题解 CF833D Red-Black Cobweb
题目大意
给出一个 \(n\) 个点的树，每条边有边权和颜色 \(0,1\) ，定义一条链合法当且仅当 \(0,1\) 颜色的边数之比小于等于 \(2\) ，求所有合法的链的边权之积的积。
\(n\le 10^5\)，答案对 \(10^9+7\) 取模。
思路
边分治板题，但是因为边界问题爆炸了。。。
首先先容斥一下，即总答案除以不合法答案，然后你发现总答案特别好求，不合法方案可是使用边分治解决。
时间复杂度 \(\Theta(n\log^2 n)\) 。
\(\texttt{Code}\)
#include <bits/stdc++.h>
using namespace std;
#define Int register int
#define mod 1000000007
#define MAXN 800005
template <typename T> inline void read (T &t){t = 0;char c = getchar();int f = 1;while (c < '0' || c > '9'){if (c == '-') f = -f;c = getchar();}while (c >= '0' && c <= '9'){t = (t << 3) + (t << 1) + c - '0';c = getchar();} t *= f;}
template <typename T,typename ... Args> inline void read (T &t,Args&... args){read (t);read (args...);}
template <typename T> inline void write (T x){if (x < 0){x = -x;putchar ('-');}if (x > 9) write (x / 10);putchar (x % 10 + '0');}
int n,ans = 1;
int qkpow (int a,int b){
int res = 1;for (;b;b >>= 1,a = 1ll * a * a % mod) if (b & 1) res = 1ll * res * a % mod;
return res;
}
int inv (int x){return qkpow (x,mod - 2);}
namespace Graph{
#define PII pair<int,int>
int cnt = 1,toop = 1,pres[MAXN],to[MAXN << 1],wei[MAXN << 1],col[MAXN << 1],nxt[MAXN << 1],head[MAXN],siz[MAXN];bool vis[MAXN];
void Add_Edge (int u,int v,int w,int c){
to[++ toop] = v,wei[toop] = w,col[toop] = c,nxt[toop] = head[u],head[u] = toop;
to[++ toop] = u,wei[toop] = w,col[toop] = c,nxt[toop] = head[v],head[v] = toop;
}
struct node{
int R,B,dis;
};
node *f,T1[MAXN],T2[MAXN];
void dfs (int u,int fa,int totr,int totb,int pre){
if (u <= n) f[++ cnt] = node {totr,totb,pre};
for (Int i = head[u];i;i = nxt[i]){
int v = to[i];
if (v == fa || vis[i]) continue;
dfs (v,u,totr + (col[i] == 0),totb + (col[i] == 1),1ll * pre * wei[i] % mod);
}
}
int ed,lim,Siz,lena,lenb;
void findedge (int u,int fa){//找重边
siz[u] = 1;
for (Int i = head[u];i;i = nxt[i]){
int v = to[i];
if (v == fa || vis[i]) continue;
findedge (v,u),siz[u] += siz[v];
int tmp = max (siz[v],Siz - siz[v]);
if (tmp < lim) ed = i,lim = tmp;
}
}
bool cmp1 (node a,node b){return 2 * a.B - a.R < 2 * b.B - b.R;}
bool cmp2 (node a,node b){return 2 * a.R - a.B < 2 * b.R - b.B;}
void Solve (int u,int S){
if (S <= 1) return ;
lim = Siz = S,findedge (u,0),vis[ed] = vis[ed ^ 1] = 1;
cnt = 0,f = T1,dfs (to[ed],0,0,0,1),lena = cnt;
cnt = 0,f = T2,dfs (to[ed ^ 1],0,0,0,1),lenb = cnt;
for (Int i = 1;i <= lenb;++ i) T2[i].R += (col[ed] == 0),T2[i].B += (col[ed] == 1),T2[i].dis = 1ll * T2[i].dis * wei[ed] % mod;
sort (T1 + 1,T1 + lena + 1,cmp1);
pres[0] = 1;for (Int i = 1;i <= lena;++ i) pres[i] = 1ll * pres[i - 1] * T1[i].dis % mod;
for (Int i = 1;i <= lenb;++ i){
int now = T2[i].R - 2 * T2[i].B,l = 1,r = lena,fuckans = 0;
while (l <= r){
int mid = (l + r) >> 1;
if (2 * T1[mid].B - T1[mid].R < now) fuckans = mid,l = mid + 1;
else r = mid - 1;
}
ans = 1ll * ans * qkpow (T2[i].dis,fuckans) % mod * pres[fuckans] % mod;
}
sort (T1 + 1,T1 + lena + 1,cmp2);
pres[0] = 1;for (Int i = 1;i <= lena;++ i) pres[i] = 1ll * pres[i - 1] * T1[i].dis % mod;
for (Int i = 1;i <= lenb;++ i){
int now = T2[i].B - 2 * T2[i].R,l = 1,r = lena,fuckans = 0;
while (l <= r){
int mid = (l + r) >> 1;
if (2 * T1[mid].R - T1[mid].B < now) fuckans = mid,l = mid + 1;
else r = mid - 1;
}
ans = 1ll * ans * qkpow (T2[i].dis,fuckans) % mod * pres[fuckans] % mod;
}
int tx = to[ed],ty = to[ed ^ 1];
if (siz[tx] > siz[ty]) siz[tx] = S - siz[ty];
else siz[ty] = S - siz[tx];
Solve (tx,siz[tx]),Solve (ty,siz[ty]);
}
}
int cnt,all = 1,toop = 1,to[MAXN << 1],wei[MAXN << 1],col[MAXN << 1],nxt[MAXN << 1],head[MAXN],las[MAXN],siz[MAXN];
void Add_Edge (int u,int v,int w,int c){
to[++ toop] = v,wei[toop] = w,col[toop] = c,nxt[toop] = head[u],head[u] = toop;
to[++ toop] = u,wei[toop] = w,col[toop] = c,nxt[toop] = head[v],head[v] = toop;
}
void dfs (int u,int fa){
siz[u] = 1;
for (Int i = head[u];i;i = nxt[i]){
int v = to[i],w = wei[i],c = col[i];
if (v == fa) continue;
if (!las[u]) las[u] = u,Graph::Add_Edge (u,v,w,c);
else ++ cnt,Graph::Add_Edge (las[u],cnt,1,-1),Graph::Add_Edge (las[u] = cnt,v,w,c);
dfs (v,u),siz[u] += siz[v],all = 1ll * all * qkpow (w,1ll * siz[v] * (n - siz[v]) % (mod - 1)) % mod;
}
}
signed main(){
read (n),cnt = n;
for (Int i = 2,u,v,w,c;i <= n;++ i) read (u,v,w,c),Add_Edge (u,v,w,c);
dfs (1,0),Graph::Solve (1,cnt),write (1ll * all * inv (ans) % mod),putchar ('\n');
return 0;
}
