A Spanner for the Day After

Authors Kevin Buchin , Sariel Har-Peled , Dániel Oláh

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Author Details

Kevin Buchin
  • Department of Mathematics and Computing Science, TU Eindhoven, The Netherlands
Sariel Har-Peled
  • Department of Computer Science, University of Illinois at Urbana-Champaign, USA
Dániel Oláh
  • Department of Mathematics and Computing Science, TU Eindhoven, The Netherlands

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Kevin Buchin, Sariel Har-Peled, and Dániel Oláh. A Spanner for the Day After. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 19:1-19:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


We show how to construct (1+epsilon)-spanner over a set P of n points in R^d that is resilient to a catastrophic failure of nodes. Specifically, for prescribed parameters theta, epsilon in (0,1), the computed spanner G has O(epsilon^{-7d} log^7 epsilon^{-1} * theta^{-6} n log n (log log n)^6) edges. Furthermore, for any k, and any deleted set B subseteq P of k points, the residual graph G \ B is (1+epsilon)-spanner for all the points of P except for (1+theta)k of them. No previous constructions, beyond the trivial clique with O(n^2) edges, were known such that only a tiny additional fraction (i.e., theta) lose their distance preserving connectivity. Our construction works by first solving the exact problem in one dimension, and then showing a surprisingly simple and elegant construction in higher dimensions, that uses the one dimensional construction in a black box fashion.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Geometric spanners
  • vertex failures
  • robustness


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