次小生成树 lca
洛谷那题solve函数不更新s的值也是可以过，数据有点弱
题意
求次小生成树。
思路
1、求最小生成树。
2、bfs预处理f和g函数，其中f（x，i）为x的第2k，g（x，i，0），g（x，i，1）为x到x+2i的最大值和，次大值。更新过程画画即可。
3、对于任意非树边，求出该两点在树上的最大值和次大值，然后更新最小的差值Min即可。
傻逼了在更新Min的时候//在数组的区间上和树的区间不一样，因为t进行变换k不能到到达进行了k次能到达t的点。
#include <iostream> #include <cmath> #include <cstdio> #include <cstring> #include <string> #include <map> #include <iomanip> #include <algorithm> #include <queue> #include <stack> #include <set> #include <vector> #define bug cout<<"--------------"<<endl #define sp ' ' using namespace std; typedef long long ll; const int maxn = 3e5+10; int b[maxn],d[maxn]; ll ans = 0,n,m,k; int tot = 0; int head[maxn],ver[maxn],edge[maxn],nextt[maxn]; void add(int x,int y,int z) { ver[++tot] = y,edge[tot] = z,nextt[tot] = head[x] , head[x] = tot; } struct node { int x, y, z; }t[maxn]; bool cmp(node a,node b) { return a.z < b.z; } int Find(int x) { return b[x]==x?x:b[x]=Find(b[x]); } int notree[maxn]; void MINtree() { for(int i = 1;i <=m;++i){ int fax = Find(t[i].x); int fay = Find(t[i].y); if(fax == fay){ notree[i] = 1; continue; } add(t[i].x,t[i].y,t[i].z); add(t[i].y,t[i].x,t[i].z); b[fax] = fay; ans += t[i].z; } } int f[maxn][30]; int g[maxn][30][2]; void bfs() { memset(g,-1,sizeof(g)); queue<int>que; que.push(1); d[1] = 1; while(que.size()){ int x = que.front();que.pop(); for(int i = head[x];i;i = nextt[i]){ int y = ver[i]; if(d[y]) continue; d[y] = d[x] + 1; f[y][0] = x; g[y][0][0] = edge[i]; que.push(y); for(int j = 1;j < k;++j){ f[y][j] = f[f[y][j-1]][j-1]; g[y][j][0] = max(g[y][j-1][0],g[f[y][j-1]][j-1][0]); if(g[y][j-1][0] == g[f[y][j-1]][j-1][0]){ g[y][j][1] = max(g[y][j-1][1],g[f[y][j-1]][j-1][1]); } else if(g[y][j-1][0] > g[f[y][j-1]][j-1][0]) { g[y][j][1] = max(g[y][j-1][1],g[f[y][j-1]][j-1][0]); } else { g[y][j][1] = max(g[y][j-1][0],g[f[y][j-1]][j-1][1]); } } } } } int Lca(int x, int y){ if(d[x] > d[y]) swap(x,y); for(int i = k;i >= 0;i--){ if(d[f[y][i]] >= d[x]) y = f[y][i]; } if(x == y) return x; for(int i = k;i >= 0;i--){ if(f[x][i] != f[y][i]){ x = f[x][i]; y = f[y][i]; } } return f[x][0]; } int Min = 1e9; void Solve(int s,int t,int w_) { int m1 = 0, m2 = 0, k = d[s] - d[t]; for(int i = 0; i <= 17; i ++) { if(k >> i & 1) { m2 = max(m2, g[s][i][1]); if(g[s][i][0] > m1) {m2 = max(m2, m1); m1 = g[s][i][0];} s = f[s][i]; } } if(m1 == w_) Min = min(Min, w_ - m2); else Min = min(Min, w_ - m1); } void Secmintree() { for(int i = 1;i <=m;++i){ if(notree[i] == 0) continue; int l = t[i].x,r = t[i].y,val = t[i].z; int L = Lca(l,r); Solve(l,L,val); Solve(r,L,val); } } int main() { //freopen("input.txt", "r", stdin); scanf("%d%d",&n,&m); for(int i = 1;i <= m;++i){ scanf("%d%d%d",&t[i].x,&t[i].y,&t[i].z); } for(int i = 1;i <= n;++i) { b[i] = i; } sort(t+1,t+1+m,cmp); MINtree(); k = log(n)/log(2) + 1; bfs(); Secmintree(); ans += Min; printf("%lld",ans ); }
#include <iostream> #include <cmath> #include <cstdio> #include <cstring> #include <string> #include <map> #include <iomanip> #include <algorithm> #include <queue> #include <stack> #include <set> #include <vector> #define bug cout<<"--------------"<<endl #define sp ' ' using namespace std; typedef long long ll; const int maxn = 3e5+10; int b[maxn],d[maxn]; ll ans = 0,n,m,k; int tot = 0; int head[maxn],ver[maxn],edge[maxn],nextt[maxn]; void add(int x,int y,int z) { ver[++tot] = y,edge[tot] = z,nextt[tot] = head[x] , head[x] = tot; } struct node { int x, y, z; }t[maxn]; bool cmp(node a,node b) { return a.z < b.z; } int Find(int x) { return b[x]==x?x:b[x]=Find(b[x]); } int notree[maxn]; void MINtree() { for(int i = 1;i <=m;++i){ int fax = Find(t[i].x); int fay = Find(t[i].y); if(fax == fay){ notree[i] = 1; continue; } add(t[i].x,t[i].y,t[i].z); add(t[i].y,t[i].x,t[i].z); b[fax] = fay; ans += t[i].z; } } int f[maxn][30]; int g[maxn][30][2]; void bfs() { memset(g,-1,sizeof(g)); queue<int>que; que.push(1); d[1] = 1; while(que.size()){ int x = que.front();que.pop(); for(int i = head[x];i;i = nextt[i]){ int y = ver[i]; if(d[y]) continue; d[y] = d[x] + 1; f[y][0] = x; g[y][0][0] = edge[i]; que.push(y); for(int j = 1;j < k;++j){ f[y][j] = f[f[y][j-1]][j-1]; g[y][j][0] = max(g[y][j-1][0],g[f[y][j-1]][j-1][0]); if(g[y][j-1][0] == g[f[y][j-1]][j-1][0]){ g[y][j][1] = max(g[y][j-1][1],g[f[y][j-1]][j-1][1]); } else if(g[y][j-1][0] > g[f[y][j-1]][j-1][0]) { g[y][j][1] = max(g[y][j-1][1],g[f[y][j-1]][j-1][0]); } else { g[y][j][1] = max(g[y][j-1][0],g[f[y][j-1]][j-1][1]); } } } } } int Lca(int x, int y){ if(d[x] > d[y]) swap(x,y); for(int i = k;i >= 0;i--){ if(d[f[y][i]] >= d[x]) y = f[y][i]; } if(x == y) return x; for(int i = k;i >= 0;i--){ if(f[x][i] != f[y][i]){ x = f[x][i]; y = f[y][i]; } } return f[x][0]; } int Min = 1e9; void Solve(int s,int t,int w_) { int m1 = 0, m2 = 0, k = d[s] - d[t]; for(int i = 0; i <= 17; i ++) { if(k >> i & 1) { m2 = max(m2, g[s][i][1]); if(g[s][i][0] > m1) {m2 = max(m2, m1); m1 = g[s][i][0];} s = f[s][i]; } } if(m1 == w_) Min = min(Min, w_ - m2); else Min = min(Min, w_ - m1); } //傻逼了，在数组的区间上和树的区间不一样，因为t进行变换k不能到到达进行了k次能到达t的点 /*void Solve(int s,int t,int val) { int k = log(d[s] - d[t] + 1) / log(2); int m1 = 0,m2 = 0; m1 = max(g[s][k][0],g[t+(1<<k)+1][k][0]); if(g[s][k][0]==g[t+(1<<k)+1][k][0]){ m2 = max(g[s][k][1],g[t+(1<<k)+1][k][1]); } else if(g[s][k][0]>g[t+(1<<k)+1][k][0]){ m2 = max(g[s][k][1],g[t+(1<<k)+1][k][0]); } else { m2 = max(g[s][k][0],g[t+(1<<k)+1][k][1]) } if(m1 == val) Min = min(Min, val - m2); else Min = min(Min, val - m1); }*/ void Secmintree() { for(int i = 1;i <=m;++i){ if(notree[i] == 0) continue; int l = t[i].x,r = t[i].y,val = t[i].z; int L = Lca(l,r); Solve(l,L,val); Solve(r,L,val); } } int main() { //freopen("input.txt", "r", stdin); scanf("%d%d",&n,&m); for(int i = 1;i <= m;++i){ scanf("%d%d%d",&t[i].x,&t[i].y,&t[i].z); } for(int i = 1;i <= n;++i) { b[i] = i; } sort(t+1,t+1+m,cmp); MINtree(); k = log(n)/log(2) + 1; bfs(); Secmintree(); ans += Min; printf("%lld",ans ); }
