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An added edge to a graph is called an inset edge. Predicting k inset edgeswhich 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 findthe single inset edge(s) of a tree with the closest change on the averagedistance to a given input. To do that we may require the effect of each insetedge for the set of inset edges. For this, we propose an algorithm with thetime 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 takesup to O(log(m)) to find the target inset edges for a custom change on theaverage distance. Using theoretical tools, the algorithm strictly avoidsrecalculating the distances with no changes, after adding a new edge to a tree.Then reduces the time complexity of calculating remaining distances using somematrix tools which first introduced in [8] with one additional technique. Thisgives us a dynamic time complexity and absolutely depends on the input treewhich is proportion to the Wiener index of the input tree.

Inset Edges Effect and Average Distance of Trees