什么是LCA
LCA就是最近公共祖先
对于有根树(Tree)的两个结点(u、v),最近公共祖先(LCA(T,u,v))表示一个结点(x),满足(x)是(u、v)的祖先且(x)的深度尽可能大。
倍增求LCA
解释明天再写
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,父亲的深度加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,看这里,解释放博客里面
}
for(int i=head[u]; i!=-1; i=e[i].next) {
int v=e[i].v;
if(v!=fa)//防止递归到爸爸
dfs(v,u);//递归下一层
}
}
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)<<'
';
}
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
",lca(x,y));
}
return 0;
}
树剖求LCA
解释同上以后再写
// 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;//初始化子树大小为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)<<'
';
}
}