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Brief Announcement: A Note on Hardness of Diameter Approximation

Authors Karl Bringmann, Sebastian Krinninger

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Karl Bringmann
Sebastian Krinninger

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Karl Bringmann and Sebastian Krinninger. Brief Announcement: A Note on Hardness of Diameter Approximation. In 31st International Symposium on Distributed Computing (DISC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 91, pp. 44:1-44:3, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)


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.
  • diameter
  • fine-grained reductions
  • conditional lower bounds


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  1. Amir Abboud, Keren Censor-Hillel, and Seri Khoury. Near-linear lower bounds for distributed distance computations, even in sparse networks. In DISC, pages 29-42, 2016. Google Scholar
  2. Massimo Cairo, Roberto Grossi, and Romeo Rizzi. New bounds for approximating extremal distances in undirected graphs. In SODA, pages 363-376, 2016. Google Scholar
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  4. Stephan Holzer, David Peleg, Liam Roditty, and Roger Wattenhofer. Brief announcement: Distributed 3/2-approximation of the diameter. In DISC, pages 562-564, 2014. Google Scholar
  5. Liam Roditty and Virginia Vassilevska Williams. Fast approximation algorithms for the diameter and radius of sparse graphs. In STOC, pages 515-524, 2013. Google Scholar
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