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$ curl repovive.com/roadmaps/graph-theory/heavy-light-decomposition/implementation-path-query

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$ curl repovive.com/roadmaps/graph-theory/heavy-light-decomposition/implementation-path-query

Repovive

Implementation - Path Query - Heavy-Light Decomposition | Graph Theory | Repovive

Repovive.

Implementation - Path Query

(Combining chain segments)

3 min

To query the maximum on path from $u$ u to $v$ v:

function path_query(u, v)
    res := -infinity
    while chain[u] != chain[v]
        if depth[head[u]] < depth[head[v]] then
            swap(u, v)
            res := max(res, seg_query(pos[head[u]], pos[u]))
            u := parent[head[u]]
            if depth[u] > depth[v] then
                swap(u, v)
                res := max(res, seg_query(pos[u], pos[v]))
                return res

This walks $O(\log n)$ O(logn) chains, querying each one and tracking the maximum.

Space complexity is $O(n)$ O(n) for the data structures used.