LCT维护生成树

把边从小到大排序，然后一条一条加边，如果成环就把环上最小的删了，我们得到的第一个生成树就是最小生成树。

然后之后每一条边都比前面的生成树的最大边大，我们用这条边的权值减去生成树里最小的，更新答案即可。

因为要维护的是最小值，用排序之后的性质，下表小的值更小来pushup

#include 
    
     #define INF 0x3f3f3f3f#define full(a, b) memset(a, b, sizeof a)#define FAST_IO ios::sync_with_stdio(false), cin.tie(0), cout.tie(0)using namespace std;typedef long long LL;inline int lowbit(int x){ return x & (-x); }inline int read(){    int ret = 0, w = 0; char ch = 0;    while(!isdigit(ch)){        w |= ch == '-', ch = getchar();    }    while(isdigit(ch)){        ret = (ret << 3) + (ret << 1) + (ch ^ 48);        ch = getchar();    }    return w ? -ret : ret;}inline int lcm(int a, int b){ return a / __gcd(a, b) * b; }template 
     
      inline A fpow(A x, B p, C lyd){    A ans = 1;    for(; p; p >>= 1, x = 1LL * x * x % lyd)if(p & 1)ans = 1LL * x * ans % lyd;    return ans;}const int N = 500005;int n, m, cnt, tot, cur, ans, val[N], fa[N], ch[N][2], rev[N], id[N], st[N];bool vis[N];struct Edge {    int u, v, w;    bool operator < (const Edge &rhs) const {        return w < rhs.w;    }} e[N];int newNode(int x){    ++ tot;    val[tot] = x, id[tot] = tot;    rev[tot] = ch[tot][0] = ch[tot][1] = fa[tot] = 0;    return tot;}bool isRoot(int x){    return ch[fa[x]][0] != x && ch[fa[x]][1] != x;}void reverse(int x){    rev[x] ^= 1, swap(ch[x][0], ch[x][1]);}void push_up(int x){    id[x] = x;    int l = ch[x][0], r = ch[x][1];    if(id[l] > n && (id[x] <= n || id[x] > id[l])) id[x] = id[l];    if(id[r] > n && (id[x] <= n || id[x] > id[r])) id[x] = id[r];}void push_down(int x){    if(rev[x]){        int l = ch[x][0], r = ch[x][1];        if(l) reverse(l);        if(r) reverse(r);        rev[x] ^= 1;    }}void rotate(int x){    int y = fa[x], z = fa[y], p = (ch[y][1] == x) ^ 1;    ch[y][p^1] = ch[x][p], fa[ch[x][p]] = y;    if(!isRoot(y)) ch[z][ch[z][1] == y] = x;    fa[x] = z, fa[y] = x, ch[x][p] = y;    push_up(y), push_up(x);}void splay(int x){    int pos = 0; st[++pos] = x;    for(int i = x; !isRoot(i); i = fa[i]) st[++pos] = fa[i];    while(pos) push_down(st[pos--]);    while(!isRoot(x)){        int y = fa[x], z = fa[y];        if(!isRoot(y)){            (ch[y][0] == x) ^ (ch[z][0] == y) ? rotate(x) : rotate(y);        }        rotate(x);    }    push_up(x);}void access(int x){    for(int p = 0; x; p = x, x = fa[x]){        splay(x), ch[x][1] = p, push_up(x);    }}void makeRoot(int x){    access(x), splay(x), reverse(x);}void link(int x, int y){    makeRoot(x);    fa[x] = y;}int findRoot(int x){    access(x), splay(x);    while(ch[x][0]) push_down(x), x = ch[x][0];    splay(x);    return x;}void split(int x, int y){    makeRoot(x), access(y), splay(y);}bool isConnect(int x, int y){    makeRoot(x);    return findRoot(y) == x;}int main(){    n = read(), m = read();    for(int i = 1; i <= m; i ++){        e[i].u = read(), e[i].v = read(), e[i].w = read();    }    ans = INF, cur = 1;    sort(e + 1, e + m + 1);    for(int i = 1; i <= n; i ++) newNode(0);    for(int i = 1; i <= m; i ++) newNode(e[i].w);    for(int i = 1; i <= m; i ++){        int u = e[i].u, v = e[i].v, w = e[i].w;        if(u == v) continue;        if(!isConnect(u, v)){            link(u, i + n), link(i + n, v);            vis[i] = true, cnt ++;        }        else{            split(u, v);            int tmp = id[v];            splay(tmp);            fa[ch[tmp][0]] = fa[ch[tmp][1]] = 0;            ch[tmp][0] = ch[tmp][1] = 0;            vis[tmp - n] = false, vis[i] = true;            link(u, i + n), link(i + n, v);            while(!vis[cur]) cur ++;        }        if(cnt == n - 1) ans = min(ans, w - e[cur].w);    }    cout << ans << endl;    return 0;}