Recently I've been looking for a good heavy-light decomposition code I can just copy-paste and continue to live happily. To my greatest disappointment, I didn't find one and had to write my own.

First, there's this blog explaining what HLD is http://blog.anudeep2011.com/heavy-light-decomposition/, but it doesn't contain full code. Also it uses LCA for queries, which is absolutely unnecessary and complicates things too much.

Second, there's this implementation https://sites.google.com/site/indy256/algo/heavy_light. OK, complete working code, and it doesn't use LCA. But it has too many arrays. Way too many! Also it's mixed with segment tree code, and I'd rather separate the logic.

Third, there's this post http://apps.topcoder.com/forums/?module=Thread&threadID=796128&start=0&mc=8. So far the best looking code, but it has only LCA method and no path query handling, which is the point of HLD. So, I refined the code and added missing parts.

The code is probably too simple to explain it, but not well tested yet. Graph can be represented for example as vector<vector<int>>. I use single segment tree for all the chains. Creating separate trees makes code harder to read and also slower (at least in my experiment). For a good segment tree implementation you can look here: http://codeforces.me/blog/entry/18051.

template <class T, int V>
class HeavyLight {
  int parent[V], heavy[V], depth[V];
  int root[V], treePos[V];
  SegmentTree<T> tree;

  template <class G>
  int dfs(const G& graph, int v) {
    int size = 1, maxSubtree = 0;
    for (int u : graph[v]) if (u != parent[v]) {
      parent[u] = v;
      depth[u] = depth[v] + 1;
      int subtree = dfs(graph, u);
      if (subtree > maxSubtree) heavy[v] = u, maxSubtree = subtree;
      size += subtree;
    }
    return size;
  }

  template <class BinaryOperation>
  void processPath(int u, int v, BinaryOperation op) {
    for (; root[u] != root[v]; v = parent[root[v]]) {
      if (depth[root[u]] > depth[root[v]]) swap(u, v);
      op(treePos[root[v]], treePos[v] + 1);
    }
    if (depth[u] > depth[v]) swap(u, v);
    op(treePos[u], treePos[v] + 1);
  }

public:
  template <class G>
  void init(const G& graph) {
    int n = graph.size();
    fill_n(heavy, n, -1);
    parent[0] = -1;
    depth[0] = 0;
    dfs(graph, 0);
    for (int i = 0, currentPos = 0; i < n; ++i)
      if (parent[i] == -1 || heavy[parent[i]] != i)
        for (int j = i; j != -1; j = heavy[j]) {
          root[j] = i;
          treePos[j] = currentPos++;
        }
    tree.init(n);
  }

  void set(int v, const T& value) {
    tree.set(treePos[v], value);
  }

  void modifyPath(int u, int v, const T& value) {
    processPath(u, v, [this, &value](int l, int r) { tree.modify(l, r, value); });
  }

  T queryPath(int u, int v) {
    T res = T();
    processPath(u, v, [this, &res](int l, int r) { res.add(tree.query(l, r)); });
    return res;
  }
};