Optimal Distance Labeling for Interval and Circular-Arc Graphs

In this paper we design a distance labeling scheme with \(\mathcal{O}(log n)\) bit labels for interval graphs and circular-arc graphs with n vertices. The set of all the labels is constructible in \(\mathcal{O}(n)\) time if the interval representation of the graph is given and sorted. As a byproduct we give a new and simpler \(\mathcal{O}(n)\) space data-structure computable after \(\mathcal{O}(n)\) 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.

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