Skip to main content
SpringerLink
Log in

A parallel bio-inspried shortest path algorithm

  • Published:
Computing Aims and scope Submit manuscript

Abstract

Physarum polycephalum is an amoeba-like organism and is able to find the shortest path in a labyrinth. Inspired by P. polycephalum, recently, a mathematical model and an algorithm (Physarum Solver) was developed. There are, however, only sequential implementations of this algorithm. In this paper, a fast and efficient parallel Physarum Solver is proposed. The proposed algorithm requires the solution of linear systems whose coefficient matrix is a symmetric M-matrix. The solution of the linear system is the most time consuming step of the Physarum Solver which is classically handled by direct solvers without taking advantage of the fact that the coefficient matrix is an M-matrix. However, direct solvers are infeasible for solving large real-world problems. In the proposed parallel Physarum Solver, an effective parallel iterative linear solver with a parallel preconditioner for M-matrices is used. The parallel scalability, solution time, and accuracy of the proposed algorithm are presented and compared to a state-of-the-art parallel implementation of \(\varDelta \)-stepping shortest path algorithm in the Parallel Boost Graph Library. Our implementation exhibits a remarkable parallel speedup with comparable accuracy for synthetic and real world applications.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. 9th DIMACS implementation challenge—shortest paths. http://www.dis.uniroma1.it/challenge9/download.shtml. Accessed Jan 2017

  2. Boost C++ Libraries. http://www.boost.org/. Accessed Apr 2016

  3. Erdos–Renyi graphs. http://cneurocvs.rmki.kfki.hu/igraph/doc/R/erdos.renyi.game.html. Accessed Jan 2017

  4. Portable, Extensible Toolkit for Scientific Computing, version 3.6.3. http://www.mcs.anl.gov/petsc. Accessed Apr 2016

  5. The Watts–Strogatz small-world model. http://cneurocvs.rmki.kfki.hu/igraph/doc/R/watts.strogatz.game.html. Accessed Jan 2017

  6. Adamatzky A, Jones J (2010) Programmable reconfiguration of Physarum machines. Nat Comput 9(1):219–237

    Article  MathSciNet  MATH  Google Scholar 

  7. Aleksandrov L, Maheshwari A, Sack JR (2005) Determining approximate shortest paths on weighted polyhedral surfaces. J ACM 52(1):25–53

    Article  MathSciNet  MATH  Google Scholar 

  8. Awerbuch B, Berger B, Cowen L, Peleg D (2006) Near-linear time construction of sparse neighborhood covers. SIAM J Comput 28(1):263–277

    Article  MathSciNet  MATH  Google Scholar 

  9. Becchetti L, Bonifaci V, Dirnberger M, Karrenbauer A, Mehlhorn K (2013) Physarum can compute shortest paths: convergence proofs and complexity bounds. In: Fomin F, Freivalds R, Kwiatkowska M, Peleg D (eds) Automata languages and programming: 40th international colloquium and ICALP 2013 Riga and Latvia and July 8–12 and 2013 and proceedings and part II. Springer, Berlin, pp 472–483

  10. Beekman M, Latty T (2015) Brainless but multi-headed: decision making by the acellular slime mould Physarum polycephalum. J Mol Biol 427(23):3734–3743. https://doi.org/10.1016/j.jmb.2015.07.007

    Article  Google Scholar 

  11. Benzi M (2002) Preconditioning techniques for large linear systems: a survey. J Comput Phys 182(2):418–477

    Article  MathSciNet  MATH  Google Scholar 

  12. Berman A, Plemmons RJ (1979) Nonnegative matrices in the mathematical sciences. Academic Press, New York. https://doi.org/10.1016/B978-0-12-092250-5.50002-3

    Book  MATH  Google Scholar 

  13. Bonifaci V (2013) Physarum can compute shortest paths: a short proof. Inf Process Lett 113(1):4–7. https://doi.org/10.1016/j.ipl.2012.09.005

    Article  MathSciNet  MATH  Google Scholar 

  14. Bonifaci V, Mehlhorn K, Varma G (2012) Physarum can compute shortest paths. J Theor Biol 309:121–133

    Article  MathSciNet  MATH  Google Scholar 

  15. Chaibou A, Sie O (2015) Improving global performance on GPU for algorithms with main loop containing a reduction operation: case of Dijkstra’s algorithm. J Comput Commun 3:41–54

    Article  Google Scholar 

  16. Chakaravarthy VT, Checconi F, Petrini F, Sabharwal Y (2014) Scalable single source shortest path algorithms for massively parallel systems. In: 2014 IEEE 28th international parallel and distributed processing symposium, pp 889–901

  17. Chakrabarti D, Zhan Y, Faloutsos C (2004) R-MAT: a recursive model for graph mining. In: Proceedings of the 2004 SIAM international conference on data mining. SIAM, Philadelphia, pp 442–446

  18. Cheng Gh, Huang Tz, Cheng Xy (2006) Preconditioned Gauss–Seidel type iterative method for solving linear systems. Appl Math Mech 27(9):1275–1279

    Article  MathSciNet  MATH  Google Scholar 

  19. Cherkassky B, Goldberg A, Radzik T (1996) Shortest path algorithms: theory and experimental evaluation. Math Program 73(2):129–174

    Article  MathSciNet  MATH  Google Scholar 

  20. Cowen LJ, Wagner CG (2004) Compact roundtrip routing in directed networks. J Algorithms 50(1):79–95

    Article  MathSciNet  MATH  Google Scholar 

  21. Crobak JR, Berry JW, Madduri K, Bader DA (2007) Advanced shortest paths algorithms on a massively-multithreaded architecture. In: 2007 IEEE international parallel and distributed processing symposium, pp 1–8

  22. Dantzig G (1963) Linear programming and extensions. Princeton University Press, Princeton

    Book  MATH  Google Scholar 

  23. Delling D, Goldberg AV, Nowatzyk A, Werneck RF (2013) PHAST: hardware-accelerated shortest path trees. J Parallel Distrib Comput 73(7):940–952

    Article  Google Scholar 

  24. Dijkstra EW (1959) A note on two problems in connexion with graphs. Numer Math 1:269–271

    Article  MathSciNet  MATH  Google Scholar 

  25. Dorigo M, Gambardella LM (1997) Ant colony system: a cooperative learning approach to the traveling salesman problem. IEEE Trans Evol Comput 1(1):53–66

    Article  Google Scholar 

  26. Edmonds N, Breuer A, Gregor D, Lumsdaine A (2006) Single-source shortest paths with the parallel boost graph library. In: The ninth DIMACS implementation challenge: the shortest path problem

  27. Elkin M (2001) Computing almost shortest paths. In: Proceedings of the twentieth annual ACM symposium on principles of distributed computing. ACM, New York, pp 53–62

  28. Ertl G (1998) Shortest path calculation in large road networks. Oper Res Spektrum 20(1):15–20

    Article  MATH  Google Scholar 

  29. Goldberg A, Kaplan H, Werneck R (2006) Reach for A*: efficient point-to-point shortest path algorithms. Technical report, Miami, FL. https://www.microsoft.com/en-us/research/publication/reach-for-a-efficient-point-to-point-shortest-path-algorithms/

  30. Gong M, Li G, Wang Z, Ma L, Tian D (2016) An efficient shortest path approach for social networks based on community structure. CAAI Trans Intell Technol 1(1):114–123

    Article  Google Scholar 

  31. Gregor D, Lumsdaine A (2005) The parallel BGL: a generic library for distributed graph computations. In: Parallel object-oriented scientific computing (POOSC)

  32. Gubichev A, Bedathur S, Seufert S, Weikum G (2010) Fast and accurate estimation of shortest paths in large graphs. In: Proceedings of the 19th ACM international conference on information and knowledge management, CIKM ’10. ACM, New York, pp 499–508

  33. Hadjidimos A, Noutsos D, Tzoumas M (2003) More on modifications and improvements of classical iterative schemes for M-matrices. Linear Algebra Appl 364:253–279

    Article  MathSciNet  MATH  Google Scholar 

  34. Ikeda T, Hsu MY, Imai H, Nishimura S, Shimoura H, Hashimoto T, Tenmoku K, Mitoh K (1994) A fast algorithm for finding better routes by AI search techniques. In: Vehicle navigation and information systems conference, 1994. Proceedings, pp 291–296

  35. Kepner J, Gilbert J (2011) Graph algorithms in the language of linear algebra. Society for Industrial and Applied Mathematics, Philadelphia. https://doi.org/10.1137/1.9780898719918

    Book  MATH  Google Scholar 

  36. Khorasani F, Vora K, Gupta R (2015) PaRMAT: a parallel generator for large R-MAT graphs. https://github.com/farkhor/PaRMAT. Accessed Jan 2017

  37. Klein P (2002) Preprocessing an undirected planar network to enable fast approximate distance queries. In: Proceedings of the thirteenth annual ACM-SIAM symposium on discrete algorithms, SODA ’02, pp 820–827

  38. Liang M, Gao C, Zhang Z (2017) A new genetic algorithm based on modified Physarum network model for bandwidth-delay constrained least-cost multicast routing. Nat Comput 16(1):85–98

    Article  MathSciNet  Google Scholar 

  39. Liu L, Song Y, Ma H, Zhang X (2012) Physarum optimization: a biology-inspired algorithm for minimal exposure path problem in wireless sensor networks. In: 2012 proceedings IEEE INFOCOM, pp 1296–1304

  40. Liu L, Song Y, Zhang H, Ma H (2015) Physarum optimization: a biology-inspired algorithm for the steiner tree problem in networks. IEEE Trans Comput 64:818–831

    Article  MathSciNet  MATH  Google Scholar 

  41. Madduri K, Bader DA, Berry JW, Crobak J (2007) An experimental study of a parallel shortest path algorithm for solving large-scale graph instances. In: Proceedings of the meeting on algorithm engineering and experiments. Society for Industrial and Applied Mathematics, Philadelphia, pp 23–35

  42. Masi L, Vasile M (2014) A multidirectional Physarum solver for the automated design of space trajectories. In: Proceedings of the 2014 IEEE congress on evolutionary computation (CEC), pp 2992–2999. https://doi.org/10.1109/CEC.2014.6900287

  43. McSherry F, Isard M, Murray DG (2015) Scalability! But at what cost? In: Proceedings of the 15th USENIX conference on hot topics in operating systems, HOTOS 15, pp 14–14

  44. Meyer U, Sanders P (1998) \(\Delta \)-stepping: a parallel single source shortest path algorithm. In: European symposium on algorithms, pp 393–404

  45. Meyer U, Sanders P (2003) \(\Delta \)-stepping: a parallelizable shortest path algorithm. J Algorithms 49:114–152

    Article  MathSciNet  MATH  Google Scholar 

  46. Miyaji T, Ohnishi I (2008) Physarum can solve the shortest path problem on Riemann surface mathematically rigorously. Int J Pure Appl Math 47:353–369

    MathSciNet  MATH  Google Scholar 

  47. Nakagaki T, Iima M, Ueda T, Nishiura Y, Saigusa T, Tero A, Kobayashi R, Showalter K (2007) Minimum-risk path finding by an adaptive amoebal network. Phys Rev Lett 99:068,104. https://doi.org/10.1103/PhysRevLett.99.068104

    Article  Google Scholar 

  48. Neumann M, Plemmons RJ (1987) Convergence of parallel multisplitting iterative methods for M-matrices. Linear Algebra Appl 88:559–573

    Article  MathSciNet  MATH  Google Scholar 

  49. Nguyen UT, Xu J (2007) Multicast routing in wireless mesh networks: minimum cost trees or shortest path trees? IEEE Commun Mag 45(11):72–77

    Article  Google Scholar 

  50. Niki H, Kohno T, Morimoto M (2008) The preconditioned Gauss–Seidel method faster than the SOR method. J Comput Appl Math 219(1):59–71

    Article  MathSciNet  MATH  Google Scholar 

  51. Peleg D, Ullman JD (1989) An optimal synchronizer for the hypercube. SIAM J Comput 18:740–747

    Article  MathSciNet  MATH  Google Scholar 

  52. Plemmons R (1977) M-matrix characterizations. I—nonsingular M-matrices. Linear Algebra Appl 18(2):175–188

    Article  MathSciNet  MATH  Google Scholar 

  53. Pohl I (1971) Bi-directional search. Mach Intell 6:124–140

    MATH  Google Scholar 

  54. Rheinboldt WC (1970) On M-functions and their application to nonlinear Gauss-Seidel iterations and to network flows. J Math Anal Appl 32:274–307

    Article  MathSciNet  MATH  Google Scholar 

  55. Saad Y (2003) Iterative methods for sparse linear systems, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia

    Book  MATH  Google Scholar 

  56. Sedgewick R, Vitter J (1986) Shortest paths in Euclidean graphs. Algorithmica 1:31–48

    Article  MathSciNet  MATH  Google Scholar 

  57. Sen S (2009) Approximating shortest paths in graphs. In: 3rd international workshop on algorithms and computation (WALCOM), pp 32–43

  58. Siriwardana J, Halgamuge SK (2012) Fast shortest path optimization inspired by shuttle streaming of Physarum polycephalum. In: 2012 IEEE congress on evolutionary computation, pp 1–8

  59. Tero A, Kobayashi R, Nakagaki T (2006) Physarum solver: a biologically inspired method of road-network navigation. Phys A Stat Mech Appl 363(1):115–119

    Article  Google Scholar 

  60. Tero A, Kobayashi R, Nakagaki T (2007) A mathematical model for adaptive transport network in path finding by true slime mold. J Theor Biol 244(4):553–564

    Article  MathSciNet  Google Scholar 

  61. Thorup M (2004) Compact oracles for reachability and approximate distances in planar digraphs. J ACM 51(6):993–1024

    Article  MathSciNet  MATH  Google Scholar 

  62. Tsuda S, Aono M, Gunji YP (2004) Robust and emergent Physarum logical-computing. Biosystems 73(1):45–55

    Article  Google Scholar 

  63. Varga RS (1962) Matrix iterative analysis. Prentice-Hall, Upper Saddle River

    MATH  Google Scholar 

  64. Young DM (1971) Iterative solution of large linear systems. Academic Press, New York

    MATH  Google Scholar 

  65. Yuster R (2012) Approximate shortest paths in weighted graphs. J Comput Syst Sci 78:632–637

    Article  MathSciNet  MATH  Google Scholar 

  66. Zhan FB, Noon CE (2000) A comparison between label-setting and label-correcting algorithms for computing one-to-one shortest paths. Nano communication networks 4. J Geogr Inf Decis Anal 4:1–11

    Google Scholar 

  67. Zhang X, Adamatzky A, Chan FTS, Deng Y, Yang H, Yang XS, Tsompanas M, Sirakoulis G, Mahadevan S (2015) A biologically inspired network design model. Sci Rep 5:10,794

    Article  Google Scholar 

  68. Zhang X, Adamatzky A, Yang H, Mahadaven S, Yang XS, Wang Q, Deng Y (2014) A bio-inspired algorithm for identification of critical components in the transportation networks. Appl Math Comput 248:18–27

    MATH  Google Scholar 

  69. Zhang X, Huang S, Hu Y, Zhang Y, Mahadevan S, Deng Y (2013) Solving 0–1 knapsack problems based on amoeboid organism algorithm. Appl Math Comput 219:9959–9970

    MathSciNet  MATH  Google Scholar 

  70. Zhang X, Zhang X, Zhang Y, Wei D, Deng Y (2013) Route selection for emergency logistics management: a bio-inspired algorithm. Saf Sci 54:87–91

    Article  Google Scholar 

  71. Zhang Z, Gao C, Liu Y, Qian T (2014) A universal optimization strategy for ant colony optimization algorithms based on the Physarum-inspired mathematical model. Bioinspiration Biomim 9(3):036,006

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by the Turkish Academy of Sciences Distinguished Young Scientist Award M.M/TUBA-GEBIP/2012-19. The numerical calculations reported in this paper were performed at the Scientific and Technological Research Council of Turkey, High Performance and Grid Computing Center (TRUBA resources). We also thank Barry Smith for his help with the PETSc.

Author information

Authors and Affiliations

  1. Department of Computer Engineering, Middle East Technical University, Ankara, Turkey

    Hilal Arslan & Murat Manguoglu

Authors
  1. Hilal Arslan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Murat Manguoglu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Murat Manguoglu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Arslan, H., Manguoglu, M. A parallel bio-inspried shortest path algorithm. Computing 101, 969–988 (2019). https://doi.org/10.1007/s00607-018-0621-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00607-018-0621-x

Keywords

Mathematics Subject Classification

Search

Navigation