Efficient Multi-robot Motion Planning for Unlabeled Discs in Simple Polygons

  • Aviv Adler
  • Mark de Berg
  • Dan Halperin
  • Kiril SoloveyEmail author
Part of the Springer Tracts in Advanced Robotics book series (STAR, volume 107)


We consider the following motion-planning problem: we are given \(m\) unit discs in a simple polygon with \(n\) vertices, each at their own start position, and we want to move the discs to a given set of \(m\) target positions. Contrary to the standard (labeled) version of the problem, each disc is allowed to be moved to any target position, as long as in the end every target position is occupied. We show that this unlabeled version of the problem can be solved in \(O\left( n\log n+mn+m^2\right) \) time, assuming that the start and target positions are at least some minimal distance from each other. This is in sharp contrast to the standard (labeled) and more general multi-robot motion planning problem for discs moving in a simple polygon, which is known to be strongly np-hard.


Target Position Simple Polygon Target Configuration Motion Graph Obstacle Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    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)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bereg, S., Dumitrescu, A., Pach, J.: Sliding disks in the plane. Int. J. Comput. Geom. Appl. 18(5), 373–387 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Dumitrescu, A., Jiang, M.: On reconfiguration of disks in the plane and related problems. Comput. Geom.: Theory Appl. 46(3), 191–202 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Goldreich, O.: Shortest move-sequence in the generalized 15-puzzle is NP-hard. Manuscript, Laboratory for Computer Science, MIT 1 (1984)Google Scholar
  5. 5.
    Goraly, G., Hassin, R.: Multi-color pebble motion on graphs. Algorithmica 58(3), 610–636 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Hearn, R.A., Demaine, E.D.: PSPACE-completeness of sliding-block puzzles and other problems through the nondeterministic constraint logic model of computation. Theor. Comput. Sci. 343(1–2), 72–96 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Hirsch, S., Halperin, D.: Hybrid motion planning: coordinating two discs moving among polygonal obstacles in the plane. In: Workshop on the Algorithmic Foundations of Robotics (WAFR), pp. 239–255. Springer, New York (2002)Google Scholar
  8. 8.
    Hopcroft, J.E., Schwartz, J.T., Sharir, M.: On the complexity of motion planning for multiple independent objects; PSPACE-hardness of the warehouseman’s problem. Int. J. Robot. Res. 3(4), 76–88 (1984)CrossRefGoogle Scholar
  9. 9.
    Kavraki, L.E., Švestka, P., Latombe, J.C., Overmars, M.H.: Probabilistic roadmaps for path planning in high dimensional configuration spaces. IEEE Trans. Robot. Autom. 12(4), 566–580 (1996)CrossRefGoogle Scholar
  10. 10.
    Kedem, K., Livne, R., Pach, J., Sharir, M.: On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles. Discret. Comput. Geom. 1, 59–70 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Kloder, S., Hutchinson, S.: Path planning for permutation-invariant multi-robot formations. In: ICRA, pp. 1797–1802 (2005)Google Scholar
  12. 12.
    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
  13. 13.
    Krontiris, A., Luna, R., Bekris, K.E.: From feasibility tests to path planners for multi-agent pathfinding. In: Symposium on Combinatorial Search (2013)Google Scholar
  14. 14.
    Kuffner, J.J., Lavalle, S.M.: RRT-connect: an efficient approach to single-query path planning. In: International Conference on Robotics and Automation (ICRA), pp. 995–1001 (2000)Google Scholar
  15. 15.
    Papadimitriou, C.H., Raghavan, P., Sudan, M., Tamaki, H.: Motion planning on a graph. In: Foundations of Computer Science, pp. 511–520 (1994)Google Scholar
  16. 16.
    Salzman, O., Hemmer, M., Halperin, D.: On the power of manifold samples in exploring configuration spaces and the dimensionality of narrow passages to appear, Workshop on the Algorithmic Foundations of Robotics (WAFR) (2012)Google Scholar
  17. 17.
    Sanchez, G., Latombe, J.C.: Using a PRM planner to compare centralized and decoupled planning for multi-robot systems. In: International Conference on Robotics and Automation (ICRA) (2002)Google Scholar
  18. 18.
    Schwartz, J.T., Sharir, M.: On the piano movers problem: II. General techniques for computing topological properties of real algebraic manifolds. Adv. Appl. Math. 4(3), 298–351 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Schwartz, J.T., Sharir, M.: On the piano movers problem: III. Coordinating the motion of several independent bodies. Int. J. Robot. Res. 2(3), 46–75 (1983)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Sharir, M., Sifrony, S.: Coordinated motion planning for two independent robots. Ann. Math. Artif. Intell. 3(1), 107–130 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Solovey, K., Halperin, D.: \(k\)-color multi-robot motion planning. Int. J. Robot. Res. (2013, in press (already appeared on-line))Google Scholar
  22. 22.
    Solovey, K., Salzman, O., Halperin, D.: Finding a needle in an exponential haystack: discrete RRT for exploration of implicit roadmaps in multi-robot motion planning. CoRR 1305.2889 (2013)
  23. 23.
    Spirakis, P.G., Yap, C.K.: Strong NP-hardness of moving many discs. Inf. Process. Lett. 19(1), 55–59 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Švestka, P., Overmars, M.H.: Coordinated path planning for multiple robots. Robot. Auton. Syst. 23, 125–152 (1998)CrossRefGoogle Scholar
  25. 25.
    Turpin, M., Michael, N., Kumar, V.: Concurrent assignment and planning of trajectories for large teams of interchangeable robots. In: International Conference on Robotics and Automation (ICRA), pp. 842–848 (2013)Google Scholar
  26. 26.
    van den Berg, J., Snoeyink, J., Lin, M.C., Manocha, D.: Centralized path planning for multiple robots: optimal decoupling into sequential plans. In: Robotics: Science and Systems (RSS) (2009)Google Scholar
  27. 27.
    Wagner, G., Choset, H.: M*: A complete multirobot path planning algorithm with performance bounds. In: International Conference on Intelligent Robots and Systems (IROS), pp. 3260–3267 (2011)Google Scholar
  28. 28.
    Wagner, G., Kang, M., Choset, H.: Probabilistic path planning for multiple robots with subdimensional expansion. In: International Conference on Robotics and Automation (ICRA), pp. 2886–2892 (2012)Google Scholar
  29. 29.
    Yap, C.K.: Coordinating the motion of several discs. Technical report, Courant Institute of Mathematical Sciences, Michigan State University, New York (1984)Google Scholar
  30. 30.
    Yu, J., LaValle, S.M.: Distance optimal formation control on graphs with a tight convergence time guarantee. In: IEEE International Conference on Decision and Control, pp. 4023–4028 (2012)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Aviv Adler
    • 1
  • Mark de Berg
    • 2
  • Dan Halperin
    • 3
  • Kiril Solovey
    • 3
    Email author
  1. 1.Department of MathematicsPrinceton UniversityNew JerseyUSA
  2. 2.Department of Mathematics and Computing ScienceTu EindhovenEindhovenThe Netherlands
  3. 3.Blavatnik School of Computer ScienceTel-Aviv UniversityTel Aviv-YafoIsrael

Personalised recommendations