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A PSO-based timing-driven Octilinear Steiner tree algorithm for VLSI routing considering bend reduction

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Abstract

Constructing a timing-driven Steiner tree is very important in VLSI performance-driven routing stage. Meanwhile, non-Manhattan architecture is supported by several manufacturing technologies and now well appreciated in the chip manufacturing circle. However, limited progress has been reported on the non-Manhattan performance-driven routing problem. In this paper, an efficient algorithm, namely, TOST_BR_MOPSO, is presented to construct the minimum-cost spanning tree with a minimum radius for performance-driven routing in Octilinear architecture (one type of the non-Manhattan architecture) based on multi-objective particle swarm optimization (MOPSO) and Elmore delay model. Edge transformation is employed in our algorithm to make the particles have the ability to achieve the optimal solution while Union-Find partition is used to prevent the generation of invalid solution. For the purpose of reducing the number of bends which is one of the key factors of chip manufacturability, we also present an edge-vertex encoding strategy combined with edge transformation. To our best knowledge, no approach has been proposed to optimize the number of bends in the process of constructing the non-Manhattan timing-driven Steiner tree. Moreover, the theorem of Markov chain is used to prove the global convergence of our proposed algorithm. Experimental results indicate that the proposed MOPSO is worthy of being studied in the field of multi-objective optimization problems, and our algorithm has a better tradeoff between the wire length and radius of the routing tree and has achieved a better delay value. Meanwhile, combining edge transformation with the encoding strategy, the proposed algorithm can significantly reduce nearly 20 % in the number of bends.

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References

  • Agrawal S, Silakari S (2013) FRPSO: Fletcher–Reeves based particle swarm optimization for multimodal function optimization. Soft Comput 1–17

  • Alfred VA, John EH, Jeffrey U (1983) Data structures and algorithms. Addison-Wesley Longman Publishing, Boston

    MATH  Google Scholar 

  • Arora T, Mose ME (2009) Ant colony optimization for power efficient routing in manhattan and non-manhattan VLSI architectures. In: Swarm intelligence symposium, pp 137–144

  • Balling R (2003) The maximin fitness function: multiobjective city and regional planning. Proceedings of the 2nd international conference on evolutionary multi-criterion optimization. Faro, Portugal, pp 1–15

    Chapter  Google Scholar 

  • Beasley JE (1990) OR-Library: distributing test problems by electronic Mail. J Oper Res Soc 41(11):1069–1072

    Article  Google Scholar 

  • Boese KD, Kahng AB, Robins G (1993) Near optimal critical sink routing tree constructions. In: Proceedings of the ACM/IEEE design automation conference, pp 182–187

  • Borah M, Owens RM, Irwin MJ (1994) An edge-based heuristic for Steiner routing. IEEE Trans Comput Aided Design 13(12):1563–1568

    Article  Google Scholar 

  • Bozorgzadeh E, Kastner R, Sarrafzadeh M (2003) Creating and exploiting flexibility in rectilinear Steiner trees. IEEE Trans Comput Aided Design 22(5):605–615

    Article  Google Scholar 

  • Carlos ACC, Gregorio TP, Maximino SL (2004) Handling multiple objectives with particle swarm optimization. IEEE Trans Evol Comput 8(3):256–279

    Article  Google Scholar 

  • Chen G, Guo W, Chen Y (2010) A PSO-based intelligent decision algorithm for VLSI floorplanning. Soft Comput 14(12):1329–1337

    Article  Google Scholar 

  • Chiang C, Chiang CS (2002) Octilinear steiner tree construction. In: Proceedings of the 45th midwest symposium on circuits and systems, pp 603–606

  • Chu C, Wong YC (2008) FLUTE: fast lookup table based rectilinear Steiner minimal tree algorithm for VLSI design. IEEE Trans Comput Aided Design 27(1):70–83

    Article  Google Scholar 

  • Clerc M (2004) Discrete particle swarm optimization, illustrated by the traveling salesman problem. In: Onwubolu GC, Babu BV (eds) New optimization techniques in engineering. Springer, Berlin, pp 219–239

    Chapter  Google Scholar 

  • Cong J, Kahng AB, Robins G, Sarrafzadeh M, Wong CK (1992) Provably good performance-driven global routing. IEEE Trans Comput Aided Design 11:739–752

    Article  Google Scholar 

  • Conover WJ (1999) Practical nonparametric statistics. Wiley, New York

    Google Scholar 

  • Costas V, Konstantinos E, Isaac E (2012) Particle swarm optimization with deliberate loss of information. Soft Comput 16(8):1373–1392

    Article  Google Scholar 

  • Coulston G (2003) Constructing exact octagonal steiner minimal trees. In: Proceedings of the 13th ACM Great Lakes symposium on VLSI, pp 1–6

  • Eberhar RC, Kennedy J (1995) A new optimizer using particles swarm theory. In: Proceedings of the 6th international symposium on micro machine and human science, Nagoya, pp 39–43

  • Garey M, Johnson D (1997) The rectilinear steiner tree problem is NP-complete. SIAM J Appl Math 32:826–834

    Article  MathSciNet  Google Scholar 

  • Guo W, Park JH, Yang LT, Vasilakos AV, Xiong N, Chen G (2011) Design and analysis of a MST-based topology control scheme with PSO for wireless sensor networks. 2011 IEEE Asia-Pacific services computing conference. IEEE, Jeju Island, pp 360–367

  • Guo W, Xiong N, Vasilakos AV, Chen G, Yu C (2012) Distributed k-connected fault-tolerant topology control algorithms with PSO in future autonomic sensor systems. Int J Sens Netw 12(1):53–62

    Article  Google Scholar 

  • Julstrom BA (2001) Encoding rectilinear Steiner trees as lists of edges. In: Proceeding of the 2001 ACM symposium on applied computing, New York, pp 356–360

  • Ho TY, Chang YW, Chen SJ (2007) Full-chip nanometer routing techniques. Springer, Berlin

    Book  Google Scholar 

  • Hou H, Hu J, Sapatnekar SS (1999) Non-hanan routing. IEEE Trans Comput Aided Des 18(4):436–444

    Article  Google Scholar 

  • Hu J, Sapatnekar S (2001) A survey on multi-net global routing for integrated circuits. Inter VLSI J 31(1):1–49

    Article  MATH  Google Scholar 

  • Kennedy J, Eberhart RC (1997) A discrete binary version of the particle swarm algorithm. In: Proceedings of the world multiconference on systemics, cybernetics and informatics, Piscataway, pp 4104–4109

  • Koh CK, Madden PH (2000) Manhattan or non-manhattan? a study of alternative VLSI routing architectures. In: Proceedings of Great Lake symposium on VLSI, pp 47–52

  • Laumanns M, Thiele L, Deb K, Zitzler E (2002) Combining convergence and diversity in evolutionary multi-objective optimization. Evol Comput 10(3):263–282

    Article  Google Scholar 

  • Liang J, Hong X, Jing T (2007) G-Tree: gravitation-direction-based rectilinear Steiner minimal tree construction considering bend reduction. Proceedings of the 7th international conference on ASIC. IEEE, Guilin, pp 1114–1117

  • Liu G, Chen G, Guo W (2012) DPSO based octagonal steiner tree algorithm for VLSI routing. 2012 IEEE fifth international conference on advanced computational intellligence. IEEE, Nanjing, pp 383–387

  • Liu G, Chen G, Guo W, Chen Z (2011) DPSO-based rectilinear steiner minimal tree construction considering bend reduction. In: Proceedings of the 7th international conference on natural computation, pp 1161–1165

  • Liu H, Cai Z, Wang Y (2010) Hybridizing particle swarm optimization with differential evolution for constrained numerical and engineering optimization. Appl Soft Comput 10(2):629–640

    Article  Google Scholar 

  • Pan QK, Tasgetiren MF, Liang YC (2006) A discrete particle swarm optimization algorithm for the permutation flowshop sequecing problem with makespan criteria. In: Proceedings of the 26th SGAI international conference on innovative techniques and applications of artificial intelligence, Cambridge, pp 19–31

  • Peng S, Chen G, Guo W (2010) A multi-objective algorithm based on discrete PSO for VLSI partitioning problem. In: Proceedings of the 2nd international conference on quantitative logic and soft computing, Jimei, pp 651–660

  • Rada-Vilela J, Zhang M, Seah W (2013) A performance study on synchronicity and neighborhood size in particle swarm optimization. Soft Comput 17(6):1019–1030

    Article  Google Scholar 

  • Ratnaweera A, Halgamuge SK, Watson HC (2004) Self-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficients. IEEE Trans Evol Comput 8(3):240–255

    Article  Google Scholar 

  • Samanta T, Ghosal P, Rahaman H, Dasgupta PS (2006) A heuristic methiod for constructing hexagonal steiner minimal trees for routing in VLSI. In: 2006 IEEE international symposium on circuits and systems, pp 1788–1791

  • Samanta T, Rahaman H, Dasgupta PS (2011) Near-optimal Y-routed delay trees in nanometric interconnect design. IET Comput Digital Tech 5(1):36–48

    Article  Google Scholar 

  • Sarrafzadeh M, Feng LK, Wong CK (1994) Single-layer global routing. IEEE Trans Comput Aided Design 13(1):38–47

    Article  Google Scholar 

  • Seo DY, Lee DT (1999) On the complexity of bicriteria spanning tree problems for a set of points in the plane. PhD Dissertation, Northwestern University

  • Shen Y, Liu Q, Guo W (2011) Obstacle-avoiding rectilinear steiner minimum tree construction based on discrete particle swarm optimization. In: Proceedings of the 2011 seventh international conference on natural computation, pp 2179–2183

  • Shi YH, Eberhart RC (1998) A modified particle swarm optimizer. In: Proceedings of the IEEE international conference of evolutionary computation, Piscataway, pp 69–73

  • Deb K, Pratap A, Agarwal S, Meyarivan T (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 6(2):182–197

    Article  Google Scholar 

  • Teig S (2002) The X architecture: not your fathers diagonal wiring. In: Proceedings of the ACM international workshop system level interconnect prediction, pp 33–37

  • Warme DM, Winter P, Zachariasen M (1998) Exact algorithms for plane steiner tree problems: a computational study., Advances in Steiner TreesKluwer Academic Publishers, Dordrecht

    Google Scholar 

  • Yan JT (2006) Dynamic tree reconstruction with application to timing-constrained congestion-driven global routing. IEE Proc Comput Digital Tech 153(2):117–129

    Article  Google Scholar 

  • Yan JT (2008) Timing-driven octilinear steiner tree construction based on steiner-point reassignment and path reconstruction. ACM Trans Design Autom Electron Syst 13(2):26

    Article  Google Scholar 

  • Yehea II, Eby GF, Jose LN (1999) Equivalent elmore delay for RLC trees. In: Proceedings of the 36th design automation conference, pp 715–720

  • Zhu Q, Zhou H, Jing T, Hong X, Yang Y (2005) Spanning graph-based nonrectilinear steiner tree algorithms. IEEE Trans Comput Aided Design 24(7):1066–1075

    Article  Google Scholar 

  • Zitzler E (1999) Evolutionary algorithms for multiobjective optimization: methods and applications. Swiss Federal Institute of Technology, Zurich

  • Zitzler E, Laumanns M, and Thiele L (2001) SPEA2: improving the strength pareto evolutionary algorithm. In: Giannakoglou KC, Tsahalis DT, Periaux J, Papailiou KD, Fogarty T (eds) Evolutionary methods for design optimization and control with applications to industrial problems. International Center for Numerical Methods in Engineering, pp 95–100

  • Zitzler E, Thiele L, Laumanns M et al (2003) Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans Evol Comput 7:117–132

    Article  Google Scholar 

Download references

Acknowledgments

This work was supported in part by the National Basic Research Program of China (2011CB808000), the National Natural Science Foundation of China (Grant Nos. 11271002 and 61300102), the Fujian Province High School Science Fund for Distinguished Young Scholars (JA12016), and the Program for New Century Excellent Talents in Fujian Province University (JA13021). The authors would like to thank Prof. Yan Jin-Tai from Department of Computer Science and Information Engineering, Chung-Hua University, for comments and helpful discussions.

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Authors and Affiliations

  1. College of Mathematics and Computer Science, Fuzhou University, Fuzhou, 350116, China

    Genggeng Liu, Wenzhong Guo, Yuzhen Niu, Guolong Chen & Xing Huang

  2. Fujian Provincial Key Laboratory of Network Computing and Intelligent Information Processing, Fuzhou University, Fuzhou, 350116, China

    Wenzhong Guo & Guolong Chen

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  1. Genggeng Liu
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  2. Wenzhong Guo
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  3. Yuzhen Niu
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  4. Guolong Chen
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  5. Xing Huang
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Correspondence to Wenzhong Guo.

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Communicated by V. Loia.

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Liu, G., Guo, W., Niu, Y. et al. A PSO-based timing-driven Octilinear Steiner tree algorithm for VLSI routing considering bend reduction. Soft Comput 19, 1153–1169 (2015). https://doi.org/10.1007/s00500-014-1329-2

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