  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.
