Optimal Distance Labeling for Interval Graphs and Related Graph Families

A distance labeling scheme is a distributed graph representation that assigns labels to the vertices and enables answering distance queries between any pair $(x,y)$ of vertices by using only the labels of $x$ and $y$. This paper presents an optimal distance labeling scheme with labels of $\mathcal{O}(\log n)$ bits for the $n$-vertex interval graphs family. It improves by $\log n$ factor the best known upper bound of [M. Katz, N. A. Katz, and D. Peleg, Distance labeling schemes for well-separated graph classes, in Proceedings of the 17th Annual Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Comput. Sci. 1770, Springer-Verlag, Berlin, 2000, pp. 516-528]. Moreover, the scheme supports constant time distance queries, and if the interval representation of the input graph is given and the intervals are sorted, then the set of labels can be computed in $\mathcal{O}(n)$ time. Our result is tight as we show that the length of any label is at least $3\log n-\mathcal{O}(\log\log n)$ bits. This lower bound derives from a new estimator of the number of unlabeled $n$-vertex interval graphs, that is, $2^{\Omega(n \log n)}$. To our knowledge, interval graphs are thereby the first known nontrivial hereditary family with $2^{\Omega(n f(n))}$ unlabeled elements and with a distance labeling scheme with $f(n)$ bit labels.