2 research outputs found
Near-optimal labeling schemes for nearest common ancestors
We consider NCA labeling schemes: given a rooted tree , label the nodes of
with binary strings such that, given the labels of any two nodes, one can
determine, by looking only at the labels, the label of their nearest common
ancestor.
For trees with nodes we present upper and lower bounds establishing that
labels of size , are both sufficient and
necessary. (All logarithms in this paper are in base 2.)
Alstrup, Bille, and Rauhe (SIDMA'05) showed that ancestor and NCA labeling
schemes have labels of size . Our lower bound
increases this to for NCA labeling schemes. Since
Fraigniaud and Korman (STOC'10) established that labels in ancestor labeling
schemes have size , our new lower bound separates
ancestor and NCA labeling schemes. Our upper bound improves the
upper bound by Alstrup, Gavoille, Kaplan and Rauhe (TOCS'04), and our
theoretical result even outperforms some recent experimental studies by Fischer
(ESA'09) where variants of the same NCA labeling scheme are shown to all have
labels of size approximately
Distributed Computation for Swapping a Failing Edge
We consider the problem of computing the best swap edges of a shortest-path tree Tr rooted in r. That is, given a single link failure: if the path is not affected by the failed link, then the message will be delivered through that path; otherwise, we want to guarantee that, when the message reaches the edge (u, v) where the failure has occurred, the message will then be re-routed using the computed swap edge. There exist highly efficient serial solutions for the problem, but unfortunately because of the structures they use, there is no known (nor foreseeable) efficient distributed implementation for them. A distributed protocol exists only for finding swap edges, not necessarily optimal ones. In [6], distributed solutions to compute the swap edge that minimizes the distance from u to r have been presented. In contrast, in this paper we focus on selecting, efficiently and distributively, the best swap edge according to an objective function suggested in [13]: we choose the swap edge that minimizes the distance from u to v