洛谷P4234 最小差值生成树

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 <bits/stdc++.h>
#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 <typename A, typename B, typename C>
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;
}