BZOJ#3786. 星系探索

Solution

子树加，换fatherfatherfather（保证还是树），询问到根路径和。
树上路径求和不好动态维护，于是转化到序列上，维护一个括号序，dfndfndfn处贡献为www，fnsfnsfns处贡献为−w-w−w，一段路径(x,y)(x,y)(x,y)的和相当于序列上dfnxdfn_xdfnx到dfnydfn_ydfny的贡献和。

于是子树加相当于区间加，交换子树相当于在括号序上取出一个区间插入到其他地方去。
这个可以直接用平衡树维护，这里用的是fhq−treapfhq-treapfhq−treap。

我们在fhq−treapfhq-treapfhq−treap中节点的下标对应的是原树的括号序编号，即dfnxdfn_xdfnx在treaptreaptreap上对应dfnxdfn_xdfnx号节点，在splitsplitsplit时，当前的括号序为中序遍历treaptreaptreap后得到的节点编号序列。需要求出节点在treaptreaptreap上的rankrankrank，然后再按rankrankrank分成两棵子树。

剩下的就是treaptreaptreap的区间标记维护区间和了（注意dfnx+=Δdfn_x+=\Deltadfnx+=Δ，则fnsx−=Δfns_x-=\Deltafnsx−=Δ，因此我们的区间和要带符号维护）。

时间复杂度O(nlgn)O(nlgn)O(nlgn)。

Code

#include <vector>
#include <list>
#include <map>
#include <set>
#include <deque>
#include <queue>
#include <stack>
#include <bitset>
#include <algorithm>
#include <functional>
#include <numeric>
#include <utility>
#include <sstream>
#include <iostream>
#include <iomanip>
#include <cstdio>
#include <cmath>
#include <cstdlib>
#include <cctype>
#include <string>
#include <cstring>
#include <ctime>
#include <cassert>
#include <string.h>
//#include <unordered_set>
//#include <unordered_map>
//#include <bits/stdc++.h>#define MP(A,B) make_pair(A,B)
#define PB(A) push_back(A)
#define SIZE(A) ((int)A.size())
#define LEN(A) ((int)A.length())
#define FOR(i,a,b) for(int i=(a);i<(b);++i)
#define fi first
#define se secondusing namespace std;template<typename T>inline bool upmin(T &x,T y) { return y<x?x=y,1:0; }
template<typename T>inline bool upmax(T &x,T y) { return x<y?x=y,1:0; }typedef long long ll;
typedef unsigned long long ull;
typedef long double lod;
typedef pair<int,int> PR;
typedef vector<int> VI;const lod eps=1e-11;
const lod pi=acos(-1);
const int oo=1<<30;
const ll loo=1ll<<62;
const int mods=998244353;
const int MAXN=600005;
const int INF=0x3f3f3f3f;//1061109567
/*--------------------------------------------------------------------*/
inline int read()
{int f=1,x=0; char c=getchar();while (c<'0'||c>'9') { if (c=='-') f=-1; c=getchar(); }while (c>='0'&&c<='9') { x=(x<<3)+(x<<1)+(c^48); c=getchar(); }return x*f;
}
struct fhq_treap
{ll a[MAXN],sum[MAXN],tag[MAXN];int b[MAXN],sz[MAXN],p[MAXN],num[MAXN],fa[MAXN],ch[MAXN][2],rt;void up(int x) { sz[x]=sz[ch[x][0]]+sz[ch[x][1]]+1;num[x]=num[ch[x][0]]+num[ch[x][1]]+p[x];sum[x]=sum[ch[x][0]]+sum[ch[x][1]]+a[x]; if (ch[x][0]) fa[ch[x][0]]=x;if (ch[x][1]) fa[ch[x][1]]=x;}void puttag(int x,ll y) { if (!x) return; tag[x]+=y,a[x]+=y*p[x],sum[x]+=y*num[x]; }void down(int x) { if (!x||tag[x]==0) return;puttag(ch[x][0],tag[x]),puttag(ch[x][1],tag[x]),tag[x]=0;}int nwnode(int id,int opt,int x) { p[id]=num[id]=opt;a[id]=sum[id]=opt*x;b[id]=rand(),sz[id]=1; return id; }void split(int nw,int k,int &x,int &y){if (!nw) x=y=0;else{down(nw);if (k<=sz[ch[nw][0]]) y=nw,split(ch[nw][0],k,x,ch[nw][0]);else x=nw,split(ch[nw][1],k-sz[ch[nw][0]]-1,ch[nw][1],y);up(nw);}}int merge(int x,int y){if (!x||!y) return x|y;down(x),down(y);if (b[x]<b[y]) return ch[x][1]=merge(ch[x][1],y),up(x),x;else return ch[y][0]=merge(x,ch[y][0]),up(y),y;}int rank(int x){int rk=sz[ch[x][0]]+1;while (fa[x]) {if (x==ch[fa[x]][1]) rk+=sz[ch[fa[x]][0]]+1;x=fa[x];}return rk;}ll Query(int x){int A,B;split(rt,rank(x),A,B);ll ans=sum[A];rt=merge(A,B);return ans;}void Swap(int l,int r,int x){int A,B,C,D;l=rank(l),r=rank(r),x=rank(x);if (x<=l-1)  //1...x...l...r...n{split(rt,r,A,D);split(A,l-1,A,B);split(A,x,A,C);rt=merge(merge(merge(A,B),C),D);}else{split(rt,x,A,D); //1...l...r...x...nsplit(A,r,A,B);split(A,l-1,A,C);rt=merge(merge(merge(A,B),C),D);}}void Update(int l,int r,int y){int A,B,C;l=rank(l),r=rank(r);split(rt,r,A,C);split(A,l-1,A,B);puttag(B,y);rt=merge(merge(A,B),C);}
} T;
vector<int> e[MAXN];
int dfn[MAXN],fns[MAXN],w[MAXN],DFN=0;
void dfs(int x,int father)
{dfn[x]=++DFN;T.rt=T.merge(T.rt,T.nwnode(DFN,1,w[x]));for (auto v:e[x]) if (v!=father) dfs(v,x);fns[x]=++DFN;T.rt=T.merge(T.rt,T.nwnode(DFN,-1,w[x]));
}
signed main()
{srand(time(0));int n=read();for (int i=2;i<=n;i++) e[read()].PB(i);for (int i=1;i<=n;i++) w[i]=read();dfs(1,0);int Case=read();while (Case--){int x,y; char c; scanf("%c",&c);if (c=='Q') x=read(),printf("%lld\n",T.Query(dfn[x]));else if (c=='C') x=read(),y=read(),T.Swap(dfn[x],fns[x],dfn[y]);else x=read(),y=read(),T.Update(dfn[x],fns[x],y);}return 0;
}