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Directed Graph Exploration

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Principles of Distributed Systems (OPODIS 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7702))

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Abstract

We study the problem of exploring all nodes of an unknown directed graph. A searcher has to construct a tour that visits all nodes, but only has information about the parts of the graph it already visited. The goal is to minimize the cost of such a tour. In this paper, we present upper and lower bounds for both the deterministic and the randomized online version of exploring all nodes of directed graphs. Our bounds are sharp or sharp up to a small constant, depending on the specific model. Essentially, exploring a directed graph has a multiplicative overhead linear in the number of nodes. If one wants to search for just a node in unweighted directed graphs, a greedy algorithm with quadratic multiplicative overhead can only be improved by a factor of at most two. We were also able to show that randomly choosing a starting point does not improve lower bounds beyond a small constant factor.

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References

  1. Albers, S., Henzinger, M.R.: Exploring Unknown Environments. SIAM J. Comput. 29(4), 1164–1188 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Asadpour, A., Goemans, M.X., Madry, A., Gharan, S.O., Saberi, A.: An O(log n/log log n)-approximation Algorithm for the Asymmetric Traveling Salesman Problem. In: Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, pp. 379–389. Society for Industrial and Applied Mathematics, Philadelphia (2010)

    Google Scholar 

  3. Ausiello, G., Bonifaci, V., Laura, L.: The on-line asymmetric traveling salesman problem. J. Discrete Algorithms 6(2), 290–298 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Baldoni, R., Bonnet, F., Milani, A., Raynal, M.: Anonymous graph exploration without collision by mobile robots. Inf. Process. Lett. 109(2), 98–103 (2008)

    Article  MathSciNet  Google Scholar 

  5. Flocchini, P., Fraigniaud, P., Santoro, N.: Capture of an intruder by mobile agents. In: Proceedings of the Fourteenth Annual ACM Symposium on Parallel Algorithms and Architectures, SPAA 2002, pp. 200–209. ACM, New York (2002)

    Google Scholar 

  6. Bender, M.A., Fernandez, A., Ron, D., Sahai, A., Vadhan, S.P.: The Power of a Pebble: Exploring and Mapping Directed Graphs. Inf. Comput. 176(1), 1–21 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brass, P., Gasparri, A., Cabrera-Mora, F., Xiao, J.: Multi-robot tree and graph exploration. In: Proceedings of the 2009 IEEE International Conference on Robotics and Automation, ICRA 2009, pp. 495–500. IEEE Press, Piscataway (2009)

    Google Scholar 

  8. Bläser, M.: A new approximation algorithm for the asymmetric TSP with triangle inequality. ACM Transactions on Algorithms 4(4), 47:1–47:15 (2008)

    Google Scholar 

  9. Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  10. Bermann, P.: On-line Searching and Navigation. In: Fiat, A., Woeginger, G.J. (eds.) Online Algorithms 1996. LNCS, vol. 1442, pp. 232–241. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  11. Burgard, W., Moors, M., Fox, D., Simmons, R.G., Thrun, S.: Collaborative Multi-Robot Exploration. In: Proceedings of the 2000 IEEE International Conference on Robotics and Automation, ICRA 2000, pp. 476–481. IEEE, San Francisco (2000)

    Google Scholar 

  12. Chalopin, J., Flocchini, P., Mans, B., Santoro, N.: Network Exploration by Silent and Oblivious Robots. In: Thilikos, D.M. (ed.) WG 2010. LNCS, vol. 6410, pp. 208–219. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  13. Das, S., Flocchini, P., Kutten, S., Nayak, A., Santoro, N.: Map construction of unknown graphs by multiple agents. Theor. Comput. Sci. 385(1-3), 34–48 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Deng, X., Papadimitriou, C.H.: Exploring an unknown graph (Extended Abstract). In: Proceedings of the 31st Annual Symposium on Foundations of Computer Science, FOCS 1990, vol. I, pp. 355–361. IEEE Computer Society, St. Louis (1990)

    Chapter  Google Scholar 

  15. Deng, X., Papadimitriou, C.H.: Exploring an unknown graph. J. Graph Theory 32(3), 265–297 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dobrev, S., Královic̆, R., Markou, E.: Online Graph Exploration with Advice. In: Even, G., Halldórsson, M.M. (eds.) SIROCCO 2012. LNCS, vol. 7355, pp. 267–278. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  17. Dynia, M., Łopuszański, J., Schindelhauer, C.: Why Robots Need Maps. In: Prencipe, G., Zaks, S. (eds.) SIROCCO 2007. LNCS, vol. 4474, pp. 41–50. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  18. Engebretsen, L.: An Explicit Lower Bound for TSP with Distances One and Two. Algorithmica 35(4), 301–318 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Feige, U., Singh, M.: Improved Approximation Ratios for Traveling Salesperson Tours and Paths in Directed Graphs. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds.) APPROX and RANDOM 2007. LNCS, vol. 4627, pp. 104–118. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  20. Fraigniaud, P., Ilcinkas, D.: Digraphs Exploration with Little Memory. In: Diekert, V., Habib, M. (eds.) STACS 2004. LNCS, vol. 2996, pp. 246–257. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  21. Fraigniaud, P., Gasieniec, L., Kowalski, D.R., Pelc, A.: Collective tree exploration. Networks 48(3), 166–177 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Frieze, A.M., Galbiati, G., Maffioli, F.: On the worst-case performance of some algorithms for the asymmetric traveling salesman problem. Networks 12(1), 23–39 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  23. Fleischer, R., Kamphans, T., Klein, R., Langetepe, E., Trippen, G.: Competitive Online Approximation of the Optimal Search Ratio. SIAM J. Comput. 38(3), 881–898 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Fleischer, R., Trippen, G.: Experimental Studies of Graph Traversal Algorithms. In: Jansen, K., Margraf, M., Mastrolli, M., Rolim, J.D.P. (eds.) WEA 2003. LNCS, vol. 2647, pp. 120–133. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  25. Fleischer, R., Trippen, G.: Exploring an Unknown Graph Efficiently. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 11–22. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  26. Hurkens, C.A.J., Woeginger, G.J.: On the nearest neighbor rule for the traveling salesman problem. Oper. Res. Lett. 32(1), 1–4 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kalyanasundaram, B., Pruhs, K.: Constructing Competitive Tours from Local Information. Theor. Comput. Sci. 130(1), 125–138 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kaplan, H., Lewenstein, M., Shafrir, N., Sviridenko, M.: Approximation Algorithms for Asymmetric TSP by Decomposing Directed Regular Multigraphs. In: Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2003, pp. 56–65. IEEE Computer Society, Washington, DC (2003)

    Chapter  Google Scholar 

  29. Kuhn, F., Oshman, R.: The Complexity of Data Aggregation in Directed Networks. In: Peleg, D. (ed.) DISC 2011. LNCS, vol. 6950, pp. 416–431. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  30. Kutten, S.: Stepwise construction of an efficient distributed traversing algorithm for general strongly connected directed networks or: Traversing one way streets with no map. In: Computer Communication Technologies for the 90’s, Proceedings of the Ninth International Conference on Computer Communication, ICCC 1988, pp. 446–452. International Council for Computer Communication, Elsevier (1988)

    Google Scholar 

  31. Megow, N., Mehlhorn, K., Schweitzer, P.: Online Graph Exploration: New Results on Old and New Algorithms. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part II. LNCS, vol. 6756, pp. 478–489. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  32. Miyazaki, S., Morimoto, N., Okabe, Y.: The Online Graph Exploration Problem on Restricted Graphs. IEICE Transactions 92-D(9), 1620–1627 (2009)

    Google Scholar 

  33. Prakash, R.: Unidirectional links prove costly in wireless ad hoc networks. In: Proceedings of the 3rd International Workshop on Discrete Algorithms and Methods for Mobile Computing and Communications, DIALM 1999, pp. 15–22. ACM, New York (1999)

    Chapter  Google Scholar 

  34. Ribeiro, B.F., Wang, P., Murai, F., Towsley, D.: Sampling directed graphs with random walks. In: Proceedings of the IEEE INFOCOM 2012, pp. 1692–1700. IEEE, Orlando (2012)

    Chapter  Google Scholar 

  35. Rosenkrantz, D.J., Stearns, R.E., Lewis II, P.M.: An Analysis of Several Heuristics for the Traveling Salesman Problem. SIAM J. Comput. 6(3), 563–581 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  36. Sedgewick, R., Vitter, J.S.: Shortest Paths in Euclidean Graphs. Algorithmica 1(1), 31–48 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  37. Vishwanathan, S.: An Approximation Algorithm for the Asymmetric Travelling Salesman Problem with Distances One and Two. Inf. Process. Lett. 44(6), 297–302 (1992)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Computer Engineering and Networks Laboratory, ETH Zurich, 8092, Zurich, Switzerland

    Klaus-Tycho Förster & Roger Wattenhofer

Authors
  1. Klaus-Tycho Förster
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  2. Roger Wattenhofer
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Editor information

Editors and Affiliations

  1. Dipartimento di Informatica, Automatica e Gestionale, Università degli Studi di Roma “La Sapienza”, Via Ariosto 25, 00168, Rome, Italy

    Roberto Baldoni

  2. School of Electrical Engineering and Computer Science, University of Ottawa, 800 King Edward Street, K1N 6N5, Ottawa, ON, Canada

    Paola Flocchini

  3. Electrical and Computing Engineering Dept., Virginia Technical University, 302 Whittemore Street, 24061, Blacksburg, VA, USA

    Ravindran Binoy

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Förster, KT., Wattenhofer, R. (2012). Directed Graph Exploration. In: Baldoni, R., Flocchini, P., Binoy, R. (eds) Principles of Distributed Systems. OPODIS 2012. Lecture Notes in Computer Science, vol 7702. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35476-2_11

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  • DOI: https://doi.org/10.1007/978-3-642-35476-2_11

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-35475-5

  • Online ISBN: 978-3-642-35476-2

  • eBook Packages: Computer ScienceComputer Science (R0)

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