Acute Tours in the Plane

Author Ahmad Biniaz

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Ahmad Biniaz
  • School of Computer Science, University of Windsor, Canada


I am very grateful to the anonymous reviewer who meticulously verified our proof, and provided valuable feedback that reduced the number of subcases to two (which was three in our original proof) and improved the bound on n to 20 (which was 36 originally).

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Ahmad Biniaz. Acute Tours in the Plane. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 16:1-16:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


We confirm the following conjecture of Fekete and Woeginger from 1997: for any sufficiently large even number n, every set of n points in the plane can be connected by a spanning tour (Hamiltonian cycle) consisting of straight-line edges such that the angle between any two consecutive edges is at most π/2. Our proof is constructive and suggests a simple O(nlog n)-time algorithm for finding such a tour. The previous best-known upper bound on the angle is 2π/3, and it is due to Dumitrescu, Pach and Tóth (2009).

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • planar points
  • acute tour
  • Hamiltonian cycle
  • equitable partition


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  1. Eyal Ackerman, Oswin Aichholzer, and Balázs Keszegh. Improved upper bounds on the reflexivity of point sets. Computational Geometry: Theory and Applications, 42(3):241-249, 2009. Google Scholar
  2. Alok Aggarwal, Don Coppersmith, Sanjeev Khanna, Rajeev Motwani, and Baruch Schieber. The angular-metric traveling salesman problem. SIAM Journal on Computing, 29(3):697-711, 1999. Also in SODA'97. Google Scholar
  3. Oswin Aichholzer, Anja Fischer, Frank Fischer, J. Fabian Meier, Ulrich Pferschy, Alexander Pilz, and Rostislav Staněk. Minimization and maximization versions of the quadratic travelling salesman problem. Optimization, 66(4):521-546, 2017. Google Scholar
  4. Oswin Aichholzer, Thomas Hackl, Michael Hoffmann, Clemens Huemer, Attila Pór, Francisco Santos, Bettina Speckmann, and Birgit Vogtenhuber. Maximizing maximal angles for plane straight-line graphs. Computational Geometry: Theory and Applications, 46(1):17-28, 2013. Google Scholar
  5. Esther M. Arkin, Sándor P. Fekete, Ferran Hurtado, Joseph S. B. Mitchell, Marc Noy, Vera Sacristán, and Saurabh Sethia. On the reflexivity of point sets. In B. Aronov, S. Basu, J. Pach, and M. Sharir, editors, Discrete and Computational Geometry: The Goodman-Pollack Festschrift, pages 139-156. Springer, 2003. Google Scholar
  6. Rom Aschner and Matthew J. Katz. Bounded-angle spanning tree: Modeling networks with angular constraints. Algorithmica, 77(2):349-373, 2017. Also in ICALP'14. Google Scholar
  7. Rom Aschner, Matthew J. Katz, and Gila Morgenstern. Do directional antennas facilitate in reducing interferences? In Proceedings of the 13th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT), pages 201-212, 2012. Google Scholar
  8. Imre Bárány, Attila Pór, and Pavel Valtr. Paths with no small angles. SIAM Journal on Discrete Mathematics, 23(4):1655-1666, 2009. Also in LATIN'08. Google Scholar
  9. Ahmad Biniaz, Prosenjit Bose, Anna Lubiw, and Anil Maheshwari. Bounded-angle minimum spanning trees. Algorithmica, 84(1):150-175, 2022. Also in SWAT'20. Google Scholar
  10. Ahmad Biniaz, Majid Daliri, and Amir Hossein Moradpour. A 10-approximation of the π/2-MST. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS), 2022. Google Scholar
  11. Paz Carmi, Matthew J. Katz, Zvi Lotker, and Adi Rosén. Connectivity guarantees for wireless networks with directional antennas. Computational Geometry: Theory and Applications, 44(9):477-485, 2011. Google Scholar
  12. R. Courant and H. Robbins. What is Mathematics? An Elementary Approach to Ideas and Methods. Oxford University Press, New York, 1979. Google Scholar
  13. Adrian Dumitrescu, János Pach, and Géza Tóth. Drawing Hamiltonian cycles with no large angles. The Electronic Journal of Combinatorics, 19(2):P31, 2012. Also in GD'09. Google Scholar
  14. Sándor P. Fekete. Geometry and the Traveling Salesman Problem. Phd thesis, University of Waterloo, 1992. Google Scholar
  15. Sándor P. Fekete and Gerhard J. Woeginger. Angle-restricted tours in the plane. Computational Geometry: Theory and Applications, 8:195-218, 1997. Google Scholar
  16. Jan Kynčl. Personal communication, 2019. Google Scholar
  17. Olimjoni Pirahmad, Alexandr Polyanskii, and Alexey Vasilevskii. On a Tverberg graph. CoRR, abs/2108.09795, 2021. URL:
  18. Sambuddha Roy and William Steiger. Some combinatorial and algorithmic applications of the borsuk-ulam theorem. Graphs and Combinatorics, 23(Supplement-1):331-341, 2007. Google Scholar
  19. Tien Tran, Min Kyung An, and Dung T. Huynh. Antenna orientation and range assignment algorithms in directional WSNs. IEEE/ACM Transaction on Networking, 25(6):3368-3381, 2017. Also in INFOCOM'16. Google Scholar