淺談LCA

目錄

• 什麼是LCA
• 倍增求LCA
  • dfs
  • bfs
• 樹剖求LCA

什麼是LCA

LCA就是最近公共祖先

對於有根樹\(Tree\)的兩個結點\(u、v\)，最近公共祖先\(LCA(T,u,v)\)表示一個結點\(x\)，知足\(x\)是\(u、v\)的祖先且\(x\)的深度儘量大。

倍增求LCA

解釋明天再寫

dfs

#include<cstdio>
#include<iostream>
#include<cstring>
using namespace std;
const int maxn=500000+2;
int n,m,root,k;
int head[maxn],d[maxn];
struct node {
    int v,next;
} e[maxn<<1];
void add(int u,int v) {
    e[k].v=v;
    e[k].next=head[u];
    head[u]=k++;
}
//________________________________________
int f[maxn][21];
void dfs(int u,int fa) {
    d[u]=d[fa]+1;//更新深度,父親的深度加1
    f[u][0]=fa;//跳2^0步到爸爸
    for(int i=1; (1<<i)/*即2^i*/<=d[u]; ++i) {
        f[u][i]=f[f[u][i-1]][i-1];//敲黑板,看這裏,解釋放博客裏面
    }
    for(int i=head[u]; i!=-1; i=e[i].next) {
        int v=e[i].v;
        if(v!=fa)//防止遞歸到爸爸
            dfs(v,u);//遞歸下一層
    }
}
int lca(int a,int b) {
    if(d[a]>d[b]) swap(a,b);
    for(int i=20; i>=0; --i) {
        if(d[a]<=d[b]-(1<<i)) b=f[b][i];
    }
    if(a==b) return a;
    for(int i=20; i>=0; --i) {
        if(f[a][i]==f[b][i]) continue;
        else a=f[a][i],b=f[b][i];
    }
    return f[a][0];
}
int main() {
    memset(head,-1,sizeof(head));
    cin>>n>>m>>root;
    for(int i=1; i<n; ++i) {
        int x,y;
        scanf("%d%d",&x,&y);
        add(x,y);
        add(y,x);
    }
    dfs(root,0);
    while(m--) {
        int a,b;
        scanf("%d%d",&a,&b);
        cout<<lca(a,b)<<'\n';
    }
    return 0;
}

bfs

#include<iostream>
#include<cstdio>
#include<algorithm>
#include<cmath>
#include<queue>
#include<stack>
#include<vector>
#include<map>
#include<string>
#include<cstring>
using namespace std;
inline int read() {
    char c = getchar();
    int x = 0, f = 1;
    while(c < '0' || c > '9') {
        if(c == '-') f = -1;
        c = getchar();
    }
    while(c >= '0' && c <= '9') x = x * 10 + c - '0', c = getchar();
    return x * f;
}
const int maxn=500005;
const int inf=0x7fffffff;
struct Edge {
    int ne,to;
} edge[maxn<<1];
int h[maxn],num_edge=0,t;
inline void add_edge(int f,int to) {
    edge[++num_edge].ne=h[f];
    edge[num_edge].to=to;
    h[f]=num_edge;
    return ;
}
int f[maxn][35],d[maxn];
int n,m,s;
void bfs() {
    int u,v;
    memset(d,0,sizeof(d));
    queue <int>q;
    q.push(s);
    d[s]=1;
    while(q.size()) {
        u=q.front();
        q.pop();
        for(int  i=h[u]; i; i=edge[i].ne) {
            v=edge[i].to;
            if(d[v])continue;
            f[v][0]=u,d[v]=d[u]+1;
        
            for(int  j=1; j<=t; j++) {
                f[v][j]=f[f[v][j-1]][j-1];
            }
            q.push(v);
        }
    }
    return ;
}
inline int lca(int x,int y) {
    if(d[x]<d[y])swap(x,y);
    if(x==y)return x;
    for(int  i=t; i>=0; i--) {
        if(d[f[x][i]]>=d[y])x=f[x][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() {
    int x,y,z;
    n=read(),m=read(),s=read();
    memset(f,0,sizeof(f));
    t=(int)(log(n)/log(2))+1;
    for(int  i=1; i<n; i++) {
        x=read();
        y=read();
        add_edge(x,y);
        add_edge(y,x);
    }
    bfs();
    for(int  i=1; i<=m; i++) {
            x=read();
        y=read();
        printf("%d\n",lca(x,y));
    }
    return 0;
}

樹剖求LCA

解釋同上之後再寫

// luogu-judger-enable-o2
#include<iostream>
#include<cstdio>
#include<algorithm>
#include<cmath>
#include<queue>
#include<stack>
#include<vector>
#include<map>
#include<string>
#include<cstring>
using namespace std;

inline int read() {
    char c = getchar();
    int x = 0, f = 1;
    while(c < '0' || c > '9') {
        if(c == '-') f = -1;
        c = getchar();
    }
    while(c >= '0' && c <= '9') x = x * 10 + c - '0', c = getchar();
    return x * f;
}
const int N=500001;
int n,m,root,cnt;
struct node
{
    int u,v,next;
}e[N<<1];
int head[N];
int son[N];
int f[N],d[N],s[N],c[N];
void add(int x,int y)
{
    e[++cnt].u=x;
    e[cnt].v=y;
    e[cnt].next=head[x];
    head[x]=cnt;
}
void dfs1(int n,int F)
{
    s[n]=1;//初始化子樹大小爲1
    d[n]=d[F]+1;
    f[n]=F; 
    int V;
    for(int i=head[n];i;i=e[i].next)
    {
        V=e[i].v;
        if(V!=F)
        {
            dfs1(V,n);
            s[n]+=s[V];
            if(s[son[n]]<s[V]) son[n]=V;
        }
    }
}
void dfs2(int n,int C)
{
    c[n]=C;
    if(son[n]) dfs2(son[n],C);
    else return;
    int V;
    for(int i=head[n];i;i=e[i].next)
    {
        V=e[i].v;
        if(V!=f[n] && V!=son[n]) dfs2(V,V);
    }
}
int lca(int x,int y)
{
    for(;c[x]!=c[y];x=f[c[x]])
    {
        if(d[c[x]]<d[c[y]]) swap(x,y);
    }
    return d[x]<d[y]? x: y;
}
int main() 
{
    n=read();m=read();root=read();
    for(int i=1;i<n;++i)
    {
        int x,y;
        x=read();
        y=read();
        add(x,y);
        add(y,x);   
    }
    dfs1(root,0);
    dfs2(root,root);
    for(int i=1;i<=m;++i)
    {
        int x,y;
        x=read();
        y=read();
        cout<<lca(x,y)<<'\n';
    }
}