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A Self-stabilizing Algorithm for the Median Problem in Partial Rectangular Grids and Their Relatives

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Given a graph G=(V,E), a vertex v of G is a median vertex if it minimizes the sum of the distances to all other vertices of G. The median problem consists of finding the set of all median vertices of G. In this note, we present self-stabilizing algorithms for the median problem in partial rectangular grids and relatives. Our algorithms are based on the fact that partial rectangular grids can be isometrically embedded into the Cartesian product of two trees, to which we apply the algorithm proposed by Antonoiu and Srimani (J. Comput. Syst. Sci. 58:215–221, 1999) and Bruell et al. (SIAM J. Comput. 29:600–614, 1999) for computing the medians in trees. Then we extend our approach from partial rectangular grids to a more general class of plane quadrangulations. We also show that the characterization of medians of trees given by Gerstel and Zaks (Networks 24:23–29, 1994) extends to cube-free median graphs, a class of graphs which includes these quadrangulations.

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  1. 1.

    Afek, Y., Kutten, S., Yung, M.: The local detection paradigm and its applications to self-stabilization. Theor. Comput. Sci. 186, 199–229 (1997)

  2. 2.

    Aggarwal, S., Kutten, S.: Time optimal self-stabilizing spanning tree algorithm. In: FSTTCS’93. LNCS, vol. 761, pp. 400–410. Springer, Berlin (1993)

  3. 3.

    Antonoiu, G., Srimani, P.K.: A self-stabilizing leader election algorithm for tree graphs. J. Parallel Distrib. Comput. 34, 227–232 (1996)

  4. 4.

    Antonoiu, G., Srimani, P.K.: Distributed self-stabilizing algorithm for minimum spanning tree construction. In: Euro-Par’97. LNCS, vol. 1300, pp. 480–487. Springer, Berlin (1997)

  5. 5.

    Antonoiu, G., Srimani, P.K.: A self-stabilizing distributed algorithm to find the median of a tree graph. J. Comput. Syst. Sci. 58, 215–221 (1999)

  6. 6.

    Antonoiu, G., Srimani, P.K.: Self-stabilizing protocol for mutual exclusion among neighboring nodes in a tree structured distributed system. Parallel Algorithms Appl. 14, 1–18 (1999)

  7. 7.

    Arora, A., Dolev, S., Gouda, M.G.: Maintaining digital clocks in step. Parallel Proc. Let. 1, 11–18 (1991)

  8. 8.

    Awerbuch, B., Kutten, S., Mansour, Y., Patt-Shamir, B., Varghese, G.: Time optimal self-stabilizing synchronization. In: STOC’93, pp. 652–661. ACM, New York (1993)

  9. 9.

    Balakrishnan, K., Brešar, B., Changat, M., Klavzar, S., Kovše, M., Subhamathi, A.R.: Computing median and antimedian sets in median graphs. Algorithmica 57, 207–216 (2010)

  10. 10.

    Bandelt, H.-J., Barthélemy, J.-P.: Medians in median graphs. Discrete Appl. Math. 8, 131–142 (1984)

  11. 11.

    Bandelt, H.-J., Chepoi, V.: Graphs with connected medians. SIAM J. Discrete Math. 15, 268–282 (2002)

  12. 12.

    Bandelt, H.-J., Chepoi, V.: Metric graph theory and geometry: a survey. In: Goodman, J.E., Pach, J., Pollack, R. (eds.) Surveys on Discrete and Computational Geometry. Twenty Years Later. Contemp. Math., vol. 453, pp. 49–86. AMS, Providence (2008)

  13. 13.

    Bandelt, H.-J., Chepoi, V., Eppstein, D.: Combinatorics and geometry of finite and infinite squaregraphs. SIAM J. Discrete Math. (to appear). Electronic preprint arXiv:0905.4537 (2009)

  14. 14.

    Bandelt, H.-J., Chepoi, V., Eppstein, D.: Ramified rectilinear polygons: coordinatization by dendrons. arXiv:1005.1721

  15. 15.

    Barcucci, E., Del Lungo, A., Nivat, M., Pinzani, R.: Medians of polyominoes: a property for reconstruction. Int. J. Imaging Syst. Technol. 9, 69–77 (1998)

  16. 16.

    Barthélemy, J.P., Janowitz, M.F.: A formal theory of consensus SIAM J. Discrete Math. 94, 305–322 (1991)

  17. 17.

    Barthélemy, J.P., Monjardet, B.: The median procedure in cluster analysis and social choice theory. Math. Soc. Sci. 1, 235–268 (1981)

  18. 18.

    Barthélemy, J.P., Leclerc, B., Monjardet, B.: On the use of ordered sets in problems of comparison and consensus of classifications. J. Classif. 3, 187–224 (1986)

  19. 19.

    Buckley, F., Harary, F.: Distances in Graphs. Addison–Wesley, Redwood City (1990)

  20. 20.

    Blair, J.R.S., Manne, F.: Efficient self-stabilizing algorithms for tree networks. In: ICDCS’03, pp. 20–26. IEEE Computer Society, Los Alamitos (2003)

  21. 21.

    Bruell, S.B., Ghosh, S., Karaata, M.H., Pemmaraju, S.V.: Self-stabilizing algorithms for finding centers and medians of trees. SIAM J. Comput. 29, 600–614 (1999)

  22. 22.

    Boldi, P., Vigna, S.: An effective characterization of computability in anonymous networks. In: DISC 2001. LNCS, vol. 2180, pp. 33–47. Springer, Berlin (2001)

  23. 23.

    Boldi, P., Vigna, S.: Universal dynamic synchronous self-stabilization. Distrib. Comput. 15, 137–153 (2002)

  24. 24.

    Chalopin, J.: Algorithmique Distribuée, Calculs Locaux et Homomorphismes de Graphes. PhD Thesis Université Bordeaux I (2006)

  25. 25.

    Chepoi, V.: Graphs of some CAT(0) complexes. Adv. Appl. Math. 24, 125–179 (2000)

  26. 26.

    Chepoi, V., Dragan, F., Vaxès, Y.: Center and diameter problem in planar quadrangulations and triangulations. In: SODA’02, pp. 346–355. ACM/SIAM, New York/Philadelphia (2002)

  27. 27.

    Chepoi, V., Fanciullini, C., Vaxès, Y.: Median problem in some plane triangulations and quadrangulations. Comput. Geom. 27, 193–210 (2004)

  28. 28.

    Datta, A.K., Derby, J.L., Lawrence, J.E., Tixeuil, S.: Stabilizing hierarchical routing. J. Interconnet. Netw. 1, 283–302 (2000)

  29. 29.

    Datta, A., Larmore, L., Vemula, P.: Self-stabilizing leader election in optimal space. In: SSS 2008. LNCS, vol. 5340, pp. 109–123. Springer, Berlin (2008)

  30. 30.

    Das Sharma, D., Pradhan, D.: Submesh allocation in mesh multicomputers using busy-lists: a best-fit approach with complete recognition capability. J. Parallel Distrib. Comput. 36, 106–118 (1996)

  31. 31.

    Daurat, A., Del Lungo, A., Nivat, M.: Medians of discrete sets according to a linear distance. Discrete Comput. Geom. 23, 465–483 (2000)

  32. 32.

    Delaët, S., Nguyen, D., Tixeuil, S.: Stabilité et auto-stabilisation du routage inter-domaine dans internet. In: RIVF, pp. 139–144. Editions Suger, Paris (2003)

  33. 33.

    Dijkstra, E.W.: Self-stabilizing systems in spite of distributed control. Commun. ACM 17, 643–644 (1974)

  34. 34.

    Dolev, S.: Self-stabilization. MIT Press, Cambridge (2000)

  35. 35.

    Dolev, S., Israeli, A., Moran, S.: Self-stabilization of dynamic systems assuming only read/write atomicity. In: PODC’90, pp. 103–118. ACM, New York (1990)

  36. 36.

    Dress, A., Scharlau, R.: Gated sets in metric spaces. Aequat. Math. 34, 112–120 (1987)

  37. 37.

    Fraigniaud, P., Vial, S.: One-to-all and all-to-all communications in partial meshes. Parallel Proc. Lett. 9, 9–20 (1999)

  38. 38.

    Gärtner, F.: A survey of self-stabilizing spanning-tree construction algorithms. Technical Report IC/2003/38, Swiss Federal Institute of Technology (2003)

  39. 39.

    Gerstel, O., Zaks, S.: A new characterization of tree medians with applications to distributed algorithms. Networks 24, 23–29 (1994)

  40. 40.

    Ghosh, S., Karaata, M.H.: A self-stabilizing algorithm for coloring planar graphs. Distrib. Comput. 7, 55–59 (1993)

  41. 41.

    Ghosh, S., Gupta, A., Pemmaraju, S.V.: A self-stabilizing algorithm for the maximum flow problem. Distrib. Comput. 10, 167–180 (1997)

  42. 42.

    Goldman, A.J., Witzgall, C.J.: A localization theorem for optimal facility placement. Transp. Sci. 4, 406–409 (1970)

  43. 43.

    Hedetniemi, S.T., Jacobs, D.P., Srimani, P.K.: Maximal matching stabilizes in time o(m). Inf. Proc. Lett. 80, 221–223 (2001)

  44. 44.

    Herman, T., Ghosh, S.: Stabilizing phase-clocks. Inf. Proc. Lett. 54, 259–265 (1995)

  45. 45.

    Huang, T.C., Lin, J., Chen, H.: A self-stabilizing algorithm which finds a 2-center of a tree. Comput. Math. Appl. 40, 607–624 (2000)

  46. 46.

    Kariv, O., Hakimi, S.L.: An algorithmic approach to network location problems, I. SIAM J. Appl. Math. 37, 513–538 (1979)

  47. 47.

    Kariv, O., Hakimi, S.L.: An algorithmic approach to network location problems, II. SIAM J. Appl. Math. 37, 539–560 (1979)

  48. 48.

    Korach, E., Rotem, D., Santoro, N.: Distributed algorithms for finding centers and medians in networks. ACM Trans. Program. Lang. Syst. 6, 380–401 (1984)

  49. 49.

    Métivier, Y., Saheb, N.: Medians and centers of polyominoes. Inf. Proc. Lett. 57, 175–181 (1981)

  50. 50.

    Mulder, H.M.: The Interval Function of a Graph, Math. Centre Tracts 132, Amsterdam (1980)

  51. 51.

    Nesterenko, M., Mizuno, M.: A quorum-based self-stabilizing distributed mutual exclusion algorithm. J. Parallel Distrib. Comput. 62, 284–305 (2002)

  52. 52.

    Soltan, P., Chepoi, V.: The solution of the Weber problem for discrete median metric spaces. Tr. Tbil. Mat. Inst. 85, 52–76 (1987) (in Russian). English translation: in Chogoshvili G.S. (ed.) Topological, Projective and Combinatorial Properties of Spaces, Nova Science, pp. 89–129 (1992)

  53. 53.

    Tansel, B.C, Francis, R.L., Lowe, T.J.: Location on networks: a survey. Manag. Sci. 29, 482–511 (1983)

  54. 54.

    van de Vel, M.L.J.: Theory of Convex Structures. North-Holland, Amsterdam (1993)

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Author information

Correspondence to Yann Vaxès.

Additional information

An extended abstract of this paper appeared in the proceedings of the 14th International Colloquium on Structural Information and Communication Complexity, SIROCCO’07. The first and the fourth authors were partly supported by the ANR grant BLAN06-1-138894 (projet OPTICOMB). The second and the third authors were supported by the ACI grant “Jeunes Chercheurs” (projet TAGADA).

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Chepoi, V., Fevat, T., Godard, E. et al. A Self-stabilizing Algorithm for the Median Problem in Partial Rectangular Grids and Their Relatives. Algorithmica 62, 146–168 (2012). https://doi.org/10.1007/s00453-010-9447-4

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  • Self-stabilizing algorithm
  • Median problem
  • Isometric embedding
  • Rectangular grid