Skip to main content
SpringerLink
Log in

Fast Multipole Networks

  • Conference paper
  • First Online:
Complex Networks & Their Applications IX (COMPLEX NETWORKS 2020 2020)

Part of the book series: Studies in Computational Intelligence ((SCI,volume 944))

Included in the following conference series:

  • 2553 Accesses

Abstract

Two prerequisites for robotic multiagent systems are mobility and communication. Fast multipole networks (FMNs) enable both ends within a unified framework. FMNs can be organized very efficiently in a distributed way from local information and are ideally suited for motion planning using artificial potentials. We compare FMNs to conventional communication topologies, and find that FMNs offer competitive communication performance (including higher network efficiency per edge at marginal energy cost) in addition to advantages for mobility.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 259.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 329.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 329.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    For the calculations in this paper, we used the very user-friendly library FMMLIB2D, available at https://cims.nyu.edu/cmcl/fmm2dlib/fmm2dlib.html.

  2. 2.

    Two clusters of points \(\{x_j\}\) and \(\{y_k\}\) are well-separated iff there exist \(x_0, y_0\) such that \(\{x_j\} \subset B^\circ _{x_0}(r)\) and \(\{y_k\} \subset B^\circ _{y_0}(r)\) with \(|x_0 - y_0| > 3r\): here \(^\circ \) denotes interior. Two squares with side length r are well-separated iff they are at distance \(\ge r\).

  3. 3.

    Though in principle the desired level of accuracy can be affected by charge values, this situation is sufficiently pathological that we can safely disregard it in practice.

  4. 4.

    The key difference between FMNs and the networks considered in [48] is that the latter are formed by inserting and permanently linking nearby charges, then dynamically evolving to obtain small-world features, whereas FMNs are (re)formed by linking nearby charges in a way that partially anticipates the next timestep of dynamical evolution. However, both types of networks exhibit aspects of small-world behavior (see Sect. 5 and [25]).

  5. 5.

    Limiting permission for direct communication in FMNs can be enforced by, e.g., cognitive radios [46] whose spectrum allocation cooperates with the FMM tree.

  6. 6.

    For \(\xi _j\) in general position, the Delaunay graph is unique.

References

  1. Avin, C.: Fast and efficient restricted Delaunay triangulation in random geometric graphs. Internet Math. 5, 195 (2008)

    Article  MathSciNet  Google Scholar 

  2. Barrat, A., Barthélemy, M., Vespignani, A.: Dynamical Processes on Complex Networks. Cambridge (2008)

    Google Scholar 

  3. Beatson, R., Greengard, L.: A short course on fast multipole methods. In: Ainsworth, M., et al. (eds.) Wavelets, Multilevel Methods, and Elliptic PDEs, Oxford (1997)

    Google Scholar 

  4. Board, J., Schulten, L.: The fast multipole algorithm. Comp. Sci. Eng. 2, 76 (2000)

    Article  Google Scholar 

  5. Chen, R., Gotsman, C.: Localizing the delaunay triangulation and its parallel implementation. In: ISVD (2012)

    Google Scholar 

  6. Chung, S.-J., et al.: A survey on aerial swarm robotics. IEEE Trans. Robot. 34, 837 (2018)

    Article  Google Scholar 

  7. Connolly, C.I., Burns, J.B., Weiss, R.: Path planning using Laplace’s equation. In: ICRA (1990)

    Google Scholar 

  8. DeCleene, B., Huntsman, S.: Wireless resilient routing reconfiguration. arXiv:1904.04865 (2019)

  9. Fuetterling, V., Lojewski, C., Pfreundt, F.-J.: High-performance \(d\)-D Delaunay triangulations for many-core computers. In: HPG (2014)

    Google Scholar 

  10. Funke, D., Sanders, P.: Parallel \(d\)-D delaunay triangulations in shared and distributed memory. In: ALENEX (2017)

    Google Scholar 

  11. Ghaffarkhah, A., Mostofi, Y.: Communication-aware motion planning in mobile networks. IEEE Trans. Auto. Control 56, 2478 (2011)

    Article  MathSciNet  Google Scholar 

  12. Greengard, L., Rokhlin, V.: A fast algorithm for particle simulations. J. Comp. Phys. 73, 325 (1987)

    Article  MathSciNet  Google Scholar 

  13. Greengard, L., Gropp, W.D.: A parallel version of the fast multipole method. Comp. Math. Appl. 20, 63 (1990)

    Article  MathSciNet  Google Scholar 

  14. Haenggi, M.: Stochastic Geometry for Wireless Networks. Cambridge (2013)

    Google Scholar 

  15. Hollinger, G., Singh, S.: Multi-robot coordination with periodic connectivity. In: ICRA (2010)

    Google Scholar 

  16. Jackson, J.D.: Classical Electrodynamics. 3rd ed. Wiley (1998)

    Google Scholar 

  17. Kantaros, Y., Zavlanos, M.M.: Distributed communication-aware coverage control by mobile sensor networks. Automatica 63, 209 (2016)

    Article  MathSciNet  Google Scholar 

  18. Kantaros, Y., Zavlanos, M.M.: Global planning for multi-robot communication networks in complex environments. IEEE Trans. Robotics 32, 1045 (2016)

    Article  Google Scholar 

  19. Kantaros, Y., Guo, M., Zavlanos, M.M.: Temporal logic task planning and intermittent connectivity control of mobile robot networks. IEEE Trans. Auto. Control 64, 4105 (2019)

    Article  MathSciNet  Google Scholar 

  20. Khatib, O.: Real-time obstacle avoidance for manipulators and mobile robots. In: ICRA (1985)

    Google Scholar 

  21. Kim, J.-O., Khosla, P.K.: Real-time obstacle avoidance using harmonic potential functions. IEEE Trans. Robot. Automat. 8, 501 (1992)

    Article  Google Scholar 

  22. Koren, Y., Borenstein, J.: Potential field method and their inherent limitations for mobile robot navigation. In: ICRA (1991)

    Google Scholar 

  23. Knorn, S., Chen, Z., Middleton, R.H.: Overview: collective control of multiagent systems. IEEE Trans. Cont. Net. Sys. 3, 334 (2016)

    Article  MathSciNet  Google Scholar 

  24. Krupke, D., et al.: Distributed cohesive control for robot swarms: maintaining good connectivity in the presence of exterior forces. In: IROS (2015)

    Google Scholar 

  25. Latora, V., Marchiori, M.: Efficient behavior of small-world networks. Phys. Rev. Lett. 87, 198701 (2001)

    Article  Google Scholar 

  26. Létourneau, P.-D., Cecka, C., Darve, E.: Cauchy fast multipole method for general analytic kernels. SIAM J. Sci. Comp. 36, A396 (2014)

    Article  MathSciNet  Google Scholar 

  27. Loo, J., Mauri, J.L., Ortiz, J.H. (eds.) Mobile Ad Hoc Networks. CRC (2016)

    Google Scholar 

  28. Majcherczyk, N., et al.: Decentralized connectivity-preserving deployment of large-scale robot swarms. In: IROS (2018)

    Google Scholar 

  29. Mesbahi, M., Egerstedt, M.: Graph Theoretic Methods in Multiagent Networks. Princeton (2010)

    Google Scholar 

  30. Minelli, M., et al.: Stop, think and roll: online gain optimization for resilient multi-robot topologies. In: DARS (2019)

    Google Scholar 

  31. Norrenbrock, C.: Percolation threshold on planar Euclidean Gabriel graphs. Eur. Phys. J. B 89, 111 (2016)

    Article  Google Scholar 

  32. Ohno, Y., et al.: Petascale molecular dynamics simulation using the fast multipole method on K computer. Comp. Phys. Comm. 185, 2575 (2014)

    Article  Google Scholar 

  33. Penrose, M.: Random Geometric Graphs. Oxford (2003)

    Google Scholar 

  34. Pimenta, L.C.A., et al.: On computing complex navigation functions. In: ICRA (2005)

    Google Scholar 

  35. Potter, D., Stadel, J., Teyssier, R.: PKDGRAV3: beyond trillion particle cosmological simulations for the next era of galaxy surveys. Comp. Astrophys. Cosmol. 4, 2 (2017)

    Article  Google Scholar 

  36. Rimon, E., Koditschek, D.E.: Exact robot navigation using artificial potential functions. IEEE Trans. Robot. Automat. 8, 501 (1992)

    Article  Google Scholar 

  37. Stephan, J., et al.: Concurrent control of mobility and communication in multirobot systems. IEEE Trans. Robot. 33, 1248 (2017)

    Article  Google Scholar 

  38. Taylor, M.E.: Partial Differential Equations: Basic Theory. Springer, Cham (1996)

    Google Scholar 

  39. Varadharajan, V.S., Adams, B., Beltrame, G.: The unbroken telephone game: keeping systems connected. In: AAMAS (2019)

    Google Scholar 

  40. Wang, Y., et al.: R3: resilient routing reconfiguration. In: SIGCOMM (2010)

    Google Scholar 

  41. Yan, Y., Mostofi, Y.: Co-optimization of communication and motion planning of a robotic operation in fading environments. In: ACSSC (2011)

    Google Scholar 

  42. Ying, L., Biros, G., Zorin, D.: A kernel-independent adaptive fast multipole algorithm in two and three dimensions. J. Comp. Phys. 196, 591 (2004)

    Article  MathSciNet  Google Scholar 

  43. Ying, L.: A kernel-independent fast multipole algorithm for radial basis functions. J. Comp. Phys. 213, 457 (2006)

    Article  Google Scholar 

  44. Yokota, R., Barba, L.A.: A tuned and scalable fast multipole method as a preeminent algorithm for exascale systems. Int. J. High Perf. Comp. Appl. 26, 337 (2012)

    Article  Google Scholar 

  45. Yokota, R., et al.: Petascale turbulence simulation using a highly parallel fast multipole method on GPUs. Comp. Phys. Comm. 184, 445 (2013)

    Article  MathSciNet  Google Scholar 

  46. Yu, R.F. (ed.) Cognitive Radio Mobile Ad Hoc Networks. Springer (2011)

    Google Scholar 

  47. Zavlanos, M.M., Pappas, G.J.: Potential fields for maintaining connectivity of mobile networks. IEEE Trans. Robotics 23, 812 (2007)

    Article  Google Scholar 

  48. Zitin, A., et al.: Spatially embedded growing small-world networks. Sci. Rep. 4, 7047 (2015)

    Article  Google Scholar 

Download references

Acknowledgements

We thank Brendan Fong, Marco Pravia, and David Spivak for their comments.

Author information

Authors and Affiliations

  1. Virginia, USA

    Steve Huntsman

Authors
  1. Steve Huntsman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Steve Huntsman .

Editor information

Editors and Affiliations

  1. Grupo de Sistemas Complejos, Universidad Politécnica de Madrid, Madrid, Madrid, Spain

    Rosa M. Benito

  2. IUT Lumière, University of Lyon, Bron Cedex, France

    Chantal Cherifi

  3. LE2I, UFR Sciences et Techniques, Université de Bourgogne, Dijon, France

    Hocine Cherifi

  4. Grupo Interdisciplinar de Sistemas Complejos, Departamento de Matematicas, Universidad Carlos III de Madrid, Leganés, Madrid, Spain

    Esteban Moro

  5. Center for Social and Biomedical Complexity, Luddy School of Informatics, Computing, and Engineering, Indiana University, Bloomington, IN, USA

    Luis Mateus Rocha

  6. Department of Chemical Engineering, Universitat Rovira i Virgili, Tarragona, Tarragona, Spain

    Marta Sales-Pardo

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Huntsman, S. (2021). Fast Multipole Networks. In: Benito, R.M., Cherifi, C., Cherifi, H., Moro, E., Rocha, L.M., Sales-Pardo, M. (eds) Complex Networks & Their Applications IX. COMPLEX NETWORKS 2020 2020. Studies in Computational Intelligence, vol 944. Springer, Cham. https://doi.org/10.1007/978-3-030-65351-4_34

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-65351-4_34

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-65350-7

  • Online ISBN: 978-3-030-65351-4

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation