LCA求解的四種模板
或許更好的閱讀體驗
樹剖在線求解LCA
思想
樹剖這裏就不多解釋了，求解LCA的過程就是輕重鏈的跳轉，跟樹剖求任意兩點間的距離一樣的操作，只不過不用線段樹去維護了，那就直接上代碼吧。
代碼
#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
const int N = 1e6 + 10;
int head[N], to[N << 1], nex[N << 1], cnt = 1;
int sz[N], dep[N], fa[N], son[N], top[N];
int n, m;
inline int read() {
int f = 1, x = 0;
char c = getchar();
while(c > '9' || c < '0') {
if(c == '-') f = -1;
c = getchar();
}
while(c >= '0' && c <= '9') {
x = (x << 1) + (x << 3) + (c ^ 48);
c = getchar();
}
return f * x;
}
void add(int x, int y) {
to[cnt] = y;
nex[cnt] = head[x];
head[x] = cnt++;
}
void dfs1(int rt, int f) {
dep[rt] = dep[f] + 1;
sz[rt] = 1, fa[rt] = f;
for(int i = head[rt]; i; i = nex[i]) {
if(to[i] == f) continue;
dfs1(to[i], rt);
if(!son[rt] || sz[to[i]] > sz[son[rt]])
son[rt] = to[i];
sz[rt] += sz[to[i]];
}
}
void dfs2(int rt, int t) {
top[rt] = t;
if(!son[rt]) return ;
dfs2(son[rt], t);
for(int i = head[rt]; i; i = nex[i]) {
if(to[i] == fa[rt] || to[i] == son[rt]) continue;
dfs2(to[i], to[i]);
}
}
int solve(int x, int y) {
while(top[x] != top[y]) {
if(dep[top[x]] < dep[top[y]]) swap(x, y);
x = fa[top[x]];
}
return dep[x] < dep[y] ? x : y;
}
int main() {
// freopen("in.txt", "r", stdin);
// freopen("out.txt", "w", stdout);
n = read(), m = read();
int rt = read();
int x, y;
for(int i = 1; i < n; i++) {
x = read(), y = read();
add(x, y);
add(y, x);
}
dfs1(rt, 0);
dfs2(rt, rt);
for(int i = 1; i <= m; i++) {
x = read(), y = read();
printf("%d\n", solve(x, y));
}
return 0;
}
Tarjan離線求解
思想
本質就是利用了dfs的節點順序，當我們正在遞歸兩個節點的最近公共祖先時，顯然這兩個點是屬於其子樹的節點，那麼當我們第一次遍歷完兩個需要求解的兩個點時，其最近的尚未被完全遍歷完子節點的節點就是他們兩個的最近公共祖先。
代碼
#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
const int N = 5e5 + 10;
int head[N], to[N << 1], nex[N << 1], cnt = 1;
int visit[N], fa[N], n, m;
int qhead[N], qto[N << 1], qnex[N << 1], qcnt = 1, qid[N << 1], ans[N];
inline int read() {
int f = 1, x = 0;
char c = getchar();
while(c > '9' || c < '0') {
if(c == '-') f = -1;
c = getchar();
}
while(c >= '0' && c <= '9') {
x = (x << 1) + (x << 3) + (c ^ 48);
c = getchar();
}
return f * x;
}
void add_edge(int x, int y) {
to[cnt] = y;
nex[cnt] = head[x];
head[x] = cnt++;
}
void add_query(int x, int y, int w) {
qto[qcnt] = y;
qnex[qcnt] = qhead[x];
qid[qcnt] = w;
qhead[x] = qcnt++;
}
int find(int rt) {
return rt == fa[rt] ? rt : fa[rt] = find(fa[rt]);
}
void tarjan(int rt, int f) {
for(int i = head[rt]; i; i = nex[i]) {
if(to[i] == f) continue;
tarjan(to[i], rt);
fa[to[i]] = rt;
}
visit[rt] = 1;
for(int i = qhead[rt]; i; i = qnex[i]) {
if(!visit[qto[i]]) continue;
ans[qid[i]] = find(qto[i]);
}
}
int main() {
// freopen("in.txt", "r", stdin);
n = read(), m = read();
int rt = read();
for(int i = 1; i < n; i++) {
int x = read(), y = read();
add_edge(x, y);
add_edge(y, x);
}
for(int i = 1; i <= n; i++) fa[i] = i;
for(int i = 1; i <= m; i++) {
int x = read(), y = read();
add_query(x, y, i);
add_query(y, x, i);
}
tarjan(rt, 0);
for(int i = 1; i <= m; i++)
printf("%d\n", ans[i]);
return 0;
}
ST表 + RMQ在線求解
思想
利用dfs的遍歷，在遍歷兩個點的時候，一定會在中間返回到其最近公共祖先，這個時候的公共祖先也就是這兩個點的遍歷中的最小值。
代碼
#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
inline ll read() {
ll f = 1, x = 0;
char c = getchar();
while(c > '9' || c < '0') {
if(c == '-') f = -1;
c = getchar();
}
while(c >= '0' && c <= '9') {
x = (x << 1) + (x << 3) + (c ^ 48);
c = getchar();
}
return f * x;
}
const int N = 5e5 + 10;
int head[N], to[N << 1], nex[N << 1], cnt = 1;
int id[N], tot, last;
int st[N << 2][30];
void add(int x, int y) {
to[cnt] = y;
nex[cnt] = head[x];
head[x] = cnt++;
}
void dfs(int rt, int fa) {
id[rt] = last = ++tot;
st[tot][0] = rt;
for(int i = head[rt]; i; i = nex[i]) {
if(to[i] == fa) continue;
dfs(to[i], rt);
st[++tot][0] = rt;
}
}
int MIN(int a, int b) {
return id[a] < id[b] ? a : b;
}
int main() {
// freopen("in.txt", "r", stdin);
int 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);
int k = log(last) / log(2);
for(int j = 1; j <= k; j++)
for(int i = 1; i + (1 << j) - 1 <= last; i++)
st[i][j] = MIN(st[i][j - 1], st[i + (1 << (j - 1))][j - 1]);
for(int i = 1; i <= m; i++) {
int x = read(), y = read();
x = id[x], y = id[y];
if(x > y) swap(x, y);
int k = log(y - x + 1) / log(2);
printf("%d\n", MIN(st[x][k], st[y - (1 << k) + 1][k]));
}
return 0;
}
倍增
思想
類似於快速冪，通過二進制數的組合來達到級別的優化，但是需要注意其中進制的枚舉大小順序。
代碼
#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
inline ll read() {
ll f = 1, x = 0;
char c = getchar();
while(c > '9' || c < '0') {
if(c == '-') f = -1;
c = getchar();
}
while(c >= '0' && c <= '9') {
x = (x << 1) + (x << 3) + (c ^ 48);
c = getchar();
}
return f * x;
}
const int N = 5e5 + 10;
int head[N], to[N << 1], nex[N << 1], cnt = 1;
int fa[N][21], dep[N], n, m;
void add(int x, int y) {
to[cnt] = y;
nex[cnt] = head[x];
head[x] = cnt++;
}
void dfs(int rt, int f) {
dep[rt] = dep[f] + 1;
fa[rt][0] = f;
for(int i = 1; 1 << i <= dep[rt]; i++)//進制由小到大遞推
fa[rt][i] = fa[fa[rt][i - 1]][i - 1];
for(int i = head[rt]; i; i = nex[i]) {
if(to[i] == f) continue;
dfs(to[i], rt);
}
}
int LCA(int x, int y) {
if(dep[x] < dep[y]) swap(x, y);
for(int i = 20; i >= 0; i--)//進制由大到小開始組合，
if(dep[fa[x][i]] >= dep[y])
x = fa[x][i];
if(x == y) return x;//注意特判
for(int i = 20; i >= 0; i--)//進制從小到大開始組合，
if(fa[x][i] != fa[y][i])
x = fa[x][i], y = fa[y][i];
return fa[x][0];//這一步尤其考慮，爲什麼x, y不知LCA,而其父節點就一定是LCA，
}
int main() {
// freopen("in.txt", "r", stdin);
int 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);
for(int i = 1; i <= m; i++) {
int x = read(), y = read();
printf("%d\n", LCA(x, y));
}
return 0;
}