Skip to main content

Finding a Needle in an Exponential Haystack: Discrete RRT for Exploration of Implicit Roadmaps in Multi-robot Motion Planning

  • Chapter
  • First Online:
Algorithmic Foundations of Robotics XI

Part of the book series: Springer Tracts in Advanced Robotics ((STAR,volume 107))


We present a sampling-based framework for multi-robot motion planning which combines an implicit representation of a roadmap with a novel approach for pathfinding in geometrically embedded graphs tailored for our setting. Our pathfinding algorithm, discrete-RRT (dRRT), is an adaptation of the celebrated RRT algorithm for the discrete case of a graph, and it enables a rapid exploration of the high-dimensional configuration space by carefully walking through an implicit representation of a tensor product of roadmaps for the individual robots. We demonstrate our approach experimentally on scenarios of up to 60 degrees of freedom where our algorithm is faster by a factor of at least ten when compared to existing algorithms that we are aware of.

This work has been supported in part by the 7th Framework Programme for Research of the European Commission, under FET-Open grant number 255827 (CGL—Computational Geometry Learning), by the Israel Science Foundation (grant no. 1102/11), by the German-Israeli Foundation (grant no. 1150-82.6/2011), and by the Hermann Minkowski–Minerva Center for Geometry at Tel Aviv University. K. Solovey and O. Salzman contributed equally to this paper.

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

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
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

Similar content being viewed by others


  1. 1.

    We mention that we are not the first to consider RRTs in discrete domains. Branicky et al. [9] applied the RRT algorithm to a discrete graph. However, a key difference between the approaches is that we assume that the graph is geometrically embedded, hence we use random points as samples while they use nodes of the graph as samples. Additionally, their technique requires that all the neighbors of a visited vertex will be considered—a costly operation in our setting, as mentioned above.

  2. 2.

    There is wide consensus on the term tensor product as defined here, and less so on the term Cartesian product. As the latter has already been used before in the context of motion planning, we will keep using it here as well.


  1. PQP—A Proximity Query Package.

  2. Graph Product: Wikipedia, The Free Encyclopedia. (2013)

  3. Adler, A., de Berg, M., Halperin, D., Solovey, K.: Efficient multi-robot motion planning for unlabeled discs in simple polygons. CoRR arXiv:1312.1038 (2013)

  4. Aronov, B., de Berg, M., van der Stappen, A.F., Švestka, P., Vleugels, J.: Motion planning for multiple robots. Discret. Comput. Geom. 22(4), 505–525 (1999)

    Article  MATH  Google Scholar 

  5. Auletta, V., Monti, A., Parente, M., Persiano, P.: A linear time algorithm for the feasibility of pebble motion on trees. In: SWAT, pp. 259–270 (1996)

    Google Scholar 

  6. van den Berg, J., Overmars, M.: Prioritized motion planning for multiple robots. In: IROS, pp. 430–435 (2005)

    Google Scholar 

  7. van den Berg, J., Snoeyink, J., Lin, M., Manocha, D.: Centralized path planning for multiple robots: optimal decoupling into sequential plans. In: RSS (2009)

    Google Scholar 

  8. de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer, Heidelberg (2008)

    Book  Google Scholar 

  9. Branicky, M.S., Curtiss, M.M., Levine, J.A., Morgan, S.B.: RRTs for nonlinear, discrete, and hybrid planning and control. In: Decision and Control, pp. 9–12 (2003)

    Google Scholar 

  10. Choset, H., Lynch, K., Hutchinson, S., Kantor, G., Burgard, G., Kavraki, L., Thrun, S.: Principles of Robot Motion: Theory, Algorithms, and Implementations. MIT Press, Cambridge (2005)

    Google Scholar 

  11. Şucan, I.A., Moll, M., Kavraki, L.E.: The open motion planning library. IEEE Robot. Autom. Mag. 19(4), 72–82 (2012)

    Article  Google Scholar 

  12. Goraly, G., Hassin, R.: Multi-color pebble motion on graphs. Algorithmica 58(3), 610–636 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hirsch, S., Halperin, D.: Hybrid motion planning: coordinating two discs moving among polygonal obstacles in the plane. In: WAFR, pp. 239–255. Springer, New York (2002)

    Google Scholar 

  14. Hopcroft, J., Schwartz, J., Sharir, M.: On the complexity of motion planning for multiple independent objects; PSPACE-hardness of the “Warehouseman’s Problem”. IJRR 3(4), 76–88 (1984)

    Google Scholar 

  15. Karaman, S., Frazzoli, E.: Sampling-based algorithms for optimal motion planning. IJRR 30(7), 846–894 (2011)

    Google Scholar 

  16. Kavraki, L.E., Švestka, P., Latombe, J.C., Overmars, M.: probabilistic roadmaps for path planning in high dimensional configuration spaces. IEEE Trans. Robot. Autom. 12(4), 566–580 (1996)

    Article  Google Scholar 

  17. Kloder, S., Hutchinson, S.: Path planning for permutation-invariant multi-robot formations. In: ICRA, pp. 1797–1802 (2005)

    Google Scholar 

  18. Kornhauser, D.: Coordinating Pebble motion on graphs, the diameter of permutation groups, and applications. M.Sc. thesis, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology (1984)

    Google Scholar 

  19. Kuffner, J.J., LaValle, S.M.: RRT-connect: an efficient approach to single-query path planning. In: ICRA, pp. 995–1001 (2000)

    Google Scholar 

  20. Kuffner, J.J.: Effective sampling and distance metrics for 3D rigid body path planning. In: ICRA, pp. 3993–3998 (2004)

    Google Scholar 

  21. LaValle, S.M.: Planning Algorithms. Cambridge University Press, Cambridge (2006)

    Google Scholar 

  22. Leroy, S., Laumond, J.P., Simeon, T.: Multiple path coordination for mobile robots: a geometric algorithm. In: IJCAI, pp. 1118–1123 (1999)

    Google Scholar 

  23. Luna, R., Bekris, K.E.: Push and swap: fast cooperative path-finding with completeness guarantees. In: IJCAI, pp. 294–300 (2011)

    Google Scholar 

  24. Muja, M., Lowe, D.G.: Fast approximate nearest neighbors with automatic algorithm configuration. In: VISSAPP, pp. 331–340. INSTICC Press (2009)

    Google Scholar 

  25. Pearl, J.: Heuristics: Intelligent Search Strategies for Computer Problem Solving. Addison-Wesley, Reading (1984)

    Google Scholar 

  26. Salzman, O., Hemmer, M., Halperin, D.: On the power of manifold samples in exploring configuration spaces and the dimensionality of narrow passages. In: WAFR, pp. 313–329 (2012)

    Google Scholar 

  27. Sanchez, G., Latombe, J.C.: Using a PRM planner to compare centralized and decoupled planning for multi-robot systems. In: ICRA, pp. 2112–2119 (2002)

    Google Scholar 

  28. Schwartz, J.T., Sharir, M.: On the piano movers’ problem: III. Coordinating the motion of several independent bodies. IJRR 2(3), 46–75 (1983)

    Google Scholar 

  29. Sharir, M., Sifrony, S.: Coordinated motion planning for two independent robots. Ann. Math. Artif. Intell. 3(1), 107–130 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  30. Solovey, K., Halperin, D.: \(k\)-color multi-robot motion planning. In: WAFR, pp. 191–207 (2012)

    Google Scholar 

  31. Spirakis, P.G., Yap, C.K.: Strong NP-hardness of moving many discs. Inf. Process. Lett. 19(1), 55–59 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  32. Turpin, M., Michael, N., Kumar, V.: Computationally efficient trajectory planning and task assignment for large teams of unlabeled robots. In: ICRA, pp. 834–840 (2013)

    Google Scholar 

  33. Švestka, P., Overmars, M.: Coordinated path planning for multiple robots. Robot. Auton. Syst. 23, 125–152 (1998)

    Article  Google Scholar 

  34. Wagner, G., Choset, H.: M*: a complete multirobot path planning algorithm with performance bounds. In: IROS, pp. 3260–3267. IEEE (2011)

    Google Scholar 

  35. Wagner, G., Kang, M., Choset, H.: Probabilistic path planning for multiple robots with subdimensional expansion. In: ICRA, pp. 2886–2892 (2012)

    Google Scholar 

  36. Yap, C.: Coordinating the motion of several discs. Technical report, Courant Institute of Mathematical Sciences, Michigan State University, New York (1984)

    Google Scholar 

Download references


We wish to thank Glenn Wagner for advising on the M* algorithm and Ariel Felner for advice regarding pathfinding algorithms on graphs. We note that the title “Finding a Needle in an Exponential Haystack” has been previously used in a talk by Joel Spencer in a different context.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Kiril Solovey .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Solovey, K., Salzman, O., Halperin, D. (2015). Finding a Needle in an Exponential Haystack: Discrete RRT for Exploration of Implicit Roadmaps in Multi-robot Motion Planning. In: Akin, H., Amato, N., Isler, V., van der Stappen, A. (eds) Algorithmic Foundations of Robotics XI. Springer Tracts in Advanced Robotics, vol 107. Springer, Cham.

Download citation

  • DOI:

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-16594-3

  • Online ISBN: 978-3-319-16595-0

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics