Inset Edges Effect and Average Distance of Trees

TL;DR

This paper presents an efficient algorithm for identifying a single inset edge in a tree that minimally impacts the average distance, improving computational efficiency by avoiding recalculations and leveraging matrix tools.

Contribution

It introduces a novel algorithm with sub-quadratic average complexity for finding the optimal inset edge in trees, avoiding recalculations and using advanced matrix techniques.

Findings

Algorithm achieves less than O(m log m) average complexity.

Effectively finds inset edges with minimal average distance change.

Reduces computational time by avoiding distance recalculations.

Abstract

An added edge to a graph is called an inset edge. Predicting k inset edges which minimize the average distance of a graph is known to be NP-Hard. When k = 1 the complexity of the problem is polynomial. In this paper, we further find the single inset edge(s) of a tree with the closest change on the average distance to a given input. To do that we may require the effect of each inset edge for the set of inset edges. For this, we propose an algorithm with the time complexity between O(m) and O(m/m) and an average of less than O( m.log(m)), where m stands for the number of possible inset edges. Then it takes up to O(log(m)) to find the target inset edges for a custom change on the average distance. Using theoretical tools, the algorithm strictly avoids recalculating the distances with no changes, after adding a new edge to a tree. Then reduces the time complexity of calculating remaining distances using some matrix tools which first introduced in [8] with one additional technique. This gives us a dynamic time complexity and absolutely depends on the input tree which is proportion to the Wiener index of the input tree.