\n| dp[i][3]<\/td>\n | \u4ece a[i] \u5f00\u59cb，\u8fde\u7eed 8 \u4e2a\u6570\u7684\u6700\u5927\u516c\u7ea6\u6570<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \u56e0\u6b64，\u7b2c 25 \u9898\u7684\u6b63\u786e\u9009\u9879\u662f：<\/p>\n B：\u4ece a[i] \u5f00\u59cb\u8fde\u7eed 2j2^j \u4e2a\u6570\u7684\u6700\u5927\u516c\u7ea6\u6570\u3002<\/p>\n
\n2. \u5982\u4f55\u4ece\u5c0f\u533a\u95f4\u62fc\u51fa\u5927\u533a\u95f4？<\/h3>\n\u5047\u8bbe\u6211\u4eec\u8981\u8ba1\u7b97\u957f\u5ea6\u4e3a 4 \u7684\u533a\u95f4：<\/p>\n a[i], a[i+1], a[i+2], a[i+3]<\/p>\n \u53ef\u4ee5\u628a\u5b83\u62c6\u6210\u4e24\u4e2a\u957f\u5ea6\u4e3a 2 \u7684\u533a\u95f4：<\/p>\n \u5de6\u8fb9：a[i], a[i+1]
\n\u53f3\u8fb9：a[i+2], a[i+3]<\/p>\n \u5148\u5206\u522b\u6c42\u51fa\u4e24\u8fb9\u7684\u6700\u5927\u516c\u7ea6\u6570，\u518d\u6c42\u5b83\u4eec\u7684\u6700\u5927\u516c\u7ea6\u6570：<\/p>\n dp[i][2] = gcd(dp[i][1], dp[i + 2][1]);<\/p>\n \u4e00\u822c\u60c5\u51b5\u4e0b：<\/p>\n dp[i][j] = gcd(
\n dp[i][j – 1],
\n dp[i + pw[j – 1]][j – 1]
\n);<\/p>\n \u8fd9\u91cc\u7684 pw[j – 1] \u5c31\u662f\u00a0 \u00a0![\"2^{j-1}\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/09\/20260923185651-6ab420f37260b.png\") ，\u8868\u793a\u53f3\u534a\u6bb5\u7684\u8d77\u70b9\u3002<\/p>\n \u8bb0\u5fc6\u65b9\u6cd5：\u5927\u533a\u95f4\u62c6\u6210\u4e24\u4e2a\u4e00\u6837\u957f\u7684\u5c0f\u533a\u95f4，\u518d\u7528 gcd() \u5408\u8d77\u6765\u3002<\/p>\n
\n\u4e94\u3001\u9010\u9898\u8bb2\u89e3<\/h2>\n\u7b2c 22 \u9898：\u533a\u95f4 [2,5] \u7684\u7b54\u6848\u662f\u4e0d\u662f 1？<\/h3>\n\u9898\u76ee\u7ed9\u51fa：<\/p>\n n = 5
\na = {4, 2, 6, 3, 9}
\nL = 2
\nR = 5<\/p>\n \u533a\u95f4 [2,5] \u4e2d\u7684\u6570\u5b57\u662f：<\/p>\n 2, 6, 3, 9<\/p>\n \u8ba1\u7b97：<\/p>\n gcd(2, 6) = 2
\ngcd(2, 3) = 1
\ngcd(1, 9) = 1<\/p>\n \u6240\u4ee5\u8f93\u51fa\u786e\u5b9e\u662f 1\u3002<\/p>\n \u7b54\u6848：\u221a（\u5bf9）\u3002<\/p>\n
\n\u7b2c 23 \u9898：\u5982\u679c\u533a\u95f4\u957f\u5ea6\u4e3a 1，\u8f93\u51fa\u4e00\u5b9a\u7b49\u4e8e a[L] \u5417？<\/h3>\n\u5f53：<\/p>\n L = R<\/p>\n \u533a\u95f4\u957f\u5ea6\u4e3a：<\/p>\n R\u2212L+1=1<\/p>\n \u8fd9\u4e2a\u533a\u95f4\u53ea\u6709\u4e00\u4e2a\u6570：<\/p>\n a[L]<\/p>\n \u7a0b\u5e8f\u67e5\u8be2\u65f6，\u4f7f\u7528：<\/p>\n lg[R – L + 1]<\/p>\n \u6b64\u65f6：<\/p>\n lg[1] = 0<\/p>\n \u56e0\u6b64\u67e5\u8be2\u5c31\u76f8\u5f53\u4e8e：<\/p>\n gcd(dp[L][0], dp[L][0])<\/p>\n \u53c8\u56e0\u4e3a：<\/p>\n dp[L][0] = a[L];<\/p>\n \u6240\u4ee5：<\/p>\n gcd(a[L], a[L]) = a[L]<\/p>\n \u56e0\u6b64\u8f93\u51fa\u4e00\u5b9a\u7b49\u4e8e a[L]\u3002<\/p>\n \u7b54\u6848：\u221a（\u5bf9）\u3002<\/p>\n
\n\u7b2c 24 \u9898：\u533a\u95f4\u6700\u5927\u516c\u7ea6\u6570\u4e00\u5b9a\u4e0d\u5c0f\u4e8e\u533a\u95f4\u6700\u5c0f\u503c\u5417？<\/h3>\n\u9898\u76ee\u8bf4：<\/p>\n \u4efb\u610f\u4e00\u6b21\u67e5\u8be2\u7684\u8f93\u51fa\u7ed3\u679c\u4e00\u5b9a\u4e0d\u5c0f\u4e8e\u8be5\u67e5\u8be2\u533a\u95f4\u5185\u7684\u6700\u5c0f\u503c\u3002<\/p>\n \u6211\u4eec\u7528\u4e00\u4e2a\u53cd\u4f8b\u5c31\u80fd\u5224\u65ad\u3002<\/p>\n \u5047\u8bbe\u533a\u95f4\u4e2d\u53ea\u6709：<\/p>\n 4, 6<\/p>\n \u533a\u95f4\u6700\u5c0f\u503c\u662f：<\/p>\n 4<\/p>\n \u4f46：<\/p>\n gcd(4, 6) = 2<\/p>\n \u663e\u7136：<\/p>\n 2 < 4\u00a0<\/p>\n \u6240\u4ee5\u6700\u5927\u516c\u7ea6\u6570\u53ef\u80fd\u5c0f\u4e8e\u533a\u95f4\u6700\u5c0f\u503c\u3002<\/p>\n \u5b9e\u9645\u4e0a，\u6b63\u6574\u6570\u7684\u6700\u5927\u516c\u7ea6\u6570\u662f\u533a\u95f4\u4e2d\u6bcf\u4e2a\u6570\u7684\u7ea6\u6570，\u56e0\u6b64\u5b83\u4e0d\u4f1a\u5927\u4e8e\u533a\u95f4\u6700\u5c0f\u503c\u3002<\/p>\n \u7b54\u6848：\u00d7（\u9519）\u3002<\/p>\n
\n\u516d\u3001\u7b2c 26 \u9898：\u5efa\u8868\u7684\u65f6\u95f4\u590d\u6742\u5ea6<\/h2>\n\u9898\u76ee\u8be2\u95ee\u7b2c 17～22 \u884c\u5efa\u8868\u8fc7\u7a0b\u7684\u65f6\u95f4\u590d\u6742\u5ea6，\u5e76\u8bf4\u660e\u4e00\u6b21 gcd() \u8fd0\u7b97\u89c6\u4e3a O(1)O(1)\u3002<\/p>\n \u5efa\u8868\u4ee3\u7801\u7684\u7ed3\u6784\u662f：<\/p>\n for (j = 1; j <= lg[n]; j++)
\n for (i = 1; i + pw[j] – 1 <= n; i++) {
\n dp[i][j] = gcd(…);
\n }<\/p>\n 1. \u5916\u5c42\u5faa\u73af<\/h3>\nj \u4ece 1 \u5230 lg[n]\u3002<\/p>\n \u800c：<\/p>\n ![\"lg[n]\\\\approx](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/09\/20260923185651-6ab420f38267c.png\") <\/p>\n
\u56e0\u6b64\u5916\u5c42\u5faa\u73af\u5927\u7ea6\u6267\u884c：<\/p>\n O(log\u2061n)\u6b21\u3002<\/p>\n
\n2. \u5185\u5c42\u5faa\u73af<\/h3>\n\u5bf9\u4e8e\u6bcf\u4e00\u4e2a j，i \u6700\u591a\u904d\u5386\u7ea6 n \u4e2a\u4f4d\u7f6e，\u56e0\u6b64\u5185\u5c42\u5faa\u73af\u662f：<\/p>\n O(n)<\/p>\n \u6bcf\u6b21\u5faa\u73af\u53ea\u505a\u4e00\u6b21 gcd()，\u9898\u76ee\u5df2\u7ecf\u544a\u8bc9\u6211\u4eec\u628a\u5b83\u770b\u6210 O(1)\u3002<\/p>\n \u6240\u4ee5\u603b\u590d\u6742\u5ea6\u4e3a：<\/p>\n O(nlog\u2061n)<\/p>\n \u9009\u9879\u4e2d\u5bf9\u5e94\u7684\u662f：<\/p>\n B：![\"\\\\Theta(n\\\\log](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/09\/20260923185651-6ab420f392dee.png\") \u00a0\u3002<\/p>\n \u7b54\u6848：B\u3002<\/p>\n
\n\u4e03\u3001\u91cd\u70b9\u8bb2\u89e3\u7b2c 27 \u9898：lg[x] = 5 \u65f6，x \u7684\u8303\u56f4\u662f\u591a\u5c11？<\/h2>\n\u8fd9\u9898\u7684\u5173\u952e\u662f\u7406\u89e3 lg[] \u662f\u5982\u4f55\u8ba1\u7b97\u7684\u3002<\/p>\n 1. \u5148\u770b\u6211\u62ff\u5230\u7684\u8bd5\u5377\u4e2d\u4ee3\u7801<\/h3>\n\u8bd5\u5377\u7b2c 14～16 \u884c\u662f：<\/p>\n for (i = 1; i <= 100000; i++)
\n if (pw[t + 1] > i)
\n lg[i] = t;
\n else
\n t++, lg[i] = t;<\/p>\n \u8bf7\u7279\u522b\u6ce8\u610f：<\/p>\n pw[t + 1] > i<\/p>\n \u8fd9\u91cc\u662f\u4e25\u683c\u5927\u4e8e\u53f7 >\u3002<\/p>\n
\n2. \u7528\u8fb9\u754c\u503c\u8ddf\u8e2a<\/h3>\n\u4e00\u5f00\u59cb：<\/p>\n t = 0;<\/p>\n \u524d\u9762\u5df2\u7ecf\u77e5\u9053：<\/p>\n pw[1] = 2
\npw[2] = 4
\npw[3] = 8
\npw[4] = 16
\npw[5] = 32
\npw[6] = 64<\/p>\n \u5f53 i = 32<\/h4>\n\u6b64\u65f6 t = 4，\u6240\u4ee5：<\/p>\n pw[t + 1] = pw[5] = 32;<\/p>\n \u5224\u65ad\u6761\u4ef6\u662f：<\/p>\n 32 > 32<\/p>\n \u8fd9\u4e0d\u6210\u7acb\u3002<\/p>\n \u4e8e\u662f\u6267\u884c：<\/p>\n t++;
\nlg[i] = t;<\/p>\n \u5f97\u5230：<\/p>\n t = 5
\nlg[32] = 5<\/p>\n \u5f53 i = 33<\/h4>\n\u73b0\u5728 t = 5：<\/p>\n pw[t + 1] = pw[6] = 64;<\/p>\n \u5224\u65ad：<\/p>\n 64 > 33<\/p>\n \u6210\u7acb\u3002<\/p>\n \u6240\u4ee5：<\/p>\n lg[33] = 5<\/p>\n \u5f53 i = 64<\/h4>\n\u6b64\u65f6\u4ecd\u7136\u662f t = 5：<\/p>\n 64 > 64<\/p>\n \u4e0d\u6210\u7acb\u3002<\/p>\n \u4e8e\u662f t \u589e\u52a0\u4e3a 6：<\/p>\n lg[64] = 6<\/p>\n \u56e0\u6b64，\u6309\u7167 \u672cPDF \u4e2d\u7684 >：<\/p>\n lg[x]=5\u2005\u200a\u27fa\u2005\u200a32\u2264x\u226463\u00a0\u00a0<\/p>\n \u6240\u4ee5\u8bd5\u5377\u539f\u4ee3\u7801\u5bf9\u5e94 C：[32,63]\u3002<\/p>\n
\n3. \u90a3\u4e48，\u6709\u7684\u8bd5\u5377，\u6761\u4ef6\u4e3a\u00a0\u00a0>= \u5c31\u4f1a\u5f97\u5230 D\u3002<\/h3>\n\u5982\u679c\u4ee3\u7801\u5b9e\u9645\u662f：<\/p>\n if (pw[t + 1] >= i)
\n lg[i] = t;
\nelse
\n t++, lg[i] = t;<\/p>\n \u90a3\u4e48\u8fb9\u754c\u4f1a\u53d8\u5316\u3002<\/p>\n \n- \n
i = 32：32 >= 32 \u6210\u7acb，\u56e0\u6b64 lg[32] = 4\u3002<\/p>\n<\/li>\n - \n
i = 33：32 >= 33 \u4e0d\u6210\u7acb，t \u589e\u52a0\u4e3a 5，\u56e0\u6b64 lg[33] = 5\u3002<\/p>\n<\/li>\n - \n
i = 64：64 >= 64 \u6210\u7acb，\u56e0\u6b64 lg[64] = 5\u3002<\/p>\n<\/li>\n - \n
i = 65：64 >= 65 \u4e0d\u6210\u7acb，t \u589e\u52a0\u4e3a 6，\u56e0\u6b64 lg[65] = 6\u3002<\/p>\n<\/li>\n<\/ul>\n \u4e8e\u662f：<\/p>\n lg[x]=5\u2005\u200a\u27fa\u2005\u200a33 \u2264 x \u2264 64\u00a0<\/p>\n \u8fd9\u65f6\u5bf9\u5e94\u7b54\u6848\u5c31\u662f：<\/p>\n D：[33,64]\u3002<\/p>\n
\n\u7ed9\u540c\u5b66\u4eec\u7684\u8fb9\u754c\u8bb0\u5fc6\u6cd5<\/h4>\n\u5224\u65ad\u6761\u4ef6\u4e2d\u7684\u7b49\u53f7\u5f88\u91cd\u8981！<\/p>\n > \u548c >= \u53ea\u5dee\u4e00\u4e2a\u7b49\u53f7，\u4f46\u5728\u4e34\u754c\u503c\u5904，\u7a0b\u5e8f\u7684\u6267\u884c\u7ed3\u679c\u5c31\u53ef\u80fd\u4e0d\u540c\u3002<\/p>\n \u6240\u4ee5\u672c\u9898\u9700\u8981\u628a\u4e24\u4ef6\u4e8b\u5206\u6e05：<\/p>\n \n\u5224\u65ad\u6761\u4ef6lg[x] = 5 \u7684\u8303\u56f4\u9009\u9879<\/tr>\n \n\n| pw[t + 1] > i（\u672c\u8bd5\u5377 PDF \u539f\u4ee3\u7801）<\/td>\n | [32,63]<\/td>\n | C<\/td>\n<\/tr>\n | \n| pw[t + 1] >= i（\u5176\u4ed6\u8bd5\u5377\u6539\u52a8\u540e\u7684\u4ee3\u7801）<\/td>\n | [33,64]<\/td>\n | D<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \u56e0\u6b64，\u8981\u4e25\u683c\u6309\u7167\u8fd9\u4efd PDF \u8bb2\u89e3，\u5e94\u5199 C；<\/p>\n \u5982\u679c\u4f60\u770b\u5230\u7684\u53e6\u4e00\u4efd\u9898\u76ee\u4ee3\u7801，\u5982\u679c\u4f7f\u7528 >=，\u7b54\u6848\u5e94\u5199 D\u3002<\/p>\n
\n\u516b\u3001\u6574\u9053\u9898\u7684\u77e5\u8bc6\u70b9\u603b\u7ed3<\/h2>\n1. \u6700\u5927\u516c\u7ea6\u6570<\/h3>\n\u4f7f\u7528\u8f97\u8f6c\u76f8\u9664\u6cd5：<\/p>\n int gcd(int x, int y) {
\n if (y == 0) return x;
\n return gcd(y, x % y);
\n}<\/p>\n
\n2. ST \u8868<\/h3>\ndp[i][j]<\/p>\n \u8868\u793a\u4ece a[i] \u5f00\u59cb，\u8fde\u7eed 2 ^ j \u4e2a\u6570\u7684\u6700\u5927\u516c\u7ea6\u6570\u3002<\/p>\n \u5927\u533a\u95f4\u53ef\u4ee5\u62c6\u6210\u4e24\u4e2a\u957f\u5ea6\u76f8\u540c\u7684\u5c0f\u533a\u95f4，\u518d\u901a\u8fc7 gcd() \u5408\u5e76\u3002<\/p>\n
\n3. \u5efa\u8868\u590d\u6742\u5ea6<\/h3>\n<\/p>\n
\n4. lg[] \u7684\u8fb9\u754c<\/h3>\n\u5fc5\u987b\u4ed4\u7ec6\u68c0\u67e5\u4ee3\u7801\u4e2d\u7684\u6bd4\u8f83\u7b26\u53f7，\u5c24\u5176\u662f > \u548c >=\u3002<\/p>\n
\n\u6700\u7ec8\u7b54\u6848<\/h3>\n\u6309\u7167\u672c\u6587\u4e2d，\u4e0a\u4f20\u7684 PDF \u4e2d\u7b2c 15 \u884c\u7684\u771f\u5b9e\u4ee3\u7801 pw[t + 1] > i：<\/p>\n \u221a\u3001\u221a\u3001\u00d7\u3001B\u3001B\u3001C<\/p>\n \u5982\u679c\u9898\u76ee\u4ee3\u7801\u6539\u6210 pw[t + 1] >= i：<\/p>\n \u221a\u3001\u221a\u3001\u00d7\u3001B\u3001B\u3001D<\/p>\n
\n\u8fd9\u9053\u9898\u6700\u503c\u5f97\u5b66\u751f\u4eec\u8bb0\u4f4f\u7684，\u4e0d\u53ea\u662f ST \u8868，\u66f4\u662f\u4e00\u4e2a\u9605\u8bfb\u7a0b\u5e8f\u7684\u597d\u4e60\u60ef：<\/p>\n \u4e0d\u8981\u53ea\u8bb0\u7ed3\u8bba，\u8981\u56de\u5230\u4ee3\u7801\u91cc\u68c0\u67e5\u6bcf\u4e00\u4e2a\u8fb9\u754c\u6761\u4ef6\u3002\u4e00\u4e2a\u7b49\u53f7，\u5c31\u53ef\u80fd\u51b3\u5b9a\u4e00\u9053\u9898\u7684\u7b54\u6848\u3002<\/p>\n
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