Localized and compact data-structure for comparability graphs

We show that every comparability graph of any two-dimensional poset over n elements (a.k.a. permutation graph) can be preprocessed in O ( n ) time, if two linear extensions of the poset are given, to produce an O ( n ) space data-structure supporting distance queries in constant time. The data-structure is localized and given as a distance labeling, that is each vertex receives a label of O ( log n )  bits so that distance queries between any two vertices are answered by inspecting their labels only. This result improves the previous scheme due to Katz, Katz and Peleg [M. Katz, N.A. Katz, D. Peleg, Distance labeling schemes for well-separated graph classes, Discrete Applied Mathematics 145 (2005) 384–402] by a log n factor. yproduct, our data-structure supports all-pair shortest-path queries in O ( d ) time for distance- d pairs, and so identifies in constant time the first edge along a shortest path between any source and destination. undamentally, we show that this optimal space and time data-structure cannot be extended for higher dimension posets. More precisely, we prove that for comparability graphs of three-dimensional posets, every distance labeling scheme requires Ω ( n 1 / 3 ) bit labels.