[学习笔记]树链剖分

开始复习(NOIP)啦！

树链剖分是一种非常实用的算法，树的重链最多只有(logn)条，所以它的复杂度是(O(nlogn))

如果树再套上一个线段树，那么复杂度多乘一个(logn)

那么线段树怎么表示区间呢？可以用(dfs)序修改，多开一个(id[i])表示(i)节点在线段树的位置

我们记录一个节点的深度(dep[i])，链首数组(top[i])，重儿子(son[i])，父亲(fa[i])，一个子树的大小(siz[i])

然后就可以愉快的解决掉【模板】树链剖分

(Code Below:)

#include <bits/stdc++.h>
#define ll long long
using namespace std;
const ll maxn=100000+10;
ll n,m,r,p;
ll sum[maxn<<2],lazy[maxn<<2],a[maxn];
ll head[maxn],from[maxn<<1],to[maxn<<1],tot;
ll top[maxn],id[maxn],dep[maxn],son[maxn],siz[maxn],mp[maxn],fa[maxn],tim;

void add(ll x,ll y){
	to[++tot]=y;
	from[tot]=head[x];
	head[x]=tot;
}
void pushup(ll rt){sum[rt]=(sum[rt<<1]+sum[rt<<1|1])%p;}
void pushdown(ll rt,ll len){
	if(lazy[rt]){
		sum[rt<<1]=(sum[rt<<1]+lazy[rt]*(len-(len>>1)))%p;
		sum[rt<<1|1]=(sum[rt<<1|1]+lazy[rt]*(len>>1))%p;
		lazy[rt<<1]=(lazy[rt<<1]+lazy[rt])%p;
		lazy[rt<<1|1]=(lazy[rt<<1|1]+lazy[rt])%p;
		lazy[rt]=0;
	}
}
void build(ll l,ll r,ll rt){
	if(l==r){
		sum[rt]=mp[l]%p;
		return;
	}
	ll mid=(l+r)>>1;
	build(l,mid,rt<<1);
	build(mid+1,r,rt<<1|1);
	pushup(rt);
}
void update(ll L,ll R,ll C,ll l,ll r,ll rt){
	if(L<=l&&r<=R){
		sum[rt]=(sum[rt]+(r-l+1)*C)%p;
		lazy[rt]=(lazy[rt]+C)%p;
		return;
	}
	ll mid=(l+r)>>1;
	pushdown(rt,r-l+1);
	if(L<=mid) update(L,R,C,l,mid,rt<<1);
	if(R>mid) update(L,R,C,mid+1,r,rt<<1|1);
	pushup(rt);
}
ll query(ll L,ll R,ll l,ll r,ll rt){
	if(L<=l&&r<=R) return sum[rt];
	ll mid=(l+r)>>1,ans=0;
	pushdown(rt,r-l+1);
	if(L<=mid) ans=(ans+query(L,R,l,mid,rt<<1))%p;
	if(R>mid) ans=(ans+query(L,R,mid+1,r,rt<<1|1))%p;
	return ans;
	
}
void dfs1(ll x,ll f,ll deep){
	dep[x]=deep;
	fa[x]=f;
	siz[x]=1;
	ll maxson=-1;
	for(ll i=head[x];i;i=from[i]){
		ll y=to[i];
		if(y==f) continue;
		dfs1(y,x,deep+1);
		siz[x]+=siz[y];
		if(siz[y]>maxson){
			son[x]=y;
			maxson=siz[y];
		}
	}
}
void dfs2(ll x,ll f){
	id[x]=++tim;
	top[x]=f;
	mp[tim]=a[x];
	if(son[x]) dfs2(son[x],f);
	for(ll i=head[x];i;i=from[i]){
		ll y=to[i];
		if(y==fa[x]||y==son[x]) continue;
		dfs2(y,y);
	}
}

int main()
{
	scanf("%lld%lld%lld%lld",&n,&m,&r,&p);
	for(ll i=1;i<=n;i++)
		scanf("%lld",&a[i]);
	for(ll i=1;i<n;i++){
		ll x,y;
		scanf("%lld%lld",&x,&y);
		add(x,y);add(y,x);
	}
	dfs1(r,0,1);
	dfs2(r,r);
	build(1,n,1);
	for(ll i=1;i<=m;i++){
		ll op,x,y,z;
		scanf("%lld",&op);
		if(op==1){
			scanf("%lld%lld%lld",&x,&y,&z);
			while(top[x]!=top[y]){
				if(dep[top[x]]<dep[top[y]]) swap(x,y);
				update(id[top[x]],id[x],z,1,n,1);
				x=fa[top[x]];
			}
			if(dep[x]>dep[y]) swap(x,y);
			update(id[x],id[y],z,1,n,1);
		}
		if(op==2){
			scanf("%lld%lld",&x,&y);
			ll ans=0;
			while(top[x]!=top[y]){
				if(dep[top[x]]<dep[top[y]]) swap(x,y);
				ans=(ans+query(id[top[x]],id[x],1,n,1))%p;
				x=fa[top[x]];
			}
			if(dep[x]>dep[y]) swap(x,y);
			ans=(ans+query(id[x],id[y],1,n,1))%p;
			printf("%lld
",ans%p);
		}
		if(op==3){
			scanf("%lld%lld",&x,&z);
			update(id[x],id[x]+siz[x]-1,z,1,n,1);
		}
		if(op==4){
			scanf("%lld",&x);
			printf("%lld
",query(id[x],id[x]+siz[x]-1,1,n,1)%p);
		}
	}
	return 0;
}

[ZJOI2008]树的统计

其实就是开两个线段树，一个维护最大值，一个维护权值和，一道考验码力的题

(Code Below:)

#include <bits/stdc++.h>
#define ll long long
using namespace std;
const ll maxn=100000+10;
const ll inf=-100000;
ll n,m;
ll sum[maxn<<2],sumv[maxn<<2],lazy[maxn<<2],lazyv[maxn<<2],a[maxn];
ll head[maxn],to[maxn<<1],nxt[maxn<<1],tot;
ll dep[maxn],top[maxn],son[maxn],siz[maxn],fa[maxn],id[maxn],mp[maxn],tim;

void add(ll x,ll y){
	to[++tot]=y;
	nxt[tot]=head[x];
	head[x]=tot;
}
void dfs1(ll x,ll f,ll deep){
	dep[x]=deep;
	fa[x]=f;
	siz[x]=1;
	int maxson=-1;
	for(ll i=head[x];i;i=nxt[i]){
		ll y=to[i];
		if(y==f) continue;
		dfs1(y,x,deep+1);
		siz[x]+=siz[y];
		if(siz[y]>maxson){
			son[x]=y;
			maxson=siz[y];
		}
	}
}
void dfs2(ll x,ll f){
	id[x]=++tim;
	mp[tim]=a[x];
	top[x]=f;
	if(son[x]) dfs2(son[x],f);
	for(ll i=head[x];i;i=nxt[i]){
		ll y=to[i];
		if(y==fa[x]||y==son[x]) continue;
		dfs2(y,y);
	}
}
void pushup(ll rt){
	sum[rt]=sum[rt<<1]+sum[rt<<1|1];
	sumv[rt]=max(sumv[rt<<1],sumv[rt<<1|1]);
}
void pushdown(ll rt,ll len){
	if(lazy[rt]){
		lazy[rt<<1]+=lazy[rt];
		lazy[rt<<1|1]+=lazy[rt];
		sum[rt<<1]+=lazy[rt]*(len-(len>>1));
		sum[rt<<1|1]+=lazy[rt]*(len>>1);
		lazy[rt]=0;
	}
	if(lazyv[rt]){
		lazyv[rt<<1]=lazyv[rt];
		lazyv[rt<<1|1]=lazyv[rt];
		sumv[rt]=lazyv[rt];
		sumv[rt]=lazyv[rt];
		lazyv[rt]=0;
	}
}
void build(ll l,ll r,ll rt){
	if(l==r){
		sum[rt]=sumv[rt]=mp[l];
		return;
	}
	ll mid=(l+r)>>1;
	build(l,mid,rt<<1);
	build(mid+1,r,rt<<1|1);
	pushup(rt);
}
void update(ll L,ll C,ll l,ll r,ll rt){
	if(l==L&&r==L){
		sum[rt]=sumv[rt]=C;
		lazy[rt]+=C;lazyv[rt]=C;
		return;
	}
	ll mid=(l+r)>>1;
	pushdown(rt,r-l+1);
	if(L<=mid) update(L,C,l,mid,rt<<1);
	if(L>mid) update(L,C,mid+1,r,rt<<1|1);
	pushup(rt);
}
ll queryM(ll L,ll R,ll l,ll r,ll rt){
	if(L<=l&&r<=R) return sumv[rt];
	ll mid=(l+r)>>1,ans=inf;
	pushdown(rt,r-l+1);
	if(L<=mid) ans=max(ans,queryM(L,R,l,mid,rt<<1));
	if(R>mid)  ans=max(ans,queryM(L,R,mid+1,r,rt<<1|1));
	return ans;
}
ll queryS(ll L,ll R,ll l,ll r,ll rt){
	if(L<=l&&r<=R) return sum[rt];
	ll mid=(l+r)>>1,ans=0;
	pushdown(rt,r-l+1);
	if(L<=mid) ans+=queryS(L,R,l,mid,rt<<1);
	if(R>mid)  ans+=queryS(L,R,mid+1,r,rt<<1|1);
	return ans;
}
int main()
{
	scanf("%lld",&n);
	for(ll i=1;i<n;i++){
		ll x,y;
		scanf("%lld%lld",&x,&y);
		add(x,y);add(y,x);
	}
	for(ll i=1;i<=n;i++)
		scanf("%lld",&a[i]);
	scanf("%lld",&m);
	dfs1(1,0,1);
	dfs2(1,1);
	build(1,n,1);
	for(ll i=1;i<=m;i++){
		char op[10];ll x,y;
		scanf("%s%lld%lld",&op,&x,&y);
		if(op[0]=='C'){
			update(id[x],y,1,n,1);
		}
		else if(op[0]=='Q'&&op[1]=='M'){
			ll ans=inf;
			while(top[x]!=top[y]){
				if(dep[top[x]]<dep[top[y]]) swap(x,y);
				ans=max(ans,queryM(id[top[x]],id[x],1,n,1));
				x=fa[top[x]];
			}
			if(dep[x]>dep[y]) swap(x,y);
			ans=max(ans,queryM(id[x],id[y],1,n,1));
			printf("%lld
",ans);
		}
		else {
			ll ans=0;
			while(top[x]!=top[y]){
				if(dep[top[x]]<dep[top[y]]) swap(x,y);
				ans+=queryS(id[top[x]],id[x],1,n,1);
				x=fa[top[x]];
			}
			if(dep[x]>dep[y]) swap(x,y);
			ans+=queryS(id[x],id[y],1,n,1);
			printf("%lld
",ans);
		}
	}
	return 0;	
}

[HAOI2015]树上操作

太特殊了吧，直接暴力都行啊

但是上树剖更快(qwq)

(Code Below:)

// luogu-judger-enable-o2
#pragma GCC optimize(2)
#include <bits/stdc++.h>
#define ll long long
using namespace std;
const ll maxn=100000+10;
ll n,m,r,p;
ll sum[maxn<<2],lazy[maxn<<2],a[maxn];
ll head[maxn],from[maxn<<1],to[maxn<<1],tot;
ll top[maxn],id[maxn],dep[maxn],son[maxn],siz[maxn],mp[maxn],fa[maxn],tim;

void add(ll x,ll y){
	to[++tot]=y;
	from[tot]=head[x];
	head[x]=tot;
}
void pushup(ll rt){sum[rt]=(sum[rt<<1]+sum[rt<<1|1])%p;}
void pushdown(ll rt,ll len){
	if(lazy[rt]){
		sum[rt<<1]=(sum[rt<<1]+lazy[rt]*(len-(len>>1)))%p;
		sum[rt<<1|1]=(sum[rt<<1|1]+lazy[rt]*(len>>1))%p;
		lazy[rt<<1]=(lazy[rt<<1]+lazy[rt])%p;
		lazy[rt<<1|1]=(lazy[rt<<1|1]+lazy[rt])%p;
		lazy[rt]=0;
	}
}
void build(ll l,ll r,ll rt){
	if(l==r){
		sum[rt]=mp[l]%p;
		return;
	}
	ll mid=(l+r)>>1;
	build(l,mid,rt<<1);
	build(mid+1,r,rt<<1|1);
	pushup(rt);
}
void update(ll L,ll R,ll C,ll l,ll r,ll rt){
	if(L<=l&&r<=R){
		sum[rt]=(sum[rt]+(r-l+1)*C)%p;
		lazy[rt]=(lazy[rt]+C)%p;
		return;
	}
	ll mid=(l+r)>>1;
	pushdown(rt,r-l+1);
	if(L<=mid) update(L,R,C,l,mid,rt<<1);
	if(R>mid) update(L,R,C,mid+1,r,rt<<1|1);
	pushup(rt);
}
ll query(ll L,ll R,ll l,ll r,ll rt){
	if(L<=l&&r<=R) return sum[rt];
	ll mid=(l+r)>>1,ans=0;
	pushdown(rt,r-l+1);
	if(L<=mid) ans=(ans+query(L,R,l,mid,rt<<1))%p;
	if(R>mid) ans=(ans+query(L,R,mid+1,r,rt<<1|1))%p;
	return ans;
	
}
void dfs1(ll x,ll f,ll deep){
	dep[x]=deep;
	fa[x]=f;
	siz[x]=1;
	ll maxson=-1;
	for(ll i=head[x];i;i=from[i]){
		ll y=to[i];
		if(y==f) continue;
		dfs1(y,x,deep+1);
		siz[x]+=siz[y];
		if(siz[y]>maxson){
			son[x]=y;
			maxson=siz[y];
		}
	}
}
void dfs2(ll x,ll f){
	id[x]=++tim;
	top[x]=f;
	mp[tim]=a[x];
	if(son[x]) dfs2(son[x],f);
	for(ll i=head[x];i;i=from[i]){
		ll y=to[i];
		if(y==fa[x]||y==son[x]) continue;
		dfs2(y,y);
	}
}

int main()
{
	scanf("%lld%lld",&n,&m);r=1;p=1e15;
	for(ll i=1;i<=n;i++)
		scanf("%lld",&a[i]);
	for(ll i=1;i<n;i++){
		ll x,y;
		scanf("%lld%lld",&x,&y);
		add(x,y);add(y,x);
	}
	dfs1(r,0,1);
	dfs2(r,r);
	build(1,n,1);
	for(ll i=1;i<=m;i++){
		ll op,x,y,z;
		scanf("%lld",&op);
		if(op==1){
			scanf("%lld%lld",&x,&z);
			y=x;
			while(top[x]!=top[y]){
				if(dep[top[x]]<dep[top[y]]) swap(x,y);
				update(id[top[x]],id[x],z,1,n,1);
				x=fa[top[x]];
			}
			if(dep[x]>dep[y]) swap(x,y);
			update(id[x],id[y],z,1,n,1);
		}
		if(op==2){
			scanf("%lld%lld",&x,&z);
			update(id[x],id[x]+siz[x]-1,z,1,n,1);
		}
		if(op==3){
			scanf("%lld",&x);
			ll ans=0;y=1;
			while(top[x]!=top[y]){
				if(dep[top[x]]<dep[top[y]]) swap(x,y);
				ans=(ans+query(id[top[x]],id[x],1,n,1))%p;
				x=fa[top[x]];
			}
			if(dep[x]>dep[y]) swap(x,y);
			ans=(ans+query(id[x],id[y],1,n,1))%p;
			printf("%lld
",ans);
		}
	}
	return 0;
}

[NOI2015]软件包管理器

解释操作：

1、安装操作：先查询(i)节点到根节点的和，将(i)节点到根节点的值从(0)改成(1)

2、卸载操作：先查询(i)节点子树的和，再修改(i)节点子树的值从(1)改成(0)

那么直接上裸的树剖就好吧

(Code Below:)

#include <bits/stdc++.h>
#define ll long long
using namespace std;
const ll maxn=100000+10;
ll n,m,r,p;
ll sum[maxn<<2],lazy[maxn<<2],a[maxn];
ll head[maxn],from[maxn<<1],to[maxn<<1],tot;
ll top[maxn],id[maxn],dep[maxn],son[maxn],siz[maxn],mp[maxn],fa[maxn],tim;

void add(ll x,ll y){
    to[++tot]=y;
    from[tot]=head[x];
    head[x]=tot;
}
void pushup(ll rt){sum[rt]=(sum[rt<<1]+sum[rt<<1|1])%p;}
void pushdown(ll rt,ll len){
    if(lazy[rt]!=-1){
        sum[rt<<1]=lazy[rt]*(len-(len>>1))%p;
        sum[rt<<1|1]=lazy[rt]*(len>>1)%p;
        lazy[rt<<1]=lazy[rt]%p;
        lazy[rt<<1|1]=lazy[rt]%p;
        lazy[rt]=-1;
    }
}
void build(ll l,ll r,ll rt){
    if(l==r){
        sum[rt]=mp[l]%p;
        return;
    }
    ll mid=(l+r)>>1;
    build(l,mid,rt<<1);
    build(mid+1,r,rt<<1|1);
    pushup(rt);
}
void update(ll L,ll R,ll C,ll l,ll r,ll rt){
    if(L<=l&&r<=R){
        sum[rt]=(r-l+1)*C%p;
        lazy[rt]=C%p;
        return;
    }
    ll mid=(l+r)>>1;
    pushdown(rt,r-l+1);
    if(L<=mid) update(L,R,C,l,mid,rt<<1);
    if(R>mid) update(L,R,C,mid+1,r,rt<<1|1);
    pushup(rt);
}
ll query(ll L,ll R,ll l,ll r,ll rt){
    if(L<=l&&r<=R) return sum[rt];
    ll mid=(l+r)>>1,ans=0;
    pushdown(rt,r-l+1);
    if(L<=mid) ans=(ans+query(L,R,l,mid,rt<<1))%p;
    if(R>mid) ans=(ans+query(L,R,mid+1,r,rt<<1|1))%p;
    return ans;
    
}
void dfs1(ll x,ll f,ll deep){
    dep[x]=deep;
    fa[x]=f;
    siz[x]=1;
    ll maxson=-1;
    for(ll i=head[x];i;i=from[i]){
        ll y=to[i];
        if(y==f) continue;
        dfs1(y,x,deep+1);
        siz[x]+=siz[y];
        if(siz[y]>maxson){
            son[x]=y;
            maxson=siz[y];
        }
    }
}
void dfs2(ll x,ll f){
    id[x]=++tim;
    top[x]=f;
    mp[tim]=a[x];
    if(son[x]) dfs2(son[x],f);
    for(ll i=head[x];i;i=from[i]){
        ll y=to[i];
        if(y==fa[x]||y==son[x]) continue;
        dfs2(y,y);
    }
}

int main()
{
    memset(lazy,-1,sizeof(lazy));
    scanf("%lld",&n);r=1;p=1e15;
    for(ll i=1;i<n;i++){
        ll x,y;
        scanf("%lld",&x);
        add(x+1,i+1);add(i+1,x+1);
    }
    dfs1(r,0,1);
    dfs2(r,r);
    build(1,n,1);
    scanf("%lld",&m);
    for(ll i=1;i<=m;i++){
        char op[20];ll x,y,z;
        scanf("%s",op);
        if(op[0]=='i'){
            scanf("%lld",&x);y=1;z=1;
            ll Ans=0;x++;ll nx=x;
            while(top[x]!=top[y]){
                if(dep[top[x]]<dep[top[y]]) swap(x,y);
                Ans=(Ans+query(id[top[x]],id[x],1,n,1))%p;
                x=fa[top[x]];
            }
            if(dep[x]>dep[y]) swap(x,y);
            Ans=(Ans+query(id[x],id[y],1,n,1))%p;
            x=nx;y=1;
            while(top[x]!=top[y]){
                if(dep[top[x]]<dep[top[y]]) swap(x,y);
                update(id[top[x]],id[x],z,1,n,1);
                x=fa[top[x]];
            }
            if(dep[x]>dep[y]) swap(x,y);
            update(id[x],id[y],z,1,n,1);
            ll ans=dep[nx];
            printf("%lld
",ans-Ans);
        }
        else {
            scanf("%lld",&x);z=0;x++;
            printf("%lld
",query(id[x],id[x]+siz[x]-1,1,n,1)%p);
            update(id[x],id[x]+siz[x]-1,z,1,n,1);
        }
    }
    return 0;
}

[SHOI2012]魔法树

其实就是软件包管理器的加强版

(Code Below:)

#include <bits/stdc++.h>
#define ll long long
#define res register int
#define mid (l+r>>1)
using namespace std;
const int maxn=100000+10;
int n,q,head[maxn],to[maxn<<1],nxt[maxn<<1],tot;
ll sum[maxn<<2],lazy[maxn<<2];
int top[maxn],son[maxn],dep[maxn],fa[maxn],id[maxn],siz[maxn],tim;

inline int read(){
	res x=0,f=1;char ch=getchar();
	while(ch<'0'||ch>'9'){if(ch=='-')f=-1;ch=getchar();}
	while(ch>='0'&&ch<='9'){x=(x<<3)+(x<<1)+ch-'0';ch=getchar();}
	return (f==1)?x:-x;
}
void print(int x){
	if(x<0){putchar('-');x=-x;}
	if(x>=10) print(x/10);
	putchar(x%10+'0');
}
inline void swap(int &a,int &b){res t=a;a=b;b=t;}
inline void pushup(int rt){sum[rt]=sum[rt<<1]+sum[rt<<1|1];}
inline void pushdown(int rt,int len){
	if(lazy[rt]){
		sum[rt<<1]+=lazy[rt]*(len-(len>>1));
		sum[rt<<1|1]+=lazy[rt]*(len>>1);
		lazy[rt<<1]+=lazy[rt];
		lazy[rt<<1|1]+=lazy[rt];
		lazy[rt]=0;
	}
}
void update(int L,int R,int C,int l,int r,int rt){
	if(L <= l && r <= R){
		sum[rt]+=(r-l+1)*C;
		lazy[rt]+=C;
		return ;
	}
	pushdown(rt,r-l+1);
	if(L <= mid) update(L,R,C,l,mid,rt<<1);
	if(R > mid) update(L,R,C,mid+1,r,rt<<1|1);
	pushup(rt);
}
ll query(int L,int R,int l,int r,int rt)
{
	if(L <= l && r <= R){
		return sum[rt];
	}
	ll ans=0;
	pushdown(rt,r-l+1);
	if(L <= mid) ans+=query(L,R,l,mid,rt<<1);
	if(R > mid) ans+=query(L,R,mid+1,r,rt<<1|1);
	return ans;
}
inline void add(int x,int y){
	to[++tot]=y;
	nxt[tot]=head[x];
	head[x]=tot;
}
void dfs1(int x,int f){
	dep[x]=dep[f]+1;
	siz[x]=1;
	fa[x]=f;
	ll maxson=-1;
	for(res i=head[x];i;i=nxt[i]){
		res y=to[i];
		if(y==f) continue;
		dfs1(y,x);
		siz[x]+=siz[y];
		if(maxson<siz[y]){
			son[x]=y;
			maxson=siz[y];
		}
	}
}
void dfs2(int x,int topf){
	id[x]=++tim;
	top[x]=topf;
	if(son[x]) dfs2(son[x],topf);
	for(res i=head[x];i;i=nxt[i]){
		res y=to[i];
		if(y==fa[x]||y==son[x]) continue;
		dfs2(y,y);
	}
}

int main()
{
	n=read();
	res x,y;
	for(res i=1;i<n;i++){
		x=read(),y=read();
		add(x+1,y+1);add(y+1,x+1);
	}
	dfs1(1,1);
	dfs2(1,1);
	q=read();
	char ch;res u,v,d;
	while(q--){
		ch=getchar();
		while(ch!='A'&&ch!='Q') ch=getchar();
		if(ch=='A'){
			u=read()+1,v=read()+1,d=read();
			for(;top[u]!=top[v];){
				if(dep[top[u]]<dep[top[v]]) swap(u,v);
				update(id[top[u]],id[u],d,1,n,1);
				u=fa[top[u]];
			}
			if(dep[u]>dep[v]) swap(u,v);
			update(id[u],id[v],d,1,n,1); 
		}
		else {
			u=read()+1;
			printf("%lld
",query(id[u],id[u]+siz[u]-1,1,n,1));
		}
	}
	return 0;
}