洛谷 - P4180 严格次小生成树（LCT维护边权）

传送门：洛谷 - P4180

思路分析

先在LCT上建立初始的最小生成树，那么考虑如何维护边权，把一条边看成一个点
然后遍历不在生成树的边，加入这条边一定形成一个环，那么删去这个环的最大值，如果最大值和边权相等，那么删去次大值

AC代码

#include <bits/stdc++.h>
#define fi first
#define se second
#define ll long long
#define pb push_back
#define mp make_pair
#define fun function
#define sz(x) (x).size()
#define lowbit(x) (x)&(-x)
#define all(x) (x).begin(),(x).end()
#define mem(a, b) memset(a,b,sizeof(a))

namespace FastIO {
#define BUF_SIZE 100000
#define OUT_SIZE 100000
    bool IOerror = 0;

    inline char nc() {
        static char buf[BUF_SIZE], *p1 = buf + BUF_SIZE, *pend = buf + BUF_SIZE;
        if (p1 == pend) {
            p1 = buf;
            pend = buf + fread(buf, 1, BUF_SIZE, stdin);
            if (pend == p1) {
                IOerror = 1;
                return -1;
            }
        }
        return *p1++;
    }

    inline bool blank(char ch) {
        return ch == ' ' || ch == '\n' || ch == '\r' || ch == '\t';
    }

    template<class T>
    inline bool read(T &x) {
        bool sign = 0;
        char ch = nc();
        x = 0;
        for (; blank(ch); ch = nc());
        if (IOerror)return false;
        if (ch == '-')sign = 1, ch = nc();
        for (; ch >= '0' && ch <= '9'; ch = nc())x = x * 10 + ch - '0';
        if (sign)x = -x;
        return true;
    }

    template<class T, class... U>
    bool read(T &h, U &... t) {
        return read(h) && read(t...);
    }

#undef OUT_SIZE
#undef BUF_SIZE
};
using namespace std;
using namespace FastIO;

mt19937 rng(chrono::steady_clock::now().time_since_epoch().count());

const int INF = 0x3f3f3f3f;
const int N = 1e6 + 10;
#define int long long
namespace LCT {
#define fa(x) (tree[x].fa)
#define ls(x) (tree[x].ch[0])
#define rs(x) (tree[x].ch[1])
#define ident(x, f) (rs(f)==x)
#define connect(x, f, s) tree[f].ch[s]=x,tree[x].fa=f
#define reverse(x) swap(ls(x),rs(x)),tree[x].tag^=1
#define notroot(x) (ls(fa(x)) == x || rs(fa(x)) == x)

    struct node {
        int fa, ch[2], val, mx1, mx2;
        bool tag;
    } tree[N];

    inline void pp(int x) {
        tree[x].mx1 = max(tree[x].val,max(tree[ls(x)].mx1,tree[rs(x)].mx1));
        tree[x].mx2 = max(tree[ls(x)].mx2,tree[rs(x)].mx2);
        if(tree[x].val != tree[x].mx1) tree[x].mx2 = max(tree[x].mx2,tree[x].val);
        if(tree[ls(x)].mx1 != tree[x].mx1) tree[x].mx2 = max(tree[x].mx2,tree[ls(x)].mx1);
        if(tree[rs(x)].mx1 != tree[x].mx1) tree[x].mx2 = max(tree[x].mx2,tree[rs(x)].mx1);
    }

    inline void pushdown(int x) {
        if (tree[x].tag) {
            if (ls(x)) reverse(ls(x));
            if (rs(x)) reverse(rs(x));
        }
        tree[x].tag = 0;
    }

    inline void pushall(int x) {
        if (notroot(x)) pushall(fa(x));
        pushdown(x);
    }

    inline void rotate(int x) {
        int f = fa(x), ff = fa(f), fs = ident(x, f), ffs = ident(f, ff);
        connect(tree[x].ch[fs ^ 1], f, fs);
        fa(x) = ff;
        if (notroot(f)) tree[ff].ch[ffs] = x;
        connect(f, x, fs ^ 1);
        pp(f), pp(x);
    }

    inline void splaying(int x) {
        pushall(x);
        while (notroot(x)) {
            int f = fa(x), ff = fa(f);
            if (notroot(f)) ident(f, ff) ^ ident(x, f) ? rotate(x) : rotate(f);
            rotate(x);
        }
    }

    inline void access(int x) {
        for (int y = 0; x; x = fa(x)) {
            splaying(x);
            rs(x) = y;
            pp(x);
            y = x;
        }
    }

    inline void mkroot(int x) {
        access(x);
        splaying(x);
        reverse(x);
    }

    inline int findroot(int x) {
        access(x);
        splaying(x);
        while (ls(x)) {
            pushdown(x);
            x = ls(x);
        }
        splaying(x);
        return x;
    }

    inline void link(int x, int y) {
        mkroot(x);
        if (findroot(y) == x) return;
        fa(x) = y;
    }

    inline void cut(int x, int y) {
        mkroot(x);
        if (findroot(y) != x || fa(y) != x || ls(y)) return;
        fa(y) = rs(x) = 0;
        pp(x);
    }

    inline void split(int x, int y) {
        mkroot(x);
        access(y);
        splaying(y);
    }

#undef fa
#undef ls
#undef rs
#undef ident
#undef reverse
#undef connect
#undef notroot
};

struct node {
    int u, v, w;
} e[N];

int vis[N], f[N];

int find(int x) {
    return f[x] == x ? x : f[x] = find(f[x]);
}


signed main() {

#ifdef xiaofan
    freopen("1.in","r",stdin);
    freopen("1.out","w",stdout);
#endif

    int n, m;
    read(n, m);
    for (int i = 1; i <= m; i++) read(e[i].u, e[i].v, e[i].w);
    for (int i = 1; i <= n; i++) f[i] = i;
    sort(e + 1, e + 1 + m, [](node x, node y) {
        return x.w < y.w;
    });
    int ans = 0;
    for (int i = 1; i <= m; i++) {
        LCT::tree[i+n].val = e[i].w;
        int u = e[i].u, v = e[i].v, w = e[i].w;
        int fu = find(u), fv = find(v);
        if (fu == fv) continue;
        f[fu] = fv;
        vis[i] = 1;
        ans += w;
        LCT::link(u, i + n);
        LCT::link(i + n, v);
    }
    int res = INF;
    for (int i = 1; i <= m; i++) {
        if (vis[i]) continue;
        int u = e[i].u, v = e[i].v, w = e[i].w;
        LCT::split(u, v);
        if (w > LCT::tree[v].mx1) res = min(res,w-LCT::tree[v].mx1);
        else res = min(res,w-LCT::tree[v].mx2);
    }
    printf("%lld\n",ans+res);


    return 0;
}