We consider NCA labeling schemes: given a rooted tree T, label the nodes of
T 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 n nodes we present upper and lower bounds establishing that
labels of size (2±ϵ)logn, ϵ<1 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 logn+Ω(loglogn). Our lower bound
increases this to logn+Ω(logn) for NCA labeling schemes. Since
Fraigniaud and Korman (STOC'10) established that labels in ancestor labeling
schemes have size logn+Θ(loglogn), our new lower bound separates
ancestor and NCA labeling schemes. Our upper bound improves the 10logn
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 8logn