倍增

洛谷 P3379 【模板】最近公共祖先（LCA）

#include <bits/stdc++.h>

using namespace std;
using ll = long long;
using p = pair<int, int>;
const int maxn(5e5 + 10);
const int maxm(1e6 + 10);
int ecnt, head[maxn];
int dep[maxn], f[maxn][20];

struct edge {
    int to, nxt;
} edges[maxm];

template<typename T = int>
inline const T read()
{
    T x = 0, f = 1;
    char ch = getchar();
    while (ch < '0' || ch > '9') {
        if (ch == '-') f = -1;
        ch = getchar();
    }
    while (ch >= '0' && ch <= '9') {
        x = (x << 3) + (x << 1) + ch - '0';
        ch = getchar();
    }
    return x * f;
}

template<typename T>
inline void write(T x, bool ln)
{
    if (x < 0) {
        putchar('-');
        x = -x;
    }
    if (x > 9) write(x / 10, false);
    putchar(x % 10 + '0');
    if (ln) putchar(10);
}

void addEdge(int u, int v)
{
    edges[ecnt].to = v;
    edges[ecnt].nxt = head[u];
    head[u] = ecnt++;
}

void bfs(int root)
{
    queue<int> q;
    q.push(root);
    dep[root] = 1;
    while (not q.empty()) {
        int u = q.front();
        q.pop();
        for (int i = head[u]; compl i; i = edges[i].nxt) {
            int v = edges[i].to;
            if (dep[v]) {
                continue;
            }
            q.push(v);
            dep[v] = dep[u] + 1;
            f[v][0] = u;
            for (int j = 1; j < 20; ++j) {
                f[v][j] = f[f[v][j - 1]][j - 1];
            }
        }
    }
}

int lca(int u, int v)
{
    if (dep[u] < dep[v]) {
        swap(u, v);
    }
    for (int i = 19; i >= 0; --i) {
        if (dep[f[u][i]] >= dep[v]) {
            u = f[u][i];
        }
    }
    if (u == v) {
        return u;
    }
    for (int i = 19; i >= 0; --i) {
        if (f[u][i] not_eq f[v][i]) {
            u = f[u][i];
            v = f[v][i];
        }
    }
    return f[u][0];
}

int main()
{
#ifdef ONLINE_JUDGE
#else
    freopen("input.txt", "r", stdin);
#endif
    ios::sync_with_stdio(false);
    memset(head, -1, sizeof head);
    int n = read(), m = read(), s = read();
    for (int i = 0; i < n - 1; ++i) {
        int u = read(), v = read();
        addEdge(u, v);
        addEdge(v, u);
    }
    bfs(s);
    while (m--) {
        int u = read(), v = read();
        write(lca(u, v), true);
    }
    return 0;
}

洛谷 P4180 [BJWC2010]严格次小生成树

#include <bits/stdc++.h>

using namespace std;
using ll = long long;
using p = pair<int, int>;
const double pi(acos(-1));
const int inf(0x3f3f3f3f);
const int maxn(1e6 + 10);
const int maxm(3e6 + 10);
int ecnt, head[maxn];
int pre[maxn], dep[maxn];
int f[maxn][17], d1[maxn][17], d2[maxn][17];

struct edge {
    int u, v, w;
    bool used;
    
    bool operator <(const edge& rhs) const
    {
        return w < rhs.w;
    }
} edges[maxm];

struct _edge {
    int to, wt, nxt;
} _edges[maxm];

template<typename T = int>
inline const T read()
{
    T x = 0, f = 1;
    char ch = getchar();
    while (ch < '0' || ch > '9') {
        if (ch == '-') f = -1;
        ch = getchar();
    }
    while (ch >= '0' && ch <= '9') {
        x = (x << 3) + (x << 1) + ch - '0';
        ch = getchar();
    }
    return x * f;
}

template<typename T>
inline void write(T x, bool ln)
{
    if (x < 0) {
        putchar('-');
        x = -x;
    }
    if (x > 9) write(x / 10, false);
    putchar(x % 10 + '0');
    if (ln) putchar(10);
}

void addEdge(int u, int v, int w)
{
    _edges[ecnt].to = v;
    _edges[ecnt].wt = w;
    _edges[ecnt].nxt = head[u];
    head[u] = ecnt++;
}

int Find(int x)
{
    return pre[x] == x ? x : pre[x] = Find(pre[x]);
}

void Union(int u, int v)
{
    pre[Find(v)] = Find(u);
}

ll kruskal(int n, int m)
{
    for (int i = 1; i <= n; ++i) {
        pre[i] = i;
    }
    sort(edges, edges + m);
    ll res = 0;
    for (int i = 0; i < m; ++i) {
        int u = edges[i].u, v = edges[i].v, w = edges[i].w;
        if (Find(u) != Find(v)) {
            Union(u, v);
            res += w;
            edges[i].used = true;
        }
    }
    return res;
}

void build(int m)
{
    memset(head, -1, sizeof head);
    for (int i = 0; i < m; ++i) {
        if (edges[i].used) {
            int u = edges[i].u, v = edges[i].v, w = edges[i].w;
            addEdge(u, v, w);
            addEdge(v, u, w);
        }
    }
}

void bfs(int root)
{
    dep[root] = 1;
    queue<int> q;
    q.push(root);
    while (not q.empty()) {
        int u = q.front();
        q.pop();
        for (int i = head[u]; compl i; i = _edges[i].nxt) {
            int v = _edges[i].to, w = _edges[i].wt;
            if (dep[v]) {
                continue;
            }
            dep[v] = dep[u] + 1;
            q.push(v);
            f[v][0] = u;
            d1[v][0] = w;
            d2[v][0] = -inf;
            for (int j = 1; j < 17; ++j) {
                int anc = f[v][j - 1];
                f[v][j] = f[anc][j - 1];
                int dis[4] = { d1[v][j - 1], d2[v][j - 1], d1[anc][j - 1], d2[anc][j - 1] };
                d1[v][j] = d2[v][j] = -inf;
                for (int k = 0; k < 4; ++k) {
                    int d = dis[k];
                    if (d > d1[v][j]) {
                        d2[v][j] = d1[v][j];
                        d1[v][j] = d;
                    } else if (d not_eq d1[v][j] and d > d2[v][j]) {
                        d2[v][j] = d;
                    }
                }
            }
        }
    }
}

int lca(int u, int v, int w)
{
    static int dis[maxn * 2];
    int cnt = 0;
    if (dep[u] < dep[v]) {
        swap(u, v);
    }
    for (int i = 16; i >= 0; --i) {
        if (dep[f[u][i]] >= dep[v]) {
            dis[cnt++] = d1[u][i];
            dis[cnt++] = d2[u][i];
            u = f[u][i];
        }
    }
    if (u not_eq v) {
        for (int i = 16; i >= 0; --i) {
            if (f[u][i] not_eq f[v][i]) {
                dis[cnt++] = d1[u][i];
                dis[cnt++] = d2[u][i];
                dis[cnt++] = d1[v][i];
                dis[cnt++] = d2[v][i];
                u = f[u][i];
                v = f[v][i];
            }
        }
        dis[cnt++] = d1[u][0];
        dis[cnt++] = d1[v][0];
    }
    int dis1 = -inf, dis2 = -inf;
    for (int i = 0; i < cnt; ++i) {
        int d = dis[i];
        if (d > dis1) {
            dis2 = dis1;
            dis1 = d;
        } else if (d not_eq dis1 and d > dis2) {
            dis2 = d;
        }
    }
    if (w > dis1) {
        return w - dis1;
    }
    if (w > dis2) {
        return w - dis2;
    }
    return inf;
}

int main()
{
#ifdef ONLINE_JUDGE
#else
    freopen("input.txt", "r", stdin);
#endif
    ios::sync_with_stdio(false);
    int n = read(), m = read();
    for (int i = 0; i < m; ++i) {
        int u = read(), v = read(), w = read();
        addEdge(u, v, w);
        edges[i] = edge{ u, v, w };
    }
    ll sum = kruskal(n, m);
    build(m);
    bfs(1);
    ll res = 1e18;
    for (int i = 0; i < m; ++i) {
        if (not edges[i].used) {
            int u = edges[i].u, v = edges[i].v, w = edges[i].w;
            res = min(res, sum + lca(u, v, w));
        }
    }
    write(res, true);
    return 0;
}