计蒜客 青云的机房组网方案

题面:

有一棵点数105的树，每个节点有一个权值，权值范围是[1,105]的，问所有两个权值互质的节点之间距离的和

题解

---

关于虚树，记下一些key point以后要是忘了可以回来看看

虚树是保留原树的所选定的一些节点，保留这些选定节点两两的LCA，按照这些保留点在原树中的关系所建的一个新树。

可以保证如果选定节点是k个，那么虚树的节点不会超过2×k。考虑每次拿走最深的两个相邻节点并且加入这两个节点的LCA，这个结论应该是显然的。

建立虚树的方法是将选定节点按照dfs序排序之后，维护一个深度单调的栈，栈中保存的是可能和下一个节点产生父子关系的点。这些点满足在同一条路径上。

每次加入一个点的时候，将这个点和栈顶的点(如果栈空的话将这个点直接入栈就可以了)的lca计算出来，找到lca在栈中维护的路径的链的位置去更新栈。具体见代码。

---

得到虚树之后每个边的贡献就是这个边两遍的点的个数的乘积，累加就好了。

#include<bits/stdc++.h>using namespace std;#define LL long longconst int maxn = 212345,maxm = 212345,max_log = 18;const int ROOT = 1;vector<pair<int,LL> > vedge[maxn];vector<int> edge[maxn];LL siz[maxn];int is[maxn];void dffs(int st,int fa){    siz[st] = is[st];    for(auto x : vedge[st]){        if(x.first != fa){            dffs(x.first,st);            siz[st] += siz[x.first];        }    }}LL dfss(int st,int fa,int n){    LL ret = 0;    for(auto it : vedge[st]){        int x = it.first; LL v = it.second;        if(x != fa){            ret += dfss(x,st,n);            ret += v * siz[x] * (n-siz[x]);        }    }    return ret;}LL dpfs(int n){    dffs(0,-1);    return dfss(0,-1,siz[0]);}int dfn[maxn],_cnt;int deep[maxn],fa[maxn][max_log];void dfs(int st,int Fa,int Deep=1){    dfn[st] = _cnt++;    for(int i=1;i<max_log;i++)        fa[st][i] = -1;    fa[st][0] = Fa,deep[st] = Deep;    for(auto x : edge[st]){        if(x == Fa) continue;        dfs(x,st,Deep+1);    }}void init(int n){    _cnt = 1;    memset(fa,-1,sizeof(fa));    dfs(ROOT,-1);    for (int j = 1;j < max_log;j++){        for (int i = 1;i <= n;i++){            if (fa[i][j-1] != -1){                fa[i][j] = fa[fa[i][j-1]][j-1];            }        }    }}int Lca(int x,int y){    LL ix,iy;    ix = iy = 0;    if (deep[x] < deep[y]) swap(x,y);    for (int i = max_log-1;i >= 0;i--){        if (deep[fa[x][i]] >= deep[y]){            x = fa[x][i];        }    }    if (x == y) return x;     for (int i = max_log-1;i >= 0;i--){        if (fa[x][i] != fa[y][i]){            x = fa[x][i];            y = fa[y][i];        }    }    return fa[x][0];}int val[maxn],cid[maxn],sid[maxn];int cnt;int tcid(int v){    if(sid[v] != -1) return sid[v];    sid[v] = cnt, cid[cnt] = v;    return cnt++;}void Link(int st,int ed,int v = -1){    if(v == -1) v = abs(deep[st] - deep[ed]);    if(v==0) return;    st = tcid(st),ed = tcid(ed);    vedge[st].push_back(make_pair(ed,v));    vedge[ed].push_back(make_pair(st,v));}struct Stack{    int sta[maxn],_cnt;    int last;    void init(){ _cnt = 0, last = -1; }    bool empty(){ return _cnt==0; }    int top(){ return sta[_cnt-1]; }    void push(int x){        if(last != -1)             Link(last,x);        sta[_cnt++] = x;        last = -1;    }    void pop(){        if(last != -1)            Link(this->top(),last);        last = this->top();        _cnt--;    } }S;vector<int> hav[maxn];LL bvtree(int k){    sort(hav[k].begin(),hav[k].end(),[](int x,int y){return dfn[x] < dfn[y];});    cnt = 0;    S.init();    for(auto x : hav[k]){        if(S.empty()){             S.push(x);            continue;        }        int lca = Lca(S.top(),x);        while(S.empty()==false && deep[S.top()] >= deep[lca])            S.pop();        if(S.empty() || deep[S.top()] < deep[lca])            S.push(lca);        if(lca != x)            S.push(x);    }    while(S.empty()==false) S.pop();    for(int i=0;i<cnt;i++)        is[i] = val[cid[i]] % k == 0;    LL ret = dpfs(cnt);    for(int i=0;i<cnt;i++){        vedge[i].clear();        sid[cid[i]] = -1;        cid[i] = -1;    }    return ret;}vector<int> fac[maxn];int mu[maxn];int main(){    int n;    for(int i=1;i<maxn;i++){        for(int j=i;j<maxn;j+=i)            fac[j].push_back(i);    }    memset(mu,0,sizeof(mu));    mu[1] = 1;    for(int i=1;i<maxn;i++){        for(int j=2*i;j<maxn;j+=i)            mu[j] -= mu[i];    }    while(~scanf("%d",&n)){        for(int i=0;i<=n;i++)            edge[i].clear(),hav[i].clear(),vedge[i].clear();        int v;        for(int i=1;i<=n;i++){            scanf("%d",&v);            val[i] = v;            for(auto x : fac[v]){                if(mu[x]) hav[x].push_back(i);            }        }        int l,r;        for(int i=1;i<n;i++){            scanf("%d %d",&l,&r);            edge[l].push_back(r);            edge[r].push_back(l);        }        init(n);        memset(sid,-1,sizeof(sid));        LL ans = 0;        for(int i=1;i<maxn;i++){            if(hav[i].empty() == false)                ans += mu[i] * bvtree(i);        }        printf("%lld\n",ans);    }    return 0;}

附上一个比赛时的弱鸡代码

#include<bits/stdc++.h>using namespace std;const int maxn = 112345,max_log = 18;vector<int> edge[maxn];vector<pair<int,int> >vedge[maxn];int dis[maxn];void pdfs(int st,int fa){    for(auto it : vedge[st]){        if(it.first == fa) continue;        dis[it.first] = dis[st] + it.second;        pdfs(it.first,st);    }}pair<int,int> getPair(int n,int st){    dis[st] = 0;    pdfs(st,-1);    st = max_element(dis,dis+n) - dis;    dis[st] = 0;    pdfs(st,-1);    return make_pair(st,max_element(dis,dis+n) - dis);}//Need checkconst int ROOT = 1;int deep[maxn],dfn[maxn],fa[maxn][max_log];int _cnt;void dfs(int st,int Fa,int Deep=1){    for(int i=1;i<max_log;i++)        fa[st][i] = -1;    dfn[st] = _cnt++;    fa[st][0] = Fa,deep[st] = Deep;    for(auto x : edge[st]){        if(x == Fa) continue;        dfs(x,st,Deep+1);    }}void init(int n){    memset(fa,-1,sizeof(fa));    _cnt = 1;    dfs(ROOT,-1);    for (int j = 1;j < max_log;j++){        for (int i = 1;i <= n;i++){            if (fa[i][j-1] != -1){                fa[i][j] = fa[fa[i][j-1]][j-1];            }        }    }}int Lca(int x,int y){    if (deep[x] < deep[y]) swap(x,y);    for (int i = max_log-1;i >= 0;i--){        if (deep[fa[x][i]] >= deep[y]){            x = fa[x][i];        }    }    if (x == y) return x;    for (int i = max_log-1;i >= 0;i--){        if (fa[x][i] != fa[y][i]){            x = fa[x][i];            y = fa[y][i];        }    }    return fa[x][0];}int cid[maxn],sid[maxn]; //assert sid is -1 initint vnt;int tcid(int v){    if(sid[v] != -1) return sid[v];    sid[v] = vnt, cid[vnt] = v;    return vnt++;}void Link(int st,int ed){    if(ed == -1 || st == ed) return;    int v = abs(deep[st]-deep[ed]);    st = tcid(st),ed = tcid(ed);    vedge[st].push_back(make_pair(ed,v));    vedge[ed].push_back(make_pair(st,v));}stack<int> S;vector<int> inp;pair<int,int> bvtree(){    if(inp.size() ==1) return make_pair(inp[0],inp[0]);    sort(inp.begin(),inp.end(),[](int x,int y){return dfn[x] < dfn[y];});    vnt = 0;    int last = -1;    while(S.empty()==false) S.pop();    for(auto x : inp){        if(S.empty()){            Link(x,last) ,last = -1 ,S.push(x) ;            continue;        }        int lca = Lca(S.top(),x);        while(!S.empty() && deep[S.top()] >= deep[lca])            Link(S.top(),last), last = S.top(), S.pop();        if(    S.empty() || deep[S.top()] <  deep[lca])            Link(lca    ,last), last = -1  ,S.push(lca);        if(lca != x) S.push(x) ;    }    while(!S.empty())        Link(S.top(),last), last = S.top(), S.pop();    auto ret = getPair(vnt,0);    ret.first = cid[ret.first] , ret.second = cid[ret.second];    for(int i=0;i<vnt;i++)        vedge[i].clear(), sid[cid[i]] = -1, cid[i] = -1;    return ret;}map<string,vector<int> >store;map<string,pair<int,int> >lis;int dist(int x,int y){    return deep[x] + deep[y] - deep[Lca(x,y)] * 2 + 1;}char name[10];int query(){    scanf("%s",name);    if(lis.find(name) == lis.end()){        scanf("%s",name);        return -1;    }    auto a = lis[name];    scanf("%s",name);    if(lis.find(name) == lis.end()){        return -1;    }    auto b = lis[name];    return max( max(dist(a.first,b.first) , dist(a.first,b.second)),                max(dist(a.second,b.first), dist(a.second,b.second)));}int main(){    int n,m;    while(~scanf("%d %d",&n,&m)){        store.clear(),lis.clear();        for(int i=1;i<=n;i++){            scanf("%s",name);            store[name].push_back(i);            edge[i].clear();        }        int l,r;        for(int i=1;i<n;i++){            scanf("%d %d",&l,&r);            edge[l].push_back(r);            edge[r].push_back(l);        }        init(n);        memset(sid,-1,sizeof(sid));        for(auto it : store){            inp = it.second;             lis[it.first] = bvtree();        }        while(m--){            printf("%d\n",query());        }    }    return 0;}