5 research outputs found
Efficient and Simple Algorithms for Fault Tolerant Spanners
It was recently shown that a version of the greedy algorithm gives a
construction of fault-tolerant spanners that is size-optimal, at least for
vertex faults. However, the algorithm to construct this spanner is not
polynomial-time, and the best-known polynomial time algorithm is significantly
suboptimal. Designing a polynomial-time algorithm to construct (near-)optimal
fault-tolerant spanners was given as an explicit open problem in the two most
recent papers on fault-tolerant spanners ([Bodwin, Dinitz, Parter, Vassilevka
Williams SODA '18] and [Bodwin, Patel PODC '19]). We give a surprisingly simple
algorithm which runs in polynomial time and constructs fault-tolerant spanners
that are extremely close to optimal (off by only a linear factor in the
stretch) by modifying the greedy algorithm to run in polynomial time. To
complement this result, we also give simple distributed constructions in both
the LOCAL and CONGEST models.Comment: 15 pages. Appeared at PODC 2020. This revision improves the running
time slightly and incorporates reviewer comment
Tight Bounds for Maximal Identifiability of Failure Nodes in Boolean Network Tomography
We study maximal identifiability, a measure recently introduced in Boolean
Network Tomography to characterize networks' capability to localize failure
nodes in end-to-end path measurements. We prove tight upper and lower bounds on
the maximal identifiability of failure nodes for specific classes of network
topologies, such as trees and -dimensional grids, in both directed and
undirected cases. We prove that directed -dimensional grids with support
have maximal identifiability using monitors; and in the
undirected case we show that monitors suffice to get identifiability of
. We then study identifiability under embeddings: we establish relations
between maximal identifiability, embeddability and graph dimension when network
topologies are model as DAGs. Our results suggest the design of networks over
nodes with maximal identifiability using
monitors and a heuristic to boost maximal identifiability on a given network by
simulating -dimensional grids. We provide positive evidence of this
heuristic through data extracted by exact computation of maximal
identifiability on examples of small real networks