Subquadratic-time Algorithm for the Diameter and all Eccentricities on Median Graphs

Abstract

On sparse graphs, Roditty and Williams [2013] proved that no O(n 2−ε )-time algorithm achieves an approximation factor smaller than 3 2 for the diameter problem unless SETH fails. We answer here an open question formulated in the literature: can we use the structural properties of median graphs to break this global quadratic barrier?We propose the first combinatorial algorithm computing exactly all eccentricities of a median graph in truly subquadratic time. Median graphs constitute the family of graphs which is the most studied in metric graph theory because their structure represents many other discrete and geometric concepts, such as CAT(0) cube complexes. Our result generalizes a recent one, stating that there is a linear-time algorithm for computing all eccentricities in median graphs with bounded dimension d, i.e. the dimension of the largest induced hypercube (note that 1-dimensional median graphs are exactly the forests). This prerequisite on d is not necessarily anymore to determine all eccentricities in subquadratic time. The execution time of our algorithm is O(n 1.6456 log O(1) n).