UVA 1728 Toll Management IV

一个n(n≤104)个点m(m≤105)条带权(0≤Ci≤1000)边的无向图，给出原图的一个最小生成树(输入的前n−1条边)

对于第i条边，定义Ai和Bi为在不改变最小生成树形态下增加和减少的最大的值

求 ∑i=1mAi×i+Bi×i×i

特别的，如果Ai或者Bi为∞，在上式中用−1代替

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首先，最小生成树的树边是可以无限制减少的，非树边是可以无限制增加的

然后对于树边增加的上限是所有跨过这条边的非树边的权值最小的那个，如果没有跨过这条边的非树边，这条树边是无限制增加的

非树边减少的下限是他跨过的所有的树边的权值中最大的那个

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以上的几部分都可以树链剖分+线段树维护出来，于是就做完了

需要注意输入的下标，节点的编号和树剖之后点的重编号的对应关系

具体见代码

#include<bits/stdc++.h>using namespace std;const int maxn = 11234,INF = 0x3f3f3f3f;#define LL long long #define root 1,1,n#define Now int o,int l,int r#define lson o<<1,l,m#define rson o<<1|1,m+1,r#define Mid int m = (l + r)>>1int ma[maxn*4],mi[maxn*4],lazy[maxn*4];int arr[maxn];void seter(int o,int v){    lazy[o] = min(lazy[o],v);    mi[o] = min(mi[o],v);}void push_down(Now){    if(l != r){        seter(o<<1,lazy[o]);        seter(o<<1|1,lazy[o]);    }    lazy[o] = INF;}int query(Now,int pos){    if(l == r){        return mi[o];    }    if(lazy[o] != INF) push_down(o,l,r);    Mid;    if(pos <= m)         return query(lson,pos);    else         return query(rson,pos);}int handle(Now,int ul,int ur,int v){    if(ul <= l && r <= ur){        seter(o,v);        return ma[o];    }    if(lazy[o] != INF) push_down(o,l,r);    int ret = 0;    Mid;    if(ul <= m) ret = max(ret,handle(lson,ul,ur,v));     if(m+1<=ur) ret = max(ret,handle(rson,ul,ur,v));     return ret;} void segInit(Now){    lazy[o] = mi[o] = INF;    if(l == r){        ma[o] = arr[l];        return;    }    Mid;    segInit(lson),segInit(rson);    ma[o] = max(ma[o<<1],ma[o<<1|1]);}vector<int> edge[maxn];int fa[maxn],son[maxn],siz[maxn];int deep[maxn],top[maxn],tid[maxn];int _cnt;void dffs(int st,int Fa,int Deep){    siz[st] = 1,deep[st] = Deep;    fa[st] = Fa,son[st] = -1;    for(auto x : edge[st]){        if(x == Fa) continue;        dffs(x,st,Deep+1);        siz[st] += siz[x];        if(son[st] == -1 || siz[son[st]] < siz[x])            son[st] = x;    }}void dfss(int st,int Top){    top[st] = Top,tid[st] = _cnt++;    if(son[st] != -1) dfss(son[st],Top);    for(auto x : edge[st]){        if(son[st] != x &&  fa[st] != x)             dfss(x,x);    }}void splite(){    _cnt = 1;    dffs(1,-1,1);    dfss(1,1);}int n;int handle(int x,int y,int v){    int ix,iy;    ix = iy = 0;    int tx = top[x],ty = top[y];    while(tx != ty){        if(deep[tx] < deep[ty])            swap(tx,ty),swap(x,y),swap(ix,iy);        ix = max(handle(root,tid[tx],tid[x],v),ix);        x = fa[tx],tx = top[x];    }    if(deep[x] < deep[y])        swap(x,y);    return max(max(ix,iy),handle(root,tid[y]+1,tid[x],v));}int x[maxn],y[maxn],v[maxn];int main(){    int T,m;    scanf("%d",&T);    int icase = 1;    while(T-- && ~scanf("%d %d",&n,&m)){        LL ans = 0;        for(int i=1;i<=n;i++) edge[i].clear();        for(int i=1;i<n;i++){            scanf("%d %d %d",&x[i],&y[i],&v[i]);            edge[x[i]].push_back(y[i]);            edge[y[i]].push_back(x[i]);        }        splite();        for(int i=1;i<n;i++){            if(deep[x[i]] < deep[y[i]]) swap(x[i],y[i]);            arr[tid[x[i]]] = v[i];        }        segInit(root);        int l,r,w;        for(int i=n;i<=m;i++){            scanf("%d %d %d",&l,&r,&w);            int ops =  w - handle(l,r,w);            ans -= 1ll * i;            ans += 1ll * i * i * ops;        }        for(int i=1;i<n;i++){            int temp = query(root,tid[x[i]]);            if(temp == INF) temp = -1;            else temp = temp - v[i];            ans += 1ll * i * temp;            ans -= 1ll * i * i;        }        printf("Case %d: %lld\n",icase++,ans);     }    return 0;}