The scaling of the minimum sum of edge lengths in uniformly random trees

TL;DR

This paper investigates how the minimum sum of edge lengths in optimal linear arrangements of uniformly random trees grows with tree size, revealing a logarithmic growth pattern contrasting with linear growth in star trees.

Contribution

It applies polynomial-time algorithms to analyze the growth of minimal edge sums in random trees, providing new insights into their structural properties.

Findings

Sum is bounded by star tree configuration

Mean edge length grows logarithmically with tree size

Contrasts with linear growth in star trees or random arrangements

Abstract

The minimum linear arrangement problem on a network consists of finding the minimum sum of edge lengths that can be achieved when the vertices are arranged linearly. Although there are algorithms to solve this problem on trees in polynomial time, they have remained theoretical and have not been implemented in practical contexts to our knowledge. Here we use one of those algorithms to investigate the growth of this sum as a function of the size of the tree in uniformly random trees. We show that this sum is bounded above by its value in a star tree. We also show that the mean edge length grows logarithmically in optimal linear arrangements, in stark contrast to the linear growth that is expected on optimal arrangements of star trees or on random linear arrangements.