We revisit the hardness of approximating the diameter of a network. In the CONGEST model, ~Omega(n) rounds are necessary to compute the diameter [Frischknecht et al. SODA'12]. Abboud et al. [DISC 2016] extended this result to sparse graphs and, at a more fine-grained level, showed that, for any integer 1 <= l <= polylog(n) , distinguishing between networks of diameter 4l + 2 and 6l + 1 requires ~Omega(n) rounds. We slightly tighten this result by showing that even distinguishing between diameter 2l + 1 and 3l + 1 requires ~Omega(n) rounds. The reduction of Abboud et al. is inspired by recent conditional lower bounds in the RAM model, where the orthogonal vectors problem plays a pivotal role. In our new lower bound, we make the connection to orthogonal vectors explicit, leading to a conceptually more streamlined exposition. This is suited for teaching both the lower bound in the CONGEST model and the conditional lower bound in the RAM model.
@InProceedings{bringmann_et_al:LIPIcs.DISC.2017.44, author = {Bringmann, Karl and Krinninger, Sebastian}, title = {{Brief Announcement: A Note on Hardness of Diameter Approximation}}, booktitle = {31st International Symposium on Distributed Computing (DISC 2017)}, pages = {44:1--44:3}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-053-8}, ISSN = {1868-8969}, year = {2017}, volume = {91}, editor = {Richa, Andr\'{e}a}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2017.44}, URN = {urn:nbn:de:0030-drops-79874}, doi = {10.4230/LIPIcs.DISC.2017.44}, annote = {Keywords: diameter, fine-grained reductions, conditional lower bounds} }
Feedback for Dagstuhl Publishing