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

  • Conference paper
Fun with Algorithms (FUN 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7288))

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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.

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References

  1. Alpern, S.: Rendezvous search on labeled networks. Naval Research Logistics 49(3), 256–274 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alpern, S., Baston, V., Essegaier, S.: Rendezvous search on a graph. Journal of Applied Probability 36(1), 223–231 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arkin, E., Hassin, R.: Approximation algorithms for the geometric covering salesman problem. Discrete Applied Mathematics 55(3), 197–218 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arkin, R.: Motor schema-based mobile robot navigation. In: Proc. IEEE Conference on Robotics and Automation, pp. 264–271 (1987)

    Google Scholar 

  5. Balch, T., Arkin, R.: Behavior-based formation control for multirobot teams. IEEE Transactions on Robotics and Automation 14(6), 926–939 (1998)

    Article  Google Scholar 

  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. 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. 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. 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)

    Chapter  Google Scholar 

  10. Culberson, J., Schaeffer, J.: Efficiently searching the 15-puzzle. Technical report, Department of Computing Science, University of Alberta (1994)

    Google Scholar 

  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. 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)

    Chapter  Google Scholar 

  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. 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)

    Chapter  Google Scholar 

  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. 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. Gudmundsson, J., Levcopoulos, C.: A fast approximation algorithm for TSP with neighborhoods. Nordic Journal of Computing 6(4), 469 (1999)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  20. Hearn, R.A., Demaine, E.D.: Games, Puzzles, and Computation. A K Peters, Ltd. (2009)

    Google Scholar 

  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)

    Article  Google Scholar 

  22. Hordern, E.: Sliding Piece Puzzles. Oxford University Press (1986)

    Google Scholar 

  23. Howard, A., Matarić, M., Sukhatme, G.: An incremental self-deployment algorithm for mobile sensor networks. Autonomous Robots 13(2), 113–126 (2002)

    Article  MATH  Google Scholar 

  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. 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. Hwang, Y., Ahuja, N.: Gross motion planning a survey. ACM Computing Surveys (CSUR) 24(3), 219–291 (1992)

    Article  Google Scholar 

  27. Karlemo, F., Östergård, P.: On sliding block puzzles. Journal of Combinatorial Mathematics and Combinatorial Computing (2000)

    Google Scholar 

  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)

    Chapter  Google Scholar 

  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. Leighton, F.T.: Introduction to Parallel Algorithms and Architectures. Morgan Kaufmann Publishers (1992)

    Google Scholar 

  31. Nassimi, D., Sahni, S.: Bitonic sort on a mesh-connected parallel computer. IEEE Transactions on Computers 100(1), 2–7 (1979)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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. 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. Scherson, I., Sen, S.: Parallel sorting in two-dimensional VLSI models of computation. IEEE Transactions on Computers 38(2), 238–249 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. Suzuki, I., Yamashita, M.: Distributed anonymous mobile robots: Formation of geometric patterns. SIAM J. Comput. 28(4), 1347–1363 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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. Wilson, R.M.: Graph puzzles, homotopy, and the alternating group. Journal of Combinatorial Theory, Series B 16(1), 86–96 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  42. Yamauchi, B.: Frontier-based exploration using multiple robots. In: Proc. 2nd International Conference on Autonomous Agents, pp. 47–53 (1998)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, Stony Brook University, NY, 11794-4400, USA

    Michael A. Bender, Ritwik Bose, Rezaul Chowdhury & Samuel McCauley

  2. Tokutek, Inc., USA

    Michael A. Bender

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  1. Michael A. Bender
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  2. Ritwik Bose
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  3. Rezaul Chowdhury
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  4. Samuel McCauley
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Editors and Affiliations

  1. School of Computer Science, Carleton University, 1125 Col. By Dr., K1S 5B6, Ottawa, ON, Canada

    Evangelos Kranakis

  2. Department of Mathematics and Computer Science, Wesleyan University, 06459, Middleton, CT, USA

    Danny Krizanc

  3. Dipartimento di Scienze Ambientali, Informatica e Statistica, Università Ca’ Foscari, Via Torino 155, 30172, Mestre (Ve), Italy

    Flaminia Luccio

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Bender, M.A., Bose, R., Chowdhury, R., McCauley, S. (2012). The Kissing Problem: How to End a Gathering When Everyone Kisses Everyone Else Goodbye. In: Kranakis, E., Krizanc, D., Luccio, F. (eds) Fun with Algorithms. FUN 2012. Lecture Notes in Computer Science, vol 7288. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30347-0_6

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  • DOI: https://doi.org/10.1007/978-3-642-30347-0_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-30346-3

  • Online ISBN: 978-3-642-30347-0

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