Deterministic Treasure Hunt in the Plane with Angular Hints

Authors Sébastien Bouchard, Yoann Dieudonné, Andrzej Pelc, Franck Petit

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Author Details

Sébastien Bouchard
  • Sorbonne Université, CNRS, INRIA, LIP6, F-75005 Paris, France
Yoann Dieudonné
  • Laboratoire MIS, Université de Picardie Jules Verne, Amiens, France
Andrzej Pelc
  • Département d'informatique, Université du Québec en Outaouais, Gatineau, Canada
Franck Petit
  • Sorbonne Université, CNRS, INRIA, LIP6, F-75005 Paris, France

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Sébastien Bouchard, Yoann Dieudonné, Andrzej Pelc, and Franck Petit. Deterministic Treasure Hunt in the Plane with Angular Hints. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 48:1-48:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


A mobile agent equipped with a compass and a measure of length has to find an inert treasure in the Euclidean plane. Both the agent and the treasure are modeled as points. In the beginning, the agent is at a distance at most D>0 from the treasure, but knows neither the distance nor any bound on it. Finding the treasure means getting at distance at most 1 from it. The agent makes a series of moves. Each of them consists in moving straight in a chosen direction at a chosen distance. In the beginning and after each move the agent gets a hint consisting of a positive angle smaller than 2 pi whose vertex is at the current position of the agent and within which the treasure is contained. We investigate the problem of how these hints permit the agent to lower the cost of finding the treasure, using a deterministic algorithm, where the cost is the worst-case total length of the agent's trajectory. It is well known that without any hint the optimal (worst case) cost is Theta(D^2). We show that if all angles given as hints are at most pi, then the cost can be lowered to O(D), which is optimal. If all angles are at most beta, where beta<2 pi is a constant unknown to the agent, then the cost is at most O(D^{2-epsilon}), for some epsilon>0. For both these positive results we present deterministic algorithms achieving the above costs. Finally, if angles given as hints can be arbitrary, smaller than 2 pi, then we show that cost Theta(D^2) cannot be beaten.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Computing methodologies → Mobile agents
  • treasure hunt
  • deterministic algorithm
  • mobile agent
  • hint
  • plane


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  1. Oswin Aichholzer, Franz Aurenhammer, Christian Icking, Rolf Klein, Elmar Langetepe, and Günter Rote. Generalized self-approaching curves. Discrete Applied Mathematics, 109(1-2):3-24, 2001. Google Scholar
  2. Steve Alpern and Shmuel Gal. The Theory of Search Games and Rendezvous. Kluwer Academic Publications, 2003. Google Scholar
  3. Ricardo A. Baeza-Yates, Joseph C. Culberson, and Gregory J. E. Rawlins. Searching in the Plane. Inf. Comput., 106(2):234-252, 1993. Google Scholar
  4. Anatole Beck and D.J. Newman. Yet more on the linear search problem. Israel J. Math., 8:419-429, 1970. Google Scholar
  5. Lucas Boczkowski, Amos Korman, and Yoav Rodeh. Searching a Tree with Permanently Noisy Advice. In 26th Annual European Symposium on Algorithms, ESA 2018, August 20-22, 2018, Helsinki, Finland, pages 54:1-54:13, 2018. Google Scholar
  6. Anthony Bonato and Richard Nowakowski. The Game of Cops and Robbers on Graphs. American Mathematical Society, 2011. Google Scholar
  7. Timothy H. Chung, Geoffrey A. Hollinger, and Volkan Isler. Search and pursuit-evasion in mobile robotics - A survey. Auton. Robots, 31(4):299-316, 2011. Google Scholar
  8. Erik D. Demaine, Sándor P. Fekete, and Shmuel Gal. Online searching with turn cost. Theor. Comput. Sci., 361(2-3):342-355, 2006. Google Scholar
  9. Yuval Emek, Tobias Langner, David Stolz, Jara Uitto, and Roger Wattenhofer. How many ants does it take to find the food? Theor. Comput. Sci., 608:255-267, 2015. Google Scholar
  10. G. Matthew Fricke, Joshua P. Hecker, Antonio D. Griego, Linh T. Tran, and Melanie E. Moses. A distributed deterministic spiral search algorithm for swarms. In 2016 IEEE/RSJ International Conference on Intelligent Robots and Systems, IROS 2016, Daejeon, South Korea, October 9-14, 2016, pages 4430-4436, 2016. Google Scholar
  11. Branko Grünbaum. Partitions of mass-distributions and convex bodies by hyperplanes. Pacific J. Math., 10:1257-1261, 1960. Google Scholar
  12. Artur Jez and Jakub Lopuszanski. On the two-dimensional cow search problem. Inf. Process. Lett., 109(11):543-547, 2009. Google Scholar
  13. Ming-Yang Kao, John H. Reif, and Stephen R. Tate. Searching in an Unknown Environment: An Optimal Randomized Algorithm for the Cow-Path Problem. Inf. Comput., 131(1):63-79, 1996. Google Scholar
  14. Elmar Langetepe. On the Optimality of Spiral Search. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, Austin, Texas, USA, January 17-19, 2010, pages 1-12, 2010. Google Scholar
  15. Elmar Langetepe. Searching for an axis-parallel shoreline. Theor. Comput. Sci., 447:85-99, 2012. Google Scholar
  16. Tobias Langner, Barbara Keller, Jara Uitto, and Roger Wattenhofer. Overcoming Obstacles with Ants. In 19th International Conference on Principles of Distributed Systems, OPODIS 2015, December 14-17, 2015, Rennes, France, pages 9:1-9:17, 2015. Google Scholar
  17. Avery Miller and Andrzej Pelc. Tradeoffs between cost and information for rendezvous and treasure hunt. J. Parallel Distrib. Comput., 83:159-167, 2015. Google Scholar
  18. Kevin Spieser and Emilio Frazzoli. The Cow-Path Game: A competitive vehicle routing problem. In Proceedings of the 51th IEEE Conference on Decision and Control, CDC 2012, December 10-13, 2012, Maui, HI, USA, pages 6513-6520, 2012. Google Scholar
  19. Amnon Ta-Shma and Uri Zwick. Deterministic Rendezvous, Treasure Hunts, and Strongly Universal Exploration Sequences. ACM Trans. Algorithms, 10(3):12:1-12:15, 2014. Google Scholar