树的最近公共祖先

LCA（Lowest Common Ancestors），即​​最近公共祖先​​，是指在有根树中，找出某两个结点u和v最近的公共祖先。

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倍增算法（在线）

//n结点的树，输出任意两个结点间的最小距离
#include<iostream>
#include<cstdio>
#include<cstring>
#include<algorithm>
#include<queue>
#include<cmath>
using namespace std;
const int SIZE = 50010;
int f[SIZE][20], d[SIZE], dist[SIZE];
int ver[2 * SIZE], Next[2 * SIZE], edge[2 * SIZE], head[SIZE];
int T, n, m, tot, t;
queue<int> q;

void add(int x, int y, int z) {
  ver[++tot] = y; edge[tot] = z; Next[tot] = head[x]; head[x] = tot;
}

// 预处理
void bfs() {
  q.push(1); d[1] = 1;
  while (q.size()) {
    int x = q.front(); q.pop();
    for (int i = head[x]; i; i = Next[i]) {
      int y = ver[i];
      if (d[y]) continue;
      d[y] = d[x] + 1;
      dist[y] = dist[x] + edge[i];
      f[y][0] = x;
      for (int j = 1; j <= t; j++)
        f[y][j] = f[f[y][j - 1]][j - 1];
      q.push(y);
    }
  }
}

// 回答一个询问
int lca(int x, int y) {
  if (d[x] > d[y]) swap(x, y);
  for (int i = t; i >= 0; i--)
    if (d[f[y][i]] >= d[x]) y = f[y][i];
  if (x == y) return x;
  for (int i = t; i >= 0; i--)
    if (f[x][i] != f[y][i]) x = f[x][i], y = f[y][i];
  return f[x][0];
}

int main() {
  cin >> T;
  while (T--) {
    cin >> n >> m;
    t = (int)(log(n) / log(2)) + 1;
    // 清空
    for (int i = 1; i <= n; i++) head[i] = d[i] = 0;
    tot = 0;
    // 读入一棵树
    for (int i = 1; i < n; i++) {
      int x, y, z;
      scanf("%d%d%d", &x, &y, &z);
      add(x, y, z), add(y, x, z);
    }
    bfs();
    // 回答问题
    for (int i = 1; i <= m; i++) {
      int x, y;
      scanf("%d%d", &x, &y);
      printf("%d\n", dist[x] + dist[y] - 2 * dist[lca(x, y)]);
    }
  }
}

Tarjan离线算法

//n结点的树，输出任意两个结点间的最小距离
#include<iostream>
#include<cstdio>
#include<cstring>
#include<algorithm>
#include<vector>
using namespace std;
const int SIZE = 50010;
int ver[2 * SIZE], Next[2 * SIZE], edge[2 * SIZE], head[SIZE];
int fa[SIZE], d[SIZE], v[SIZE], lca[SIZE], ans[SIZE];
vector<int> query[SIZE], query_id[SIZE];
int T, n, m, tot, t;

void add(int x, int y, int z) {
  ver[++tot] = y; edge[tot] = z; Next[tot] = head[x]; head[x] = tot;
}

void add_query(int x, int y, int id) {
  query[x].push_back(y), query_id[x].push_back(id);
  query[y].push_back(x), query_id[y].push_back(id);
}

int get(int x) {
  if (x == fa[x]) return x;
  return fa[x] = get(fa[x]);
}

void tarjan(int x) {
  v[x] = 1;
  for (int i = head[x]; i; i = Next[i]) {
    int y = ver[i];
    if (v[y]) continue;
    d[y] = d[x] + edge[i];
    tarjan(y);
    fa[y] = x;
  }
  for (int i = 0; i < query[x].size(); i++) {
    int y = query[x][i];
    int id = query_id[x][i];
    if (v[y] == 2) {
      int lca = get(y);
      ans[id] = min(ans[id], d[x] + d[y] - 2 * d[lca]);
    }
  }
  v[x] = 2;
}

int main() {
  cin >> T;
  while (T--) {
    cin >> n >> m;
    for (int i = 1; i <= n; i++) {
      head[i] = 0;
      query[i].clear(), query_id[i].clear();
      fa[i] = i, v[i] = 0;
    }
    tot = 0;
    for (int i = 1; i < n; i++) {
      int x, y, z;
      scanf("%d%d%d", &x, &y, &z);
      add(x, y, z), add(y, x, z);
    }
    for (int i = 1; i <= m; i++) {
      int x, y;
      scanf("%d%d", &x, &y);
      if (x == y) ans[i] = 0;
      else {
        add_query(x, y, i);
        ans[i] = 1 << 30;
      }
    }
    tarjan(1);
    for (int i = 1; i <= m; i++) printf("%d\n", ans[i]);
  }
}

DFS + ST在线算法

//在一棵树上，有n个节点，n−1条边，给定两个节点p,q，求它们的最近公共祖先
const int MAXN = 10010;
int rmq[2 * MAXN];          //  rmq数组,就是欧拉序列对应的深度序列

struct ST
{
    int mm[2 * MAXN];
    int dp[2 * MAXN][20];     //  最小值对应的下标
    void init(int n)
    {
        mm[0] = -1;
        for (int i = 1; i <= n; i++)
        {
            mm[i] = ((i & (i - 1)) == 0) ? mm[i - 1] + 1 : mm[i - 1];
            dp[i][0] = i;
        }
        for (int j = 1; j <= mm[n]; j++)
        {
            for (int i = 1; i + (1 << j) - 1 <= n; i++)
            {
                dp[i][j] = rmq[dp[i][j - 1]] < rmq[dp[i + (1 << (j - 1))][j - 1]] ? dp[i][j - 1] : dp[i + (1 << (j - 1))][j - 1];
            }
        }
    }
    int query(int a,int b)  //  查询[a,b]之间最小值的下标
    {
        if (a > b)
        {
            swap(a, b);
        }
        int k = mm[b - a + 1];
        return rmq[dp[a][k]] <= rmq[dp[b - (1 << k) + 1][k]] ? dp[a][k] : dp[b - (1 << k) + 1][k];
    }
};

//  边的结构体定义
struct Edge
{
    int to, next;
};

Edge edge[MAXN * 2];

int tot, head[MAXN];
int F[MAXN * 2];        //  欧拉序列,就是dfs遍历的顺序,长度为2*n-1,下标从1开始
int P[MAXN];            //  P[i]表示点i在F中第一次出现的位置
int cnt;
ST st;

void init()
{
    tot = 0;
    memset(head, -1, sizeof(head));
}

void addedge(int u, int v)   //  加边,无向边需要加两次
{
    edge[tot].to = v;
    edge[tot].next = head[u];
    head[u] = tot++;
}

void dfs(int u, int pre, int dep)
{
    F[++cnt] = u;
    rmq[cnt] = dep;
    P[u] = cnt;
    for (int i = head[u]; i != -1; i = edge[i].next)
    {
        int v = edge[i].to;
        if (v == pre)
        {
            continue;
        }
        dfs(v, u, dep + 1);
        F[++cnt] = u;
        rmq[cnt] = dep;
    }
}

void LCA_init(int root, int node_num)   //  查询LCA前的初始化
{
    cnt = 0;
    dfs(root, root, 0);
    st.init(2 * node_num - 1);
}

int query_lca(int u, int v)             //  查询u,v的lca编号
{
    return F[st.query(P[u], P[v])];
}

bool flag[MAXN];

int main()
{
    int T;
    int N;
    int u, v;
    scanf("%d", &T);
    while(T--)
    {
        scanf("%d", &N);
        init();
        memset(flag, false, sizeof(flag));
        for (int i = 1; i < N; i++)
        {
            scanf("%d%d", &u, &v);
            addedge(u, v);
            addedge(v, u);
            flag[v] = true;
        }
        int root;
        for (int i = 1; i <= N; i++)
        {
            if (!flag[i])
            {
                root = i;
                break;
            }
        }
        LCA_init(root, N);
        scanf("%d%d", &u, &v);
        printf("%d\n", query_lca(u, v));
    }
    return 0;
}