分类目录归档：最近公共祖先

洛谷 P3379 【模板】最近公共祖先（LCA）重写

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朴素算法：洛谷70分,O2 80分

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#include<iostream>
#include<vector>
using namespace std;
int n, m, s;
vector<int> edges[500005];
int father[500005],depth[500005];
void dfs(int node, int fath) {
    father[node] = fath;
    depth[node] = depth[fath] + 1;
    for (int i = 0; i < edges[node].size(); i++) {
        if (edges[node][i] == fath)continue;
        dfs(edges[node][i], node);
    }
}
int query(int a, int b) {
    while (a != b) {
        if (depth[a] >= depth[b])a = father[a];
        else b = father[b];
    }
    return a;
}
int main() {
    ios::sync_with_stdio(false);
    cin.tie(0);
    cin >> n >> m >> s;
    for (int i = 1; i <= n - 1; i++) {
        int x, y;
        cin >> x >> y;
        edges[x].push_back(y);
        edges[y].push_back(x);
    }
    dfs(s, s);
    for (int i = 1; i <= m; i++) {
        int a, b;
        cin >> a >> b;
        cout << query(a, b) << "\n";
    }
}

倍增：洛谷80分，O2 100分

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#include<iostream>
#include<vector>
#include<algorithm>
#define MAXLOG 20
using namespace std;
int n, m, s;
vector<int> edges[500005];
int depth[500005];
int ancestor[500005][MAXLOG+2];
void dfs(int node, int fath) {
    ancestor[node][0] = fath;
    depth[node] = depth[fath] + 1;
    for (int i = 1; i <= MAXLOG; i++)ancestor[node][i] = ancestor[ancestor[node][i - 1]][i - 1];
    for (int i = 0; i < edges[node].size(); i++) {
        if (edges[node][i] == fath)continue;
        dfs(edges[node][i], node);
    }
}
int query(int a, int b) {
    if (depth[b] > depth[a])swap(a, b);
    for (int i = MAXLOG; ~i; i--)if (depth[ancestor[a][i]] >= depth[b])a = ancestor[a][i];
    if (a == b)return a;
    for (int i = MAXLOG; ~i; i--) if(ancestor[a][i]!=ancestor[b][i]){
        a = ancestor[a][i];
        b = ancestor[b][i];
    }
    return ancestor[a][0];
}
int main() {
    ios::sync_with_stdio(false);
    cin.tie(0);
    cin >> n >> m >> s;
    for (int i = 1; i <= n - 1; i++) {
        int x, y;
        cin >> x >> y;
        edges[x].push_back(y);
        edges[y].push_back(x);
    }
    dfs(s, s);
    for (int i = 1; i <= m; i++) {
        int a, b;
        cin >> a >> b;
        cout << query(a, b) << "\n";
    }
}

Tarjan:洛谷70分，O2 100分

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#include<iostream>
#include<vector>
#include<algorithm>
#include<cstring>
using namespace std;
int n, m, s;
vector<int> edges[500005],askPair[500005];
bool visited[500005];
int father[500005];
inline int _find(int node) {
    if (father[node]==-1)return node;
    return father[node] = _find(father[node]);
}
inline void _union(int x, int y) {
    if(_find(x)!=_find(y))father[_find(x)] = y;
}
struct ask {
    int u, v;
    int ans;
}asks[500005];
inline int checkV(int id,int node) {
    if (asks[id].u != node)return asks[id].u;
    return asks[id].v;
}
void dfs(int node,int fath) {
    for (int i = 0; i < edges[node].size(); i++) {
        if (edges[node][i] == fath)continue;
        dfs(edges[node][i],node);
        _union(edges[node][i], node);
    }
    visited[node] = true;
    for (int i = 0; i < askPair[node].size(); i++) {
        int v = checkV(askPair[node][i], node);
        if (visited[v])asks[askPair[node][i]].ans = _find(v);
    }
}
int main() {
    memset(father, -1, sizeof(father));
    ios::sync_with_stdio(false);
    cin.tie(0);
    cin >> n >> m >> s;
    for (int i = 1; i <= n - 1; i++) {
        int x, y;
        cin >> x >> y;
        edges[x].push_back(y);
        edges[y].push_back(x);
    }
    for (int i = 1; i <= m; i++) {
        cin >> asks[i].u >> asks[i].v;
        askPair[asks[i].u].push_back(i);
        askPair[asks[i].v].push_back(i);
    }
    dfs(s,s);
    for (int i = 1; i <= m; i++)cout << asks[i].ans << "\n";
}

洛谷 【模板】最近公共祖先（LCA）

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调了半天才调出来的东西啊……233，在写tarjan的循环时写错了，把_v_ask写成了_v，太容易写错了，一定要多加注意，这个模版还需要练习，线段树也是。。
Tarjan::

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#include<bits/stdc++.h>
#define MAXN 1000005
#define MAXM 1000005
#define EdgeLoop(e,node) for(int e=_first[node];e!=-1;e=_next[e])
#define EdgeAskLoop(e,node) for(int e=_first_ask[node];e!=-1;e=_next_ask[e])
int n, m, s;
int _father[MAXN];
int graph_m_ptr = 0, graph_m_ptr_ask = 0;
int _u[MAXN], _v[MAXN], _next[MAXN], _first[MAXN];
bool _visited[MAXN];
int _u_ask[MAXN], _v_ask[MAXN], _next_ask[MAXN], _first_ask[MAXN];
int _ans[MAXM];
int _find(int x){
    if (_father[x] == x)return x;
    return _father[x] = _find(_father[x]);
}
void _merge(int x, int y){//merge y branch to x branch
    _father[_find(x)] = _find(y);
}
void insert_edge(int s, int t){
    _u[graph_m_ptr] = s;
    _v[graph_m_ptr] = t;
    _next[graph_m_ptr] = _first[s];
    _first[s] = graph_m_ptr;
    graph_m_ptr++;
}
void insert_edge_ask(int s, int t){
    _u_ask[graph_m_ptr_ask] = s;
    _v_ask[graph_m_ptr_ask] = t;
    _next_ask[graph_m_ptr_ask] = _first_ask[s];
    _first_ask[s] = graph_m_ptr_ask;
    graph_m_ptr_ask++;
}
void tarjan(int u){
    _visited[u] = true;
    EdgeLoop(e, u){
        if (!_visited[_v[e]]){
            tarjan(_v[e]);
            _merge(_v[e],u);
        }
    }
    EdgeAskLoop(e, u){
        if (_visited[_v_ask[e]]){
            _ans[e/2] = _find(_v_ask[e]);
        }
    }
}
using namespace std;
int main(){
    ios::sync_with_stdio(false);
    cin >> n >> m >> s;
    for (int i = 1; i <= n; i++){
        _father[i] = i;
    }
    memset(_first, -1, sizeof(_first));
    memset(_first_ask, -1, sizeof(_first_ask));
    for (int i = 1; i <= n - 1; i++){
        int x, y;
        cin >> x >> y;
        insert_edge(x, y);
        insert_edge(y, x);
    }
    for (int i = 1; i <= m; i++){
        int a, b;
        cin >> a >> b;
        insert_edge_ask(a, b);
        insert_edge_ask(b, a);
    }
    tarjan(s);
    for (int i = 0; i < m; i++){
        cout << _ans[i] << endl;
    }
}

树上倍增：

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#include<bits/stdc++.h>
using namespace std;
int edge_ptr = 0;
struct edge{
    int v;
    int next;
}edges[1000005];
int head[500005];
int n, m, s;
int father[500005][25],depth[500005];
bool visited[500005];
inline void insert_edge(int s,int t){
    edge_ptr++;
    edges[edge_ptr].v = t;
    edges[edge_ptr].next = head[s];
    head[s] = edge_ptr;
}
inline void dfs(int u){
    visited[u] = true;
    for (int i = 1, len = 1; len <= depth[u]; i++, len <<= 1){
        father[u][i] = father[father[u][i - 1]][i - 1];
    }
    for (int e = head[u]; e != -1;e=edges[e].next){
        if (!visited[edges[e].v]){
            depth[edges[e].v] = depth[u] + 1;
            father[edges[e].v][0] = u;
            dfs(edges[e].v);
        }
    }
}
inline int query(int x, int y){
    if (depth[x] > depth[y])swap(x,y);
    int diff = depth[y] - depth[x];
    for (int i = 0, two = 1; diff != 0; two <<= 1, i++){
        if (diff&two){
            y = father[y][i];
            diff -= two;
        }
    }
    if (x == y)return x;
    for (int i = log2(depth[x]); i >= 0; i--){
        if (father[x][i] != father[y][i]){
            x = father[x][i];
            y = father[y][i];
        }
    }
    return father[x][0];
}
int main(){
    ios::sync_with_stdio(false);
    memset(head, -1, sizeof(head));
    cin >> n >> m >> s;
    for (int i = 1; i <= n - 1; i++){
        int x, y;
        cin >> x >> y;
        insert_edge(x, y);
        insert_edge(y, x);
    }
    dfs(s);
    for (int i = 1; i <= m; i++){
        int x, y;
        cin >> x >> y;
        cout << query(x, y) << endl;
    }
}