{"id":157,"date":"2026-05-16T11:22:53","date_gmt":"2026-05-16T03:22:53","guid":{"rendered":"https:\/\/huxiaole.cloud\/?p=157"},"modified":"2026-05-16T11:26:10","modified_gmt":"2026-05-16T03:26:10","slug":"cf1614c-divan-and-bitwise-operations","status":"publish","type":"post","link":"https:\/\/huxiaole.cloud\/index.php\/2026\/05\/16\/cf1614c-divan-and-bitwise-operations\/","title":{"rendered":"CF1614C Divan and bitwise operations"},"content":{"rendered":"\n

CF1614C Divan and bitwise operations<\/h1>\n\n\n\n

\u9898\u76ee\u63cf\u8ff0<\/h2>\n\n\n\n

\u6709\u4e00\u6b21\uff0cDivan \u5206\u6790\u4e86\u4e00\u4e2a\u7531 n \u4e2a\u975e\u8d1f\u6574\u6570\u7ec4\u6210\u7684\u5e8f\u5217 a_1, a_2, \\ldots, a_n\u3002\u4ed6\u5bf9\u8be5\u5e8f\u5217\u7684\u6bcf\u4e00\u4e2a\u975e\u7a7a\u5b50\u5e8f\u5217\uff0c\u8ba1\u7b97\u5176\u6240\u6709\u5143\u7d20\u7684\u6309\u4f4d\u5f02\u6216\uff08bitwise XOR\uff09\uff0c\u5e76\u5c06\u6240\u6709\u5f02\u6216\u503c\u76f8\u52a0\uff0c\u5f97\u5230\u8be5\u5e8f\u5217\u7684\u201c\u8212\u9002\u5ea6\u201d\uff08coziness\uff09\u3002<\/p>\n\n\n\n

\u5982\u679c\u5e8f\u5217 c \u53ef\u4ee5\u901a\u8fc7\u4ece\u5e8f\u5217 d \u4e2d\u5220\u9664\u82e5\u5e72\uff08\u53ef\u80fd\u4e3a\u96f6\u6216\u5168\u90e8\uff09\u5143\u7d20\u5f97\u5230\uff0c\u5219\u79f0 c \u662f d \u7684\u4e00\u4e2a\u5b50\u5e8f\u5217\u3002\u4f8b\u5982\uff0c[1,\\,2,\\,3,\\,4]\u3001[2,\\,4] \u548c [2] \u90fd\u662f [1,\\,2,\\,3,\\,4] \u7684\u5b50\u5e8f\u5217\uff0c\u4f46 [4,\\,3] \u548c [0] \u4e0d\u662f\u3002<\/p>\n\n\n\n

Divan \u5bf9\u81ea\u5df1\u7684\u5206\u6790\u975e\u5e38\u81ea\u8c6a\uff0c\u4f46\u73b0\u5728\u4ed6\u4e22\u5931\u4e86\u5e8f\u5217 a\uff0c\u4e5f\u5fd8\u8bb0\u4e86\u8212\u9002\u5ea6\u7684\u503c\uff01\u4e0d\u8fc7\uff0cDivan \u8fd8\u8bb0\u5f97\u5173\u4e8e\u5e8f\u5217 a \u7684 m \u4e2a\u8fde\u7eed\u5b50\u6bb5\u7684\u6309\u4f4d\u6216\uff08bitwise OR\uff09\u503c\u3002\u5e76\u4e14\uff0c\u539f\u5e8f\u5217\u7684\u6bcf\u4e2a\u5143\u7d20\u90fd\u81f3\u5c11\u5305\u542b\u5728\u8fd9 m \u4e2a\u533a\u95f4\u4e2d\u7684\u67d0\u4e00\u4e2a\u3002<\/p>\n\n\n\n

Divan \u8bf7\u4f60\u5e2e\u5fd9\uff0c\u6839\u636e\u4ed6\u8bb0\u5f97\u7684\u4fe1\u606f\uff0c\u8ba1\u7b97\u51fa\u5e8f\u5217 a \u7684\u8212\u9002\u5ea6\u3002\u5982\u679c\u53ef\u80fd\u6709\u591a\u4e2a\u8212\u9002\u5ea6\u503c\uff0c\u8f93\u51fa\u4efb\u610f\u4e00\u4e2a\u5373\u53ef\u3002<\/p>\n\n\n\n

\u7531\u4e8e\u7ed3\u679c\u53ef\u80fd\u975e\u5e38\u5927\uff0c\u8bf7\u8f93\u51fa\u5bf9 10^9+7 \u53d6\u6a21\u540e\u7684\u503c\u3002<\/p>\n\n\n\n

\u8f93\u5165\u683c\u5f0f<\/h2>\n\n\n\n

\u7b2c\u4e00\u884c\u5305\u542b\u4e00\u4e2a\u6574\u6570 t\uff081 \\le t \\le 10^3\uff09\uff0c\u8868\u793a\u6d4b\u8bd5\u7528\u4f8b\u7684\u6570\u91cf\u3002<\/p>\n\n\n\n

\u6bcf\u4e2a\u6d4b\u8bd5\u7528\u4f8b\u7684\u7b2c\u4e00\u884c\u5305\u542b\u4e24\u4e2a\u6574\u6570 n \u548c m\uff081 \\le n, m \\le 2 \\times 10^5\uff09\uff0c\u5206\u522b\u8868\u793a\u5e8f\u5217\u7684\u957f\u5ea6\u548c\u5df2\u77e5\u7684\u8fde\u7eed\u533a\u95f4\u6570\u91cf\u3002<\/p>\n\n\n\n

\u63a5\u4e0b\u6765\u7684 m \u884c\uff0c\u6bcf\u884c\u63cf\u8ff0\u4e00\u4e2a\u533a\u95f4\uff0c\u5305\u542b\u4e09\u4e2a\u6574\u6570 l\u3001r\u3001x\uff081 \\le l \\le r \\le n\uff0c0 \\le x \\le 2^{30} – 1\uff09\uff0c\u8868\u793a\u533a\u95f4 [l, r] \u7684\u6309\u4f4d\u6216\u503c\u4e3a x\u3002<\/p>\n\n\n\n

\u4fdd\u8bc1\u6bcf\u4e2a\u5e8f\u5217\u5143\u7d20\u81f3\u5c11\u88ab\u5305\u542b\u5728\u67d0\u4e00\u4e2a\u533a\u95f4\u5185\u3002\u4fdd\u8bc1\u4e00\u5b9a\u5b58\u5728\u6ee1\u8db3\u6240\u6709\u6761\u4ef6\u7684\u5e8f\u5217\u3002<\/p>\n\n\n\n

\u4fdd\u8bc1\u6240\u6709\u6d4b\u8bd5\u7528\u4f8b\u4e2d n \u7684\u603b\u548c\u4e0e m \u7684\u603b\u548c\u4e0d\u8d85\u8fc7 2 \\times 10^5\u3002<\/p>\n\n\n\n

\u8f93\u51fa\u683c\u5f0f<\/h2>\n\n\n\n

\u5bf9\u4e8e\u6bcf\u4e2a\u6d4b\u8bd5\u7528\u4f8b\uff0c\u8f93\u51fa\u4efb\u610f\u4e00\u4e2a\u6ee1\u8db3\u6761\u4ef6\u7684\u5e8f\u5217 a \u7684\u8212\u9002\u5ea6\uff0c\u5bf9 10^9+7 \u53d6\u6a21\u540e\u7684\u503c\u3002<\/p>\n\n\n\n

\u8f93\u5165\u8f93\u51fa\u6837\u4f8b #1<\/h2>\n\n\n\n

\u8f93\u5165 #1<\/h3>\n\n\n\n
3\n2 1\n1 2 2\n3 2\n1 3 5\n2 3 5\n5 4\n1 2 7\n3 3 7\n4 4 0\n4 5 2\n<\/code><\/pre>\n\n\n\n
\u8f93\u51fa #1<\/h3>\n\n\n\n
4\n20\n112\n<\/code><\/pre>\n\n\n\n
\u8bf4\u660e\/\u63d0\u793a<\/h2>\n\n\n\n
\u5728\u7b2c\u4e00\u4e2a\u6837\u4f8b\u4e2d\uff0c\u6ee1\u8db3\u6761\u4ef6\u7684\u5e8f\u5217\u4e4b\u4e00\u662f [0, 2]\u3002\u8003\u8651\u5b83\u6240\u6709\u7684\u975e\u7a7a\u5b50\u5e8f\u5217\uff1a<\/p>\n\n\n\n
    \n
  • [0]\uff1a\u8be5\u5b50\u5e8f\u5217\u7684\u6309\u4f4d\u5f02\u6216\u4e3a 0\uff1b<\/li>\n\n\n\n
  • [2]\uff1a\u8be5\u5b50\u5e8f\u5217\u7684\u6309\u4f4d\u5f02\u6216\u4e3a 2\uff1b<\/li>\n\n\n\n
  • [0, 2]\uff1a\u8be5\u5b50\u5e8f\u5217\u7684\u6309\u4f4d\u5f02\u6216\u4e3a 2\u3002<\/li>\n<\/ul>\n\n\n\n
    \u6240\u6709\u7ed3\u679c\u4e4b\u548c\u4e3a 4\uff0c\u6240\u4ee5\u7b54\u6848\u4e3a 4\u3002<\/p>\n\n\n\n
    \u5728\u7b2c\u4e8c\u4e2a\u6837\u4f8b\u4e2d\uff0c\u6ee1\u8db3\u6761\u4ef6\u7684\u5e8f\u5217\u4e4b\u4e00\u662f [0,\\,5,\\,5]\u3002<\/p>\n\n\n\n
    \u5728\u7b2c\u4e09\u4e2a\u6837\u4f8b\u4e2d\uff0c\u6ee1\u8db3\u6761\u4ef6\u7684\u5e8f\u5217\u4e4b\u4e00\u662f [5,\\,6,\\,7,\\,0,\\,2]\u3002<\/p>\n\n\n\n
    \u7531 ChatGPT 4.1 \u7ffb\u8bd1<\/p>\n\n\n\n
    \u601d\u8def<\/h2>\n\n\n\n
    1. \u4e3a\u4ec0\u4e48\u201c\u6bcf\u4e00\u4f4d\u5355\u72ec\u5206\u6790\u201d\uff1f<\/h3>\n\n\n\n
    \u5728\u5904\u7406\u5f02\u6216\uff08XOR\uff09\u3001\u6216\uff08OR\uff09\u7b49\u4f4d\u8fd0\u7b97\u9898\u76ee\u65f6\uff0c\u4e0d\u540c\u4e8c\u8fdb\u5236\u4f4d\u4e4b\u95f4\u662f\u6ca1\u6709\u4efb\u4f55\u8fdb\u4f4d\u3001\u501f\u4f4d\u76f8\u4e92\u5f71\u54cd\u7684\u3002\u56e0\u6b64\uff0c\u89e3\u9898\u7684\u6700\u4f18\u7b56\u7565\u901a\u5e38\u662f\u62c6\u4f4d\u8ba1\u7b97<\/strong>\u3002\u6211\u4eec\u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6bcf\u4e2a\u6570\u5b57\u770b\u4f5c\u5b8c\u5168\u72ec\u7acb\u7684\u4e8c\u8fdb\u5236\u4f4d\uff08\u9898\u76ee\u6570\u636e\u8303\u56f4\u5230 2^{30}-1\uff0c\u5373\u6700\u591a 30 \u4f4d\uff09\uff0c\u7136\u540e\u53ea\u76ef\u7740\u5176\u4e2d\u67d0\u4e00\u4e2a\u7279\u5b9a\u7684\u4f4d\u53bb\u63a8\u5bfc\u3002<\/p>\n\n\n\n
    2. \u5b50\u5e8f\u5217\u5728\u4ec0\u4e48\u60c5\u51b5\u4e0b\uff0c\u8be5\u4f4d\u5bf9\u7b54\u6848\u6709\u8d21\u732e\uff1f<\/h3>\n\n\n\n
    \u9898\u89e3\u4e2d\u8bf4\uff1a\u201c\u82e5 sum(1)%2=1 \u65f6\u8be5\u4f4d\u624d\u5bf9\u7b54\u6848\u6709\u8d21\u732e\u201d\u3002<\/p>\n\n\n\n
      \n
    • \u5f02\u6216\u7684\u672c\u8d28\u89c4\u5219<\/strong>\uff1a\u76f8\u540c\u4e3a 0\uff0c\u4e0d\u540c\u4e3a 1\u3002\u591a\u4e2a\u6570\u4e00\u8d77\u5f02\u6216\u65f6\uff0c\u53ea\u6709\u5f53 1 \u7684\u4e2a\u6570\u4e3a\u5947\u6570\u65f6\uff0c\u7ed3\u679c\u7684\u8fd9\u4e00\u4f4d\u624d\u662f 1<\/strong>\uff1b\u5982\u679c\u6709\u5076\u6570\u4e2a 1\uff0c\u5f02\u6216\u7ed3\u679c\u5c31\u662f 0\u3002
      <\/li>\n\n\n\n
    • \u56e0\u6b64\uff0c\u5bf9\u4e8e\u67d0\u4e00\u4e2a\u7279\u5b9a\u7684\u4e8c\u8fdb\u5236\u4f4d\uff0c\u6211\u4eec\u53ea\u9700\u8981\u4ece\u539f\u5e8f\u5217\u4e2d\u6311\u51fa\u5947\u6570\u4e2a 1<\/strong>\uff0c\u518d\u642d\u914d\u4e0a\u4efb\u610f\u4e2a 0<\/strong>\uff0c\u7ec4\u6210\u7684\u5b50\u5e8f\u5217\u5728\u8fd9\u4e00\u4f4d\u4e0a\u7684\u5f02\u6216\u7ed3\u679c\u5c31\u662f 1\u3002\u53ea\u6709\u7ed3\u679c\u4e3a 1\uff0c\u624d\u4f1a\u5bf9\u6700\u7ec8\u7684\u201c\u8212\u9002\u5ea6\u201d\u6570\u503c\u4ea7\u751f\u8d21\u732e\u3002
      <\/li>\n<\/ul>\n\n\n\n
      3. \u6700\u96be\u61c2\u7684\u4e00\u6b65\uff1a\u7ec4\u5408\u6570\u5b66\u63a8\u5bfc\u516c\u5f0f<\/h3>\n\n\n\n
      \u9898\u89e3\u76f4\u63a5\u7ed9\u51fa\u4e86\u516c\u5f0f\uff1a<\/p>\n\n\n\n
      2^{n-x} \\cdot (\\sum_{k=0}^{2k+1 \\le x} C_x^{2k+1}) = 2^{n-1}<\/p>\n\n\n\n
      \u5b83\u7684\u63a8\u5bfc\u903b\u8f91\u5982\u4e0b\uff1a<\/p>\n\n\n\n
      \u5047\u8bbe\u5728\u957f\u5ea6\u4e3a n \u7684\u539f\u5e8f\u5217\u4e2d\uff0c\u5f53\u524d\u89c2\u5bdf\u7684\u8fd9\u4e00\u4f4d\u4e0a\uff0c\u603b\u5171\u6709 x \u4e2a 1\uff0c\u90a3\u4e48\u5269\u4e0b\u7684 n-x \u4e2a\u6570\u5728\u8fd9\u4e00\u4f4d\u4e0a\u5fc5\u5b9a\u90fd\u662f 0\u3002<\/p>\n\n\n\n
        \n
      1. \u9009 1 \u7684\u65b9\u6848\u6570<\/strong>\uff1a\u6211\u4eec\u9700\u8981\u4ece\u8fd9 x \u4e2a 1 \u4e2d\u9009\u51fa\u5947\u6570\u4e2a\uff081\u4e2a\u30013\u4e2a\u30015\u4e2a\u2026\u2026\uff09\u3002\u5728\u7ec4\u5408\u6570\u5b66\u4e2d\uff0c\u6839\u636e\u4e8c\u9879\u5f0f\u5b9a\u7406\uff0c\u4ece x \u4e2a\u5143\u7d20\u4e2d\u9009\u51fa\u5947\u6570\u4e2a\u5143\u7d20\u7684\u7ec4\u5408\u6570\u4e4b\u548c\uff0c\u6c38\u8fdc\u7b49\u4e8e\u603b\u5b50\u96c6\u6570\u7684\u4e00\u534a\uff1a \\sum_{k=0}^{2k+1 \\le x} C_x^{2k+1} = 2^{x-1}<\/li>\n\n\n\n
      2. \u9009 0 \u7684\u65b9\u6848\u6570<\/strong>\uff1a0 \u65e0\u8bba\u600e\u4e48\u5f02\u6216\u90fd\u4e0d\u4f1a\u6539\u53d8\u7ed3\u679c\u3002\u6240\u4ee5\u8fd9 n-x \u4e2a 0\uff0c\u6bcf\u4e2a\u90fd\u6709\u201c\u9009\u5165\u5b50\u5e8f\u5217\u201d\u6216\u201c\u4e0d\u9009\u5165\u201d\u4e24\u79cd\u72b6\u6001\uff0c\u65b9\u6848\u6570\u4e3a 2^{n-x}\u3002<\/li>\n\n\n\n
      3. \u4e58\u6cd5\u539f\u7406\u5408\u5e76<\/strong>\uff1a\u5c06\u4e24\u8005\u76f8\u4e58\uff1a 2^{x-1} \\times 2^{n-x} = 2^{n-1}<\/li>\n<\/ol>\n\n\n\n
        \u8fd9\u4e2a\u7ed3\u8bba\u975e\u5e38\u91cd\u8981\uff1a<\/strong>\u53ea\u8981\u8fd9\u4e00\u4f4d\u4e0a\u81f3\u5c11\u5b58\u5728\u4e00\u4e2a 1<\/strong>\uff08\u5373 x > 0\uff09\uff0c\u4e0d\u7ba1 1 \u7684\u5177\u4f53\u6570\u91cf\u662f\u591a\u5c11\uff0c\u80fd\u8ba9\u5b50\u5e8f\u5217\u5f02\u6216\u7ed3\u679c\u4e3a 1 \u7684\u5b50\u5e8f\u5217\u603b\u6570\uff0c\u6c38\u8fdc\u90fd\u662f\u56fa\u5b9a\u7684 2^{n-1} \u4e2a<\/strong>\u3002<\/p>\n\n\n\n
        4. \u4e3a\u4ec0\u4e48\u7b54\u6848\u662f\u5168\u5c40\u6216\u548c\u4e58\u4ee5 2^{n-1}\uff1f<\/h3>\n\n\n\n
        \u65e2\u7136\u6bcf\u4e00\u4f4d\u53ea\u8981\u5b58\u5728 1\uff0c\u5b83\u5bf9\u7b54\u6848\u4ea7\u751f\u8d21\u732e\u7684\u6b21\u6570\u5c31\u56fa\u5b9a\u662f 2^{n-1} \u6b21\u3002<\/p>\n\n\n\n
        \u90a3\u4e48\u95ee\u9898\u5c31\u53d8\u6210\u4e86\uff1a\u5230\u5e95\u54ea\u4e9b\u4f4d\u4e0a\u5b58\u5728 1\uff1f<\/strong><\/p>\n\n\n\n
          \n
        • \u9898\u76ee\u7ed9\u51fa\u4e86 m \u4e2a\u533a\u95f4\u7684\u6309\u4f4d\u6216\uff08OR\uff09\u503c\u3002\u56e0\u4e3a\u6240\u6709\u533a\u95f4\u8986\u76d6\u4e86\u6574\u4e2a\u6570\u7ec4\uff0c\u628a\u8fd9 m \u4e2a\u533a\u95f4\u7684\u503c\u518d\u5168\u90e8\u201c\u6216\u201d\u8d77\u6765\uff0c\u7b49\u4ef7\u4e8e\u628a\u539f\u6570\u7ec4\u6240\u6709\u7684\u6570\u5b57\u5168\u90e8\u201c\u6216\u201d\u8d77\u6765\uff08\u5373\u5168\u5c40\u6216\u548c S\uff09\u3002
          <\/li>\n\n\n\n
        • \u5168\u5c40\u6216\u548c S \u7684\u67d0\u4e00\u4f4d\u5982\u679c\u662f 1\uff0c\u5c31\u610f\u5473\u7740\u539f\u6570\u7ec4\u5728\u8fd9\u4e00\u4f4d\u4e0a\u81f3\u5c11\u6709\u4e00\u4e2a\u6570\u5b57\u662f 1<\/strong>\u3002
          <\/li>\n\n\n\n
        • \u56e0\u6b64\uff0cS \u672c\u8eab\u5c31\u4ee3\u8868\u4e86\u6240\u6709\u6709\u6548\u4f4d\u7684\u503c\u7684\u603b\u548c\u3002\u5c06\u5176\u6574\u4f53\u4e58\u4ee5\u51fa\u73b0\u6b21\u6570 2^{n-1}\uff0c\u5c31\u662f\u6700\u7ec8\u7684\u7b54\u6848\uff1a \\text{Coziness} = S \\times 2^{n-1} \\pmod{10^9+7}<\/li>\n<\/ul>\n\n\n\n
          \u4ee3\u7801<\/h2>\n\n\n\n
          const int mod=1000000007;  \nint qpow(int base,int exp){  \n    int res=1;  \n    base%=mod;  \n    while(exp){  \n        if(exp&1) res=(res*base)%mod;  \n        base=(base*base)%mod;  \n        exp\/=2;  \n    }  \n    return res;  \n}  \n  \nvoid solve() {  \n    int n,m;  \n    cin>>n>>m;  \n        int ans=0;  \n    while(m--){  \n        int l,r,x;  \n        cin>>l>>r>>x;  \n        ans|=x;  \n    }  \n    cout<<(ans*qpow(2,n-1))%mod<<endl;  \n}\n<\/code><\/pre>\n","protected":false},"excerpt":{"rendered":"
          CF1614C Divan and bitwise operations \u9898\u76ee\u63cf\u8ff0 \u6709\u4e00\u6b21\uff0cDivan \u5206\u6790\u4e86 […]<\/p>\n","protected":false},"author":1,"featured_media":99,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[20,14],"tags":[],"class_list":["post-157","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-codeforces","category-14"],"_links":{"self":[{"href":"https:\/\/huxiaole.cloud\/index.php\/wp-json\/wp\/v2\/posts\/157","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/huxiaole.cloud\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/huxiaole.cloud\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/huxiaole.cloud\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/huxiaole.cloud\/index.php\/wp-json\/wp\/v2\/comments?post=157"}],"version-history":[{"count":1,"href":"https:\/\/huxiaole.cloud\/index.php\/wp-json\/wp\/v2\/posts\/157\/revisions"}],"predecessor-version":[{"id":158,"href":"https:\/\/huxiaole.cloud\/index.php\/wp-json\/wp\/v2\/posts\/157\/revisions\/158"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/huxiaole.cloud\/index.php\/wp-json\/wp\/v2\/media\/99"}],"wp:attachment":[{"href":"https:\/\/huxiaole.cloud\/index.php\/wp-json\/wp\/v2\/media?parent=157"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/huxiaole.cloud\/index.php\/wp-json\/wp\/v2\/categories?post=157"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/huxiaole.cloud\/index.php\/wp-json\/wp\/v2\/tags?post=157"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}