Let C be the unit circle in R^2. We can view C as a plane graph whose vertices are all the points on C, and the distance between any two points on C is the length of the smaller arc between them. We consider a graph augmentation problem on C, where we want to place k >= 1 shortcuts on C such that the diameter of the resulting graph is minimized. We analyze for each k with 1 <= k <= 7 what the optimal set of shortcuts is. Interestingly, the minimum diameter one can obtain is not a strictly decreasing function of k. For example, with seven shortcuts one cannot obtain a smaller diameter than with six shortcuts. Finally, we prove that the optimal diameter is 2 + Theta(1/k^(2/3)) for any k.
@InProceedings{bae_et_al:LIPIcs.ISAAC.2017.9, author = {Bae, Sang Won and de Berg, Mark and Cheong, Otfried and Gudmundsson, Joachim and Levcopoulos, Christos}, title = {{Shortcuts for the Circle}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {9:1--9:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-054-5}, ISSN = {1868-8969}, year = {2017}, volume = {92}, editor = {Okamoto, Yoshio and Tokuyama, Takeshi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.9}, URN = {urn:nbn:de:0030-drops-82133}, doi = {10.4230/LIPIcs.ISAAC.2017.9}, annote = {Keywords: Computational geometry, graph augmentation problem, circle, shortcut, diameter} }
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