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Competitive Search in the Line and the Star with Predictions

Author Spyros Angelopoulos

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Spyros Angelopoulos
  • CNRS and LIP6, Sorbonne University, Paris, France


I am thankful to Shahin Kamali for several helpful discussions.

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Spyros Angelopoulos. Competitive Search in the Line and the Star with Predictions. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 12:1-12:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


We study the classic problem of searching for a hidden target in the line and the m-ray star, in a setting in which the searcher has some prediction on the hider’s position. We first focus on the main metric for comparing search strategies under predictions; namely, we give positive and negative results on the consistency-robustness tradeoff, where the performance of the strategy is evaluated at extreme situations in which the prediction is either error-free, or adversarially generated, respectively. For the line, we show tight bounds concerning this tradeoff, under the untrusted advice model, in which the prediction is in the form of a k-bit string which encodes the responses to k binary queries. For the star, we give tight, and near-tight tradeoffs in the positional and the directional models, in which the prediction is related to the position of the target within the star, and to the ray on which the target hides, respectively. Last, for all three prediction models, we show how to generalize our study to a setting in which the performance of the strategy is evaluated as a function of the searcher’s desired tolerance to prediction errors, both in terms of positive and inapproximability results.

Subject Classification

ACM Subject Classification
  • Theory of computation → Online algorithms
  • Theory of computation → Theory and algorithms for application domains
  • Search problems
  • line and star search
  • competitive ratio
  • predictions
  • consistency and robustness


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  1. Steve Alpern and Shmuel Gal. The theory of search games and rendezvous, volume 55 of International series in operations research and management science. Kluwer, 2003. Google Scholar
  2. Steve Alpern and Thomas Lidbetter. Mining coal or finding terrorists: The expanding search paradigm. Operations Research, 61(2):265-279, 2013. Google Scholar
  3. Spyros Angelopoulos. Online search with a hint. In Proceedings of the 12th Innovations in Theoretical Computer Science Conference (ITCS), pages 51:1-51:16, 2021. Google Scholar
  4. Spyros Angelopoulos. Further connections between contract-scheduling and ray-searching problems. Journal of Scheduling, 25(2):139-155, 2022. Google Scholar
  5. Spyros Angelopoulos, Christoph Dürr, and Shendan Jin. Best-of-two-worlds analysis of online search. In Proceedings of the 36th International Symposium on Theoretical Aspects of Computer Science (STACS), pages 7:1-7:17, 2019. Google Scholar
  6. Spyros Angelopoulos, Christoph Dürr, and Thomas Lidbetter. The expanding search ratio of a graph. In Proceedings of the 33rd Symposium on Theoretical Aspects of Computer Science (STACS), pages 9:1-9:14, 2016. Google Scholar
  7. Spyros Angelopoulos and Shahin Kamali. Contract scheduling with predictions. Journal of Artificial Intelligence Research, 77:395-426, 2023. Google Scholar
  8. Ricardo A. Baeza-Yates, Joseph C. Culberson, and Gregory G.E. Rawlins. Searching in the plane. Information and Computation, 106:234-244, 1993. Google Scholar
  9. Siddhartha Banerjee, Vincent Cohen-Addad, Anupam Gupta, and Zhouzi Li. Graph searching with predictions. In Proceedings of the 14th Conference on Innovations in Theoretical Computer Science (ITCS), volume 251 of LIPIcs, pages 12:1-12:24, 2023. Google Scholar
  10. Anatole Beck. On the linear search problem. Naval Research Logistics, 2:221-228, 1964. Google Scholar
  11. Anatole Beck and Donald J. Newman. Yet more on the linear search problem. Israel Journal of Mathematics, 8:419-429, 1970. Google Scholar
  12. Richard Bellman. An optimal search problem. SIAM Review, 5:274, 1963. Google Scholar
  13. Daniel S. Bernstein, Lev Finkelstein, and Shlomo Zilberstein. Contract algorithms and robots on rays: Unifying two scheduling problems. In Proceedings of the 18th International Joint Conference on Artificial Intelligence (IJCAI), pages 1211-1217, 2003. Google Scholar
  14. Lucas Boczkowski, Amos Korman, and Yoav Rodeh. Searching a tree with permanently noisy advice. In ESA 2018-26th Annual European Symposium on Algorithms, pages 1-32, 2018. Google Scholar
  15. Anthony Bonato, Konstantinos Georgiou, Calum MacRury, and Pawel Pralat. Probabilistically faulty searching on a half-line - (Extended abstract). In LATIN 2020: Theoretical Informatics - 14th Latin American Symposium, São Paulo, Brazils, volume 12118 of Lecture Notes in Computer Science, pages 168-180. Springer, 2020. Google Scholar
  16. Prosenjit Bose, Jean-Lou De Carufel, and Stephane Durocher. Searching on a line: A complete characterization of the optimal solution. Theoretical Computer Science, 569:24-42, 2015. Google Scholar
  17. Marek Chrobak, Leszek Gasieniec, Thomas Gorry, and Russell Martin. Group search on the line. In International Conference on Current Trends in Theory and Practice of Informatics, pages 164-176. Springer, 2015. Google Scholar
  18. Anne Condon, Amol Deshpande, Lisa Hellerstein, and Ning Wu. Algorithms for distributional and adversarial pipelined filter ordering problems. ACM Transaction on Algorithms, 5(2):24:1-24:34, 2009. Google Scholar
  19. Jurek Czyzowicz, Evangelos Kranakis, Danny Krizanc, Lata Narayanan, and Jaroslav Opatrny. Search on a line with faulty robots. Distributed Comput., 32(6):493-504, 2019. Google Scholar
  20. Erik D Demaine, Sándor P Fekete, and Shmuel Gal. Online searching with turn cost. Theoretical Computer Science, 361:342-355, 2006. Google Scholar
  21. Stefan Dobrev, Rastislav Královič, and Euripides Markou. Online graph exploration with advice. In Structural Information and Communication Complexity: 19th International Colloquium, SIROCCO 2012, pages 267-278. Springer, 2012. Google Scholar
  22. Franziska Eberle, Alexander Lindermayr, Nicole Megow, Lukas Nölke, and Jens Schlöter. Robustification of online graph exploration methods. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 36(9), pages 9732-9740, 2022. Google Scholar
  23. Pierre Fraigniaud, David Ilcinkas, and Andrzej Pelc. Tree exploration with advice. Information and Computation, 206(11):1276-1287, 2008. Google Scholar
  24. Shmuel Gal. A general search game. Israel Journal of Mathematics, 12:32-45, 1972. Google Scholar
  25. Shmuel Gal. Minimax solutions for linear search problems. SIAM Journal on Applied Mathematics, 27:17-30, 1974. Google Scholar
  26. Shmuel Gal. Search Games. Academic Press, 1980. Google Scholar
  27. Subir Kumar Ghosh and Rolf Klein. Online algorithms for searching and exploration in the plane. Comput. Sci. Rev., 4(4):189-201, 2010. Google Scholar
  28. Barun Gorain and Andrzej Pelc. Deterministic graph exploration with advice. ACM Transactions on Algorithms, 15(1):1-17, 2018. Google Scholar
  29. C. Hipke, C. Icking, R. Klein, and E. Langetepe. How to find a point in the line within a fixed distance. Discrete Applied Mathematics, 93:67-73, 1999. Google Scholar
  30. Sungjin Im, Ravi Kumar, Mahshid Montazer Qaem, and Manish Purohit. Online knapsack with frequency predictions. In Proceedings of the 34th Annual Conference on Neural Information Processing Systems (NeurIPS), pages 2733-2743, 2021. Google Scholar
  31. Parick Jaillet and Matthew Stafford. Online searching. Operations Research, 49:234-244, 1993. Google Scholar
  32. Ming-Yang Kao, Yuan Ma, Michael Sipser, and Yiqun Yin. Optimal constructions of hybrid algorithms. Journal of Algorithms, 29(1):142-164, 1998. Google Scholar
  33. Ming-Yang Kao, John H Reif, and Stephen R Tate. Searching in an unknown environment: an optimal randomized algorithm for the cow-path problem. Information and Computation, 131(1):63-80, 1996. Google Scholar
  34. David G. Kirkpatrick. Hyperbolic dovetailing. In Proceedings of the 17th Annual European Symposium on Algorithms (ESA), pages 616-627, 2009. Google Scholar
  35. Dennis Komm, Rastislav Královič, Richard Královič, and Jasmin Smula. Treasure hunt with advice. In Structural Information and Communication Complexity: 22nd International Colloquium, SIROCCO 2015, pages 328-341. Springer, 2015. Google Scholar
  36. E. Koutsoupias, C.H. Papadimitriou, and M. Yannakakis. Searching a fixed graph. In Proc. of the 23rd Int. Colloq. on Automata, Languages and Programming (ICALP), pages 280-289, 1996. Google Scholar
  37. Andrey Kupavskii and Emo Welzl. Lower bounds for searching robots, some faulty. In Proceedings of the 37th ACM Symposium on Principles of Distributed Computing (PODC), pages 447-453, 2018. Google Scholar
  38. Elmar Langetepe. On the optimality of spiral search. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1-12. SIAM, 2010. Google Scholar
  39. Russell Lee, Jessica Maghakian, Mohammad H. Hajiesmaili, Jian Li, Ramesh K. Sitaraman, and Zhenhua Liu. Online peak-aware energy scheduling with untrusted advice. In Proceedings of the 12th ACM International Conference on Future Energy Systems (eEnergy), pages 107-123. ACM, 2021. Google Scholar
  40. Tongxin Li, Ruixiao Yang, Guannan Qu, Guanya Shi, Chenkai Yu, Adam Wierman, and Steven H. Low. Robustness and consistency in linear quadratic control with untrusted predictions. Proc. ACM Meas. Anal. Comput. Syst., 6(1):18:1-18:35, 2022. Google Scholar
  41. Thomas Lidbetter. Search and rescue in the face of uncertain threats. Eur. J. Oper. Res., 285(3):1153-1160, 2020. Google Scholar
  42. Alexander Lindermayr and Nicole Megow. Repository of works on algorithms with predictions., 2023. Accessed: 2023-04-01.
  43. Alejandro López-Ortiz and Svem Schuierer. The ultimate strategy to search on m rays? Theoretical Computer Science, 261(2):267-295, 2001. Google Scholar
  44. Alejandro López-Ortiz and Sven Schuierer. On-line parallel heuristics, processor scheduling and robot searching under the competitive framework. Theoretical Computer Science, 310(1-3):527-537, 2004. Google Scholar
  45. Thodoris Lykouris and Sergei Vassilvitskii. Competitive caching with machine learned advice. J. ACM, 68(4):24:1-24:25, 2021. Google Scholar
  46. Andrew McGregor, Krzysztof Onak, and Rina Panigrahy. The oil searching problem. In Proceedings of the 17th European Symposium on Algorithms (ESA), pages 504-515, 2009. Google Scholar
  47. Michael Mitzenmacher and Sergei Vassilvitskii. Algorithms with predictions. In Beyond the Worst-Case Analysis of Algorithms, pages 646-662. Cambridge University Press, 2020. Google Scholar
  48. Andrzej Pelc and Ram Narayan Yadav. Advice complexity of treasure hunt in geometric terrains. Information and Computation, 281:104705, 2021. Google Scholar
  49. Manish Purohit, Zoya Svitkina, and Ravi Kumar. Improving online algorithms via ML predictions. In Proceedings of the 32nd Conference on Neural Information Processing Systems (NeurIPS), pages 9661-9670, 2018. Google Scholar
  50. Ronald L. Rivest, Albert R. Meyer, Daniel J. Kleitman, Karl Winklmann, and Joel Spencer. Coping with errors in binary search procedures. J. Comput. Syst. Sci., 20(3):396-404, 1980. Google Scholar
  51. Sven Schuierer. Lower bounds in online geometric searching. Computational Geometry: Theory and Applications, 18(1):37-53, 2001. Google Scholar
  52. Alexander Wei and Fred Zhang. Optimal robustness-consistency trade-offs for learning-augmented online algorithms. In Proceedings of the 33rd Conference on Neural Information Processing Systems (NeurIPS), 2020. Google Scholar
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