DeepAI AI Chat
Log In Sign Up

Medians in median graphs in linear time

by   Laurine Bénéteau, et al.

The median of a graph G is the set of all vertices x of G minimizing the sum of distances from x to all other vertices of G. It is known that computing the median of dense graphs in subcubic time refutes the APSP conjecture and computing the median of sparse graphs in subquadratic time refutes the HS conjecture. In this paper, we present a linear time algorithm for computing medians of median graphs, improving over the existing quadratic time algorithm. Median graphs constitute the principal class of graphs investigated in metric graph theory, due to their bijections with other discrete and geometric structures (CAT(0) cube complexes, domains of event structures, and solution sets of 2-SAT formulas). Our algorithm is based on the known majority rule characterization of medians in a median graph G and on a fast computation of parallelism classes of edges (Θ-classes) of G. The main technical contribution of the paper is a linear time algorithm for computing the Θ-classes of a median graph G using Lexicographic Breadth First Search (LexBFS). Namely, we show that any LexBFS ordering of the vertices of a median graph G has the following fellow traveler property: the fathers of any two adjacent vertices of G are also adjacent. Using the fast computation of the Θ-classes of a median graph G, we also compute the Wiener index (total distance) of G in linear time.


page 1

page 2

page 3

page 4


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

Median graphs form the class of graphs which is the most studied in metr...

Subquadratic-time algorithm for the diameter and all eccentricities on median graphs

On sparse graphs, Roditty and Williams [2013] proved that no O(n^2-ε)-ti...

On the computational complexity of the Steiner k-eccentricity

The Steiner k-eccentricity of a vertex v of a graph G is the maximum Ste...

A linear-time algorithm and analysis of graph Relative Hausdorff distance

Graph similarity metrics serve far-ranging purposes across many domains ...

Computing the Yolk in Spatial Voting Games without Computing Median Lines

The yolk is an important concept in spatial voting games as it generalis...

Identifying Consistent Statements about Numerical Data with Dispersion-Corrected Subgroup Discovery

Existing algorithms for subgroup discovery with numerical targets do not...