Path Queries II - Implementation

Here is the solution:

function preprocess()
    dfs_init(1, -1, 0)
    dfs_hld(1, -1, 0)
    for v from 1 to n
        seg_update(pos[v], val[v])

function update(v, x)
    val[v] := x
    seg_update(pos[v], x)

function query(u, v)
    return path_query(u, v)

This runs in $O(n)$ time and uses $O(n)$ space.