【题解】Luogu P4069 [SDOI2016]游戏

原题传送门

看到这种题，想都不用想，先写一个树链剖分

然后发现修改操作增加的是等差数列，这使我们想到了李超线段树

先进性树剖，然后用李超线段树维护区间最小，这样就做完了（写码很容易出错）

复杂度为\(O(n\log^3n)\)，少见的复杂度啊qaq，但常数不用怕

#include <bits/stdc++.h>
#define ll long long
#define N 100005
#define M 100005
#define inf 123456789123456789LL
#define getchar nc
using namespace std;
inline char nc(){
    static char buf[100000],*p1=buf,*p2=buf;
    return p1==p2&&(p2=(p1=buf)+fread(buf,1,100000,stdin),p1==p2)?EOF:*p1++;
}
inline ll read()
{
    register ll x=0,f=1;register 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 x*f;
}
inline void write(register ll x)
{
    if(!x)putchar('0');if(x<0)x=-x,putchar('-');
    static int sta[25];register int tot=0;
    while(x)sta[tot++]=x%10,x/=10;
    while(tot)putchar(sta[--tot]+48);
}
inline ll Min(register ll a,register ll b)
{
    return a<b?a:b;
}
struct edge{
    int to,next;
    ll v;
}e[M<<1];
int head[N],cnt=0;
inline void add(register int u,register int v,register ll w)
{
    e[++cnt]=(edge){v,head[u],w};
    head[u]=cnt;
}
inline ll f(register ll x,register ll k,register ll b)
{
    return x*k+b;
}
struct node{
    ll a,b,minn;
}tr[N<<2];
int n,m;
int size[N],fa[N],son[N],dep[N];
int pl[N],top[N],tot=0;
ll dis[N],pre[N];
inline void build(register int x,register int l,register int r)
{
    tr[x].a=0,tr[x].b=tr[x].minn=inf;
    if(l==r)
        return;
    int mid=l+r>>1;
    build(x<<1,l,mid);
    build(x<<1|1,mid+1,r);
}
inline void dfs1(register int x)
{
    size[x]=1;
    for(register int i=head[x];i;i=e[i].next)
        if(e[i].to!=fa[x])
        {
            fa[e[i].to]=x;
            dep[e[i].to]=dep[x]+1;
            dis[e[i].to]=dis[x]+e[i].v;
            dfs1(e[i].to);
            if(size[e[i].to]>size[son[x]])
                son[x]=e[i].to;
            size[x]+=size[e[i].to];
        }
}
inline void dfs2(register int x,register int t)
{
    pl[x]=++tot,pre[tot]=dis[x],top[x]=t;
    if(son[x])
        dfs2(son[x],t);
    for(register int i=head[x];i;i=e[i].next)
        if(e[i].to!=son[x]&&e[i].to!=fa[x])
            dfs2(e[i].to,e[i].to);
}
inline int getlca(register int x,register int y)
{
    while(top[x]!=top[y])
    {
        if(dep[top[x]]<dep[top[y]])
            x^=y^=x^=y;
        x=fa[top[x]];
    }
    if(dep[x]>dep[y])
        x^=y^=x^=y;
    return x;
}
inline void Change(register int x,register int l,register int r,register ll a,register ll b)
{
    int mid=l+r>>1,fl,fr,fm;
    fl=(f(pre[l],tr[x].a,tr[x].b)>f(pre[l],a,b));
    fr=(f(pre[r],tr[x].a,tr[x].b)>f(pre[r],a,b));
    fm=(f(pre[mid],tr[x].a,tr[x].b)>f(pre[mid],a,b));
    if(fl&&fr&&fm)
    {
        tr[x].a=a,tr[x].b=b,tr[x].minn=Min(tr[x].minn,Min(f(pre[l],a,b),f(pre[r],a,b)));
        return;
    }
    if(!(fl|fr|fm))
        return;
    if(fm)
    {
        if(fr)
            Change(x<<1,l,mid,tr[x].a,tr[x].b);
        else
            Change(x<<1|1,mid+1,r,tr[x].a,tr[x].b);
        tr[x].a=a,tr[x].b=b,tr[x].minn=Min(tr[x].minn,Min(f(pre[l],a,b),f(pre[r],a,b)));
    }
    else
    {
        if(!fr)
            Change(x<<1,l,mid,a,b);
        else
            Change(x<<1|1,mid+1,r,a,b);
    }
    tr[x].minn=Min(tr[x].minn,Min(tr[x<<1].minn,tr[x<<1|1].minn));
}
inline void change(register int x,register int l,register int r,register int L,register int R,register ll a,register ll b)
{
    if(L<=l&&r<=R)
    {
        Change(x,l,r,a,b);
        return;
    }
    int mid=l+r>>1;
    if(L<=mid)
        change(x<<1,l,mid,L,R,a,b);
    if(R>mid)
        change(x<<1|1,mid+1,r,L,R,a,b);
    tr[x].minn=Min(tr[x].minn,Min(tr[x<<1].minn,tr[x<<1|1].minn));
}
inline void cal1(register int s,register int t,register ll a,register ll b)
{
    int lca=getlca(s,t);
    int x=s,y=t;
    while(top[x]!=top[lca])
    {
        change(1,1,n,pl[top[x]],pl[x],-a,a*dis[s]+b);
        x=fa[top[x]];
    }
    change(1,1,n,pl[lca],pl[x],-a,a*dis[s]+b);
    while(top[y]!=top[lca])
    {
        change(1,1,n,pl[top[y]],pl[y],a,dis[s]*a-dis[lca]*2*a+b);
        y=fa[top[y]];
    }
    if(y!=lca)
        change(1,1,n,pl[lca]+1,pl[y],a,dis[s]*a-dis[lca]*2*a+b);
}
inline ll query(register int x,register int l,register int r,register int L,register int R)
{
    ll res=Min(f(pre[L],tr[x].a,tr[x].b),f(pre[R],tr[x].a,tr[x].b));
    if(L==l&&r==R)
        return Min(res,tr[x].minn);
    int mid=l+r>>1;
    if(R<=mid)
        return Min(res,query(x<<1,l,mid,L,R));
    else if(L>mid)
        return Min(res,query(x<<1|1,mid+1,r,L,R));
    else
        return Min(res,Min(query(x<<1,l,mid,L,mid),query(x<<1|1,mid+1,r,mid+1,R)));
}
inline ll cal2(register int s,register int t)
{
    int x=s,y=t;
    ll res=inf;
    while(top[x]!=top[y])
    {
        if(dep[top[x]]<dep[top[y]])
            x^=y^=x^=y;
        res=Min(res,query(1,1,n,pl[top[x]],pl[x]));
        x=fa[top[x]];
    }
    if(dep[x]>dep[y])
        x^=y^=x^=y;
    res=Min(res,query(1,1,n,pl[x],pl[y]));
    return res;
}
int main()
{
    n=read(),m=read();
    for(register int i=1;i<=n-1;++i)
    {
        int u=read(),v=read();
        ll w=read();
        add(u,v,w),add(v,u,w);
    }
    dfs1(1);
    dfs2(1,1);
    build(1,1,n);
    while(m--)
    {
        int opt=read();
        if(opt==1)
        {
            int s=read(),t=read();
            ll a=read(),b=read();
            cal1(s,t,a,b);
        }
        else
        {
            int s=read(),t=read();
            write(cal2(s,t)),puts("");
        }
    }
    return 0;
}