Directed Graph Exploration

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7702)


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.


online algorithms graph exploration mobile agents and autonomous robots 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Albers, S., Henzinger, M.R.: Exploring Unknown Environments. SIAM J. Comput. 29(4), 1164–1188 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 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. 3.
    Ausiello, G., Bonifaci, V., Laura, L.: The on-line asymmetric traveling salesman problem. J. Discrete Algorithms 6(2), 290–298 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 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)MathSciNetCrossRefGoogle Scholar
  5. 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. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 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. 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. 9.
    Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  10. 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)CrossRefGoogle Scholar
  11. 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. 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)CrossRefGoogle Scholar
  13. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 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)CrossRefGoogle Scholar
  15. 15.
    Deng, X., Papadimitriou, C.H.: Exploring an unknown graph. J. Graph Theory 32(3), 265–297 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 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)CrossRefGoogle Scholar
  17. 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)CrossRefGoogle Scholar
  18. 18.
    Engebretsen, L.: An Explicit Lower Bound for TSP with Distances One and Two. Algorithmica 35(4), 301–318 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 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)CrossRefGoogle Scholar
  20. 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)CrossRefGoogle Scholar
  21. 21.
    Fraigniaud, P., Gasieniec, L., Kowalski, D.R., Pelc, A.: Collective tree exploration. Networks 48(3), 166–177 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 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)CrossRefGoogle Scholar
  25. 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)CrossRefGoogle Scholar
  26. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Kalyanasundaram, B., Pruhs, K.: Constructing Competitive Tours from Local Information. Theor. Comput. Sci. 130(1), 125–138 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 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)CrossRefGoogle Scholar
  29. 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)CrossRefGoogle Scholar
  30. 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. 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)CrossRefGoogle Scholar
  32. 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. 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)CrossRefGoogle Scholar
  34. 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)CrossRefGoogle Scholar
  35. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Sedgewick, R., Vitter, J.S.: Shortest Paths in Euclidean Graphs. Algorithmica 1(1), 31–48 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 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)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Computer Engineering and Networks LaboratoryETH ZurichZurichSwitzerland

Personalised recommendations