TY - GEN
T1 - Sometimes reliable spanners of almost linear size
AU - Buchin, Kevin
AU - Har-Peled, Sariel
AU - Oláh, Dániel
N1 - Publisher Copyright:
© Kevin Buchin, Sariel Har-Peled, and Dániel Oláh
PY - 2020/8/1
Y1 - 2020/8/1
N2 - Reliable spanners can withstand huge failures, even when a linear number of vertices are deleted from the network. In case of failures, some of the remaining vertices of a reliable spanner may no longer admit the spanner property, but this collateral damage is bounded by a fraction of the size of the attack. It is known that Ω(n log n) edges are needed to achieve this strong property, where n is the number of vertices in the network, even in one dimension. Constructions of reliable geometric (1 + ε)spanners, for n points in Rd, are known, where the resulting graph has O(n log n loglog6n) edges. Here, we show randomized constructions of smaller size spanners that have the desired reliability property in expectation or with good probability. The new construction is simple, and potentially practical – replacing a hierarchical usage of expanders (which renders the previous constructions impractical) by a simple skip list like construction. This results in a 1-spanner, on the line, that has linear number of edges. Using this, we present a construction of a reliable spanner in Rd with O(n loglog2n logloglog n) edges.
AB - Reliable spanners can withstand huge failures, even when a linear number of vertices are deleted from the network. In case of failures, some of the remaining vertices of a reliable spanner may no longer admit the spanner property, but this collateral damage is bounded by a fraction of the size of the attack. It is known that Ω(n log n) edges are needed to achieve this strong property, where n is the number of vertices in the network, even in one dimension. Constructions of reliable geometric (1 + ε)spanners, for n points in Rd, are known, where the resulting graph has O(n log n loglog6n) edges. Here, we show randomized constructions of smaller size spanners that have the desired reliability property in expectation or with good probability. The new construction is simple, and potentially practical – replacing a hierarchical usage of expanders (which renders the previous constructions impractical) by a simple skip list like construction. This results in a 1-spanner, on the line, that has linear number of edges. Using this, we present a construction of a reliable spanner in Rd with O(n loglog2n logloglog n) edges.
KW - Geometric spanners
KW - Reliability
KW - Vertex failures
UR - http://www.scopus.com/inward/record.url?scp=85092474435&partnerID=8YFLogxK
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U2 - 10.4230/LIPIcs.ESA.2020.27
DO - 10.4230/LIPIcs.ESA.2020.27
M3 - Conference contribution
AN - SCOPUS:85092474435
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 28th Annual European Symposium on Algorithms, ESA 2020
A2 - Grandoni, Fabrizio
A2 - Herman, Grzegorz
A2 - Sanders, Peter
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 28th Annual European Symposium on Algorithms, ESA 2020
Y2 - 7 September 2020 through 9 September 2020
ER -