这里没有水表
树链剖分是一个通过轻重链剖分、dfs序和线段树的使用,在有根树上维护路径和子树信息的有效算法
其预处理时间复杂度为\(O(nlog^2n)\),单次操作时间复杂度为\(O(log^2n)\),常数优秀
主要是防止自己忘记,所以只作简要说明,要学习的可以参考友链里大佬的博客
预处理
第一遍dfs
需要处理的数组有\(fa(父亲),son(重儿子),sz(子树大小)和dep(深度)\)
ll fa[N],son[N],sz[N],dep[N];
il void dfs1(int u,int f){sz[u]=1;fa[u]=f;dep[u]=dep[f]+1;for(RG int i=head[u];i;i=nxt[i]){RG int v=to[i];if(v==f)continue;dfs1(v,u);sz[u]+=sz[v];if(!son[u]||sz[v]>sz[son[u]])son[u]=v;}
}第二遍dfs
需要处理的数组有\(w(原树节点在线段树节点上的编号),fw(线段树节点在原树节点上的编号),top(所在重链的顶部)\)
优先搜索重儿子,因为需要将重链上的节点放在线段树上相邻的位置
ll top[N],w[N],fw[N],cntw;
il void dfs2(int u,int f){top[u]=(son[fa[u]]==u)?top[fa[u]]:u;w[u]=++cntw;fw[w[u]]=u;if(son[u])dfs2(son[u],u);for(RG int i=head[u];i;i=nxt[i]){RG int v=to[i];if(v==f||v==son[u])continue;dfs2(v,u);}
}其实到这里树链剖分就做完了
build线段树
线段树上节点的值通过\(fw\)数组和原树对应起来(这里以luogu树链剖分模板中的操作为例)
#define ls (i<<1)
#define rs (i<<1|1)
#define mid ((l+r)>>1)
ll sum[N<<2],lz[N<<2];
il void update(int i){sum[i]=(sum[ls]+sum[rs])%mod;}
il void build(ll i,ll l,ll r){if(l==r){sum[i]=val[fw[l]]%mod;return;}build(ls,l,mid);build(rs,mid+1,r);update(i);
}维护子树信息
以\(x\)为根的子树在线段树上表现出来的区间为\((w[x],w[x]+sz[x]-1)\)
修改
il void modify_tree(){RG int x=read(),k=read();modify(1,1,n,w[x],w[x]+sz[x]-1,k);
}查询
il void query_tree(){RG int x=read();printf("%lld\n",query(1,1,n,w[x],w[x]+sz[x]-1));
}维护路径信息
对于一条\(u->v\)的路径,我们根据之前的重链剖分可以将其分成不超过\(log(n)\)条重链,
每条重链分别在线段树上进行维护,这时重链的合并也就成为维护路径信息的一大难点
修改
il void modify_road(){RG int u=read(),v=read();RG ll k=read();while(top[u]!=top[v]){if(dep[top[u]]<dep[top[v]])swap(u,v);modify(1,1,n,w[top[u]],w[u],k);u=fa[top[u]];}if(dep[u]>dep[v])swap(u,v);modify(1,1,n,w[u],w[v],k);
}查询
il void query_road(){RG int u=read(),v=read();RG ll ret=0;while(top[u]!=top[v]){if(dep[top[u]]<dep[top[v]])swap(u,v);ret=(ret+query(1,1,n,w[top[u]],w[u]))%mod;u=fa[top[u]];}if(dep[u]>dep[v])swap(u,v);printf("%lld\n",(ret+query(1,1,n,w[u],w[v]))%mod);
}luogu P3384 【模板】树链剖分
囊括了上面的操作.
#include<algorithm>
#include<iostream>
#include<cstdlib>
#include<iomanip>
#include<cstring>
#include<complex>
#include<vector>
#include<cstdio>
#include<string>
#include<bitset>
#include<cmath>
#include<queue>
#include<stack>
#include<map>
#include<set>
#define mp make_pair
#define pb push_back
#define RG register
#define il inline
using namespace std;
typedef unsigned long long ull;
typedef vector<int>VI;
typedef long long ll;
typedef double dd;
const dd eps=1e-10;
const int N=100010;
const int M=50010*2;
il ll read(){RG ll data=0,w=1;RG char ch=getchar();while(ch!='-'&&(ch<'0'||ch>'9'))ch=getchar();if(ch=='-')w=-1,ch=getchar();while(ch<='9'&&ch>='0')data=data*10+ch-48,ch=getchar();return data*w;
}ll n,m,r,mod,val[N];
ll head[N],nxt[N<<1],to[N<<1],cnt;
il void add(int u,int v){to[++cnt]=v;nxt[cnt]=head[u];head[u]=cnt;
}ll fa[N],son[N],sz[N],dep[N];
il void dfs1(int u,int f){sz[u]=1;fa[u]=f;dep[u]=dep[f]+1;for(RG int i=head[u];i;i=nxt[i]){RG int v=to[i];if(v==f)continue;dfs1(v,u);sz[u]+=sz[v];if(!son[u]||sz[v]>sz[son[u]])son[u]=v;}
}ll top[N],w[N],fw[N],cntw;
il void dfs2(int u,int f){top[u]=(son[fa[u]]==u)?top[fa[u]]:u;w[u]=++cntw;fw[w[u]]=u;if(son[u])dfs2(son[u],u);for(RG int i=head[u];i;i=nxt[i]){RG int v=to[i];if(v==f||v==son[u])continue;dfs2(v,u);}
}#define ls (i<<1)
#define rs (i<<1|1)
#define mid ((l+r)>>1)
ll sum[N<<2],lz[N<<2];
il void update(int i){sum[i]=(sum[ls]+sum[rs])%mod;}
il void build(ll i,ll l,ll r){if(l==r){sum[i]=val[fw[l]]%mod;return;}build(ls,l,mid);build(rs,mid+1,r);update(i);
}il void pushdown(ll i,ll l,ll r){if(!lz[i])return;sum[ls]=(sum[ls]+lz[i]*(mid-l+1)%mod)%mod;sum[rs]=(sum[rs]+lz[i]*(r-mid)%mod)%mod;lz[ls]=(lz[ls]+lz[i])%mod;lz[rs]=(lz[rs]+lz[i])%mod;lz[i]=0;
}il void modify(ll i,ll l,ll r,ll x,ll y,ll k){if(x<=l&&r<=y){sum[i]=(sum[i]+k*(r-l+1)%mod)%mod;lz[i]=(lz[i]+k)%mod;return;}pushdown(i,l,r);if(x<=mid)modify(ls,l,mid,x,y,k);if(y>=mid+1)modify(rs,mid+1,r,x,y,k);update(i);
}il ll query(ll i,ll l,ll r,ll x,ll y){if(x<=l&&r<=y)return sum[i]%mod;pushdown(i,l,r);RG ll s=0;if(x<=mid)s=query(ls,l,mid,x,y)%mod;if(y>=mid+1)s=(s+query(rs,mid+1,r,x,y))%mod;return s;
}il void modify_tree(){RG int x=read(),k=read();modify(1,1,n,w[x],w[x]+sz[x]-1,k);
}il void query_tree(){RG int x=read();printf("%lld\n",query(1,1,n,w[x],w[x]+sz[x]-1));
}il void modify_road(){RG int u=read(),v=read();RG ll k=read();while(top[u]!=top[v]){if(dep[top[u]]<dep[top[v]])swap(u,v);modify(1,1,n,w[top[u]],w[u],k);u=fa[top[u]];}if(dep[u]>dep[v])swap(u,v);modify(1,1,n,w[u],w[v],k);
}il void query_road(){RG int u=read(),v=read();RG ll ret=0;while(top[u]!=top[v]){if(dep[top[u]]<dep[top[v]])swap(u,v);ret=(ret+query(1,1,n,w[top[u]],w[u]))%mod;u=fa[top[u]];}if(dep[u]>dep[v])swap(u,v);printf("%lld\n",(ret+query(1,1,n,w[u],w[v]))%mod);
}int main()
{n=read();m=read();r=read();mod=read();for(RG int i=1;i<=n;i++)val[i]=read()%mod;for(RG int j=1,u,v;j<n;j++){u=read();v=read();add(u,v);add(v,u);}dfs1(r,0);dfs2(r,0);build(1,1,n);for(RG int i=1,opt;i<=m;i++){opt=read();if(opt==1)modify_road();if(opt==2)query_road();if(opt==3)modify_tree();if(opt==4)query_tree();}return 0;
}[SDOI2011]染色
询问树上的路径有多少条颜色段,区间修改
记录区间左颜色段,右颜色段,和颜色段数,合并重链时需要留意
#include<algorithm>
#include<iostream>
#include<cstdlib>
#include<iomanip>
#include<cstring>
#include<complex>
#include<vector>
#include<cstdio>
#include<string>
#include<bitset>
#include<cmath>
#include<queue>
#include<stack>
#include<map>
#include<set>
#define mp make_pair
#define pb push_back
#define RG register
#define il inline
using namespace std;
typedef unsigned long long ull;
typedef vector<int>VI;
typedef long long ll;
typedef double dd;
const dd eps=1e-10;
const int mod=1e9+7;
const int N=100100;
const int M=50010*2;
il ll read(){RG ll data=0,w=1;RG char ch=getchar();while(ch!='-'&&(ch<'0'||ch>'9'))ch=getchar();if(ch=='-')w=-1,ch=getchar();while(ch<='9'&&ch>='0')data=data*10+ch-48,ch=getchar();return data*w;
}il void file(){freopen("a.in","r",stdin);freopen("a.out","w",stdout);
}int n,m,c[N];char ch;
int head[N],nxt[N<<1],to[N<<1],cnt;
il void add(int u,int v){to[++cnt]=v;nxt[cnt]=head[u];head[u]=cnt;
}int fa[N],son[N],sz[N],dep[N];
void dfs1(int u){sz[u]=1;for(RG int i=head[u];i;i=nxt[i]){RG int v=to[i];if(v==fa[u])continue;fa[v]=u;dep[v]=dep[u]+1;dfs1(v);sz[u]+=sz[v];if(!son[u]||sz[v]>sz[son[u]])son[u]=v;}
}int top[N],w[N],fw[N],cntw;
void dfs2(int u,int tp){top[u]=tp;w[u]=++cntw;fw[w[u]]=u;if(son[u])dfs2(son[u],tp);for(RG int i=head[u];i;i=nxt[i]){RG int v=to[i];if(v==fa[u]||v==son[u])continue;dfs2(v,v);}
}#define ls (i<<1)
#define rs (i<<1|1)
#define mid ((l+r)>>1)
int sum[N<<2],lc[N<<2],rc[N<<2],s[N<<2];
il void update(int i){sum[i]=sum[ls]+sum[rs]-(rc[ls]==lc[rs]);lc[i]=lc[ls];rc[i]=rc[rs];
}
void build(int i,int l,int r){if(l==r){sum[i]=1;lc[i]=rc[i]=c[fw[l]];return;}build(ls,l,mid);build(rs,mid+1,r);update(i);
}
il void pushdown(int i){if(!s[i])return;sum[ls]=sum[rs]=1;s[ls]=s[rs]=lc[ls]=rc[ls]=lc[rs]=rc[rs]=s[i];s[i]=0;
}void modify(int i,int l,int r,int x,int y,int w){if(x<=l&&r<=y){sum[i]=1;s[i]=lc[i]=rc[i]=w;return;}pushdown(i);if(x<=mid)modify(ls,l,mid,x,y,w);if(y>mid)modify(rs,mid+1,r,x,y,w);update(i);
}int query(int i,int l,int r,int x,int y,int &L,int &R){if(x<=l&&r<=y){L=lc[i];R=rc[i];return sum[i];}pushdown(i);if(x<=mid&&y<=mid)return query(ls,l,mid,x,y,L,R);if(x>mid&&y>mid)return query(rs,mid+1,r,x,y,L,R);RG int lr=0,rl=0,s=0;s=query(ls,l,mid,x,y,L,lr)+query(rs,mid+1,r,x,y,rl,R);if(lr==rl&&lr)s--;return s;
}il void modify_road(){RG int u=read(),v=read(),k=read();while(top[u]!=top[v]){if(dep[top[u]]<dep[top[v]])swap(u,v);modify(1,1,n,w[top[u]],w[u],k);u=fa[top[u]];}if(dep[u]>dep[v])swap(u,v);modify(1,1,n,w[u],w[v],k);
}il void query_road(){RG int u=read(),v=read(),tu=-1,tv=-1,l=0,r=0,sum=0,ret;while(top[u]!=top[v]){if(dep[top[u]]<dep[top[v]])swap(u,v),swap(tu,tv);ret=query(1,1,n,w[top[u]],w[u],l,r);sum+=ret;if(r==tu)sum--;tu=l;u=fa[top[u]];}if(dep[u]>dep[v])swap(u,v),swap(tu,tv);ret=query(1,1,n,w[u],w[v],l,r);sum+=ret;if(l==tu)sum--;if(r==tv)sum--;printf("%d\n",sum);
}int main()
{n=read();m=read();for(RG int i=1;i<=n;i++)c[i]=read();for(RG int i=1,u,v;i<n;i++){u=read();v=read();add(u,v);add(v,u);}dfs1(1);dfs2(1,1);build(1,1,n);for(RG int i=1;i<=m;i++){ch=0;while(ch!='C'&&ch!='Q')ch=getchar();if(ch=='C')modify_road();else query_road();}return 0;
}[NOI2015]软件包管理器
其实就是弱化版的模板。
#include<algorithm>
#include<iostream>
#include<cstdlib>
#include<iomanip>
#include<cstring>
#include<complex>
#include<vector>
#include<cstdio>
#include<string>
#include<bitset>
#include<cmath>
#include<queue>
#include<stack>
#include<map>
#include<set>
#define mp make_pair
#define pb push_back
#define RG register
#define il inline
using namespace std;
typedef unsigned long long ull;
typedef vector<int>VI;
typedef long long ll;
typedef double dd;
const dd eps=1e-10;
const int mod=1e9+7;
const int N=100010;
const int M=50010*2;
il ll read(){RG ll data=0,w=1;RG char ch=getchar();while(ch!='-'&&(ch<'0'||ch>'9'))ch=getchar();if(ch=='-')w=-1,ch=getchar();while(ch<='9'&&ch>='0')data=data*10+ch-48,ch=getchar();return data*w;
}il void file(){freopen("a.in","r",stdin);freopen("a.out","w",stdout);
}int n,q;char ch[10];
int head[N],nxt[N<<1],to[N<<1],cnt;
il void add(int u,int v){to[++cnt]=v;nxt[cnt]=head[u];head[u]=cnt;
}int sz[N],fa[N],dep[N],son[N];
void dfs1(int u){sz[u]=1;for(RG int i=head[u];i;i=nxt[i]){RG int v=to[i];if(v==fa[u])continue;fa[v]=u;dep[v]=dep[u]+1;dfs1(v);sz[u]+=sz[v];if(!son[u]||sz[v]>sz[son[u]])son[u]=v;}
}int top[N],w[N],fw[N],cntw;
void dfs2(int u,int tp){top[u]=tp;w[u]=++cntw;fw[w[u]]=u;if(son[u])dfs2(son[u],tp);for(RG int i=head[u];i;i=nxt[i]){RG int v=to[i];if(v==fa[u]||v==son[u])continue;dfs2(v,v);}
}#define ls (i<<1)
#define rs (i<<1|1)
#define mid ((l+r)>>1)
int sum[2][N<<2],lz[N<<2];
//初始时lz=-1
il void update(int i){sum[0][i]=sum[0][ls]+sum[0][rs];sum[1][i]=sum[1][ls]+sum[1][rs];
}
il void pushdown(int i,int l,int r){if(lz[i]==-1)return;lz[ls]=lz[rs]=lz[i];sum[lz[ls]^1][ls]=0;sum[lz[ls]][ls]=mid-l+1;sum[lz[rs]^1][rs]=0;sum[lz[rs]][rs]=r-mid;lz[i]=-1;
}
il void build(int i,int l,int r){lz[i]=-1;if(l==r){sum[0][i]=1;sum[1][i]=0;return;}build(ls,l,mid);build(rs,mid+1,r);update(i);
}il void modify(int i,int l,int r,int x,int y,int k){if(x<=l&&r<=y){lz[i]=k;sum[k][i]=r-l+1;sum[k^1][i]=0;return;}pushdown(i,l,r);if(x<=mid)modify(ls,l,mid,x,y,k);if(y>mid)modify(rs,mid+1,r,x,y,k);update(i);
}il int query(int i,int l,int r,int x,int y,int k){if(x<=l&&r<=y)return sum[k][i];pushdown(i,l,r);RG int s=0;if(x<=mid)s=query(ls,l,mid,x,y,k);if(y>mid)s+=query(rs,mid+1,r,x,y,k);return s;
}il void modify_tree(int u,int k){modify(1,1,n,w[u],w[u]+sz[u]-1,k);
}
il void query_tree(int u,int k){printf("%d\n",query(1,1,n,w[u],w[u]+sz[u]-1,k));
}il void modify_road(int u,int v,int k){while(top[u]!=top[v]){if(dep[top[u]]<dep[top[v]])swap(u,v);modify(1,1,n,w[top[u]],w[u],k);u=fa[top[u]];}if(dep[u]>dep[v])swap(u,v);modify(1,1,n,w[u],w[v],k);
}il void query_road(int u,int v,int k){RG int s=0;while(top[u]!=top[v]){if(dep[top[u]]<dep[top[v]])swap(u,v);s+=query(1,1,n,w[top[u]],w[u],k);u=fa[top[u]];}if(dep[u]>dep[v])swap(u,v);printf("%d\n",s+query(1,1,n,w[u],w[v],k));
}int main()
{n=read();for(RG int i=2;i<=n;i++){fa[i]=read()+1;add(i,fa[i]);add(fa[i],i);}dfs1(1);dfs2(1,1);build(1,1,n);q=read();for(RG int i=1,x;i<=q;i++){scanf("%s",ch+1);x=read()+1;if(ch[1]=='i')query_road(1,x,0),modify_road(1,x,1);else query_tree(x,1),modify_tree(x,0);}return 0;
}[JLOI2014]松鼠的新家
本来是树上差分来着...被我拿树剖来练手了。
#include<algorithm>
#include<iostream>
#include<cstdlib>
#include<iomanip>
#include<cstring>
#include<complex>
#include<vector>
#include<cstdio>
#include<string>
#include<bitset>
#include<cmath>
#include<queue>
#include<stack>
#include<map>
#include<set>
#define mp make_pair
#define pb push_back
#define RG register
#define il inline
using namespace std;
typedef unsigned long long ull;
typedef vector<int>VI;
typedef long long ll;
typedef double dd;
const dd eps=1e-10;
const int mod=1e9+7;
const int N=300010;
const int M=50010*2;
il ll read(){RG ll data=0,w=1;RG char ch=getchar();while(ch!='-'&&(ch<'0'||ch>'9'))ch=getchar();if(ch=='-')w=-1,ch=getchar();while(ch<='9'&&ch>='0')data=data*10+ch-48,ch=getchar();return data*w;
}il void file(){freopen("a.in","r",stdin);freopen("a.out","w",stdout);
}int n,a[N];
int head[N],nxt[N<<1],to[N<<1],cnt;
il void add(int u,int v){to[++cnt]=v;nxt[cnt]=head[u];head[u]=cnt;
}int sz[N],fa[N],son[N],dep[N];
void dfs1(int u){sz[u]=1;for(RG int i=head[u];i;i=nxt[i]){RG int v=to[i];if(v==fa[u])continue;dep[v]=dep[u]+1;fa[v]=u;dfs1(v);sz[u]+=sz[v];if(!son[u]||sz[son[u]]<sz[v])son[u]=v;}
}int top[N],w[N],fw[N],cntw;
void dfs2(int u,int tp){top[u]=tp;w[u]=++cntw;fw[w[u]]=u;if(son[u])dfs2(son[u],tp);for(RG int i=head[u];i;i=nxt[i]){RG int v=to[i];if(v==fa[u]||v==son[u])continue;dfs2(v,v);}
}#define ls (i<<1)
#define rs (i<<1|1)
#define mid ((l+r)>>1)
ll sum[N<<2],lz[N<<2];
il void update(int i){sum[i]=sum[ls]+sum[rs];}
il void pushdown(int i,int l,int r){if(!lz[i])return;sum[ls]+=lz[i]*(mid-l+1);sum[rs]+=lz[i]*(r-mid);lz[ls]+=lz[i];lz[rs]+=lz[i];lz[i]=0;
}il void modify(int i,int l,int r,int x,int y,int k){if(x<=l&&r<=y){lz[i]+=k;sum[i]+=k*(r-l+1);return;}pushdown(i,l,r);if(x<=mid)modify(ls,l,mid,x,y,k);if(y>mid)modify(rs,mid+1,r,x,y,k);update(i);
}il ll query(int i,int l,int r,int x,int y){if(x<=l&&r<=y)return sum[i];pushdown(i,l,r);RG ll s=0;if(x<=mid)s=query(ls,l,mid,x,y);if(y>mid)s+=query(rs,mid+1,r,x,y);return s;
}il void modify_road(int u,int v){while(top[u]!=top[v]){if(dep[top[u]]<dep[top[v]])swap(u,v);modify(1,1,n,w[top[u]],w[u],1);u=fa[top[u]];}if(dep[u]>dep[v])swap(u,v);modify(1,1,n,w[u],w[v],1);
}int main()
{n=read();for(RG int i=1;i<=n;i++)a[i]=read();for(RG int i=1,u,v;i<n;i++){u=read();v=read();add(u,v);add(v,u);}dfs1(1);dfs2(1,1);for(RG int i=2;i<=n;i++){modify_road(a[i-1],a[i]);modify(1,1,n,w[a[i]],w[a[i]],-1);}for(RG int i=1;i<=n;i++)printf("%lld\n",query(1,1,n,w[i],w[i]));return 0;
}未完待续......(其实是我写的题太少了)
pdf)
