The total Steiner $k$-distance for $b$-ary recursive trees and linear recursive trees

Götz Olaf Munsonius

doi:10.46298/dmtcs.2779

Abstract

We prove a limit theorem for the total Steiner $k$-distance of a random $b$-ary recursive tree with weighted edges. The total Steiner $k$-distance is the sum of all Steiner $k$-distances in a tree and it generalises the Wiener index. The limit theorem is obtained by using a limit theorem in the general setting of the contraction method. In order to use the contraction method we prove a recursion formula and determine the asymptotic expansion of the expectation using the so-called Master Theorem by Roura (2001). In a second step we prove a transformation of the total Steiner $k$-distance of $b$-ary trees with weighted edges to arbitrary recursive trees. This transformation yields the limit theorem for the total Steiner $k$-distance of the linear recursive trees when the parameter of these trees is a non-negative integer.

Highlights

The analysis of algorithms often leads to the consideration of functionals that deal with distances between nodes in trees
For example the internal path length of a binary search tree, which is the sum of all distances between the nodes and the root, equals the number of comparisons that are needed to build up the tree
For a set of k nodes in a network the size of the smallest subtree containing this set is a quantity which is related to the communication potential between these nodes

Summary

Introduction

The analysis of algorithms often leads to the consideration of functionals that deal with distances between nodes in trees. We consider the total Steiner k-distance of random trees. This is the sum of Steiner distances for all possible choices of k nodes and it generalises the Wiener index We define the k-th internal path length as the sum of the sizes of ancestor trees for all possible choices of k nodes. The b-ary recursive tree with weighted edges can be considered as a special case of the tree model in the paper of Broutin and Devroye (2006) in discrete time where the lifetimes of the edges are independent exponentially distributed random variables. We define the random b-ary recursive tree with weighted edges in Section 2 and state some basic properties.

The model of the random b-ary recursive tree with weighted edges

The k-th internal path length and the total Steiner k-distance

Bijection to linear recursive trees