Advertisement

The Kissing Problem: How to End a Gathering When Everyone Kisses Everyone Else Goodbye

  • Michael A. Bender
  • Ritwik Bose
  • Rezaul Chowdhury
  • Samuel McCauley
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7288)

Abstract

This paper introduces the kissing problem: given a rectangular room with n people in it, what is the most efficient way for each pair of people to kiss each other goodbye? The room is viewed as a set of pixels that form a subset of the integer grid. At most one person can stand on a pixel at once, and people move horizontally or vertically. In order to move into a pixel in time step t, the pixel must be empty in time step t − 1.

The paper gives one algorithm for kissing everyone goodbye.

(1) This algorithm is a 4 + o(1)-approximation algorithm in a crowded room (e.g., only one unoccupied pixel).

(2) It is a 10 + o(1)-approximation algorithm for kissing in a comfortable room (e.g., at most half the pixels are empty).

(3) It is a 25+o(1)-approximation for kissing in a sparse room.

Keywords

Approximation Algorithm Mobile Robot Approximation Ratio Travel Salesman Problem Travel Salesman Problem 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alpern, S.: Rendezvous search on labeled networks. Naval Research Logistics 49(3), 256–274 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Alpern, S., Baston, V., Essegaier, S.: Rendezvous search on a graph. Journal of Applied Probability 36(1), 223–231 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Arkin, E., Hassin, R.: Approximation algorithms for the geometric covering salesman problem. Discrete Applied Mathematics 55(3), 197–218 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Arkin, R.: Motor schema-based mobile robot navigation. In: Proc. IEEE Conference on Robotics and Automation, pp. 264–271 (1987)Google Scholar
  5. 5.
    Balch, T., Arkin, R.: Behavior-based formation control for multirobot teams. IEEE Transactions on Robotics and Automation 14(6), 926–939 (1998)CrossRefGoogle Scholar
  6. 6.
    Balch, T., Hybinette, M.: Behavior-based coordination of large-scale robot formations. In: Proc. 4th International Conference on MultiAgent Systems, pp. 363–364 (2000)Google Scholar
  7. 7.
    Batalin, M., Sukhatme, G.: Spreading out: A local approach to multi-robot coverage. In: Proc. 6th International Symposium on Distributed Autonomous Robotic Systems, pp. 373–382 (2002)Google Scholar
  8. 8.
    Burgard, W., Moors, M., Fox, D., Simmons, R., Thrun, S.: Collaborative multi-robot exploration. In: Proc. IEEE International Conference on Robotics and Automation (ICRA), vol. 1, pp. 476–481 (2000)Google Scholar
  9. 9.
    Cieliebak, M., Flocchini, P., Prencipe, G., Santoro, N.: Solving the Robots Gathering Problem. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 1181–1196. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  10. 10.
    Culberson, J., Schaeffer, J.: Efficiently searching the 15-puzzle. Technical report, Department of Computing Science, University of Alberta (1994)Google Scholar
  11. 11.
    Das, S., Flocchini, P., Santoro, N., Yamashita, M.: On the computational power of oblivious robots: forming a series of geometric patterns. In: Proc. 29th ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing (PODC), pp. 267–276 (2010)Google Scholar
  12. 12.
    Dessmark, A., Fraigniaud, P., Pelc, A.: Deterministic Rendezvous in Graphs. In: Di Battista, G., Zwick, U. (eds.) ESA 2003. LNCS, vol. 2832, pp. 184–195. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  13. 13.
    Flocchini, P., Prencipe, G., Santoro, N., Widmayer, P.: Distributed coordination of a set of autonomous mobile robots. In: Proc. IEEE Intelligent Vehicles Symposium (IV), pp. 480–485 (2000)Google Scholar
  14. 14.
    Flocchini, P., Prencipe, G., Santoro, N., Widmayer, P.: Gathering of Asynchronous Oblivious Robots with Limited Visibility. In: Ferreira, A., Reichel, H. (eds.) STACS 2001. LNCS, vol. 2010, pp. 247–258. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  15. 15.
    Garey, M., Graham, R., Johnson, D.: Some NP-complete geometric problems. In: Proc. 8th Annual ACM Symposium on Theory of Computing (STOC), pp. 10–22 (1976)Google Scholar
  16. 16.
    Gervasi, V., Prencipe, G.: Need a fleet? Use the force! In: Proc. 2nd International Conference on Fun With Algorithms (FUN), pp. 149–164 (2001)Google Scholar
  17. 17.
    Gudmundsson, J., Levcopoulos, C.: A fast approximation algorithm for TSP with neighborhoods. Nordic Journal of Computing 6(4), 469 (1999)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Hearn, R., Demaine, E.: PSPACE-completeness of sliding-block puzzles and other problems through the nondeterministic constraint logic model of computation. Theoretical Computer Science 343(1-2), 72–96 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Hearn, R.A., Demaine, E.D.: The Nondeterministic Constraint Logic Model of Computation: Reductions and Applications. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 401–413. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  20. 20.
    Hearn, R.A., Demaine, E.D.: Games, Puzzles, and Computation. A K Peters, Ltd. (2009)Google Scholar
  21. 21.
    Hopcroft, J., Schwartz, J., Sharir, M.: On the complexity of motion planning for multiple independent objects; PSPACE-hardness of the “warehouseman’s problem”. The International Journal of Robotics Research 3(4), 76–88 (1984)CrossRefGoogle Scholar
  22. 22.
    Hordern, E.: Sliding Piece Puzzles. Oxford University Press (1986)Google Scholar
  23. 23.
    Howard, A., Matarić, M., Sukhatme, G.: An incremental self-deployment algorithm for mobile sensor networks. Autonomous Robots 13(2), 113–126 (2002)zbMATHCrossRefGoogle Scholar
  24. 24.
    Howard, A., Matarić, M., Sukhatme, G.: Mobile sensor network deployment using potential fields: A distributed, scalable solution to the area coverage problem. In: Proc. 6th International Symposium on Distributed Autonomous Robotics Systems (DARS), pp. 299–308 (2002)Google Scholar
  25. 25.
    Hsiang, T., Arkin, E., Bender, M., Fekete, S., Mitchell, J.: Algorithms for rapidly dispersing robot swarms in unknown environments. In: Proc. 5th Workshop on Algorithmic Foundations of Robotics (WAFR), pp. 77–94 (2004)Google Scholar
  26. 26.
    Hwang, Y., Ahuja, N.: Gross motion planning a survey. ACM Computing Surveys (CSUR) 24(3), 219–291 (1992)CrossRefGoogle Scholar
  27. 27.
    Karlemo, F., Östergård, P.: On sliding block puzzles. Journal of Combinatorial Mathematics and Combinatorial Computing (2000)Google Scholar
  28. 28.
    Kowalski, D.R., Malinowski, A.: How to Meet in Anonymous Network. In: Flocchini, P., Gąsieniec, L. (eds.) SIROCCO 2006. LNCS, vol. 4056, pp. 44–58. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  29. 29.
    Kurazume, R., Nagata, S.: Cooperative positioning with multiple robots. In: Proceedings of 1994 IEEE International Conference on Robotics and Automation, pp. 1250–1257. IEEE (1994)Google Scholar
  30. 30.
    Leighton, F.T.: Introduction to Parallel Algorithms and Architectures. Morgan Kaufmann Publishers (1992)Google Scholar
  31. 31.
    Nassimi, D., Sahni, S.: Bitonic sort on a mesh-connected parallel computer. IEEE Transactions on Computers 100(1), 2–7 (1979)CrossRefGoogle Scholar
  32. 32.
    Rater, D., Warmuth, M.: Finding a shortest solution for the nxn extension of the 15-puzzle is intractable. Journal of Symbolic Computation 10, 111–137 (1990)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Ratliff, H., Rosenthal, A.: Order-picking in a rectangular warehouse: a solvable case of the traveling salesman problem. Operations Research, 507–521 (1983)Google Scholar
  34. 34.
    Rekleitis, I.M., Dudek, G., Milios, E.E.: Graph-based exploration using multiple robots. In: 5th International Symposium on Distributed and Autonomous Robotic Systems, pp. 241–250. Springer (2000)Google Scholar
  35. 35.
    Scherson, I., Sen, S.: Parallel sorting in two-dimensional VLSI models of computation. IEEE Transactions on Computers 38(2), 238–249 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Simmons, R., Apfelbaum, D., Burgard, W., Fox, D., Moors, M., Thrun, S., Younes, H.: Coordination for multi-robot exploration and mapping. In: Proceedings National Conference on Artificial Intelligence, pp. 852–858. AAAI Press, MIT Press, Menlo Park, Cambridge (1999, 2000)Google Scholar
  37. 37.
    Singh, K., Fujimura, K.: Map making by cooperating mobile robots. In: Proceedings of 1993 IEEE International Conference on Robotics and Automation (ICRA), pp. 254–259 (1993)Google Scholar
  38. 38.
    Suzuki, I., Yamashita, M.: Distributed anonymous mobile robots: Formation of geometric patterns. SIAM J. Comput. 28(4), 1347–1363 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Wagner, I., Lindenbaum, M., Bruckstein, A.: Distributed covering by ant-robots using evaporating traces. IEEE Transactions on Robotics and Automation 15(5), 918–933 (1999)CrossRefGoogle Scholar
  40. 40.
    Wang, J.: On sign-board based inter-robot communication in distributed robotic systems. In: Proc. 1994 IEEE International Conference on Robotics and Automation, pp. 1045–1050 (1994)Google Scholar
  41. 41.
    Wilson, R.M.: Graph puzzles, homotopy, and the alternating group. Journal of Combinatorial Theory, Series B 16(1), 86–96 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Yamauchi, B.: Frontier-based exploration using multiple robots. In: Proc. 2nd International Conference on Autonomous Agents, pp. 47–53 (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Michael A. Bender
    • 1
    • 2
  • Ritwik Bose
    • 1
  • Rezaul Chowdhury
    • 1
  • Samuel McCauley
    • 1
  1. 1.Department of Computer ScienceStony Brook UniversityUSA
  2. 2.Tokutek, Inc.USA

Personalised recommendations