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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. We also present a linear time algorithm to compute medians in the associated ℓ 1 -cube complexes. 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) via Lexicographic Breadth First Search (LexBFS). We show that any LexBFS ordering of the vertices of a median graph satisfies the following fellow traveler property : the parents of any two adjacent vertices are also adjacent. Using the fast computation of the Θ-classes, we also compute the Wiener index (total distance) in linear time and the distance matrix in optimal quadratic time. [ABSTRACT FROM AUTHOR]