Improve minimum spanning tree with new edge, with better running time than O(|V|)? The problem gives a MST $T$ and a series of $Q$ queries, each one with a new edge $e = \{u,v\}$ such that no edge between $u$ and $v$ exists in $T$. For every query, we have to improve $T$ with $e$ and print the new weight of $T$.

The best I can do is run a DFS ($O(|V|)$) to find the current path $P$ between $u$ and $v$, and find the heaviest edge $e_\text{max}$ in $P$. If $w(e) &gt; w(e_\text{max})$, we improve $T$ removing $e_\text{max}$ and inserting $e$. The overall running time for a test case is $O(Q|V|)$.

Does anybody know an asymptotically faster algorithm for this problem?
Using Heavy path decomposition.

The key idea is to partition all edges into heavy edges and light edges.
Consider tree edge from parent node X to child node Y:


heavy edge: size of sub tree Y is at least half of the size of sub tree X;
light edge: size of sub tree Y is smaller than half of the size of sub tree X;


There are at most $O(\log |V|)$ heavy paths (path with only heavy edges) from any leaf to root. Maintaining each heavy path using segment tree or balanced BST, and converting MST into tree of heavy paths, one can query the heaviest edge $e_{max}$ between any two vertices $u$,$v$ in $O(\log |V|)$. 

Another method is using Link cut tree which handles each query in $O(\log |V|)$ amortized time complexity. This can also handle the case that each query is applied to the result of previous implement.