TY - GEN
T1 - A spanner for the day after
AU - Buchin, Kevin
AU - Har-Peled, Sariel
AU - Oláh, Dániel
N1 - Publisher Copyright:
© Kevin Buchin, Sariel Har-Peled, and Dániel Oláh.
PY - 2019/6/1
Y1 - 2019/6/1
N2 - We show how to construct (1 + ε)-spanner over a set P of n points in ℝd that is resilient to a catastrophic failure of nodes. Specifically, for prescribed parameters ϑ, ε ∈ (0, 1), the computed spanner G has O(ε−7d log7 ε−1 · ϑ−6n log n(log log n)6) edges. Furthermore, for any k, and any deleted set B ⊆ P of k points, the residual graph G \ B is (1 + ε)-spanner for all the points of P except for (1 + ϑ)k of them. No previous constructions, beyond the trivial clique with O(n2) edges, were known such that only a tiny additional fraction (i.e., ϑ) 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.
AB - We show how to construct (1 + ε)-spanner over a set P of n points in ℝd that is resilient to a catastrophic failure of nodes. Specifically, for prescribed parameters ϑ, ε ∈ (0, 1), the computed spanner G has O(ε−7d log7 ε−1 · ϑ−6n log n(log log n)6) edges. Furthermore, for any k, and any deleted set B ⊆ P of k points, the residual graph G \ B is (1 + ε)-spanner for all the points of P except for (1 + ϑ)k of them. No previous constructions, beyond the trivial clique with O(n2) edges, were known such that only a tiny additional fraction (i.e., ϑ) 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.
KW - Geometric spanners
KW - Robustness
KW - Vertex failures
UR - http://www.scopus.com/inward/record.url?scp=85068046133&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85068046133&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.SoCG.2019.19
DO - 10.4230/LIPIcs.SoCG.2019.19
M3 - Conference contribution
AN - SCOPUS:85068046133
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 35th International Symposium on Computational Geometry, SoCG 2019
A2 - Barequet, Gill
A2 - Wang, Yusu
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 35th International Symposium on Computational Geometry, SoCG 2019
Y2 - 18 June 2019 through 21 June 2019
ER -