HYSBZ - 4016 最短路径树问题 点分治 + 最短路径最小字典序

题目传送门

题解：首先对于给定的图，需要找到那些从1好点出发然后到x号点的最短路， 如果有多条最短路就要找到字典序最小的路，这样扣完这些边之后就会有一棵树。然后再就是很普通的点分治了。

对于扣边这个问题， 我们先跑一遍最短路，这样就可以得到1号点到其他的点的距离。

然后在跑一遍dfs， 我们在跑dfs找路的时候， 可以通过 d[u] + ct[i] == d[v] 来判断是不是最短路是否可以走这条边， 然后我们再从所有可能边中的最小编号出发，这样我们就能保证字典序最小了。

代码：

 #include<bits/stdc++.h>
 using namespace std;
 #define Fopen freopen("_in.txt","r",stdin); freopen("_out.txt","w",stdout);
 #define LL long long
 #define ULL unsigned LL
 #define fi first
 #define se second
 #define pb push_back
 #define lson l,m,rt<<1
 #define rson m+1,r,rt<<1|1
 #define lch(x) tr[x].son[0]
 #define rch(x) tr[x].son[1]
 #define max3(a,b,c) max(a,max(b,c))
 #define min3(a,b,c) min(a,min(b,c))
 typedef pair<int,int> pll;
 const int inf = 0x3f3f3f3f;
 const LL INF = 0x3f3f3f3f3f3f3f3f;
 const LL mod =  (int)1e9+;
 const int N = 3e4 + ;
 vector<pll> vc[N];
 int d[N];
 void dij(){
     memset(d, inf, sizeof(d));
     priority_queue<pll, vector<pll>, greater<pll> > q;
     d[] = ;
     q.push(pll(,));
     int x, dd, v, ct;
     while(!q.empty()){
         x = q.top().se, dd = q.top().fi;
         q.pop();
         if(dd != d[x]) continue;
         for(int i = ; i < vc[x].size(); ++i){
             v = vc[x][i].fi , ct = vc[x][i].se;
             if(d[v] > d[x] + ct){
                 d[v] = d[x] + ct;
                 q.push(pll(d[v], v));
             }
         }
     }
     return;
 }
 int vis[N];
 int head[N], to[N<<], val[N<<], nt[N<<], tot = ;
 void add(int u, int v, int ct){
     to[tot] = v;
     val[tot] = ct;
     nt[tot] = head[u];
     head[u] = tot++;
     return ;
 }
 void dfs(int u){
     vis[u] = ;
     int v, dd;
     for(int i = ; i < vc[u].size(); i++){
         v = vc[u][i].fi, dd = vc[u][i].se;
         if(vis[v] || d[v] != d[u] + dd) continue;
         add(u, v, dd);
         add(v, u, dd);
         dfs(v);
     }
     return ;
 }
 int sz[N];
 int rt, minval;
 int n, m, k;
 void get_rt(int o, int u, int num){
     sz[u] = ;
     int v;
     int maxval = ;
     for(int i = head[u]; ~i; i = nt[i]){
         v = to[i];
         if(v == o || vis[v]) continue;
         get_rt(u, v, num);
         sz[u] += sz[v];
         maxval = max(maxval, sz[v]);
     }
     if(o) maxval = max(maxval, num - sz[u]);
     if(maxval < minval){
         minval = maxval;
         rt = u;
     }
 }
 int fans = , fcnt = ;
 int cnt[N], dis[N];
 void Update(int vval, int num){
     if(fans == vval) fcnt += num;
     else if(fans < vval) fans = vval, fcnt = num;
     return ;
 }
 void Dfs(int o, int u, int w, int num){
     sz[u] = ;
     if(num == k-)
         Update(w, );
     if(k >= num && dis[k-num]){
         Update(w+dis[k-num], cnt[k-num]);
     }
     for(int i = head[u]; ~i; i = nt[i]){
         int v = to[i];
         if(v == o || vis[v]) continue;
         Dfs(u, v,w+val[i], num+);
         sz[u] += sz[v];
     }
     return ;
 }
 void Change(int o, int u, int w, int num, int op){
     if(num >= k) return ;
     if(op == ) {
         if(dis[num+] < w) dis[num+] = w, cnt[num+] = ;
         else if(dis[num+] == w) cnt[num+]++;
     }
     else dis[num+] = cnt[num+] = ;
     for(int i = head[u]; ~i; i = nt[i]){
         int v = to[i];
         if(v == o || vis[v]) continue;
         Change(u, v, w+val[i], num+, op);
     }
     return ;
 }
 void solve(int x, int num){
     if(num <= ) return ;
     minval = inf;
     get_rt(, x, num);
     vis[rt] = ;
     int v;
     for(int i = head[rt]; ~i; i = nt[i]){
         v = to[i];
         if(vis[v]) continue;
         Dfs(, v, val[i], );
         Change(, v, val[i], , );
     }
     for(int i = head[rt]; ~i; i = nt[i]){
         v = to[i];
         if(vis[v]) continue;
         Change(, v, val[i], , );
     }
     for(int i = head[rt]; ~i; i = nt[i]){
         v = to[i];
         if(vis[v]) continue;
         solve(v, sz[v]);
     }
     return ;
 }
 int main(){
     int u, v, w;
     memset(head, -, sizeof(head));
     scanf("%d%d%d", &n, &m, &k);
     for(int i = ; i <= m; i++){
         scanf("%d%d%d", &u, &v, &w);
         vc[u].pb(pll(v,w));
         vc[v].pb(pll(u,w));
     }
     dij();
     dfs();
     memset(vis, , sizeof(vis));
     solve(, n);
     printf("%d %d", fans, fcnt);
     return ;
 }