TY - GEN
T1 - From hierarchical partitions to hierarchical covers
T2 - 4th Annual ACM Symposium on Theory of Computing, STOC 2014
AU - Solomon, Shay
PY - 2014
Y1 - 2014
N2 - A (1+ε)-spanner for a doubling metric (X, δ) is a subgraph H of the complete graph corresponding to (X, δ), which preserves all pairwise distances to within a factor of 1 + ε. A natural requirement from a spanner, which is essential for many applications (mainly in distributed systems or wireless networks), is to be robust against vertex and edge failures - so that even when some vertices and edges in the network fail, we still have a (1 + ε)-spanner for what remains. The spanner H is called a k-fault-tolerant (1 + ε)-spanner, for1 ≤ κ ≤ n - 2, if for any F ⊆ X with |F| ≤ κ, the graph H \ F (obtained by removing from H the vertices of F and their incident edges) is a (1 + ε)-spanner for X \ F. In this paper we devise an optimal construction of faulttolerant spanners for doubling metrics: For any n-point doubling metric, any ε > 0, and any integer 1 ≤ κ ≤ n - 2, our construction provides a κ-fault-tolerant (1+ε)-spanner with optimal degree O(κ) within optimal time O(n log n + κn). We then strengthen this result to provide near-optimal (up to a factor of log κ) guarantees on the diameter and weight of our spanners, namely, diameter O(log n) and weight O(κ2 + log n) · ω(MST), while preserving the optimal guarantees on the degree O(κ) and the runtime O(n log n + κn). Our result settles several fundamental open questions in this area, culminating a long line of research that started with the STOC'95 paper of Arya et al. and the STOC'98 paper of Levcopoulos et al. On the way to this result we develop a new technique for constructing spanners in doubling metrics. In particular, our spanner construction is based on a novel hierarchical cover of the metric, whereas most previous constructions of spanners for doubling and Euclidean metrics (such as the net-tree spanner) are based on hierarchical partitions of the metric. We demonstrate the power of hierarchical covers in the context of geometric spanners by improving the state-of-the-art results in this area.
AB - A (1+ε)-spanner for a doubling metric (X, δ) is a subgraph H of the complete graph corresponding to (X, δ), which preserves all pairwise distances to within a factor of 1 + ε. A natural requirement from a spanner, which is essential for many applications (mainly in distributed systems or wireless networks), is to be robust against vertex and edge failures - so that even when some vertices and edges in the network fail, we still have a (1 + ε)-spanner for what remains. The spanner H is called a k-fault-tolerant (1 + ε)-spanner, for1 ≤ κ ≤ n - 2, if for any F ⊆ X with |F| ≤ κ, the graph H \ F (obtained by removing from H the vertices of F and their incident edges) is a (1 + ε)-spanner for X \ F. In this paper we devise an optimal construction of faulttolerant spanners for doubling metrics: For any n-point doubling metric, any ε > 0, and any integer 1 ≤ κ ≤ n - 2, our construction provides a κ-fault-tolerant (1+ε)-spanner with optimal degree O(κ) within optimal time O(n log n + κn). We then strengthen this result to provide near-optimal (up to a factor of log κ) guarantees on the diameter and weight of our spanners, namely, diameter O(log n) and weight O(κ2 + log n) · ω(MST), while preserving the optimal guarantees on the degree O(κ) and the runtime O(n log n + κn). Our result settles several fundamental open questions in this area, culminating a long line of research that started with the STOC'95 paper of Arya et al. and the STOC'98 paper of Levcopoulos et al. On the way to this result we develop a new technique for constructing spanners in doubling metrics. In particular, our spanner construction is based on a novel hierarchical cover of the metric, whereas most previous constructions of spanners for doubling and Euclidean metrics (such as the net-tree spanner) are based on hierarchical partitions of the metric. We demonstrate the power of hierarchical covers in the context of geometric spanners by improving the state-of-the-art results in this area.
KW - Doubling metrics
KW - Fault-tolerant spanners
UR - http://www.scopus.com/inward/record.url?scp=84904356961&partnerID=8YFLogxK
U2 - 10.1145/2591796.2591864
DO - 10.1145/2591796.2591864
M3 - منشور من مؤتمر
SN - 9781450327107
T3 - Proceedings of the Annual ACM Symposium on Theory of Computing
SP - 363
EP - 372
BT - STOC 2014 - Proceedings of the 2014 ACM Symposium on Theory of Computing
PB - Association for Computing Machinery
Y2 - 31 May 2014 through 3 June 2014
ER -