New Approximation Algorithms for Touring Regions

Authors Benjamin Qi , Richard Qi

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Benjamin Qi
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Richard Qi
  • Massachusetts Institute of Technology, Cambridge, MA, USA


We thank Dhruv Rohatgi, Quanquan Liu, Erik Demaine, Danny Mittal, Spencer Compton, William Kuszmaul, Timothy Qian, Jonathan Kelner, and Chris Zhang for helpful discussions. Special thanks to Xinyang Chen for writing one of the proofs in the full version of this paper and providing inspiration.

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Benjamin Qi and Richard Qi. New Approximation Algorithms for Touring Regions. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 54:1-54:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We analyze the touring regions problem: find a (1+ε)-approximate Euclidean shortest path in d-dimensional space that starts at a given starting point, ends at a given ending point, and visits given regions R₁, R₂, R₃, … , R_n in that order. Our main result is an O (n/√ε log{1/ε} + 1/ε)-time algorithm for touring disjoint disks. We also give an O(min(n/ε, n²/√ε))-time algorithm for touring disjoint two-dimensional convex fat bodies. Both of these results naturally generalize to larger dimensions; we obtain O(n/{ε^{d-1}} log²1/ε + 1/ε^{2d-2}) and O(n/ε^{2d-2})-time algorithms for touring disjoint d-dimensional balls and convex fat bodies, respectively.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Approximation algorithms analysis
  • shortest paths
  • convex bodies
  • fat objects
  • disks


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  1. Xxii open cup, grand prix of siberia, 2022. Last accessed 9 April 2022. URL:
  2. Alok Aggarwal and Maria Klawe. Applications of generalized matrix searching to geometric algorithms. Discrete Applied Mathematics, 27(1-2):3-23, 1990. Google Scholar
  3. Arash Ahadi, Amirhossein Mozafari, and Alireza Zarei. Touring a sequence of disjoint polygons: Complexity and extension. Theoretical Computer Science, 556:45-54, 2014. Google Scholar
  4. Tetsuo Asano, David Kirkpatrick, and Chee Yap. Pseudo approximation algorithms with applications to optimal motion planning. Discrete & Computational Geometry, 31(1):139-171, 2004. Google Scholar
  5. Heinz H. Bauschke and Jonathan M. Borwein. On projection algorithms for solving convex feasibility problems. SIAM Review, 38(3):367-426, 1996. URL:
  6. Timothy M Chan. (near-) linear-time randomized algorithms for row minima in monge partial matrices and related problems. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1465-1482. SIAM, 2021. Google Scholar
  7. Chang-Chien Chou. An exact algorithm for computing the shortest path touring n circles in 2d. In International Conference on Computational Problem-Solving, pages 98-103, 2010. Google Scholar
  8. P. L. Combettes and H. J. Trussell. Method of successive projections for finding a common point of sets in metric spaces, December 1990. URL:
  9. Moshe Dror, Alon Efrat, Anna Lubiw, and Joseph S. B. Mitchell. Touring a sequence of polygons. In Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, STOC '03, pages 473-482, New York, NY, USA, 2003. Association for Computing Machinery. URL:
  10. Alon Efrat, Günter Rote, and Micha Sharir. On the union of fat wedges and separating a collection of segments by a line. Comput. Geom. Theory Appl., 3(5):277-288, November 1993. URL:
  11. Peter Gritzmann and Victor Klee. Computational convexity. CRC Press, Taylor; Francis Group, a Chapman; Hall Book, 2017. Google Scholar
  12. Matthew J. Katz. 3-d vertical ray shooting and 2-d point enclosure, range searching, and arc shooting amidst convex fat objects. Computational Geometry, 8(6):299-316, 1997. URL:
  13. Amirhossein Mozafari and Alireza Zarei. Touring polygons: An approximation algorithm. In IWOCA, 2012. Google Scholar
  14. Mark H. Overmars. Point location in fat subdivisions. Information Processing Letters, 44(5):261-265, 1992. URL:
  15. Mark H. Overmars and A.F. van der Stappen. Range searching and point location among fat objects. J. Algorithms, 21:629-656, 1996. Google Scholar
  16. R. R. Phelps. Convex sets and nearest points. Proceedings of the American Mathematical Society, 8(4):790-797, 1957. URL:
  17. Valentin Polishchuk and Joseph SB Mitchell. Touring convex bodies-a conic programming solution. In CCCG, pages 290-293, 2005. Google Scholar
  18. David J. Smith and Mavina K. Vamanamurthy. How small is a unit ball? Mathematics Magazine, 62(2):101-107, 1989. URL:
  19. Xuehou Tan and Bo Jiang. Efficient algorithms for touring a sequence of convex polygons and related problems. In T.V. Gopal, Gerhard Jäger, and Silvia Steila, editors, Theory and Applications of Models of Computation, pages 614-627, Cham, 2017. Springer International Publishing. Google Scholar