Memory Efficient Anonymous Graph Exploration

  • Leszek Gąsieniec
  • Tomasz Radzik
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5344)

Abstract

Efficient exploration of unknown or unmapped environments has become one of the fundamental problem domains in algorithm design. Its applications range from robot navigation in hazardous environments to rigorous searching, indexing and analysing digital data available on the Internet. A large number of exploration algorithms has been proposed under various assumptions about the capability of mobile (exploring) entities and various characteristics of the environment which are to be explored. This paper considers the graph model, where the environment is represented by a graph of connections in which discrete moves are permitted only along its edges. Designing efficient exploration algorithms in this model has been extensively studied under a diverse set of assumptions, e.g., directed vs undirected graphs, anonymous nodes vs nodes with distinct identities, deterministic vs probabilistic solutions, single vs multiple agent exploration, as well as in the context of different complexity measures including the time complexity, the memory consumption, and the use of other computational resources such as tokens and messages. In this work the emphasis is on memory efficient exploration of anonymous graphs. We discuss in more detail three approaches: random walk, Propp machine and basic walk, reviewing major relevant results, presenting recent developments, and commenting on directions for further research.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Albers, S., Henzinger, M.R.: Exploring unknown environments. SIAM Journal on Computing 29, 1164–1188 (2000)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Aldous, D.J.: On the time taken by random walks on finite groups to visit every state. Journal Probability Theory and Related Fields 62(3), 361–374 (1983)MathSciNetMATHGoogle Scholar
  3. 3.
    Aleliunas, R., Karp, R.M., Lipton, R.J., Lovasz, L., Rackoff, C.: Random walks, universal traversal sequences, and the complexity of maze problems. In: FOCS 1979, pp. 218–223 (1979)Google Scholar
  4. 4.
    Alon, N., Azar, Y., Ravid, Y.: Universal sequences for complete graphs. Discrete Appl. Math. 27(1-2), 25–28 (1990)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Alon, N., Avin, C., Koucky, M., Kozma, G., Lotker, Z., Tuttle, M.R.: Many random walks are faster than one. In: SPAA 2008: Proceedings of the twentieth annual symposium on Parallelism in algorithms and architectures, pp. 119–128. ACM, New York (2008)CrossRefGoogle Scholar
  6. 6.
    Awerbuch, B., Betke, M., Rivest, R.L., Singh, M.: Piecemeal graph exploration by a mobile robot. Information and Computation 152(2), 155–172 (1999)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bar-Noy, A., Borodin, A., Karchmer, M., Linial, N., Werman, M.: Bounds on universal sequences. SIAM J. Comput. 18, 268–277 (1989)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bender, M., Fernandez, A., Ron, D., Sahai, A., Vadhan, S.: The power of a pebble: Exploring and mapping directed graphs. Information and Computation 176(1), 1–21 (2002)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Bender, M., Slonim, D.K.: The power of team exploration: two robots can learn unlabeled directed graphs. In: Proc. of FOCS 1994, pp. 75–85 (1994)Google Scholar
  10. 10.
    Bhatt, S., Even, S., Greenberg, D., Tayar, R.: Traversing Directed Eulerian Mazes. Journal of Graph Algorithms and Applications 6(2), 157–173 (2002)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Blum, A., Raghavan, P., Schieber, B.: Navigating in unfamiliar geometric terrain. SIAM Journal on Computing 26, 110–137 (1997)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Blum, M., Kozen, D.: On the power of the compass (or, why mazes are easier to search than graphs). In: Proc. of FOCS 1978, pp. 132–142 (1978)Google Scholar
  13. 13.
    Bridgland, M.F.: Universal traversal sequences for paths and cycles. J. Algorithms 8(5), 395–404 (1987)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Broder, A.Z., Karlin, A.R., Raghavan, P., Upfal, E.: Trading space for time in undirected s-t connectivity. SIAM J. Comput. 23(2), 324–334 (1994)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Budach, L.: Automata and labyrinths. Math. Nachrichten, 195–282 (1978)Google Scholar
  16. 16.
    Buss, J.F., Tompa, M.: Lower bounds on universal traversal sequences based on chains of length five. Inf. Comput. 120(2), 326–329 (1995)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Chandra, A.K., Raghavan, P., Ruzzo, W.L., Smolensky, R.: The electrical resistance of a graph captures its commute and cover times. In: Proc. STOC 1989: Proceedings of the twenty-first annual ACM symposium on Theory of computing, pp. 574–586. ACM Press, New York (1989)CrossRefGoogle Scholar
  18. 18.
    Cohen, R., Fraigniaud, P., Ilcinkas, D., Korman, A., Peleg, D.: Label-guided graph exploration by a finite automaton. In: Proc. of the 32nd International Colloquium on Automata, Languages and Programming (ICALP), pp. 335–346 (2005)Google Scholar
  19. 19.
    Cook, S.A., Rackoff, C.: Space lower bounds for maze threadability on restricted machines. SIAM Journal on Computing 9(3), 636–652 (1980)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Cooper, C., Frieze, A.: The cover time of random regular graphs. SIAM J. Discret. Math. 18(4), 728–740 (2005)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Cooper, C., Frieze, A.M.: Crawling on web graphs. In: Proc. STOC 2002, pp. 419–427 (2002)Google Scholar
  22. 22.
    Cooper, C., Frieze, A.M., Radzik, T.: Multiple random walks in random regular graphs (unpublished manuscript, 2008)Google Scholar
  23. 23.
    Cooper, C., Klasing, R., Radzik, T.: A randomized algorithm for the joining protocol in dynamic distributed networks. Technical Report RR-1432-07, LaBRI, Bordeaux, France (June 2007)Google Scholar
  24. 24.
    Cooper, J.N., Spencer, J.: Simulating a random walk with constant error. Combinatorics, Probability and Computing 15, 815–822 (2006)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Cooper, J., Doerr, B., Friedrich, T., Spencer, J.H.: Deterministic random walks on regular trees. In: Proc. of SODA 2008, pp. 766–772 (2008)Google Scholar
  26. 26.
    Cooper, J., Doerr, B., Spencer, J.H., Tardos, G.: Deterministic random walks on the integers. European Journal of Combinatorics 28(8), 2072–2090 (2007)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Czyzowicz, J., Dobrev, S., Gąsieniec, L., Ilcinkas, D., Jansson, J., Klasing, R., Lignos, I., Martin, R., Sadakane, K., Sung, W.-K.: More efficient periodic traversal in anonymous undirected graphs (unpublished manuscript, 2008)Google Scholar
  28. 28.
    Deng, X., Kameda, T., Papadimitriou, C.H.: How to learn an unknown environment: The rectilinear case. Journal of the ACM 45, 215–245 (1998)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Deng, X., Papadimitriou, C.H.: Exploring an unknown graph. Journal of Graph Theory 32(3), 265–297 (1999)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Dessmark, A., Pelc, A.: Optimal graph exploration without good maps. Theoretical Computer Science 326(1-3), 343–362 (2004)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Diks, K., Fraigniaud, P., Kranakis, E., Pelc, A.: Tree exploration with little memory. Journal of Algorithms 51, 38–63 (2004)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Dobrev, S., Jansson, J., Sadakane, K., Sung, W.-K.: Finding Short Right-Hand-on-the-Wall Walks in Graphs. In: Pelc, A., Raynal, M. (eds.) SIROCCO 2005. LNCS, vol. 3499, pp. 127–139. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  33. 33.
    Doerr, B., Friedrich, T.: Deterministic Random Walks on the Two-Dimensional Grid. In: Asano, T. (ed.) ISAAC 2006. LNCS, vol. 4288, pp. 474–483. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  34. 34.
    Duncan, C.A., Kobourov, S.G., Kumar, V.S.A.: Optimal constrained graph exploration. ACM Transaction on Algorithms 2(3), 380–402 (2006)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Feige, U.: A tight upper bound on the cover time for random walks on graphs. Random Structures and Algorithms 6(1), 51–54 (1995)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Feige, U.: A tight lower bound on the cover time for random walks on graphs. Random Struct. Algorithms 6(4), 433–438 (1995)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Feige, U.: Collecting coupons on trees, and the cover time of random walks. Computational Complexity 6(4), 341–356 (1996)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Flocchini, P., Mans, B., Santoro, N.: On the impact of sense of direction on message complexity. Information Processing Letters 63(1), 23–31 (1997)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Flocchini, P., Mans, B., Santoro, N.: Sense of direction: definition, properties and classes. Networks 32(3), 165–180 (1998)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Fraigniaud, P., Gavoille, C., Mans, B.: Interval routing schemes allow broadcasting with linear message-complexity. Distributed Computing 14(4), 217–229 (2001)CrossRefMATHGoogle Scholar
  41. 41.
    Fraigniaud, P., Ilcinkas, D.: Digraph exploration with little memory. In: Diekert, V., Habib, M. (eds.) STACS 2004. LNCS, vol. 2996, pp. 246–257. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  42. 42.
    Fraigniaud, P., Ilcinkas, D., Peer, G., Pelc, A., Peleg, D.: Graph exploration by a finite automaton. Theoretical Computer Science 345(2-3), 331–344 (2005)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Garey, M.R., Johnson, D.S., Tarjan, R.E.: The planar hamiltonian circuit problem is np-complete. SIAM J. Comput. 5(4), 704–714 (1976)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Gąsieniec, L., Klasing, R., Martin, R., Navarra, A., Zhang, X.: Fast Periodic Graph Exploration with Constant Memory. J. Comput. Syst. Sci. 74(5), 808–822 (2007)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Gąsieniec, L., Pelc, A., Radzik, T., Zhang, X.: Tree exploration with logarithmic memory. In: SODA, pp. 585–594 (2007)Google Scholar
  46. 46.
    Hoory, S., Wigderson, A.: Universal traversal sequences for expander graphs. Inf. Process. Lett. 46(2), 67–69 (1993)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Ikeda, S., Kubo, I., Okumoto, N., Yamashita, M.: Impact of local topological information on random walks on finite graphs. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 1054–1067. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  48. 48.
    Ilcinkas, D.: Setting Port Numbers for Fast Graph Exploration. In: Flocchini, P., Gąsieniec, L. (eds.) SIROCCO 2006. LNCS, vol. 4056, pp. 59–69. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  49. 49.
    Impagliazzo, R., Nisan, N., Wigderson, A.: Pseudorandomness for network algorithms. In: Proc. STOC 1994: Proceedings of the twenty-sixth annual ACM symposium on Theory of computing, pp. 356–364. ACM Press, New York (1994)CrossRefGoogle Scholar
  50. 50.
    Istrail, S.: Polynomial universal traversing sequences for cycles are constructible. In: Proc. STOC 1988: Proceedings of the twentieth annual ACM symposium on Theory of computing, pp. 491–503. ACM Press, New York (1988)CrossRefGoogle Scholar
  51. 51.
    Karloff, H.J., Paturi, R., Simon, J.: Universal traversal sequences of length n O(logn) for cliques. Inf. Process. Lett. 28(5), 241–243 (1988)MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Koucký, M.: Universal traversal sequences with backtracking. J. Comput. Syst. Sci. 65(4), 717–726 (2002)MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Koucký, M.: Log-space constructible universal traversal sequences for cycles of length o(n4.03). Theor. Comput. Sci. 296(1), 117–144 (2003)MathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    Kozen, D.: Automata and planar graphs. In: Proc. of Fundations Computatial Theory (FCT 1979), pp. 243–254 (1979)Google Scholar
  55. 55.
    Law, C., Siu, K.-Y.: Distributed construction of random expander graphs. In: Proc. 22nd Annual Joint Conference of the IEEE Computer and Communications Societies (April 2003)Google Scholar
  56. 56.
    Panaite, P., Pelc, A.: Impact of topographic information on graph exploration efficiency. Networks 36, 96–103 (2000)MathSciNetCrossRefMATHGoogle Scholar
  57. 57.
    Priezzhev, V.B., Dhar, D., Dhar, A., Krishnamurthy, S.: Eulerian walkers as a model of selforganized criticality. Physics Review Letters 77, 5079–5082 (1996)CrossRefGoogle Scholar
  58. 58.
    Rabin, M.O.: Maze threading automata. Technical Report Seminar Talk, University of California at Berkeley (October 1967)Google Scholar
  59. 59.
    Rao, N., Kareti, S., Shi, W., Iyengar, S.: Robot navigation in unknown terrains: Introductory survey of length, non-heuristic algorithms. Technical Report ORNL/TM12410, Oak Ridge National Lab (1993)Google Scholar
  60. 60.
    Reingold, O.: Undirected st-connectivity in log-space. In: Proc. STOC 2005, pp. 376–385 (2005)Google Scholar
  61. 61.
    Rollik, H.A.: Automaten in planaren graphen. Acta Informatica 13, 287–298 (1980)MathSciNetCrossRefMATHGoogle Scholar
  62. 62.
    Rubinfeld, R.: The cover time of a regular expander is O(n log n). Information Processing Letters 35, 49–51 (1990)MathSciNetCrossRefMATHGoogle Scholar
  63. 63.
    Winkler, P., Zuckerman, D.: Multiple cover time. Random Structures and Algorithms 9, 403–411 (1996)MathSciNetCrossRefMATHGoogle Scholar
  64. 64.
    Yanovski, V., Wagner, I.A., Bruckstein, A.M.: A Distributed Ant Algorithm for Efficiently Patrolling a Network. Algorithmica 37, 165–186 (2003)MathSciNetCrossRefMATHGoogle Scholar
  65. 65.
    Zuckerman, D.: A technique for lower bounding the cover time. SIAM J. Discret. Math. 5(1), 81–87 (1992)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Leszek Gąsieniec
    • 1
  • Tomasz Radzik
    • 2
  1. 1.Department of Computer ScienceUniversity of LiverpoolLiverpoolUnited Kingdom
  2. 2.Department of Computer ScienceKing’s College London, StrandLondonUnited Kingdom

Personalised recommendations