# Does it make sense to apply path compression to kruskal algorithm?

Let’s say there are E edges and V vertices. Kruskal algorithm has two important time complexity equations ,

1. Sort edges- Elog(E)
2. For whether edge from v1->v2 , apply find(v1) and find(v2). If cycle exist don’t connect. Other wise connect them. - E*timecomplexity(find(v))

Now in the second step , time complexity of find(v) will be O(v) for skewed sets. We can improve it by using union by rank where we append small to talk in term a of height, O(log(v)).

Now my question is can we apply path compression here to further reduce find operation time complexity further to O(α(n)) (where α is the inverse of the Ackerman function). ? Wouldn’t path compression distort the graph itself as we will be re arranging the edges!!

So I believe best case for union and find Is -> E(log(V))

I am thinking time complexity without path compression using union by rank is , O(ElogE)(sort) +O(Elogv)(find and union) that would be upperbound O(ElogE)

Am i right here ? Or we can use path compression here ??