LCT小结

这里没有水表
这几天连树剖和LCT都不会写了
所以放下总结两篇

LCT是一个通过链剖分和Splay的使用,在无根树上维护路径信息的有效算法
其单次操作时间复杂度为O(logn),常数巨大

Splay

需要先学Splay(不会的我可不管),于是跳过Splay部分;

主要结构和思路

讲的就交给书予大佬了
这里是一些无关紧要的压行

il void access(int x){for(RG int y=0;x;y=x,x=fa[x])splay(x),s[1][x]=y,update(x);}
il int findroot(int x){access(x);splay(x);while(s[0][x])x=s[0][x];return x;}
il void makeroot(int x){access(x);splay(x);reverse(x);}
il void link(int x,int y){makeroot(x);fa[x]=y;}
il void split(int x,int y){makeroot(x);access(y);splay(y);}
il void cut(int x,int y){split(x,y);fa[x]=s[0][y]=0;}

注意一个Splay的reverse(区间翻转)操作
节点上有懒标记的时候这个节点是不是已经被修改过了?
是吧所以我们需要一个reverse的函数再pushdown

il void reverse(int i){if(!i)return;swap(s[0][i],s[1][i]);rv[i]^=1;}
il void pushdown(int i){
if(rv[i]){reverse(s[0][i]);reverse(s[1][i]);rv[i]=0;}
}

[luoguP3690] 【模板】Link Cut Tree （动态树）

注意：
split请务必保证两个点联通
cut请务必保证两个点直接相连
link请务必保证两个点不联通
不然出现玄学错误谁都救不了你了
--摘自书予大佬

#include<iostream>
#include<cstdlib>
#include<cstdio>
#include<cmath>
#include<algorithm>
#include<cstring>
#define RG register
using namespace std;
const int N=300010;

int Orz,n,m,x,y;
int s[2][N],fa[N],sum[N],val[N],rv[N];
int cal[N],top;

inline void update(int i){
    sum[i]=sum[s[0][i]]^sum[s[1][i]]^val[i];}
inline void pushdown(int i){if(!rv[i])return;rv[i]^=1;rv[s[0][i]]^=1;rv[s[1][i]]^=1;swap(s[1][i],s[0][i]);}
inline bool isroot(int i){
    return s[0][fa[i]]!=i&&s[1][fa[i]]!=i;}

inline bool isr(int i){return s[1][fa[i]]==i;}
inline void rot(int i){
    RG int j=fa[i],k=fa[j];
    RG bool b=isr(i);
    fa[i]=k;
    if(!isroot(j))s[isr(j)][k]=i;
    
    s[b][j]=s[!b][i];
    if(s[!b][i])fa[s[!b][i]]=j;
    
    s[!b][i]=j;fa[j]=i;
    update(j);
}

inline void splay(int i){
    cal[++top]=i;
    for(RG int x=i;!isroot(x);x=fa[x])cal[++top]=fa[x];//
    while(top)pushdown(cal[top--]);//
    for(RG int j=fa[i];!isroot(i);rot(i),j=fa[i])
        if(!isroot(j))isr(i)^isr(j)?rot(i):rot(j);
    update(i);
}

inline void access(int x){for(RG int y=0;x;y=x,x=fa[x])splay(x),s[1][x]=y,update(x);}
inline void makeroot(int x){access(x);splay(x);rv[x]^=1;}
inline int findroot(int x){access(x);splay(x);while(s[0][x])x=s[0][x];return x;}
inline void split(int x,int y){makeroot(x);access(y);splay(y);}//
inline void cut(int x,int y){split(x,y);fa[x]=s[0][y]=0;}//
inline void link(int x,int y){makeroot(x);fa[x]=y;}

inline void print(int i){
    if(s[0][i])print(s[0][i]);
    printf("%d ",val[i]);
    if(s[1][i])print(s[1][i]);
}

int main()
{
    scanf("%d%d",&n,&m);
    for(RG int i=1;i<=n;i++)scanf("%d",&val[i]);
    while(m--){
        scanf("%d%d%d",&Orz,&x,&y);
        if(!Orz){split(x,y);printf("%d
",sum[y]);}
        if(Orz==1)if(findroot(x)!=findroot(y))link(x,y);//
        if(Orz==2)if(findroot(x)==findroot(y))cut(x,y);
        if(Orz==3){access(x);splay(x);val[x]=y;update(x);}
    }
    return 0;
}

[luoguP1501] [国家集训队]Tree II

这里是带修改的LCT

#include<bits/stdc++.h>
#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
#define isr(i) (s[1][fa[i]]==i)
#define isroot(i) (s[0][fa[i]]!=i&&s[1][fa[i]]!=i)
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=51061;
const int N=100010;
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);
}

ll n,q;char c;
ll fa[N],s[2][N],sz[N],val[N],rv[N],sum[N],add[N],times[N],cal[N],top;
il void update(int i){
    sum[i]=(sum[s[0][i]]+sum[s[1][i]]+val[i])%mod;
    sz[i]=sz[s[0][i]]+sz[s[1][i]]+1;
}
il void reverse(int i){
    if(!i)return;swap(s[0][i],s[1][i]);rv[i]^=1;
}
il void modify(int i,int t,int a){
    sum[i]=(sum[i]*t+a*sz[i])%mod;
    val[i]=(val[i]*t+a)%mod;
    times[i]=times[i]*t%mod;
    add[i]=(add[i]*t+a)%mod;
}
il void pushdown(int i){
    if(rv[i]){reverse(s[0][i]);reverse(s[1][i]);rv[i]=0;}
    if(add[i]||times[i]!=1){
        if(s[0][i])modify(s[0][i],times[i],add[i]);
        if(s[1][i])modify(s[1][i],times[i],add[i]);
        add[i]=0;times[i]=1;
    }
}

il void rot(int i){
    RG int j=fa[i],k=fa[j];
    RG bool b=isr(i);
    if(!isroot(j))s[isr(j)][k]=i;fa[i]=k;
    if(s[!b][i])fa[s[!b][i]]=j;s[b][j]=s[!b][i];
    s[!b][i]=j;fa[j]=i;
    update(j);
}
il void splay(int i){
    cal[++top]=i;
    for(RG int x=i;!isroot(x);x=fa[x])cal[++top]=fa[x];
    while(top)pushdown(cal[top--]);
    for(RG int j=fa[i];!isroot(i);rot(i),j=fa[i])
        if(!isroot(j))(isr(i)^isr(j))?rot(i):rot(j);
    update(i);
}

il void access(int x){for(RG int y=0;x;y=x,x=fa[x])splay(x),s[1][x]=y,update(x);}
il int findroot(int x){access(x);splay(x);while(s[0][x])x=s[0][x];return x;}
il void makeroot(int x){access(x);splay(x);reverse(x);}
il void link(int x,int y){makeroot(x);fa[x]=y;}
il void split(int x,int y){makeroot(x);access(y);splay(y);}
il void cut(int x,int y){split(x,y);fa[x]=s[0][y]=0;}

il void modify_add(){
    RG int u=read(),v=read(),c=read()%mod;
    split(u,v);modify(v,1,c);
}
il void update_line(){
    RG int u1=read(),v1=read(),u2=read(),v2=read();
    cut(u1,v1);link(u2,v2);
}
il void modify_times(){
    RG int u=read(),v=read(),c=read()%mod;
    split(u,v);modify(v,c,0);
}
il void query(){
    RG int u=read(),v=read();split(u,v);
    printf("%lld
",sum[v]);
}

int main()
{
    n=read();q=read();
    for(RG int i=1;i<=n;i++)modify(i,1,1),sz[i]=1;
    for(RG int i=1,u,v;i<n;i++){
        u=read();v=read();link(u,v);
    }
    for(RG int i=1;i<=q;i++){
        c=0;while(c!='+'&&c!='-'&&c!='*'&&c!='/')c=getchar();
        if(c=='+')modify_add();
        if(c=='-')update_line();
        if(c=='*')modify_times();
        if(c=='/')query();
    }
    return 0;
}

[luoguP4172] [WC2006]水管局长

其实离线+倒序处理询问就变成魔法森林(动态维护最小生成树)了

#include<bits/stdc++.h>
#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
#define isr(i) (s[1][fa[i]]==i)
#define isroot(i) (s[0][fa[i]]!=i&&s[1][fa[i]]!=i)
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;
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,q,tot=1;
int fa[N<<1],s[2][N<<1],val[N<<1],rv[N<<1],mx[N<<1],pmx[N<<1];
il void update(int i){
    mx[i]=val[i];pmx[i]=i;
    if(s[0][i]&&mx[s[0][i]]>mx[i]){
        mx[i]=mx[s[0][i]];pmx[i]=pmx[s[0][i]];
    }
    if(s[1][i]&&mx[s[1][i]]>mx[i]){
        mx[i]=mx[s[1][i]];pmx[i]=pmx[s[1][i]];
    }
}
il void reverse(int i){swap(s[0][i],s[1][i]);rv[i]^=1;}
il void pushdown(int i){
    if(!rv[i])return;
    reverse(s[0][i]);reverse(s[1][i]);
    rv[i]=0;
}

il void rot(int i){
    RG int j=fa[i],k=fa[j];
    RG bool b=isr(i);
    if(!isroot(j))s[isr(j)][k]=i;fa[i]=k;
    if(s[!b][i])fa[s[!b][i]]=j;s[b][j]=s[!b][i];
    s[!b][i]=j;fa[j]=i;
    update(j);
}
int cal[N<<1],top;
il void splay(int i){
    cal[++top]=i;
    for(RG int x=i;!isroot(x);x=fa[x])cal[++top]=fa[x];
    while(top)pushdown(cal[top--]);
    for(RG int j=fa[i];!isroot(i);rot(i),j=fa[i])
        if(!isroot(j))(isr(i)^isr(j))?rot(i):rot(j);
    update(i);
}

int f[N<<1];
il void access(int x){for(RG int y=0;x;y=x,x=fa[x])splay(x),s[1][x]=y,update(x);}
il void makeroot(int x){access(x);splay(x);reverse(x);}
il int find(int x){return f[x]?f[x]=find(f[x]):x;}
il void split(int x,int y){makeroot(x);access(y);splay(y);}
il void link(int x,int y){
    makeroot(x);fa[x]=y;
}
il void cut(int x,int y){
    split(x,y);fa[x]=s[0][y]=0;
}

int l[N],r[N],cnt;
il int query(int u,int v){
    split(u,v);
    return pmx[v];
}
il void add(int u,int v,int w){
    if(find(u)!=find(v)){
        cnt++;val[n+cnt]=w;l[cnt]=u;r[cnt]=v;
        link(u,n+cnt);link(n+cnt,v);
        f[find(u)]=find(v);
    }
    else{
        RG int i=query(u,v);
        if(val[i]>w){
            cut(i,l[i-n]);cut(i,r[i-n]);
            l[i-n]=u;r[i-n]=v;val[i]=w;
            link(u,i);link(i,v);
        }
    }
}

struct edge{
    int u,v,t,id;
    bool operator <(const edge b)const{
        return (u==b.u)?v<b.v:u<b.u;
    }
}E[N];
bool cmp(edge a,edge b){return a.id>b.id;}
struct qry{int u,v,ans;}Q[N];
int main()
{
    n=read();m=read();q=read();
    for(RG int i=1,x,y,w;i<=m;i++){
        x=read();y=read();w=read();if(x>y)swap(x,y);
        E[i]=(edge){x,y,w,0};
    }
    
    sort(E+1,E+m+1);
    for(RG int i=1,k,x,y;i<=q;i++){
        k=read();x=read();y=read();if(x>y)swap(x,y);
        if(k==1)Q[++tot]=(qry){x,y,0};
        if(k==2){
            E[lower_bound(E+1,E+m+1,(edge){x,y,0,0})-E].id=tot;
        }
    }
    for(RG int i=1;i<=m;i++)if(!E[i].id)E[i].id=tot;
    
    sort(E+1,E+m+1,cmp);
    for(RG int i=tot,l=1;i>=2;i--){
        while(E[l].id==i&&l<=m){add(E[l].u,E[l].v,E[l].t);l++;}
        Q[i].ans=val[query(Q[i].u,Q[i].v)];
    }
    for(RG int i=2;i<=tot;i++)printf("%d
",Q[i].ans);
    return 0;
}