristics for Steiner Trees on the Plane with Obstacles. In: Zarges, C., Verel, S. (eds) Evolutionary Computation in Combinatorial Optimization. EvoCOP 2021. Lecture Notes in Computer Science(), vol 12692. Springer, Cham. https://doi.org/10.1007/978-3-030-72904-2_8
Download citation
DOI: https://doi.org/10.1007/978-3-030-72904-2_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-72903-5
Online ISBN: 978-3-030-72904-2
eBook Packages: Computer ScienceComputer Science (R0)