Diameter, radius and all eccentricities in linear time for constant-dimension median graphs

TL;DR

This paper presents a linear-time algorithm for computing all eccentricities, diameter, and radius of median graphs with constant dimension, extending efficient metric computations to broader classes of median graphs.

Contribution

It introduces a combinatorial algorithm that computes all eccentricities in median graphs of constant dimension in near-linear time, addressing a previously open problem.

Findings

All eccentricities of median graphs with constant dimension can be computed in O(2^{O(d log d)} n) time.

The algorithm enables linear-time computation of diameter and radius for median graphs with fixed dimension.

Planar median graphs, including cube-free median graphs, have all eccentricities computable in linear time.

Abstract

Median graphs form the class of graphs which is the most studied in metric graph theory. Recently, B\'en\'eteau et al. [2019] designed a linear-time algorithm computing both the $\Theta$-classes and the median set of median graphs. A natural question emerges: is there a linear-time algorithm computing the diameter and the radius for median graphs? We answer positively to this question for median graphs $G$ with constant dimension $d$, i.e. the dimension of the largest induced hypercube of $G$. We propose a combinatorial algorithm computing all eccentricities of median graphs with running time $O(2^{O(d\log d)}n)$. As a consequence, this provides us with a linear-time algorithm determining both the diameter and the radius of median graphs with $d = O(1)$, such as cube-free median graphs. As the hypercube of dimension 4 is not planar, it shows also that all eccentricities of planar median graphs can be computed in $O(n)$.