{"id":83,"date":"2020-03-27T23:30:44","date_gmt":"2020-03-27T15:30:44","guid":{"rendered":"http:\/\/47.103.223.234:100\/?p=83"},"modified":"2021-05-01T16:09:29","modified_gmt":"2021-05-01T08:09:29","slug":"%e5%90%8e%e7%bc%80%e6%95%b0%e7%bb%84","status":"publish","type":"post","link":"http:\/\/www.zyhcoding.club\/index.php\/2020\/03\/27\/%e5%90%8e%e7%bc%80%e6%95%b0%e7%bb%84\/","title":{"rendered":"\u540e\u7f00\u6570\u7ec4\u539f\u7406 \u6a21\u677f"},"content":{"rendered":"

\u540e\u7f00\u6570\u7ec4<\/h2>\n

\u987e\u540d\u601d\u4e49\uff0c\u540e\u7f00\u6570\u7ec4\u5c31\u662f\u5bf9\u4e00\u4e2a\u5b57\u7b26\u4e32\u7684\u6bcf\u4e2a\u540e\u7f00\u7684\u6392\u540d\uff0c\u8fd9\u4e2a\u6392\u540d\u662f\u4ece\u5c0f\u5230\u5927\u6392\u7684\uff0c\u663e\u7136\u4e0d\u4f1a\u6709\u4e24\u4e2a\u540e\u7f00\u76f8\u7b49\u3002
\n\u90a3\u4e48\u4ecb\u7ecd\u51e0\u4e2a\u53d8\u91cf\uff1a
\n\u9996\u5148\u6bcf\u4e2a\u540e\u7f00\u6211\u4eec\u7528\u5b83\u7b2c\u4e00\u4e2a\u5b57\u7b26\u7684\u4e0b\u6807\u6765\u552f\u4e00\u7684\u8868\u793a\u4e86\u3002<\/p>\n

\n

SA[]:\u8fd9\u5c31\u662f\u6700\u7ec8\u6240\u5f97\u5230\u7684\u6570\u7ec4\uff0cSA[i]\u8868\u793a\u7b2ci\u5c0f\u7684\u540e\u7f00\u3002\u5b83\u6ee1\u8db3Suffix(SA[i])<Suffix(SA[i+1])\u3002\u6570\u7ec4\u503c\u4e00\u76f4\u66f4\u65b0\u3002
\nRank[]:\u540d\u6b21\u6570\u7ec4\uff0cRank[i]\u8868\u793a\u7684\u662f\u4ee5i\u5f00\u5934\u7684\u540e\u7f00\u662f\u7b2c\u51e0\u5927\u3002Rank\u548cSA\u4e92\u9006\u3002\u6570\u7ec4\u503c\u4e00\u76f4\u66f4\u65b0\u3002<\/strong><\/p>\n<\/blockquote>\n

\u8fd9\u4e24\u4e2a\u6570\u7ec4\uff0c\u4e00\u4e2a\u662f\u6839\u636e\u6392\u540d\u5927\u5c0f\u5bf9\u540e\u7f00\u4e0b\u6807\u6392\u5e8f\uff0c\u4e00\u4e2a\u662f\u6839\u636e\u540e\u7f00\u4e0b\u6807\u5927\u5c0f\u5bf9\u6392\u540d\u6392\u5e8f\uff0c\u4e8c\u8005\u90fd\u662f\u4ece\u5c0f\u5230\u5927\u6392\u3002
\n\u8fd9\u662f\u4e24\u4e2a\u6700\u4e3b\u8981\u7684\u53d8\u91cf\uff0c\u4e0b\u9762\u5728\u5904\u7406\u7ec6\u8282\u65f6\u8fd8\u4f1a\u7528\u5230\u4e00\u4e9b\u4e2d\u95f4\u53d8\u91cf\uff0c\u7b49\u7528\u5230\u7684\u65f6\u5019\u5728\u5177\u4f53\u4ecb\u7ecd\u3002
\n\u62ff\u4e00\u5f20\u88ab\u76d7\u4e86\u597d\u591a\u6b21\u7684\u56fe\u6765\u89e3\u91ca\u8fd9\u4e24\u4e2a\u53d8\u91cf\uff1a
\n\"\u4e24\u6570\u7ec4\u5173\u7cfb\"<\/p>\n

\u500d\u589e<\/h4>\n

\u63a5\u4e0b\u6765\u8bb2\u8fd9\u4e2a\u7b97\u6cd5\u5927\u81f4\u5b9e\u73b0\u7684\u8fc7\u7a0b\uff1a\u4e3b\u8981\u601d\u60f3\u662f\u500d\u589e\u7b97\u6cd5\u3002
\n\u7528\u500d\u589e\u7684\u65b9\u6cd5\u5bf9\u4ece\u6bcf\u4e2a\u5b57\u7b26\u5f00\u59cb\u7684\u957f\u5ea6\u4e3a2^{k}<\/code>\u7684\u5b50\u4e32\u8fdb\u884c\u6392\u5e8f\uff0c\u6bcf\u4e2a\u5b50\u4e32\u7684\u7ed3\u5c3e\u4e0d\u8981\u8d85\u8fc7\u6bcd\u4e32\u7684\u7ed3\u5c3e\u65f6\u3002\u8fd9\u6837\u5c31\u53ef\u4ee5\u7528O(nlogn)\u7684\u590d\u6742\u5ea6\u5bf9\u6240\u6709\u540e\u7f00\u6392\u5e8f\u4e86\u3002\u5728\u6bcf\u4e00\u6b21\u5bf9\u957f\u5ea6\u4e3a2^{k-1}<\/code>\u7684\u5b50\u4e32\u6392\u597d\u5e8f\u4e4b\u540e\uff0c\u5f97\u5230\u4e00\u4e9b\u5173\u952e\u5b57(\u5373\u6392\u5e8f\u7684\u540d\u6b21)\uff0c\u8fd9\u4e9b\u5173\u952e\u5b57\u4f5c\u4e3a\u7b2c\u4e00\u5173\u952e\u5b57\uff0c\u7136\u540e\u8be5\u5b57\u7b26\u540e\u7684\u7b2c2^{k-1}<\/code>\u4e2a\u5b57\u7b26\u7684\u5173\u952e\u5b57\u4f5c\u4e3a\u7b2c\u4e8c\u5173\u952e\u5b57\uff0c\u6839\u636e\u8fd9\u4e24\u4e2a\u5173\u952e\u5b57\u8fdb\u884c\u6392\u5e8f(\u7b2c\u4e00\u5173\u952e\u5b57\u7684\u4f18\u5148\u7ea7\u5927\u4e8e\u7b2c\u4e8c)\u3002\u8fd9\u6837\u5c31\u57282^{k-1}<\/code>\u7684\u57fa\u7840\u4e0a\u5b8c\u6210\u4e862^{k}<\/code>\u7684\u6392\u5e8f\u3002\u82e5\u8d85\u8fc7\u4e86\u4e32\u5c3e\uff0c\u5219\u7b2c\u4e8c\u5173\u952e\u5b57\u4e3a0\u3002
\n\"\u500d\u589e\u65b9\u6cd5\"
\n\u8fd9\u6837\u5c31\u53ef\u4ee5\u5728O(nlogn)\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e0b\u5b8c\u6210\u6240\u6709\u540e\u7f00\u7684\u6392\u5e8f\u3002\u5bb9\u6613\u53d1\u73b0\uff0c\u5728\u7b2c\u4e00\u4e2a\u5b57\u7b26\u4e3a\u9996\u7684\u5b50\u4e32\u5230\u8fbe\u4e32\u5c3e\u65f6\uff0c\u6240\u6709\u5b50\u4e32\u7684\u540d\u6b21\u4e00\u5b9a\u4f1a\u5404\u4e0d\u76f8\u540c\uff0c\u4e2d\u95f4\u8fc7\u7a0b\u4e2d\u53ef\u80fd\u4f1a\u6709\u76f8\u540c\u7684\u51fa\u73b0\u3002<\/p>\n

\u63a5\u4e0b\u6765\u518d\u6765\u770b\u5bf9\u4e24\u4e2a\u5173\u952e\u5b57\u6392\u5e8f\u7684\u95ee\u9898\uff0c\u65b9\u6cd5\u5f88\u591a\uff0c\u6bd4\u5982\u5199\u8fdb\u4e00\u4e2a\u7ed3\u6784\u4f53\u76f4\u63a5sort\uff0c\u4f46\u662f\uff0c\u52a0\u4e0a\u9700\u8981\u6392\u5e8f\u7684\u6b21\u6570\u548c\u6bcf\u6b21\u6392\u5e8f\u7684\u65f6\u95f4\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3an(logn)^2<\/code>\uff0c\u8fd9\u6837\u8fd8\u662f\u592a\u6162\u4e86\uff0c\u7528sort\u672a\u514d\u4f1a\u6709\u4e9b\u5c0f\u9898\u5927\u505a\uff0c\u56e0\u4e3a\u53ea\u6709\u4e24\u4e2a\u5173\u952e\u5b57\uff0c\u4e14\u6570\u636e\u7684\u8303\u56f4\u4e5f\u662f\u53ef\u4ee5\u786e\u5b9a\u7684(\u6392\u540d\u7684\u540d\u6b21\uff0c\u6570\u503c\u5f53\u7136\u4e0d\u4f1a\u592a\u5927)\uff0c\u5f53\u6570\u636e\u4e0d\u5927\u65f6\uff0c\u4e0d\u7981\u60f3\u5230\u4e86\u4e00\u79cd\u4eba\u89c1\u4eba\u7231\u7684\u6392\u5e8f\u65b9\u6cd5--"\u6876"\u6392\u3002\u6240\u4ee5\u8fd9\u91cc\u4ecb\u7ecd\u4e00\u79cd\u548c\u6876\u6392\u6709\u5173\u7cfb\u7684\u65b0\u7b97\u6cd5---\u57fa\u6570\u6392\u5e8f\u3002
\n\u53d8\u91cf\u540d\uff1a<\/p>\n

\n

Rank[]\uff1a\u7b2c\u4e00\u5173\u952e\u5b57\uff0c\u8868\u793a\u7b2ci\u4e2a\u4e0b\u6807\u7684\u6392\u540d\uff0c\u5b58\u7684\u662f\u6392\u540d\u540d\u6b21\u3002\u6570\u7ec4\u503c\u4e00\u76f4\u66f4\u65b0\u3002
\ntp[]\uff1a\u5b58\u653e\u7684\u662f\u7b2c\u4e8c\u5173\u952e\u5b57\u4e2d\uff0c\u6392\u540d\u4e3ai\u7684\u4e0b\u6807\uff0c\u6ce8\u610f\u5b58\u7684\u662f\u4e0b\u6807\u3002\u6570\u7ec4\u503c\u4e00\u76f4\u66f4\u65b0\u3002<\/strong><\/p>\n<\/blockquote>\n

\u5728\u6bcf\u6b21\u57fa\u6570\u6392\u5e8f\u4e4b\u524d\uff0c\u4e00\u5b9a\u6ce8\u610f\uff0c\u5b58\u653e\u7b2c\u4e00\u5173\u952e\u5b57\u7684\u6570\u7ec4Rank\u5b58\u7684\u662f\u6392\u540d\u540d\u6b21\uff0c\u5b58\u653e\u7b2c\u4e8c\u5173\u952e\u5b57\u7684\u6570\u7ec4tp\u5b58\u653e\u7684\u662f\u4e0b\u6807\u3002
\n\u518d\u6b21\u58f0\u660eRSort\u7684\u4f5c\u7528\uff1a\u4e24\u4e2a\u5b58\u653e\u7b2c\u4e00\u4e8c\u5173\u952e\u5b57\u7684\u6570\u7ec4ab\u90fd\u6ca1\u6539\u53d8\uff0c\u6539\u53d8\u7684SA\uff0cSA\u66f4\u65b0\u4e3a\u5efa\u7acb\u5728ab\u7684\u57fa\u7840\u4e0a\u7684\u65b0\u7684\u7b2c\u4e00\u5173\u952e\u5b57\uff01<\/strong>
\n\u57fa\u6570\u6392\u5e8f\u5176\u5b9e\u8fd8\u4e0d\u662f\u7279\u522b\u6e05\u6670\uff0c\u4ee5\u540e\u518d\u5199\u5427\u3002\u3002\u3002\u3002\u3002<\/p>\n

height\u6570\u7ec4<\/h4>\n

\u8fd8\u6709\u4e00\u4e2a\u91cd\u8981\u5e94\u7528---height\u6570\u7ec4<\/p>\n

\n

height[]:height[i]\u8868\u793aSuffix[SA[i]]\u548cSuffix[SA[i-1]]\u7684LCP\uff0c\u5373\u5927\u5c0f\u76f8\u90bb\u7684\u4e24\u4e2a\u540e\u7f00\u7684\u6700\u5927\u516c\u5171\u524d\u7f00\u3002\u6ce8\u610f\u662f\u548c\u524d\u4e00\u540d\u3002
\nH[]:H[i]=Height[Rank[i]],\u5373\u4e0b\u6807\u4e3ai\u7684\u540e\u7f00\u548c\u6392\u5728\u5b83\u524d\u4e00\u4f4d\u7684\u540e\u7f00\u7684LCP<\/strong><\/p>\n<\/blockquote>\n

\u91cd\u8981\u6027\u8d28\uff1aH[i] >= H[i-1]+1
\n\u8bc1\u660e\uff1a\u6211\u4eec\u5047\u8bbeSuffix[i-1]\u7684\u6392\u540d\u4e0a\u4e00\u4f4d\u7684\u540e\u7f00\u4e3aSuffix[k1]\uff0cSuffix[i]\u7684\u6392\u540d\u4e0a\u4e00\u4f4d\u7684\u540e\u7f00\u4e3aSuffix[k2]\uff0c\u73b0\u5728\u53bb\u6389Suffix[i-1]\u7684\u7b2c\u4e00\u4e2a\u5b57\u7b26\uff0c\u53d8\u6210\u4e86Suffix[i]\uff0cSuffix[i]\u548cSuffix[k1]\u7684LCP\u4e3aH[i-1]-1\uff0c\u8fd9\u4e2a\u662f\u663e\u7136\u6613\u89c1\u7684\u3002\u6211\u4eec\u8fd8\u77e5\u9053\u4ec0\u4e48\u5462\uff1fSuffix[k2]\u53ef\u80fd\u4e0d\u7b49\u4e8eSuffix[k1]\uff0c\u4f46\u662fSuffix[k2]\u548cSuffix[i]\u7684LCP\u4e00\u5b9a\u4e0d\u6bd4Suffix[k1]\u548cSuffix[i]\u7684LCP\u5c0f\u3002\u6240\u4ee5\u6709H[i]>=H[i-1]+1...\u5c31\u8fd9\u6837\u3002\u5229\u7528\u7684\u6027\u8d28\u5c31\u662f\uff0c\u6392\u540d\u8d8a\u76f8\u8fd1\u7684\u4e24\u4e2a\u540e\u7f00\uff0c\u5b83\u4eec\u7684LCP\u5c31\u8d8a\u5927\u3002
\n\u90a3\u4e48\u6709\u521d\u59cb\u6761\u4ef6height[1] = 0,\u53ef\u4ee5\u7528O(n)\u7684\u590d\u6742\u5ea6\u6c42\u51faheight
\n\u548c\u6c42kmp\u4e2dnext\u7684\u4f5c\u6cd5\u6709\u70b9\u50cf\u3002\u3002<\/p>\n

void GetHeight(){\n    int j,k=0;\n    for (int i=1;i<=n;i++){\n        if (k)  k--;\n        j = SA[Rank[i]-1];\n        while (s[i+k]==s[j+k])  k++;\n        Height[Rank[i]] = k;\n    }\n}<\/code><\/pre>\n
\u5f88\u7ed5\uff0c\u4ee5\u540e\u8981\u591a\u6765\u770b\u770b\uff0c\u591a\u60f3\u60f3\u7ec6\u8282\uff01<\/p>\n
#include <cstdio>\n#include <cstring>\n#include <algorithm>\nusing namespace std;\nconst int maxn = 1e6+5;\nint SA[maxn],Rank[maxn],tp[maxn],tax[maxn];\nchar s[maxn];\nint n,m;\nvoid GetSA(int a[],int b[]);       \/\/a \u7b2c\u4e00\u5173\u952e\u5b57    b \u7b2c\u4e8c\u5173\u952e\u5b57\nvoid RSort(int a[],int b[]);\nint main(){\n    scanf ("%s",s+1);\n    n = strlen(s+1);\n    GetSA(Rank,tp);\n    for (int i=1;i<=n;i++){\n        printf ("%d",SA[i]);\n        if (i!=n)   putchar(' ');\n    }\n    return 0;\n}\n\nvoid GetSA(int a[],int b[]){\n    for (int i=1;i<=n;i++)      a[i] = s[i],b[i] = i;   \/\/\u7b2c\u4e00\u5173\u952e\u5b57\u5c31\u5148\u7b49\u4e8e\u8fd9\u4e2a\u5b57\u7b26\u7684\u5927\u5c0f\uff0c\u7b2c\u4e8c\u5173\u952e\u5b57\u7531\u4e8e\u90fd\u662f0\uff0c\u6240\u4ee5\u6392\u5e8f\u540e\u5c31\u6309\u4e0b\u6807\u5927\u5c0f\u7ed9\u4e86\n    m = 130;             \/\/m\u7684\u804c\u8d23\u662f \u6bcf\u6b21\u6392\u540d\u540e\u540d\u6b21\u7684\u6700\u5927\u503c\n    RSort(a,b);\n    for (int p,k=1;p!=n;k<<=1,m=p){    \/\/m==n\u540e\uff0c\u6392\u5e8f\u7ed3\u675f\uff01   \/\/\u5728\u6bcf\u4e00\u8f6e\u5faa\u73af\u7ed3\u675f\u540e\uff0cSA\u503c\u5df2\u66f4\u65b0\u4e3a\u6700\u65b0\u7684\u7b2c\u4e00\u5173\u952e\u5b57\uff0c\u7136\u540e\u5c31\u8be5\u66f4\u65b0b\u7b2c\u4e8c\u5173\u952e\u5b57\u4e86\u3002\n        p = 0;\n        for (int i=n+1-k;i<=n;i++)  b[++p] = i;\n        for (int i=1;i<=n;i++)  if (SA[i]>k)    b[++p] = SA[i]-k;     \/\/\u65e2\u7136SA\u4e2d\u6709\u5df2\u7ecf\u6392\u5e8f\u597d\u7684\u7b2c\u4e00\u5173\u952e\u5b57\uff0c\u90a3\u7b2c\u4e8c\u5173\u952e\u5b57\u81ea\u7136\u5c31\u987a\u5e8f\u7684\u4ece\u4e2d\u6311\u9009\u5c31\u884c\u4e86\u3002\n        RSort(a,b);     swap(a,b);                                              \/\/1\n        a[SA[1]] = p = 1;\n        for (int i=2;i<=n;i++)\n            a[SA[i]] = (b[SA[i]]==b[SA[i-1]] && b[SA[i]+k]==b[SA[i-1]+k])?p:++p;    \/\/1\u4e0d\u662f\u7279\u522b\u6e05\u695a\uff01\n    }\n    return;\n}\nvoid RSort(int a[],int b[]){   \/\/RSort\u524d\uff0ca\u548cSA\u90fd\u662f\u7b2c\u4e00\u5173\u952e\u5b57(\u4e0d\u540c\u7248\u672c)\uff0c\u6392\u5e8f\u540e\uff0cSA\u91cc\u7684\u7b2c\u4e00\u5173\u952e\u5b57\u66f4\u65b0\u4e86\uff0cab\u90fd\u6ca1\u53d8\u3002\n    for (int i=1;i<=m;i++)  tax[i] = 0;\n    for (int i=1;i<=n;i++)  tax[a[i]]++;       \/\/\u8fd9\u4e00\u6b65\uff0c\u7ecf\u6d4b\u8bd5\uff0ctax[a[b[i]]]++;\u4e5f\u6b63\u786e\uff0c\u636e\u8bf4\u8fd8\u662f\u6b63\u89e3\u3002\u3002\n    for (int i=1;i<=m;i++)  tax[i] += tax[i-1];\n    for (int i=n;i>=1;i--)  SA[tax[a[b[i]]]--] = b[i];               \/\/2\u4e00\u4e2a\u70b9\u672a\u641e\u5b9a\n}<\/code><\/pre>\n
\u6a21\u677f<\/h2>\n
\u8fd9\u91cc\u518d\u8d34\u4e00\u4e0b\u5b9e\u6218\u65f6\u4f1a\u4f7f\u7528\u7684\u6700\u65b0\u7684\u677f\u5b50<\/p>\n
#include <cstdio>\n#include <cstring>\n#include <algorithm>\nusing namespace std;\nconst int maxn = 2e4+5;\ntypedef struct SA{\n   int SA[maxn],Rank[maxn],tp[maxn],tax[maxn],Height[maxn];\n   int n,m;\n   char s[maxn];\n   void build(){\n      m = 130;\n      n = strlen(s+1);\n      GetSA(Rank,tp);\n      GetHeight();\n   }\n   void GetSA(int a[],int b[]);\n   void RSort(int a[],int b[]);\n   void GetHeight();\n   int check(int i);\n}SA;\nSA a;\nint ans;\nint main(){\n   scanf ("%s",a.s+1);\n   a.build();\n   return 0;\n}\nvoid SA::GetHeight(){\n   int j,k=0;\n   for (int i=1;i<=n;i++){\n      if (k)   k--;\n      j = SA[Rank[i]-1];\n      while (s[i+k]==s[j+k])  k++;\n      Height[Rank[i]] = k;\n   }\n   return;\n}\nvoid SA::GetSA(int a[],int b[]){\n   for (int i=1;i<=n;i++)  a[i] = s[i],b[i] = i;\n   RSort(a,b);\n   int p=0,k=1;\n   for (;p!=n;k<<=1,m=p){\n      p = 0;\n      for (int i=n+1-k;i<=n;i++)   b[++p] = i;\n      for (int i=1;i<=n;i++)  if (SA[i]>k)   b[++p] = SA[i]-k;\n      RSort(a,b);    swap(a,b);\n      a[SA[1]] = p = 1;\n      for (int i=2;i<=n;i++)\n         a[SA[i]] = (b[SA[i]]==b[SA[i-1]] && b[SA[i]+k]==b[SA[i-1]+k])?p:++p;\n   }\n   for (int i=1;i<=n;i++)  Rank[SA[i]] = i;          \/\/\u8fd9\u4e00\u6b65\uff01\u56e0\u4e3aGetSA\u51fd\u6570\u53ea\u80fd\u4fdd\u8bc1SA\u6570\u7ec4\u5b8c\u5168\u6b63\u786e\uff0c\u5728\u8d4b\u503c\u5b8c\u6210\u540e\u6700\u597d\u5728\u7ed9Rank\u6570\u7ec4\u518d\u8d4b\u4e00\u6b21\u503c\n   return;\n}\nvoid SA::RSort(int a[],int b[]){\n   for (int i=1;i<=m;i++)  tax[i] = 0;\n   for (int i=1;i<=n;i++)  tax[a[b[i]]]++;\n   for (int i=1;i<=m;i++)  tax[i] += tax[i-1];\n   for (int i=n;i>=1;i--)  SA[tax[a[b[i]]]--] = b[i];\n   return;\n}<\/code><\/pre>\n
\u5e94\u7528<\/h2>\n
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\u4efb\u610f\u4e24\u4e2a\u540e\u7f00\u7684LCP<\/strong><\/p>\n<\/blockquote>\n
\u5728SA\u4e0a\u5efa\u7acb\u7ebf\u6bb5\u6811\uff0c\u5047\u5b9a\u4e24\u4e2a\u540e\u7f00\u4e3ai\uff0cj\uff0c\u90a3\u4e48\u5b83\u4eec\u7684LCP\u5e94\u4e3amin{Height[Rank[i]~Rank[j]]}.\u4e5f\u5c31\u662f\u4ed6\u4fe9\u7684\u6392\u540d\u6240\u6784\u6210\u7684\u8303\u56f4\u5185\uff0cHeight\u7684\u6700\u5c0f\u503c\u3002
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\u4e0d\u53ef\u91cd\u53e0\u6700\u957f\u91cd\u590d\u5b50\u4e32<\/strong> POJ1743<\/a><\/p>\n<\/blockquote>\n
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\u672c\u8d28\u4e0d\u540c\u7684\u5b50\u4e32<\/strong><\/p>\n<\/blockquote>\n
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