最近公共祖先Lca（ST表，树剖，倍增，Tarjan， LCT）

• 树的最近公共祖先有好几种求法，这里列出4种。根据luogu P3379 【模板】最近公共祖先（LCA）的提交，按由快到慢进行了排序。
• 这是我的一些提交记录：
  • 树链剖分
  • RMQ(ST表)
  • 倍增
  • Tarjan
  • LCT

A 树链剖分求lca

• 轻重链剖分，很快，且好写

Code

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//树链剖分
#include <cstdio>
using namespace std;
const int N = 5e5+5;
struct Side {
    int t, next;
}e[N<<1];
int head[N], tot;
void Add(int x, int y) {
    e[++tot] = (Side){y, head[x]};
    head[x] = tot;
}
int siz[N], f[N], son[N], d[N], top[N];
void Dfs1(int x) {
    siz[x] = 1;
    d[x] = d[f[x]] + 1;
    for (int i = head[x]; i; i = e[i].next) {
        int y = e[i].t;
        if (y == f[x]) continue;
        f[y] = x;
        Dfs1(y);
        siz[x] += siz[y];
        if (!son[x] || siz[son[x]] < siz[y])
            son[x] = y;
    }
}
void Dfs2(int x, int tp) {
    top[x] = tp;
    if (!son[x]) return;
    Dfs2(son[x], tp);
    for (int i = head[x]; i; i = e[i].next) {
        int y = e[i].t;
        if (y == f[x] || y == son[x]) continue;
        Dfs2(y, y);
    }
}
int Lca(int x, int y) {
    while (top[x] != top[y])
        d[top[x]] > d[top[y]] ? x = f[top[x]] : y = f[top[y]];
    return d[x] < d[y] ? x : y;
}
int n, m, s;
int main() {
    scanf("%d%d%d", &n, &m, &s);
    for (int i = 1; i < n; i++) {
        int x, y;
        scanf("%d%d", &x, &y);
        Add(x, y); Add(y, x);
    }
    Dfs1(s); Dfs2(s, s);
    while (m--) {
        int x, y;
        scanf("%d%d", &x, &y);
        printf("%d
", Lca(x, y));
    }
    return 0;
}

B Lca转RMQ

• 利用的是欧拉序的一个特性——任意两个点之间（包括这两点）深度最小的节点就是两个点的最近公共祖先。

欧拉序：就是从根结点出发，按dfs的顺序在绕回原点所经过所有点的顺序

• 反证法：

  • 假设深度最小的点不是LCA，那么这个LCA点一定在两点之外。

  • 但是欧拉序在走到任意一个兄弟前，一定要经过两个距离最近的公共点（也就是LCA）

  • 所以假设不成立，证明LCA点一定在两点之间。

  • 证毕。

• 使用RMQ不是很好写，但是在询问多于点数的时候十分优秀。

Code

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//RMQ求Lca
#include <cstdio>
using namespace std;
const int N = 5e5 + 5;

#define swap(x, y) {
    int a = x; x = y; y = a;
}

int read(int x = 0, int f = 1, char c = getchar()) {
    for (; c < '0' || c > '9'; c = getchar())
        if (c == '-') f = -1;
    for (; c >='0' && c <='9'; c = getchar())
        x = x * 10 + c - '0';
    return x * f;
}

struct Edge {
    int next, t;
}e[N<<1];
int head[N], edc;

void Add(int x, int y) {
    e[++edc] = (Edge) {head[x], y};
    head[x] = edc;
}

int n, m, rt, lg[N<<1], f[21][N<<1];//f是深度最低的节点
int dep[N], dfn[N], dfc;

void Dfs(int x, int fa) {
    dfn[x] = ++dfc;
    f[0][dfc] = x;
    dep[x] = dep[fa] + 1;
    for (int i = head[x]; i; i = e[i].next) {
        int y = e[i].t;
        if (y == fa) continue;
        Dfs(y, x);
        f[0][++dfc] = x;
    }
}

int Lca(int x, int y) {
    x = dfn[x]; y = dfn[y];
    if (x > y) swap(x, y);
    int k = lg[y-x+1]; y = y - (1 << k) + 1;
    return dep[f[k][x]] < dep[f[k][y]] ? f[k][x] : f[k][y];
}

int main() {
    n = read(), m = read(), rt = read();
    for (int i = 1; i < n; ++i) {
        int x = read(), y = read();
        Add(x, y); Add(y, x);
    }
    Dfs(rt, 0);
    //ST表预处理
    for (int i = 2; i <= dfc; ++i)
        lg[i] = lg[i>>1] + 1;
    for (int i = 0; i < lg[dfc]; ++i)
        for (int x = 1; (x + (1 << i + 1)) <= dfc; ++x) {
            int y = x + (1 << i);
            f[i+1][x] = dep[f[i][x]] < dep[f[i][y]] ? f[i][x] : f[i][y];
        }
    while (m--)
        printf("%d
", Lca(read(), read()));
    return 0;
}

C 倍增求Lca

• f[x][k]表示从x向根节点走(2^k)步到达的节点
  d[x]表示树的深度
  预处理时间复杂度为(O(nlog n)),每次询问时间复杂度为(O(log n))

Code

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#include <cstdio>
#include <algorithm>

const int N = 5e5 + 5;

int read(int x = 0, int f = 1, char c = getchar()) {
    for (; c < '0' || c > '9'; c = getchar())
        if (c == '-') f = -1;
    for (; c >='0' && c <='9'; c = getchar())
        x = x * 10 + c - '0';
    return x * f;
}

struct Edge {
    int next, t;
}e[N<<1];
int head[N], edc;

void Add(int x, int y) {
    e[++edc] = (Edge) {head[x], y};
    head[x] = edc;
}

int n, m, s, f[20][N], dep[N];

void Dfs(int x) {
    dep[x] = dep[f[0][x]] + 1;
    for (int i = 0; (1 << i + 1) <= dep[x]; ++i)
        f[i+1][x] = f[i][f[i][x]];
    for (int i = head[x]; i; i = e[i].next) {
        int y = e[i].t;
        if (y == f[0][x]) continue;
        f[0][y] = x;
        Dfs(y);
    }
}

int Lca(int x, int y) {
    if (dep[x] < dep[y]) std::swap(x, y);
    for (int k = dep[x] - dep[y], i = 0; k; k >>= 1, ++i)
        if (k & 1) x = f[i][x];
    if (x == y) return x;
    for (int i = 19; i >= 0; --i)
        if (f[i][x] != f[i][y])
            x = f[i][x], y = f[i][y];
    return f[0][x];
}

int main() {
    n = read(), m = read(), s = read();
    for (int i = 1; i < n; ++i) {
        int x = read(), y = read();
        Add(x, y); Add(y, x);
    }
    Dfs(s);
    while (m--) {
        int x = read(), y = read();
        printf("%d
", Lca(x, y));
    }
    return 0;
}

D tarjan求lca

• 这种算法本质上是用并查集对向上标记法的优化，是离线算法，即一次性读入所有询问，统一计算，统一输出。
  时间复杂度(O(n+m))，但是常数比较大，可以对并查集部分进行优化。

  • v[]进行标记
    (v[x]doteq 0) --> x节点未访问过
    (v[x]doteq 1) --> x节点已经访问，但未回溯
    (v[x]doteq 2) --> x节点已经访问并回溯

Code

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#include <cstdio>
#include <vector>
#include <algorithm>
using namespace std;
const int N = 1e5+5;
struct side {
    int t, d, next;
}e[N<<1];
int head[N], tot;
void add(int x, int y, int z) {
    e[++tot].next = head[x];
    head[x] = tot;
    e[tot].t = y, e[tot].d = z;
}
int n, m, Q, d[N], f[N], v[N], ans[N];
vector<int> q[N], h[N];
int found(int x) {
    return x == f[x] ? x : (f[x] = found(f[x]));
}
void Dfs(int x) {
    v[x] = 1;
    for (int i = head[x]; i; i = e[i].next) {
        int y = e[i].t;
        if (v[y]) continue;
        d[y] = d[x] + e[i].d;
        Dfs(y);
        f[y] = x;
    }
    for (int i = 0; i < q[x].size(); i++) {
        int y = q[x][i], id = h[x][i];
        if (v[y] != 2) continue;
        ans[id] = min(ans[id], d[x] + d[y] - 2*d[found(y)]);
    }
    v[x] = 2;
}
int main() {
    scanf("%d%d", &n, &m);
    for (int i = 1; i <= n; i++) f[i] = i;
    while (m--) {
        int x, y, z;
        scanf("%d%d%d", &x, &y, &z);
        add(x, y, z); add(y, x, z);
    }
    scanf("%d", &Q);
    for (int i = 1; i <= Q; i++) {
        int x, y;
        scanf("%d%d", &x, &y);
        if (x == y) continue;
        ans[i] = 1 << 30;
        q[x].push_back(y), h[x].push_back(i);
        q[y].push_back(x), h[y].push_back(i);
    }
    Dfs(1);
    for (int i = 1; i <= Q; i++)
        printf("%d
", ans[i]);
    return 0;
}

E LCT求lca

• 常数巨大，但可以维护动态树的Lca，其他方法都不可以

Code

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#include <cstdio>
#include <algorithm>
#define ls(x) c[x][0]
#define rs(x) c[x][1]
#define Get(x) (c[f[x]][1] == x)
#define Nroot(x) (c[f[x]][Get(x)] == x)

using namespace std;
const int N = 5e5 + 5;

int read(int x = 0, int f = 1, char c = getchar()) {
    for (; c < '0' || c > '9'; c = getchar())
        if (c == '-') f = -1;
    for (; c >='0' && c <='9'; c = getchar())
        x = x * 10 + c - '0';
    return x * f;
}

int n, m, rt;
int f[N], c[N][2], tag[N], stk[N], tp;

void Rotate(int x) {
    int y = f[x], z = f[y], k = Get(x), B = c[x][k^1];
    if (Nroot(y)) c[z][Get(y)] = x; f[x] = z;
    c[x][k^1] = y; f[y] = x; c[y][k] = B; f[B] = y;
}

void Pushdown(int x) {
    if (!tag[x]) return;
    swap(ls(x), rs(x)); tag[x] = 0;
    tag[ls(x)] ^= 1; tag[rs(x)] ^= 1;
}

void Splay(int x) {
    for (int y = x; y; y = Nroot(y) ? f[y] : 0) stk[++tp] = y;
    while (tp) Pushdown(stk[tp--]);
    while (Nroot(x)) {
        int y = f[x];
        if (Nroot(y)) Get(x) == Get(y) ? Rotate(y) : Rotate(x);
        Rotate(x);
    }
}

void Access(int x) {
    for (int y = 0; x; y = x, x = f[x])
        Splay(x), c[x][1] = y;
}

void Mroot(int x) {
    Access(x); Splay(x); tag[x] ^= 1;
}

void Link(int x, int y) {
    Mroot(x); f[x] = y;
}

int Lca(int x, int y) {
    int p = 0; Mroot(rt); Access(y);
    for (; x; p = x, x = f[x])
        Splay(x), c[x][1] = p;
    return p;
}

int main() {
    n = read(); m = read(); rt = read();
    for (int i = 1; i < n; ++i) Link(read(), read());
    while (m--) printf("%d
", Lca(read(), read()));
    return 0;
}