TY - GEN
T1 - Locality-sensitive orderings and applications to reliable spanners
AU - Filtser, Arnold
AU - Le, Hung
N1 - Publisher Copyright:
© 2022 ACM.
PY - 2022/9/6
Y1 - 2022/9/6
N2 - Chan, Har-Peled, and Jones [2020] recently developed locality-sensitive ordering (LSO), a new tool that allows one to reduce problems in the Euclidean space g.,d to the 1-dimensional line. They used LSO's to solve a host of problems. Later, Buchin, Har-Peled, and Oláh [2019,2020] used the LSO of Chan et al. to construct very sparse reliable spanners for the Euclidean space. A highly desirable feature of a reliable spanner is its ability to withstand a massive failure: the network remains functioning even if 90% of the nodes fail. In a follow-up work, Har-Peled, Mendel, and Oláh [2021] constructed reliable spanners for general and topologically structured metrics. Their construction used a different approach, and is based on sparse covers. In this paper, we develop the theory of LSO's in non-Euclidean metrics by introducing new types of LSO's suitable for general and topologically structured metrics. We then construct such LSO's, as well as constructing considerably improved LSO's for doubling metrics. Afterwards, we use our new LSO's to construct reliable spanners with improved stretch and sparsity parameters. Most prominently, we construct Õ(n)-size reliable spanners for trees and planar graphs with the optimal stretch of 2. Along the way to the construction of LSO's and reliable spanners, we introduce and construct ultrametric covers, and construct 2-hop reliable spanners for the line.
AB - Chan, Har-Peled, and Jones [2020] recently developed locality-sensitive ordering (LSO), a new tool that allows one to reduce problems in the Euclidean space g.,d to the 1-dimensional line. They used LSO's to solve a host of problems. Later, Buchin, Har-Peled, and Oláh [2019,2020] used the LSO of Chan et al. to construct very sparse reliable spanners for the Euclidean space. A highly desirable feature of a reliable spanner is its ability to withstand a massive failure: the network remains functioning even if 90% of the nodes fail. In a follow-up work, Har-Peled, Mendel, and Oláh [2021] constructed reliable spanners for general and topologically structured metrics. Their construction used a different approach, and is based on sparse covers. In this paper, we develop the theory of LSO's in non-Euclidean metrics by introducing new types of LSO's suitable for general and topologically structured metrics. We then construct such LSO's, as well as constructing considerably improved LSO's for doubling metrics. Afterwards, we use our new LSO's to construct reliable spanners with improved stretch and sparsity parameters. Most prominently, we construct Õ(n)-size reliable spanners for trees and planar graphs with the optimal stretch of 2. Along the way to the construction of LSO's and reliable spanners, we introduce and construct ultrametric covers, and construct 2-hop reliable spanners for the line.
KW - $2$-hop spanners
KW - Doubling Metric
KW - Minor Free graphs
KW - Reliable Spanners
KW - Ultrametric cover
KW - ocality-Sensitive Orderings
UR - http://www.scopus.com/inward/record.url?scp=85132702554&partnerID=8YFLogxK
U2 - 10.1145/3519935.3520042
DO - 10.1145/3519935.3520042
M3 - Conference contribution
AN - SCOPUS:85132702554
T3 - Proceedings of the Annual ACM Symposium on Theory of Computing
SP - 1066
EP - 1079
BT - STOC 2022 - Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing
A2 - Leonardi, Stefano
A2 - Gupta, Anupam
PB - Association for Computing Machinery
T2 - 54th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2022
Y2 - 20 June 2022 through 24 June 2022
ER -