Abstract

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 ℓ1-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.

Highlights

  • The median problem is one of the oldest optimization problems in Euclidean geometry [49]

  • We show that the medians in median graphs can be computed in optimal O(m) time

  • We present a linear time algorithm to compute medians in median graphs and in associated 1-cube complexes

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Summary

Introduction

The median problem ( called the Fermat-Torricelli problem or the Weber problem) is one of the oldest optimization problems in Euclidean geometry [49]. We present a linear time algorithm to compute medians in median graphs and in associated 1-cube complexes. With the Θ-classes of G at hand and the majority rule for halfspaces, we can compute the weights of halfspaces of G in optimal time O(m), leading to an algorithm of the same complexity for computing the median set. The best algorithm to compute the Θ-classes of a median graph G has complexity O(m log n) [39] It was used in [39] to recognize median graphs in subquadratic time. A polynomial-time algorithm to compute the 2-distance between two points in this space was proposed in [60] This result was recently extended in [42] to all CAT(0) cube complexes. In a recent breakthrough [20], a subquadratic algorithm for the Wiener index and the diameter of planar graphs was presented

Preliminaries
Facts about median graphs
Computation of the Θ-classes
The median of G
Result
The median problem in the cube complex of G
The median problem in event structures

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