{"id":57627,"date":"2025-08-15T16:39:52","date_gmt":"2025-08-15T08:39:52","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/57627.html"},"modified":"2025-08-15T16:39:52","modified_gmt":"2025-08-15T08:39:52","slug":"%e3%80%90%e5%b0%8f%e7%99%bd%e5%ad%a6%e7%ae%97%e6%b3%95%e3%80%91lca%e5%80%8d%e5%a2%9e%e6%b3%95%e6%b1%82%e5%85%ac%e5%85%b1%e7%a5%96%e5%85%88%e8%b6%85%e8%af%a6%e7%bb%86%e8%a7%a3%e6%9e%90%e4%be%8b","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/57627.html","title":{"rendered":"\u3010\u5c0f\u767d\u5b66\u7b97\u6cd5\u3011LCA\u500d\u589e\u6cd5\u6c42\u516c\u5171\u7956\u5148\u8d85\u8be6\u7ec6\u89e3\u6790+\u4f8b\u9898[\u6d1b\u8c37]P3379\u6700\u8fd1\u516c\u5171\u7956\u5148\uff08LCA\uff09"},"content":{"rendered":"

\u7b97\u6cd5\u4ecb\u7ecd<\/h3>\n

\u4ec0\u4e48\u662fLCA？\u4e3e\u4e2a\u7b80\u5355\u7684\u4f8b\u5b50，\u60f3\u8c61\u4e00\u4e0b\u4f60\u7684\u5bb6\u65cf\u65cf\u8c31，\u4f60\u548c\u4f60\u7684\u8868\u54e5\u60f3\u627e\u5230\u4f60\u4eec\u6700\u8fd1\u7684\u516c\u5171\u7956\u5148，\u53ef\u80fd\u662f\u4f60\u7684\u7237\u7237\u5976\u5976，\u4e5f\u53ef\u80fd\u662f\u66f4\u65e9\u7684\u7956\u5148，\u800c\u6700\u8fd1\u516c\u5171\u7956\u5148\u5c31\u662f\u8ddd\u79bb\u4f60\u4eec\u6700\u8fd1\u7684\u90a3\u4e2a\u5171\u540c\u7956\u8f88，LCA\u7b97\u6cd5\u5c31\u662f\u5728\u4e00\u68f5\u6811\u4e2d，\u627e\u5230\u4e24\u4e2a\u8282\u70b9\u6700\u8fd1\u7684\u516c\u5171\u7956\u5148\u8282\u70b9，\u5bf9\u4e8e\u6811\u4e2d\u7684\u4e24\u4e2a\u8282\u70b9u\u548cv，\u4ed6\u4eec\u7684lca\u662f\u6ee1\u8db3\u4ee5\u4e0b\u4e09\u4e2a\u6761\u4ef6\u7684\u8282\u70b9w：<\/p>\n

1. w\u662fu\u7684\u7956\u5148；<\/p>\n

2. w\u662fv\u7684\u7956\u5148；<\/p>\n

3. \u5728\u6ee1\u8db3\u524d\u4e24\u4e2a\u6761\u4ef6\u7684\u6240\u6709\u8282\u70b9\u4e2d，w\u7684\u6df1\u5ea6\u6700\u5927（\u5373\u6700\u6df1\u7684\u516c\u5171\u7956\u5148）<\/p>\n

\u6838\u5fc3\u7b97\u6cd5<\/h3>\n

\u57fa\u672c\u601d\u8def<\/h4>\n

1. \u5206\u522b\u627e\u5230\u4e24\u4e2a\u8282\u70b9\u5230\u6839\u8282\u70b9\u7684\u8def\u5f84<\/p>\n

2. \u6bd4\u8f83\u4e24\u6761\u8def\u5f84，\u627e\u5230\u6700\u540e\u4e00\u4e2a\u76f8\u540c\u7684\u8282\u70b9<\/p>\n

\u5177\u4f53\u6b65\u9aa4<\/h4>\n

1\u3001\u56fe\u7684\u5904\u7406，\u5148\u7528dfs\u904d\u5386\u56fe，\u5b58\u4e0b\u5404\u4e2a\u70b9\u7684\u6df1\u5ea6，\u4ee5\u53ca\u5404\u4e2a\u70b9\u500d\u589e\u8df3\u8f6c\u6570\u7ec4<\/h5>\n

```Objective-C++<\/p>\n

void dfs(int u,int fa) \/\/ u\u4e3a\u5f53\u524d\u8282\u70b9,fa\u4e3a\u7236\u8282\u70b9
\n{
\nf[u][0] = fa;
\nd[u] = d[fa]+1;
\nst[u]=g++;
\nfor(int i=1;(1<<i)<=d[u];i++)
\nf[u][i] = f[f[u][i-1]][i-1]; \/\/\u9884\u5904\u7406f\u6570\u7ec4
\n\/\/\u5b58u\u7684\u500d\u589e\u6570\u7ec4<\/p>\n

for(int i=head[u];i!=-1;i=e[i].next){
\nint v = e[i].to;
\nif(v!=fa)
\n dfs(v,u);\/\/\u904d\u5386u\u7684\u5b69\u5b50，\u7ee7\u7eed\u8fdb\u5165dfs\u9884\u5904\u7406\u500d\u589e\u6570\u7ec4
\n}
\n}<\/p>\n

\u500d\u589e\u6cd5：\u80fd\u8df3\u8fdc\u5c31\u8df3\u8fdc，\u80fd\u8df3\u8fd1\u5c31\u8df3\u8fd1；<\/p>\n

\u9884\u5904\u7406：\u5bf9\u6bcf\u4e2a\u8282\u70b9\u9884\u8ba1\u7b97\u5b83\u7684<\/p>\n

2<\/p>\n

0<\/p>\n

2^0<\/p>\n

<\/span><\/span>2<\/span><\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3001<\/p>\n

2<\/p>\n

1<\/p>\n

2^1<\/p>\n

<\/span><\/span>2<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3001<\/p>\n

2<\/p>\n

2<\/p>\n

2^2<\/p>\n

<\/span><\/span>2<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2026\u2026\u7ea7\u7684\u7956\u5148，\u4fbf\u4e8e\u67e5\u8be2\u65f6\u5229\u7528\u4e8c\u8fdb\u5236\u601d\u60f3，\u5feb\u901f\u8df3\u5230LCA\u3002<\/p>\n

2\u3001\u6c42u\u548cv\u7684lca，\u9996\u5148\u5c06u\u548cv\u8df3\u8f6c\u5230\u540c\u4e00\u4e2a\u6df1\u5ea6，\u7279\u5224\u4e00\u4e0b\u540e，\u5f00\u59cb\u627e\u516c\u5171\u7956\u5148<\/h5>\n

```Objective-C++
\nint lca(int x,int y){
\nif(d[x]>d[y]) \/\/\u4ee4y\u4e3a\u66f4\u6df1\u8282\u70b9
\nswap(x,y);
\nfor(int i=20;i>=0;i–){
\nif(d[y]-(1<<i)>=d[x])
\ny=f[y][i];\/\/\u6539\u53d8y \u4f7fy\u4e0ex\u5728\u540c\u4e00\u6df1\u5ea6
\n}
\nif(x==y) return y;\/\/\u7279\u5224，\u6b64\u65f6\u82e5x=y，\u5219\u8bf4\u660ex\u6070\u597d\u662fy\u7684\u7956\u5148\u2014\u2014>\u7279\u6b8a\u60c5\u51b5
\nfor(int i=20;i>=0;i–){
\nif(f[x][i]!=f[y][i])
\nx=f[x][i],y=f[y][i]; \/\/\u4e0d\u76f8\u7b49\u5219\u8bf4\u660e\u8fd8\u6ca1\u6709\u5230\u516c\u5171\u7956\u5148\u90e8\u5206，\u7ee7\u7eed\u4e0a\u8df3，\u76f4\u81f3\u6709\u76f8\u540c\u90e8\u5206
\n}
\nreturn f[x][0];\/\/\u4e0a\u8df31\u6b65\u76f8\u540c，\u5373\u6700\u8fd1\u7684\u516c\u5171\u7956\u5148
\n}<\/p>\n

\u6b65\u9aa4\u89e3\u91ca<\/h3>\n

\u4e3a\u4ec0\u4e48\u8981\u5148\u5bf9\u9f50\u6df1\u5ea6\u5462？<\/p>\n

\u5047\u8bbe\u4e24\u4e2a\u4eba\u7ad9\u5728\u4e0d\u540c\u7684\u697c\u5c42\u7684\u697c\u68af\u4e0a，\u8981\u627e\u5230\u4ed6\u4eec\u6700\u8fd1\u7684\u6c47\u5408\u70b9，\u9996\u5148\u8981\u8ba9\u4ed6\u4eec\u7ad9\u5230\u540c\u4e00\u697c\u5c42；<\/p>\n

\u4e3a\u4ec0\u4e48\u8981\u4ece\u5927\u5230\u5c0f\u904d\u5386i？<\/p>\n

\u5c31\u50cf\u6211\u4eec\u53bb\u8d85\u5e02\u4e70\u4e1c\u897f，\u7528\u4e0d\u540c\u9762\u989d\u7684\u94b1\u5e01\u627e\u96f6\u4e00\u6837，\u4f18\u5148\u4f7f\u7528\u5927\u9762\u989d（\u5927\u6b65\u8df3\u8dc3），\u5269\u4f59\u7684\u7528\u5c0f\u9762\u989d\u6765\u8865\u9f50（\u5c0f\u6b65\u8c03\u6574）\u3002<\/p>\n

\u4e3a\u4ec0\u4e48\u6700\u540e\u8fd4\u56de f[u][0] ?<\/p>\n

\u7ecf\u8fc7\u7b2c\u4e8c\u6b65\u4e4b\u540e，u\u548cv\u5df2\u7ecf\u975e\u5e38\u63a5\u8fd1LCA\u4e86，\u90a3\u4e48\u4ed6\u4eec\u7684\u76f4\u63a5\u7236\u8282\u70b9\u5c31\u662f\u6211\u4eec\u8981\u627e\u7684LCA\u3002<\/p>\n

\u6a21\u677f\u4f8b\u9898<\/h3>\n

P3379 \u3010\u6a21\u677f\u3011\u6700\u8fd1\u516c\u5171\u7956\u5148（LCA）<\/p>\n

\u76f4\u63a5\u5957\u6211\u4e0a\u9762\u6240\u7ed9\u51fa\u7684\u6a21\u677f\u5373\u53ef\u5566~ \u4e0d\u6e05\u695a\u600e\u4e48\u5957\u6a21\u677f\u7684\u53ef\u4ee5\u53c2\u8003\u4e0b\u9762AC\u4ee3\u7801\u793a\u4f8b，\u6211\u4f1a\u4e3a\u5927\u5bb6\u5199\u8be6\u7ec6\u6ce8\u91ca\u561f~<\/p>\n

#include<bits\/stdc++.h>
\nusing namespace std;<\/p>\n

const int MAX_N = 500000+7, MAX_M = 500000+7;<\/p>\n

int n,m,s;
\nint d[MAX_N]; \/\/\u4fdd\u5b58\u8282\u70b9\u6df1\u5ea6
\nint f[MAX_N][21]; \/\/\u4fdd\u5b58\u8282\u70b9\u7684\u4e0d\u540c\u500d\u6570\u7684\u7236\u8282\u70b9
\nint g=1,st[MAX_N];\/\/ dfs\u5e8f||\u65f6\u95f4\u6233<\/p>\n

int head[MAX_N], cnt=-1;\/\/cnt\u8bb0\u5f55\u8fb9\u7684\u6570\u76ee
\nstruct Edge{
\n int to , next;
\n}e[MAX_N*2];<\/p>\n

void add(int u,int v){
\n e[++cnt].to = v;\/\/++cnt\u5148\u5c06cnt\u7684\u503c\u52a01，\u5f97\u5230\u4e00\u4e2a\u65b0\u7684\u7f16\u53f7，e[++cnt].to=v\u628a\u65b0\u8fb9\u7684\u91cd\u70b9\u8bbe\u4e3av
\n e[cnt].next = head[u];\/\/head[u]\u4fdd\u5b58\u7684\u662f\u4ece\u9876\u70b9u\u51fa\u53d1\u7684\u5f53\u524d\u7b2c\u4e00\u6761\u8fb9\u7684\u7f16\u53f7
\n \/\/e[cnt].next=head[u]\u8ba9\u65b0\u6dfb\u52a0\u7684\u8fb9\u6307\u5411\u539f\u6765\u4ece\u9876\u70b9u\u51fa\u53d1\u7684\u7b2c\u4e00\u6761\u8fb9，\u610f\u5473\u7740\u65b0\u8fb9\u6210\u4e3a\u4e86\u4ece\u9876\u70b9u\u51fa\u53d1\u7684\u8fb9\u94fe\u8868\u4e2d\u7684\u65b0\u5934\u90e8\u3002
\n head[u] = cnt;\/\/head[u]\u66f4\u65b0\u4e3a\u65b0\u8fb9\u7684\u7f16\u53f7，\u4e5f\u5c31\u662f\u8ba9\u65b0\u8fb9\u6210\u4e3a\u4ece\u9876\u70b9u\u51fa\u53d1\u7684\u7b2c\u4e00\u6761\u8fb9
\n}\/\/\u5934\u63d2\u6cd5\u5b58\u56fe<\/p>\n

void dfs(int u,int fa) \/\/ u\u4e3a\u5f53\u524d\u8282\u70b9,fa\u4e3a\u7236\u8282\u70b9
\n{
\n f[u][0] = fa;\/\/u\u5411\u4e0a\u8df32^0=1\u6b65\u65f6\u5c31\u662f\u5176\u7236\u8282\u70b9
\n d[u] = d[fa]+1;\/\/\u8be5\u8282\u70b9\u7684\u6df1\u5ea6\u662f\u5176\u7236\u8282\u70b9\u7684\u6df1\u5ea6\u52a0\u4e00
\n st[u]=g++;
\n for(int i=1;(1<<i)<=d[u];i++)
\n f[u][i] = f[f[u][i-1]][i-1];
\n\/\/\u9884\u5904\u7406f\u6570\u7ec4，\u5373u\u500d\u589e\u6570\u7ec4<\/p>\n

for(int i=head[u];i!=-1;i=e[i].next){
\n int v = e[i].to;
\n if(v!=fa)\/\/\u904d\u5386\u56fe，\u904d\u5386\u5f53\u524d\u8282\u70b9\u7684\u5b50\u8282\u70b9，\u7ee7\u7eed\u8fdb\u5165dfs，
\n dfs(v,u);\/\/\u76f4\u81f3\u6240\u6709\u8282\u70b9\u7684\u500d\u589e\u6570\u7ec4\u5168\u90e8\u5904\u7406\u5b8c
\n }
\n}<\/p>\n

int lca(int x,int y){
\n if(d[x]>d[y]) \/\/\u4ee4y\u4e3a\u66f4\u6df1\u8282\u70b9
\n swap(x,y);
\n for(int i=20;i>=0;i–){
\n if(d[y]-(1<<i)>=d[x])
\n y=f[y][i];\/\/\u6539\u53d8y \u4f7fy\u4e0ex\u5728\u540c\u4e00\u6df1\u5ea6
\n }
\n if(x==y) return y;\/\/\u7279\u5224，\u6b64\u65f6\u82e5x=y，\u5219\u8bf4\u660ex\u6070\u597d\u662fy\u7684\u7956\u5148\u2014\u2014>\u7279\u6b8a\u60c5\u51b5
\n for(int i=20;i>=0;i–){
\n if(f[x][i]!=f[y][i])
\n x=f[x][i],y=f[y][i]; \/\/\u4e0d\u76f8\u7b49\u5219\u4e0a\u8df3
\n }
\n return f[x][0];
\n}<\/p>\n

int main()
\n{
\n memset(head,-1,sizeof(head));
\n int a,b;
\n scanf("%d%d%d",&n,&m,&s);
\n for(int i=1;i<n;i++)
\n {
\n scanf("%d%d",&a,&b);
\n add(a,b);
\n add(b,a); \/\/\u65e0\u5411\u56fe，\u8981\u52a0\u4e24\u6b21
\n }
\n dfs(s,0);\/\/\u8fdb\u5165dfs\u8fdb\u884c\u9884\u5904\u7406<\/p>\n

for(int i=1;i<=m;i++)
\n {
\n scanf("%d%d",&a,&b);
\n printf("%d\\\\n",lca(a,b));
\n }<\/p>\n

return 0;
\n}<\/p>\n","protected":false},"excerpt":{"rendered":"

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