非嚴格次小生成樹

很簡單，先做最小生成樹
然後枚舉沒加入的邊加入，替換掉這個環內最大的邊
最後取\(min\)

嚴格次小生成樹

還是一樣的
可以考慮維護一個嚴格次大值
最大值和枚舉的邊相同就替換次大值的邊
否則替換最大值的邊
最後取\(min\)

裸題

Luogu
隨你用各種姿勢\(AC\)

\(LCT\)常數大，但是好寫，開\(O2\)可以過

# include <bits/stdc++.h>
# define RG register
# define IL inline
# define Fill(a, b) memset(a, b, sizeof(a))
using namespace
 
 std;
typedef long long ll;
const int _(4e5 + 5);

IL ll Input(){
    RG ll x = 0, z = 1; RG char c = getchar();
    for(; c < '0' || c > '9'; c = getchar()) z = c == '-' ? -1 : 1;
    for(; c >= '0' && c <= '9'; c = getchar()) x = (x << 1
 
) + (x << 3) + (c ^ 48);
    return x * z;
}

int ch[2][_], fa[_], rev[_], S[_], mx[_], _mx[_], val[_], anc[_];
int n, m, ff[_], tt[_], w[_], id[_], vis[_];
ll ans = 1e18, mst;

# define ls ch[0][x]
# define rs ch[1][x]

IL int Son(RG int x){  return ch[1][fa[x]] == x;  }

IL int Isroot(RG int x){  return
 
 ch[0][fa[x]] != x && ch[1][fa[x]] != x;  }

IL void Update(RG int x){
    mx[x] = val[x], _mx[x] = -1;
    if(mx[ls] > mx[x]) _mx[x] = mx[x], mx[x] = mx[ls];
    else _mx[x] = max(mx[ls], _mx[x]);
    if(mx[rs] > mx[x]) _mx[x] = mx[x], mx[x] = mx[rs];
    else _mx[x] = max(mx[rs], _mx[x]);
    _mx[x] = max(_mx[x], max(_mx[ls], _mx[rs]));
}

IL void Reverse(RG int x){
    if(!x) return;
    rev[x] ^= 1, swap(ls, rs);
}

IL void Pushdown(RG int x){
    if(!rev[x]) return;
    rev[x] = 0, Reverse(ls), Reverse(rs);
}

IL void Rotate(RG int x){
    RG int y = fa[x], z = fa[y], c = Son(x);
    if(!Isroot(y)) ch[Son(y)][z] = x; fa[x] = z;
    ch[c][y] = ch[!c][x]; fa[ch[c][y]] = y;
    ch[!c][x] = y; fa[y] = x;
    Update(y);
}

IL void Splay(RG int x){
    RG int top = 0; S[++top] = x;
    for(RG int y = x; !Isroot(y); y = fa[y]) S[++top] = fa[y];
    while(top) Pushdown(S[top--]);
    for(RG int y = fa[x]; !Isroot(x); Rotate(x), y = fa[x])
        if(!Isroot(y)) Son(x) ^ Son(y) ? Rotate(x) : Rotate(y);
    Update(x);
}

IL void Access(RG int x){
    for(RG int y = 0; x; y = x, x = fa[x]) Splay(x), rs = y, Update(x);
}

IL void Makeroot(RG int x){
    Access(x), Splay(x), Reverse(x);
}

IL int Find(RG int x){
    return anc[x] == x ? x : anc[x] = Find(anc[x]);
}

IL void Split(RG int x, RG int y){
    Makeroot(x), Access(y), Splay(y);
}

IL void Link(RG int x, RG int y){
    Makeroot(x), fa[x] = y;
}

IL int Cmp(RG int x, RG int y){  return w[x] < w[y];  }

int main(RG int argc, RG char* argv[]){
    n = Input(), m = Input();
    for(RG int i = 1; i <= n; ++i) val[i] = -1, anc[i] = i;
    for(RG int i = 1; i <= m; ++i)
        ff[i] = Input(), tt[i] = Input(), val[n + i] = w[i] = Input(), id[i] = i;
    sort(id + 1, id + m + 1, Cmp);
    for(RG int i = 1, j = 1; i <= m && j < n; ++i){
        RG int fx = Find(ff[id[i]]), fy = Find(tt[id[i]]);
        if(fx == fy) continue;
        anc[fx] = fy, mst += w[id[i]], ++j, vis[id[i]] = 1;
        Link(ff[id[i]], id[i] + n), Link(tt[id[i]], id[i] + n);
    }
    for(RG int i = 1; i <= m; ++i){
        if(vis[i]) continue;
        Split(ff[i], tt[i]);
        if(mx[tt[i]] != w[i]) ans = min(ans, mst - mx[tt[i]] + w[i]);
        else if(_mx[tt[i]] != w[i]) ans = min(ans, mst - _mx[tt[i]] + w[i]);
    }
    printf("%lld\n", ans);
    return 0;
}

嚴格次小生成樹(Bzoj1977：[Beijing2010組隊]次小生成樹)