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A spanner is reliable if it can withstand large, catastrophic failures in the network. More precisely, any failure of some nodes can only cause a small damage in the remaining graph in terms of the dilation. In other words, the spanner property is maintained for almost all nodes in the residual graph. Constructions of reliable spanners of near linear size are known in the low-dimensional Euclidean settings. Here, we present new constructions of reliable spanners for planar graphs, trees, and (general) metric spaces.
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Sometimes Reliable Spanners of Almost Linear Size
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(nlog 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 ℝ^d, are known, where the resulting graph has 𝒪(n log n log log⁶n) 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 ℝ^d with 𝒪(n log log²n log log log n) edges.
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- Award ID(s):
- 1907400
- PAR ID:
- 10226331
- Editor(s):
- Grandoni, Fabrizio; Herman, Grzegorz; Sanders, Peter
- Date Published:
- Journal Name:
- 28th Annual European Symposium on Algorithms (ESA)
- Volume:
- 173
- Page Range / eLocation ID:
- 27:1--27:15
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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