Medians in median graphs and their cube complexes in linear time
Résumé
The median of a set of vertices P of a graph G is the set of all
vertices x of G minimizing the sum of distances from $x$ to all
vertices of P. In this paper, we present a linear time algorithm
to compute medians in median graphs, improving over the existing
quadratic time algorithm. We also present a linear time algorithm to
compute medians in the l_1-cube complexes associated with
median graphs. Median graphs constitute the principal class of
graphs investigated in metric graph theory and have a rich geometric
and combinatorial structure, due to their bijections with CAT(0)
cube complexes and domains of event structures. Our algorithm is
based on the majority rule characterization of medians in median
graphs and on a fast computation of parallelism classes of edges
(Theta-classes or hyperplanes) via Lexicographic Breadth First
Search (LexBFS). To prove the correctness of our algorithm, we show
that any LexBFS ordering of the vertices of G satisfies the
following \emph{fellow traveler property} of independent interest:
the parents of any two adjacent vertices of G are also
adjacent. Using the fast computation of the Theta-classes, we
also compute the Wiener index (total distance) of G in linear time
and the distance matrix in optimal quadratic time.
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