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 ℓ1-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 (Θ-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 fellow traveler property of independent interest: the parents of
any two adjacent vertices of G are also adjacent. Using the fast computation
of the Θ-classes, we also compute the Wiener index (total distance) of
G in linear time and the distance matrix in optimal quadratic time