Heavy Light Decomposition

Note

1.DFS1
mark all the depth
mark fathers
mark the heavy/light children
mark the size of each subtree

void dfs1(long long pos, long long f, long long depth){
  dep[pos] = depth;
  fat[pos] = f;
  sz[pos] = 1;
  long long maxi = -1;
  for (long long v : adj[pos]){
    if (v == f) continue;
    dfs1(v,pos,depth+1);
    sz[pos]+=sz[v];
    if (maxi<sz[v]) {maxi = sz[v];son[pos] = v;}
  }
}

2.DFS2
mark the of the members in the base array
record the top of each node
traverse heavy son first then light son
mark the heavy son along the path
give the base array a value
record the id of each node

void dfs2(long long pos, long long top_pos){
  id[pos] = ++cnt;
  wt[cnt] = num[pos];
  top[pos] = top_pos;
  if (!son[pos]) return;
  dfs2(son[pos],top_pos);
  for (long long v : adj[pos]){
    if (v==fat[pos] || v==son[pos])  continue;
    dfs2(v,v);
  }
}

3.make_tree
make segment tree base on the base array

void make_tree(int way, int l, int r){
  if (l==r) {seg[way] = wt[l]%p;return;}
  int mid = (l+r)/2;
  make_tree(way*2,l,mid);
  make_tree(way*2+1,mid+1,r);
  seg[way] = (seg[way*2]+seg[way*2+1])%p;
}
//push function
void push(int way,int lenn){
    lazy[way*2]+=lazy[way];
    lazy[way*2+1]+=lazy[way];
    seg[way*2]+=lazy[way]*(lenn-(lenn>>1));
    seg[way*2+1]+=lazy[way]*(lenn>>1);
    seg[way*2]%=p;
    seg[way*2+1]%=p;
    lazy[way]=0;
}

4.query_up
query base on the segment tree

int query_up(int way, int l, int r, int qlow, int qhigh){
  push(way,l,r);
  if (qlow<= l && r<=qhigh) return seg[way]%p;
  if (l>qhigh || r<qlow) return 0;

  int mid = (l+r)/2;
  return (query_up(way*2,l,mid,qlow,qhigh) + query_up(way*2+1,mid+1,r,qlow,qhigh))%p;
}

5.query
1) check if they are on the same chain. if Not, add the distance from a node to the top of the chain, and move up
2) repeat step 1 until they are on the same chain
3) query the distance between the nodes based on the base array

int query(int x, int y){
  int ans = 0;
  while(top[x]!=top[y]){
    if (dep[top[x]]<dep[top[y]]) swap(x,y);
    ans = (ans+query_up(1,1,n,id[top[x]],id[x]))%p;
    x = fat[top[x]];
  }
  if (dep[x]>dep[y]) swap(x,y);
  ans = (ans + query_up(1,1,n,id[x],id[y]))%p;
  return ans;
}

6.query_son
query the id[x] plus its size -1 the subtree of a node is consecutive in the base array

int qSon(int x){
  return query_up(1,1,n,id[x],id[x]+sz[x]-1);
}

7.update
update normal segment tree (use id as substitution)

void update(int way, int l, int r, int qlow, int qhigh, int val){
 push(way,l,r);
 if (qlow<=l && r<=qhigh) {
   lazy[way] += val;
   push(way,l,r);
   return;
 }
 if (l>qhigh || r<qlow) return;
 int mid = (l+r)/2;
 update(way*2,l,mid,qlow,qhigh,val);
 update(way*2+1,mid+1,r,qlow,qhigh,val);
 seg[way] = (seg[way*2]+seg[way*2+1])%p;
}

8.update chain
1) check if they are on the same chain. If not, update the chain, and go to the father of the chain head
2) repeat until they are on the same chain
3) update the id of the nodes on the segment tree

void upPath(int x, int y, int val){
  while(top[x]!=top[y]){
    if (dep[top[x]]<dep[top[y]]) swap(x,y);
    update(1,1,n,id[top[x]],id[x],val);
    x = fat[top[x]];
  }
  if (dep[x]>dep[y]) swap(x,y);
  update(1,1,n,id[x],id[y],val);
}

9.update_son
update id[x] plus its size -1

void upSon(int x, int val){
  update(1,1,n,id[x],id[x]+sz[x]-1,val);
}

optional: LCA
1) check if the two nodes are on the same chain
2) if not, find the one with a deeper head, move it up
3) repeat until they are in the same chain
3) return the one with higher depth

int LCA (int x , int y ) {
    int fx = top[x] , fy = top[y] ;
    while(fx!=fy) {
        if(level[fx]< level[fy]) swap(x,y) , swap(fx ,fy) ;
        x = fat[fx] ; fx = top[x];
    }
    if( level[x] > level[y] ) swap(x,y) ;
    return x ;
}

Note to myself
Segment tree is able to support different data structure