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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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