Optimal distance labeling for interval and circular-arc graphs

Abstract

In this paper we design a distance labeling scheme with O(log n) bit labels for interval graphs and circular-arc graphs with n vertices. The set of all the labels is constructible in O(n) time if the interval representation of the graph is given and sorted. As a byproduct we give a new and simpler O(n) space data-structure computable after O(n) preprocessing time, and supporting constant worst-case time distance queries for interval and circular-arc graphs. These optimal bounds improve the previous scheme of Katz, Katz, and Peleg (STACS '00) by a log n factor. To the best of our knowledge, the interval graph family is the first hereditary family having 2Ω(n log n) unlabeled n-vertex graphs and supporting a o(log2 n) bit distance labeling scheme. © Springer-Verlag 2003.