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Erschienen in:
2005 | OriginalPaper | Buchkapitel
Localized and Compact Data-Structure for Comparability Graphs
verfasst von : Fabrice Bazzaro, Cyril Gavoille
Erschienen in: Algorithms and Computation
Verlag: Springer Berlin Heidelberg
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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 on their labels only. As a byproduct, 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. More fundamentally, we show that this optimal space and time data-structure cannot be extended for higher dimension posets (we prove that for comparability graphs of three-dimensional posets, every distance labeling scheme requires Ω(
n
1/3
) bit labels).