Luogu4069 SDOI2016 游戏 树链剖分、李超线段树

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每一次加的是一个一次函数，一些呈一次函数的线段求最小值，显然用到李超线段树。

然后把维护序列的李超线段树强行上树，就直接套上树剖就可以了。

至于李超树如何区间查询，因为一次函数线段的最小值一定会取在两端，所以对于每一个点维护它和它的子树中所有线段的最低点，递归的时候如果当前区间被询问区间包含就直接返回维护的最低点，对于经过的线段再考虑它是否能够更新答案。

复杂度\(O(nlog^3n)\)可能自带小常数就过了吧……

#include<bits/stdc++.h>
//this code is written by Itst
using namespace std;

int read(){
    int a = 0; bool f = 0; char c = getchar();
    while(!isdigit(c)){f = c == '-'; c = getchar();}
    while(isdigit(c)){a = a * 10 + c - 48; c = getchar();}
    return f ? -a : a;
}

#define int long long
#define ld long double
struct line{
    int k , b;
    line(int _k = 0 , int _b = 123456789123456789ll) : k(_k) , b(_b){}
};
ld sect(line a , line b){return 1.0 * (a.b - b.b) / (b.k - a.k);}

const int _ = 1e5 + 7;

struct Edge{
    int end , upEd , w;
}Ed[_ << 1];
int head[_] , fa[_] , dep[_] , len[_] , dfn[_] , ind[_] , top[_] , sz[_] , son[_];
int N , M , cntEd , ts;

void addEd(int a , int b , int c){
    Ed[++cntEd] = (Edge){b , head[a] , c};
    head[a] = cntEd;
}

void dfs1(int x , int p){
    sz[x] = 1; fa[x] = p; dep[x] = dep[p] + 1;
    for(int i = head[x] ; i ; i = Ed[i].upEd)
        if(Ed[i].end != p){
            len[Ed[i].end] = len[x] + Ed[i].w;
            dfs1(Ed[i].end , x); sz[x] += sz[Ed[i].end];
            if(sz[Ed[i].end] > sz[son[x]]) son[x] = Ed[i].end;
        }
}

void dfs2(int x , int t){
    top[x] = t; ind[dfn[x] = ++ts] = x;
    if(!son[x]) return;
    dfs2(son[x] , t);
    for(int i = head[x] ; i ; i = Ed[i].upEd)
        if(Ed[i].end != fa[x] && Ed[i].end != son[x])
            dfs2(Ed[i].end , Ed[i].end);
}

namespace segTree{
    line arr[_ << 2]; int mn[_ << 2] , pl[_ << 2] , pr[_ << 2];

#define mid ((l + r) >> 1)
#define lch (x << 1)
#define rch (x << 1 | 1)
    
    void init(int x , int l , int r){
        pl[x] = l; pr[x] = r; mn[x] = 123456789123456789ll;
        if(l != r){
            init(lch , l , mid); init(rch , mid + 1, r);
        }
    }

    int calc(line A , int x){return A.k * x + A.b;}
    
    void up(int x){
        if(pl[x] != pr[x])
            mn[x] = min(min(mn[lch] , mn[rch]) , min(calc(arr[x] , len[ind[pl[x]]]) , calc(arr[x] , len[ind[pr[x]]])));
        else
            mn[x] = min(calc(arr[x] , len[ind[pl[x]]]) , calc(arr[x] , len[ind[pr[x]]]));
    }
    
    void insert(int x , int l , int r , int L , int R , line now){
        if(l >= L && r <= R){
            if(arr[x].k == now.k)
                return (void)(arr[x] = arr[x].b < now.b ? arr[x] : now) , up(x);
            ld pos = sect(arr[x] , now);
            if(arr[x].k > now.k) swap(arr[x] , now);
            if(pos > len[ind[r]]) arr[x] = now;
            else
                if(!(pos < len[ind[l]]))
                    if(pos >= len[ind[mid]]){
                        swap(arr[x] , now);
                        if(l != r) insert(rch , mid + 1 , r , L , R , now);
                    }
                    else
                        if(l != r) insert(lch , l , mid , L , R , now);
            return up(x);
        }
        if(mid >= L) insert(lch , l , mid , L , R , now);
        if(mid < R) insert(rch , mid + 1 , r , L , R , now);
        up(x);
    }

    int query(int x , int l , int r , int L , int R){
        if(l >= L && r <= R) return mn[x];
        int ans = 123456789123456789ll;
        if(mid >= L) ans = query(lch , l , mid , L , R);
        if(mid < R) ans = min(ans , query(rch , mid + 1 , r , L , R));
        return min(ans , min(calc(arr[x] , len[ind[max(l , L)]]) , calc(arr[x] , len[ind[min(r , R)]])));
    }
}

int LCA(int x , int y){
    int tx = top[x] , ty = top[y];
    while(tx != ty){
        if(dep[tx] < dep[ty]){swap(tx , ty); swap(x , y);}
        x = fa[tx]; tx = top[x];
    }
    return dfn[x] > dfn[y] ? y : x;
}

void modify(int x , int y , int a , int b){
    int tx = top[x] , ty = top[y] , flgx = -1 , flgy = 1 , lx = 0 , ly = len[x] + len[y] - 2 * len[LCA(x , y)];
    while(tx != ty){
        if(dep[tx] < dep[ty]){swap(tx , ty); swap(flgx , flgy); swap(x , y); swap(lx , ly);}
        int _k = flgx * a , _b = lx * a + b - len[x] * _k;
        segTree::insert(1 , 1 , N , dfn[tx] , dfn[x] , line(_k , _b));
        lx += -flgx * (len[x] - len[fa[tx]]);
        x = fa[tx]; tx = top[x];
    }
    if(dep[x] < dep[y]){swap(flgx , flgy); swap(x , y); swap(lx , ly);}
    int _k = flgx * a , _b = lx * a + b - len[x] * _k;
    segTree::insert(1 , 1 , N , dfn[y] , dfn[x] , line(_k , _b));
}

int work(int x , int y){
    int tx = top[x] , ty = top[y] , ans = 123456789123456789ll;
    while(tx != ty){
        if(dep[tx] < dep[ty]){swap(tx , ty); swap(x , y);}
        ans = min(ans , segTree::query(1 , 1 , N , dfn[tx] , dfn[x]));
        x = fa[tx]; tx = top[x];
    }
    return min(ans , segTree::query(1 , 1 , N , min(dfn[x] , dfn[y]) , max(dfn[x] , dfn[y])));
}

signed main(){
#ifndef ONLINE_JUDGE
    freopen("in","r",stdin);
    freopen("out","w",stdout);
#endif
    N = read(); M = read();
    for(int i = 1 ; i < N ; ++i){
        int a = read() , b = read() , c = read();
        addEd(a , b , c); addEd(b , a , c);
    }
    dfs1(1 , 0); dfs2(1 , 1); segTree::init(1 , 1 , N);
    for(int i = 1 ; i <= M ; ++i)
        if(read() == 1){
            int s = read() , t = read() , a = read() , b = read();
            modify(s , t , a , b);
        }
        else{
            int s = read() , t = read();
            printf("%lld\n" , work(s , t));
        }
    return 0;
}