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Chapter 4 - Divide and Conquer, Introduction to Algorithms (CLRS)

Chapter 4 - Divide and Conquer, Introduction to Algorithms (CLRS)

アルゴリズムイントロダクション 第4章 分割統治

研究室内の非公式の輪読会で使用した資料です。
内容: 分割統治の基本的な考え方、Strassenの方法、マスター法による計算量の算出など

8462b18bc6d57cac5579ef4b2f64a6de?s=128

Keisuke Okumura | 奥村圭祐

November 29, 2020
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  1. アルゴリズムイントロダクション 第三版 4章 分割統治 2020.5.6 奥村圭祐 20年度 東工大Defago研・裏輪行 DIVIDE-AND-CONQUER

  2. /35 2 公開用の断り書き https://tanukifont.com/ フォントはたぬきフォントを使用させていただいています。 本資料は研究室内の非公式の輪読会で使用したもので、 対象書籍は「アルゴリズム イントロダクション」の第三版(通称CLRS)になります。 スライド中の擬似コード(英語)は英語版のスクリーンショットを使用しています。 厳密性に欠いているところはありますがご容赦ください。

    間違い等ございましたら okumura.k[at]coord.c.titech.ac.jp まで、 ご連絡いただけると幸いです。
  3. /35 4.1 最大部分配列問題 4.2 行列積のための Strassen のアルゴリズム 4.3 漸化式を解くための置換え法 4.4

    漸化式を解くための再帰木法 4.5 漸化式を解くためのマスター法 4.6 マスター定理を解くための証明 4章の構成 スキップ 目標: 再帰を使ったアルゴリズムの計算量が見積もれるようになること! 3
  4. /35 C <latexit sha1_base64="re76Zkt0iSC9uIARbCEdpPGTKdg=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbRRI9ELh4hkUcCGzI79MLI7OxmZtaEEL7AiweN8eonefNvHGAPClbSSaWqO91dQSK4Nq777eQ2Nre2d/K7hb39g8Oj4vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj2txvP6HSPJYPZpKgH9Gh5CFn1FipUesXS27ZXYCsEy8jJchQ7xe/eoOYpRFKwwTVuuu5ifGnVBnOBM4KvVRjQtmYDrFrqaQRan+6OHRGLqwyIGGsbElDFurviSmNtJ5Ege2MqBnpVW8u/ud1UxPe+lMuk9SgZMtFYSqIicn8azLgCpkRE0soU9zeStiIKsqMzaZgQ/BWX14nrUrZuypXGtel6l0WRx7O4BwuwYMbqMI91KEJDBCe4RXenEfnxXl3PpatOSebOYU/cD5/AJczjMs=</latexit> C <latexit sha1_base64="re76Zkt0iSC9uIARbCEdpPGTKdg=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbRRI9ELh4hkUcCGzI79MLI7OxmZtaEEL7AiweN8eonefNvHGAPClbSSaWqO91dQSK4Nq777eQ2Nre2d/K7hb39g8Oj4vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj2txvP6HSPJYPZpKgH9Gh5CFn1FipUesXS27ZXYCsEy8jJchQ7xe/eoOYpRFKwwTVuuu5ifGnVBnOBM4KvVRjQtmYDrFrqaQRan+6OHRGLqwyIGGsbElDFurviSmNtJ5Ege2MqBnpVW8u/ud1UxPe+lMuk9SgZMtFYSqIicn8azLgCpkRE0soU9zeStiIKsqMzaZgQ/BWX14nrUrZuypXGtel6l0WRx7O4BwuwYMbqMI91KEJDBCe4RXenEfnxXl3PpatOSebOYU/cD5/AJczjMs=</latexit> Q. 行列の積に必用なスカラー積の回数は? A

    <latexit sha1_base64="Lihtv2jYSe0RaYbwwPdS8141boc=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbRRI+oF4+QyCOBDZkdemFkdnYzM2tCCF/gxYPGePWTvPk3DrAHBSvppFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLR3cxvPaHSPJYPZpygH9GB5CFn1FipftMrltyyOwdZJV5GSpCh1it+dfsxSyOUhgmqdcdzE+NPqDKcCZwWuqnGhLIRHWDHUkkj1P5kfuiUnFmlT8JY2ZKGzNXfExMaaT2OAtsZUTPUy95M/M/rpCa89idcJqlByRaLwlQQE5PZ16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv7xKmpWyd1Gu1C9L1dssjjycwCmcgwdXUIV7qEEDGCA8wyu8OY/Oi/PufCxac042cwx/4Hz+AJQrjMk=</latexit> n ⇥ n <latexit sha1_base64="rU/xrREL14Gj7SkmwaoAogYXTA0=">AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Ae2oWy2m3bpZhN2J0IJ/RdePCji1X/jzX/jts1BWx8MPN6bYWZekEhh0HW/ncLa+sbmVnG7tLO7t39QPjxqmTjVjDdZLGPdCajhUijeRIGSdxLNaRRI3g7GtzO//cS1EbF6wEnC/YgOlQgFo2ilR0V6KCJuiOqXK27VnYOsEi8nFcjR6Je/eoOYpRFXyCQ1puu5CfoZ1SiY5NNSLzU8oWxMh7xrqaJ2jZ/NL56SM6sMSBhrWwrJXP09kdHImEkU2M6I4sgsezPxP6+bYnjtZ0IlKXLFFovCVBKMyex9MhCaM5QTSyjTwt5K2IhqytCGVLIheMsvr5JWrepdVGv3l5X6TR5HEU7gFM7Bgyuowx00oAkMFDzDK7w5xnlx3p2PRWvByWeO4Q+czx8D1JB8</latexit> ただし , はともに 行列 B <latexit sha1_base64="wW6OXYFoJg3RvlrYIdlxqPzQzSE=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbRRI8ELx4hkUcCGzI79MLI7OxmZtaEEL7AiweN8eonefNvHGAPClbSSaWqO91dQSK4Nq777eQ2Nre2d/K7hb39g8Oj4vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfju7nffkKleSwfzCRBP6JDyUPOqLFSo9YvltyyuwBZJ15GSpCh3i9+9QYxSyOUhgmqdddzE+NPqTKcCZwVeqnGhLIxHWLXUkkj1P50ceiMXFhlQMJY2ZKGLNTfE1MaaT2JAtsZUTPSq95c/M/rpia89adcJqlByZaLwlQQE5P512TAFTIjJpZQpri9lbARVZQZm03BhuCtvrxOWpWyd1WuNK5L1VoWRx7O4BwuwYMbqMI91KEJDBCe4RXenEfnxXl3PpatOSebOYU/cD5/AJWvjMo=</latexit> A <latexit sha1_base64="Lihtv2jYSe0RaYbwwPdS8141boc=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbRRI+oF4+QyCOBDZkdemFkdnYzM2tCCF/gxYPGePWTvPk3DrAHBSvppFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLR3cxvPaHSPJYPZpygH9GB5CFn1FipftMrltyyOwdZJV5GSpCh1it+dfsxSyOUhgmqdcdzE+NPqDKcCZwWuqnGhLIRHWDHUkkj1P5kfuiUnFmlT8JY2ZKGzNXfExMaaT2OAtsZUTPUy95M/M/rpCa89idcJqlByRaLwlQQE5PZ16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv7xKmpWyd1Gu1C9L1dssjjycwCmcgwdXUIV7qEEDGCA8wyu8OY/Oi/PufCxac042cwx/4Hz+AJQrjMk=</latexit> aik · bkj <latexit sha1_base64="EaK/j49NXiMBsiA5BvVbzMPZwN8=">AAAB+3icbVDLSsNAFJ3UV62vWJduBovgqiRV0GXRjcsK9gFtCJPJpB0zmQkzE7GE/IobF4q49Ufc+TdO2yy09cCFwzn3cu89Qcqo0o7zbVXW1jc2t6rbtZ3dvf0D+7DeUyKTmHSxYEIOAqQIo5x0NdWMDFJJUBIw0g/im5nffyRSUcHv9TQlXoLGnEYUI20k364jP6dxMcKh0DDw8/ih8O2G03TmgKvELUkDlOj49tcoFDhLCNeYIaWGrpNqL0dSU8xIURtliqQIx2hMhoZylBDl5fPbC3hqlBBGQpriGs7V3xM5SpSaJoHpTJCeqGVvJv7nDTMdXXk55WmmCceLRVHGoBZwFgQMqSRYs6khCEtqboV4giTC2sRVMyG4yy+vkl6r6Z43W3cXjfZ1GUcVHIMTcAZccAna4BZ0QBdg8ASewSt4swrrxXq3PhatFaucOQJ/YH3+AFkvlKM=</latexit> AB <latexit sha1_base64="Deg1ISqu+l+cY7wvWxGKALrr7/0=">AAAB6XicbVDLTgJBEOzFF+IL9ehlIjHxRHbRRI+IF49o5JHAhswOvTBhdnYzM2tCCH/gxYPGePWPvPk3DrAHBSvppFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLR7cxvPaHSPJaPZpygH9GB5CFn1Fjp4abWK5bcsjsHWSVeRkqQod4rfnX7MUsjlIYJqnXHcxPjT6gynAmcFrqpxoSyER1gx1JJI9T+ZH7plJxZpU/CWNmShszV3xMTGmk9jgLbGVEz1MveTPzP66QmvPYnXCapQckWi8JUEBOT2dukzxUyI8aWUKa4vZWwIVWUGRtOwYbgLb+8SpqVsndRrtxflqq1LI48nMApnIMHV1CFO6hDAxiE8Ayv8OaMnBfn3flYtOacbOYY/sD5/AEdZ40V</latexit> における の回数を求めよ. B <latexit sha1_base64="wW6OXYFoJg3RvlrYIdlxqPzQzSE=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbRRI8ELx4hkUcCGzI79MLI7OxmZtaEEL7AiweN8eonefNvHGAPClbSSaWqO91dQSK4Nq777eQ2Nre2d/K7hb39g8Oj4vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfju7nffkKleSwfzCRBP6JDyUPOqLFSo9YvltyyuwBZJ15GSpCh3i9+9QYxSyOUhgmqdddzE+NPqTKcCZwVeqnGhLIxHWLXUkkj1P50ceiMXFhlQMJY2ZKGLNTfE1MaaT2JAtsZUTPSq95c/M/rpia89adcJqlByZaLwlQQE5P512TAFTIjJpZQpri9lbARVZQZm03BhuCtvrxOWpWyd1WuNK5L1VoWRx7O4BwuwYMbqMI91KEJDBCe4RXenEfnxXl3PpatOSebOYU/cD5/AJWvjMo=</latexit> C <latexit sha1_base64="re76Zkt0iSC9uIARbCEdpPGTKdg=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbRRI9ELh4hkUcCGzI79MLI7OxmZtaEEL7AiweN8eonefNvHGAPClbSSaWqO91dQSK4Nq777eQ2Nre2d/K7hb39g8Oj4vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj2txvP6HSPJYPZpKgH9Gh5CFn1FipUesXS27ZXYCsEy8jJchQ7xe/eoOYpRFKwwTVuuu5ifGnVBnOBM4KvVRjQtmYDrFrqaQRan+6OHRGLqwyIGGsbElDFurviSmNtJ5Ege2MqBnpVW8u/ud1UxPe+lMuk9SgZMtFYSqIicn8azLgCpkRE0soU9zeStiIKsqMzaZgQ/BWX14nrUrZuypXGtel6l0WRx7O4BwuwYMbqMI91KEJDBCe4RXenEfnxXl3PpatOSebOYU/cD5/AJczjMs=</latexit> としよう cij <latexit sha1_base64="K9dOE+aEnRzt4rru0/mhaSUkfgA=">AAAB7XicbVBNSwMxEJ34WetX1aOXYBE8ld0q6LHoxWMF+wHtUrJptk2bTZYkK5Sl/8GLB0W8+n+8+W9M2z1o64OBx3szzMwLE8GN9bxvtLa+sbm1Xdgp7u7tHxyWjo6bRqWasgZVQul2SAwTXLKG5VawdqIZiUPBWuH4bua3npg2XMlHO0lYEJOB5BGnxDqpSXsZH017pbJX8ebAq8TPSRly1Hulr25f0TRm0lJBjOn4XmKDjGjLqWDTYjc1LCF0TAas46gkMTNBNr92is+d0seR0q6kxXP190RGYmMmceg6Y2KHZtmbif95ndRGN0HGZZJaJuliUZQKbBWevY77XDNqxcQRQjV3t2I6JJpQ6wIquhD85ZdXSbNa8S8r1Yercu02j6MAp3AGF+DDNdTgHurQAAojeIZXeEMKvaB39LFoXUP5zAn8Afr8Ac43j0c=</latexit> n X k=1 aik · bkj <latexit sha1_base64="pqJJkBv+1aPeIQCeCYRJSV2wUvE=">AAACCXicbVDLSsNAFJ3UV62vqEs3g0VwVZIq6EYounFZwT6giWEymbRjJpMwMxFKyNaNv+LGhSJu/QN3/o3TNgttPXDhcM693HuPnzIqlWV9G5Wl5ZXVtep6bWNza3vH3N3ryiQTmHRwwhLR95EkjHLSUVQx0k8FQbHPSM+PriZ+74EISRN+q8YpcWM05DSkGCkteSZ0ZBZ7eXRhF3c5L5CX06hwcJAo6Gv5vvDMutWwpoCLxC5JHZRoe+aXEyQ4iwlXmCEpB7aVKjdHQlHMSFFzMklShCM0JANNOYqJdPPpJwU80koAw0To4gpO1d8TOYqlHMe+7oyRGsl5byL+5w0yFZ67OeVppgjHs0VhxqBK4CQWGFBBsGJjTRAWVN8K8QgJhJUOr6ZDsOdfXiTdZsM+aTRvTuutyzKOKjgAh+AY2OAMtMA1aIMOwOARPINX8GY8GS/Gu/Exa60Y5cw++APj8wd6iJrU</latexit> aik <latexit sha1_base64="u5pUxUJjWqy/T0FxIXdwNNEZDuw=">AAAB7XicbVDLSgNBEOyNrxhfUY9eBoPgKexGQY9BLx4jmAckS5idzCZj5rHMzAphyT948aCIV//Hm3/jJNmDJhY0FFXddHdFCWfG+v63V1hb39jcKm6Xdnb39g/Kh0cto1JNaJMornQnwoZyJmnTMstpJ9EUi4jTdjS+nfntJ6oNU/LBThIaCjyULGYEWye1cD9j42m/XPGr/hxolQQ5qUCORr/81RsokgoqLeHYmG7gJzbMsLaMcDot9VJDE0zGeEi7jkosqAmz+bVTdOaUAYqVdiUtmqu/JzIsjJmIyHUKbEdm2ZuJ/3nd1MbXYcZkkloqyWJRnHJkFZq9jgZMU2L5xBFMNHO3IjLCGhPrAiq5EILll1dJq1YNLqq1+8tK/SaPowgncArnEMAV1OEOGtAEAo/wDK/w5invxXv3PhatBS+fOYY/8D5/AMyqj0Y=</latexit> … … bkj <latexit sha1_base64="q8ArhrgFqhpSxZMXe76bph+n9XE=">AAAB7XicbVBNSwMxEJ34WetX1aOXYBE8ld0q6LHoxWMF+wHtUrJptk2bTZYkK5Sl/8GLB0W8+n+8+W9M2z1o64OBx3szzMwLE8GN9bxvtLa+sbm1Xdgp7u7tHxyWjo6bRqWasgZVQul2SAwTXLKG5VawdqIZiUPBWuH4bua3npg2XMlHO0lYEJOB5BGnxDqpGfay8WjaK5W9ijcHXiV+TsqQo94rfXX7iqYxk5YKYkzH9xIbZERbTgWbFrupYQmhYzJgHUcliZkJsvm1U3zulD6OlHYlLZ6rvycyEhsziUPXGRM7NMveTPzP66Q2ugkyLpPUMkkXi6JUYKvw7HXc55pRKyaOEKq5uxXTIdGEWhdQ0YXgL7+8SprVin9ZqT5clWu3eRwFOIUzuAAfrqEG91CHBlAYwTO8whtS6AW9o49F6xrKZ07gD9DnD8+6j0g=</latexit> … … 回のスカラー積 n <latexit sha1_base64="2QmInd+nHO60uhS7AYCvgpiU4a8=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlpuyXK27VnYOsEi8nFcjR6Je/eoOYpRFKwwTVuuu5ifEzqgxnAqelXqoxoWxMh9i1VNIItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2ooszYbEo2BG/55VXSrlW9i2qteVmp3+RxFOEETuEcPLiCOtxBA1rAAOEZXuHNeXRenHfnY9FacPKZY/gD5/MH2F+M9g==</latexit> これを 個の要素に対して 実行するのだから… n2 <latexit sha1_base64="/km/1ACR5+beF22lqjYY9SbQkws=">AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGi/YA2ls120y7dbMLuRCihP8GLB0W8+ou8+W/ctjlo64OBx3szzMwLEikMuu63s7K6tr6xWdgqbu/s7u2XDg6bJk414w0Wy1i3A2q4FIo3UKDk7URzGgWSt4LRzdRvPXFtRKwecJxwP6IDJULBKFrpXj1We6WyW3FnIMvEy0kZctR7pa9uP2ZpxBUySY3peG6CfkY1Cib5pNhNDU8oG9EB71iqaMSNn81OnZBTq/RJGGtbCslM/T2R0ciYcRTYzoji0Cx6U/E/r5NieOVnQiUpcsXmi8JUEozJ9G/SF5ozlGNLKNPC3krYkGrK0KZTtCF4iy8vk2a14p1XqncX5dp1HkcBjuEEzsCDS6jBLdShAQwG8Ayv8OZI58V5dz7mrStOPnMEf+B8/gD9so2a</latexit> n3 <latexit sha1_base64="trT348r9gUOfD5dUs+ENvWYiEhE=">AAAB6nicbVDLTgJBEOzFF+IL9ehlIjHxRHbBRI9ELx4xyiOBlcwOvTBhdnYzM2tCCJ/gxYPGePWLvPk3DrAHBSvppFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLRzcxvPaHSPJYPZpygH9GB5CFn1FjpXj5We8WSW3bnIKvEy0gJMtR7xa9uP2ZphNIwQbXueG5i/AlVhjOB00I31ZhQNqID7FgqaYTan8xPnZIzq/RJGCtb0pC5+ntiQiOtx1FgOyNqhnrZm4n/eZ3UhFf+hMskNSjZYlGYCmJiMvub9LlCZsTYEsoUt7cSNqSKMmPTKdgQvOWXV0mzUvaq5crdRal2ncWRhxM4hXPw4BJqcAt1aACDATzDK7w5wnlx3p2PRWvOyWaO4Q+czx//No2b</latexit> 回は必用そうに見える 4
  5. /35 行列積を求める簡単なアルゴリズム ⇥(n3) <latexit sha1_base64="G8WRpL5exN/uz8BAetuwevHrbn8=">AAAB8nicbVBNSwMxEM3Wr1q/qh69BItQL2W3FfRY9OKxQr+gXUs2zbah2WRJZoWy9Gd48aCIV3+NN/+NabsHbX0w8Hhvhpl5QSy4Adf9dnIbm1vbO/ndwt7+weFR8fikbVSiKWtRJZTuBsQwwSVrAQfBurFmJAoE6wSTu7nfeWLacCWbMI2ZH5GR5CGnBKzU6zfHDEhZPtYuB8WSW3EXwOvEy0gJZWgMil/9oaJJxCRQQYzpeW4Mfko0cCrYrNBPDIsJnZAR61kqScSMny5OnuELqwxxqLQtCXih/p5ISWTMNApsZ0RgbFa9ufif10sgvPFTLuMEmKTLRWEiMCg8/x8PuWYUxNQSQjW3t2I6JppQsCkVbAje6svrpF2teLVK9eGqVL/N4sijM3SOyshD16iO7lEDtRBFCj2jV/TmgPPivDsfy9ack82coj9wPn8AON+Qjg==</latexit> 計算量は当然 日本語版P63 5

  6. /35 削りたい 6

  7. /35 分割して統治せよ Divide et impera Wikipediaより 古代ローマ帝国は、支配下に治めた都市相互の連帯を禁じ、 都市毎に応じて処遇に格差をつける分割統治によって、 征服した都市からの反乱を抑えることに成功した。 7

  8. /35 分割統治法 部分問題の解を組合わせて元の解を得る 結合 部分問題を再帰的に解く 統治 問題をいくつかの部分問題に分割 分割 8

  9. /35 復習: マージソートの場合 4 2 1 3 4 2 1

    3 4 2 3 1 2 4 1 3 1 2 3 4 分割 分割 分割 結合 結合 結合 統治 統治 9
  10. /35 行列積に当てはめてみる C <latexit sha1_base64="re76Zkt0iSC9uIARbCEdpPGTKdg=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbRRI9ELh4hkUcCGzI79MLI7OxmZtaEEL7AiweN8eonefNvHGAPClbSSaWqO91dQSK4Nq777eQ2Nre2d/K7hb39g8Oj4vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj2txvP6HSPJYPZpKgH9Gh5CFn1FipUesXS27ZXYCsEy8jJchQ7xe/eoOYpRFKwwTVuuu5ifGnVBnOBM4KvVRjQtmYDrFrqaQRan+6OHRGLqwyIGGsbElDFurviSmNtJ5Ege2MqBnpVW8u/ud1UxPe+lMuk9SgZMtFYSqIicn8azLgCpkRE0soU9zeStiIKsqMzaZgQ/BWX14nrUrZuypXGtel6l0WRx7O4BwuwYMbqMI91KEJDBCe4RXenEfnxXl3PpatOSebOYU/cD5/AJczjMs=</latexit> A <latexit sha1_base64="Lihtv2jYSe0RaYbwwPdS8141boc=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbRRI+oF4+QyCOBDZkdemFkdnYzM2tCCF/gxYPGePWTvPk3DrAHBSvppFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLR3cxvPaHSPJYPZpygH9GB5CFn1FipftMrltyyOwdZJV5GSpCh1it+dfsxSyOUhgmqdcdzE+NPqDKcCZwWuqnGhLIRHWDHUkkj1P5kfuiUnFmlT8JY2ZKGzNXfExMaaT2OAtsZUTPUy95M/M/rpCa89idcJqlByRaLwlQQE5PZ16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv7xKmpWyd1Gu1C9L1dssjjycwCmcgwdXUIV7qEEDGCA8wyu8OY/Oi/PufCxac042cwx/4Hz+AJQrjMk=</latexit> B <latexit

    sha1_base64="wW6OXYFoJg3RvlrYIdlxqPzQzSE=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbRRI8ELx4hkUcCGzI79MLI7OxmZtaEEL7AiweN8eonefNvHGAPClbSSaWqO91dQSK4Nq777eQ2Nre2d/K7hb39g8Oj4vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfju7nffkKleSwfzCRBP6JDyUPOqLFSo9YvltyyuwBZJ15GSpCh3i9+9QYxSyOUhgmqdddzE+NPqTKcCZwVeqnGhLIxHWLXUkkj1P50ceiMXFhlQMJY2ZKGLNTfE1MaaT2JAtsZUTPSq95c/M/rpia89adcJqlByZaLwlQQE5P512TAFTIjJpZQpri9lbARVZQZm03BhuCtvrxOWpWyd1WuNK5L1VoWRx7O4BwuwYMbqMI91KEJDBCe4RXenEfnxXl3PpatOSebOYU/cD5/AJWvjMo=</latexit> C <latexit sha1_base64="re76Zkt0iSC9uIARbCEdpPGTKdg=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbRRI9ELh4hkUcCGzI79MLI7OxmZtaEEL7AiweN8eonefNvHGAPClbSSaWqO91dQSK4Nq777eQ2Nre2d/K7hb39g8Oj4vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj2txvP6HSPJYPZpKgH9Gh5CFn1FipUesXS27ZXYCsEy8jJchQ7xe/eoOYpRFKwwTVuuu5ifGnVBnOBM4KvVRjQtmYDrFrqaQRan+6OHRGLqwyIGGsbElDFurviSmNtJ5Ege2MqBnpVW8u/ud1UxPe+lMuk9SgZMtFYSqIicn8azLgCpkRE0soU9zeStiIKsqMzaZgQ/BWX14nrUrZuypXGtel6l0WRx7O4BwuwYMbqMI91KEJDBCe4RXenEfnxXl3PpatOSebOYU/cD5/AJczjMs=</latexit> A11 <latexit sha1_base64="WtsRVvyqN1EY0jYrpA0wO4qTpYw=">AAAB7XicbVBNSwMxEJ31s9avqkcvwSJ4Kpsq6LHqxWMF+wHtUrJpto3NJkuSFcrS/+DFgyJe/T/e/Dem7R609cHA470ZZuaFieDG+v63t7K6tr6xWdgqbu/s7u2XDg6bRqWasgZVQul2SAwTXLKG5VawdqIZiUPBWuHoduq3npg2XMkHO05YEJOB5BGnxDqped3LMJ70SmW/4s+AlgnOSRly1Hulr25f0TRm0lJBjOlgP7FBRrTlVLBJsZsalhA6IgPWcVSSmJkgm107QadO6aNIaVfSopn6eyIjsTHjOHSdMbFDs+hNxf+8TmqjqyDjMkktk3S+KEoFsgpNX0d9rhm1YuwIoZq7WxEdEk2odQEVXQh48eVl0qxW8Hmlen9Rrt3kcRTgGE7gDDBcQg3uoA4NoPAIz/AKb57yXrx372PeuuLlM0fwB97nD+4JjrQ=</latexit> A12 <latexit sha1_base64="HvNqFpXTG6rWIkCLPWE0ieuTihk=">AAAB7XicbVBNSwMxEJ34WetX1aOXYBE8ld0q6LHqxWMF+wHtUrJpto3NJkuSFcrS/+DFgyJe/T/e/Dem7R609cHA470ZZuaFieDGet43WlldW9/YLGwVt3d29/ZLB4dNo1JNWYMqoXQ7JIYJLlnDcitYO9GMxKFgrXB0O/VbT0wbruSDHScsiMlA8ohTYp3UvO5lfnXSK5W9ijcDXiZ+TsqQo94rfXX7iqYxk5YKYkzH9xIbZERbTgWbFLupYQmhIzJgHUcliZkJstm1E3zqlD6OlHYlLZ6pvycyEhszjkPXGRM7NIveVPzP66Q2ugoyLpPUMknni6JUYKvw9HXc55pRK8aOEKq5uxXTIdGEWhdQ0YXgL768TJrVin9eqd5flGs3eRwFOIYTOAMfLqEGd1CHBlB4hGd4hTek0At6Rx/z1hWUzxzBH6DPH++OjrU=</latexit> A21 <latexit sha1_base64="H82LD9IAmpAW5rWkE2+2R7km4QU=">AAAB7XicbVBNSwMxEJ34WetX1aOXYBE8ld0q6LHqxWMF+wHtUrJpto3NJkuSFcrS/+DFgyJe/T/e/Dem7R609cHA470ZZuaFieDGet43WlldW9/YLGwVt3d29/ZLB4dNo1JNWYMqoXQ7JIYJLlnDcitYO9GMxKFgrXB0O/VbT0wbruSDHScsiMlA8ohTYp3UvO5lVX/SK5W9ijcDXiZ+TsqQo94rfXX7iqYxk5YKYkzH9xIbZERbTgWbFLupYQmhIzJgHUcliZkJstm1E3zqlD6OlHYlLZ6pvycyEhszjkPXGRM7NIveVPzP66Q2ugoyLpPUMknni6JUYKvw9HXc55pRK8aOEKq5uxXTIdGEWhdQ0YXgL768TJrVin9eqd5flGs3eRwFOIYTOAMfLqEGd1CHBlB4hGd4hTek0At6Rx/z1hWUzxzBH6DPH++PjrU=</latexit> A22 <latexit sha1_base64="gGMSnDhj1RvmVuBmOJuXydE//ow=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRbBU9ldC3qsevFYwX5Au5Rsmm1js8mSZIWy9D948aCIV/+PN/+NabsHbX0w8Hhvhpl5YcKZNq777RTW1jc2t4rbpZ3dvf2D8uFRS8tUEdokkkvVCbGmnAnaNMxw2kkUxXHIaTsc38789hNVmknxYCYJDWI8FCxiBBsrta77me9P++WKW3XnQKvEy0kFcjT65a/eQJI0psIQjrXuem5iggwrwwin01Iv1TTBZIyHtGupwDHVQTa/dorOrDJAkVS2hEFz9fdEhmOtJ3FoO2NsRnrZm4n/ed3URFdBxkSSGirIYlGUcmQkmr2OBkxRYvjEEkwUs7ciMsIKE2MDKtkQvOWXV0nLr3oXVf++Vqnf5HEU4QRO4Rw8uIQ63EEDmkDgEZ7hFd4c6bw4787HorXg5DPH8AfO5w/xFI62</latexit> B11 <latexit sha1_base64="8gCv0nL+iW+7P6Y4gWHTEJ+WgiI=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRbBU9lUQY+lXjxWsLXQLiWbZtvYbLIkWaEs/Q9ePCji1f/jzX9j2u5BWx8MPN6bYWZemAhurO9/e4W19Y3NreJ2aWd3b/+gfHjUNirVlLWoEkp3QmKY4JK1LLeCdRLNSBwK9hCOb2b+wxPThit5bycJC2IylDzilFgntRv9DONpv1zxq/4caJXgnFQgR7Nf/uoNFE1jJi0VxJgu9hMbZERbTgWblnqpYQmhYzJkXUcliZkJsvm1U3TmlAGKlHYlLZqrvycyEhsziUPXGRM7MsveTPzP66Y2ug4yLpPUMkkXi6JUIKvQ7HU04JpRKyaOEKq5uxXREdGEWhdQyYWAl19eJe1aFV9Ua3eXlXojj6MIJ3AK54DhCupwC01oAYVHeIZXePOU9+K9ex+L1oKXzxzDH3ifP++SjrU=</latexit> B12 <latexit sha1_base64="QuM51X7FDHygo6xFiwqXTVC2cok=">AAAB7XicbVBNSwMxEJ3Ur1q/qh69BIvgqexWQY+lXjxWsLXQLiWbZtvYbLIkWaEs/Q9ePCji1f/jzX9j2u5BWx8MPN6bYWZemAhurOd9o8La+sbmVnG7tLO7t39QPjxqG5VqylpUCaU7ITFMcMlallvBOolmJA4FewjHNzP/4Ylpw5W8t5OEBTEZSh5xSqyT2o1+5tem/XLFq3pz4FXi56QCOZr98ldvoGgaM2mpIMZ0fS+xQUa05VSwaamXGpYQOiZD1nVUkpiZIJtfO8VnThngSGlX0uK5+nsiI7Exkzh0nTGxI7PszcT/vG5qo+sg4zJJLZN0sShKBbYKz17HA64ZtWLiCKGau1sxHRFNqHUBlVwI/vLLq6Rdq/oX1drdZaXeyOMowgmcwjn4cAV1uIUmtIDCIzzDK7whhV7QO/pYtBZQPnMMf4A+fwDxF462</latexit> B21 <latexit sha1_base64="LRdxHHGIKQrElvWuyxMUGbU5xvU=">AAAB7XicbVBNSwMxEJ3Ur1q/qh69BIvgqexWQY+lXjxWsLXQLiWbZtvYbLIkWaEs/Q9ePCji1f/jzX9j2u5BWx8MPN6bYWZemAhurOd9o8La+sbmVnG7tLO7t39QPjxqG5VqylpUCaU7ITFMcMlallvBOolmJA4FewjHNzP/4Ylpw5W8t5OEBTEZSh5xSqyT2o1+VvOn/XLFq3pz4FXi56QCOZr98ldvoGgaM2mpIMZ0fS+xQUa05VSwaamXGpYQOiZD1nVUkpiZIJtfO8VnThngSGlX0uK5+nsiI7Exkzh0nTGxI7PszcT/vG5qo+sg4zJJLZN0sShKBbYKz17HA64ZtWLiCKGau1sxHRFNqHUBlVwI/vLLq6Rdq/oX1drdZaXeyOMowgmcwjn4cAV1uIUmtIDCIzzDK7whhV7QO/pYtBZQPnMMf4A+fwDxGI62</latexit> B22 <latexit sha1_base64="Tp6lmLSuM4gEcdNM4n4aPeq4+Fg=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRbBU9ldC3os9eKxgv2AdinZNNvGZpMlyQpl6X/w4kERr/4fb/4b03YP2vpg4PHeDDPzwoQzbVz32ylsbG5t7xR3S3v7B4dH5eOTtpapIrRFJJeqG2JNORO0ZZjhtJsoiuOQ0044uZ37nSeqNJPiwUwTGsR4JFjECDZWajcGme/PBuWKW3UXQOvEy0kFcjQH5a/+UJI0psIQjrXueW5iggwrwwins1I/1TTBZIJHtGepwDHVQba4doYurDJEkVS2hEEL9fdEhmOtp3FoO2NsxnrVm4v/eb3URDdBxkSSGirIclGUcmQkmr+OhkxRYvjUEkwUs7ciMsYKE2MDKtkQvNWX10nbr3pXVf++Vqk38jiKcAbncAkeXEMd7qAJLSDwCM/wCm+OdF6cd+dj2Vpw8plT+APn8wfynY63</latexit> C11 <latexit sha1_base64="GyVbAkGkl2JkUgwwoB3OD7RzOHs=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRbBU9lUQY/FXjxWsLXQLiWbZtvYbLIkWaEs/Q9ePCji1f/jzX9j2u5BWx8MPN6bYWZemAhurO9/e4W19Y3NreJ2aWd3b/+gfHjUNirVlLWoEkp3QmKY4JK1LLeCdRLNSBwK9hCOGzP/4Ylpw5W8t5OEBTEZSh5xSqyT2o1+hvG0X674VX8OtEpwTiqQo9kvf/UGiqYxk5YKYkwX+4kNMqItp4JNS73UsITQMRmyrqOSxMwE2fzaKTpzygBFSruSFs3V3xMZiY2ZxKHrjIkdmWVvJv7ndVMbXQcZl0lqmaSLRVEqkFVo9joacM2oFRNHCNXc3YroiGhCrQuo5ELAyy+vknatii+qtbvLSv0mj6MIJ3AK54DhCupwC01oAYVHeIZXePOU9+K9ex+L1oKXzxzDH3ifP/EbjrY=</latexit> C12 <latexit sha1_base64="PhDJdWnMIvK9Wdgl3YkVIwL1mMs=">AAAB7XicbVDLSgNBEOyNrxhfUY9eBoPgKexGQY/BXDxGMDGQLGF2MpuMmccyMyuEJf/gxYMiXv0fb/6Nk2QPmljQUFR1090VJZwZ6/vfXmFtfWNzq7hd2tnd2z8oHx61jUo1oS2iuNKdCBvKmaQtyyynnURTLCJOH6JxY+Y/PFFtmJL3dpLQUOChZDEj2Dqp3ehnQW3aL1f8qj8HWiVBTiqQo9kvf/UGiqSCSks4NqYb+IkNM6wtI5xOS73U0ASTMR7SrqMSC2rCbH7tFJ05ZYBipV1Ji+bq74kMC2MmInKdAtuRWfZm4n9eN7XxdZgxmaSWSrJYFKccWYVmr6MB05RYPnEEE83crYiMsMbEuoBKLoRg+eVV0q5Vg4tq7e6yUr/J4yjCCZzCOQRwBXW4hSa0gMAjPMMrvHnKe/HevY9Fa8HLZ47hD7zPH/Kgjrc=</latexit> C22 <latexit sha1_base64="cxrJJIC+l9L+ZIBe7sL1xZt+0Cw=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRbBU9ldC3os9uKxgv2AdinZNNvGZpMlyQpl6X/w4kERr/4fb/4b03YP2vpg4PHeDDPzwoQzbVz32ylsbG5t7xR3S3v7B4dH5eOTtpapIrRFJJeqG2JNORO0ZZjhtJsoiuOQ0044acz9zhNVmknxYKYJDWI8EixiBBsrtRuDzPdng3LFrboLoHXi5aQCOZqD8ld/KEkaU2EIx1r3PDcxQYaVYYTTWamfappgMsEj2rNU4JjqIFtcO0MXVhmiSCpbwqCF+nsiw7HW0zi0nTE2Y73qzcX/vF5qopsgYyJJDRVkuShKOTISzV9HQ6YoMXxqCSaK2VsRGWOFibEBlWwI3urL66TtV72rqn9fq9Rv8ziKcAbncAkeXEMd7qAJLSDwCM/wCm+OdF6cd+dj2Vpw8plT+APn8wf0Jo64</latexit> C21 <latexit sha1_base64="281MpEfJrh/xPJY7c3yuZWmIESY=">AAAB7XicbVDLSgNBEOyNrxhfUY9eBoPgKexGQY/BXDxGMDGQLGF2MpuMmccyMyuEJf/gxYMiXv0fb/6Nk2QPmljQUFR1090VJZwZ6/vfXmFtfWNzq7hd2tnd2z8oHx61jUo1oS2iuNKdCBvKmaQtyyynnURTLCJOH6JxY+Y/PFFtmJL3dpLQUOChZDEj2Dqp3ehntWDaL1f8qj8HWiVBTiqQo9kvf/UGiqSCSks4NqYb+IkNM6wtI5xOS73U0ASTMR7SrqMSC2rCbH7tFJ05ZYBipV1Ji+bq74kMC2MmInKdAtuRWfZm4n9eN7XxdZgxmaSWSrJYFKccWYVmr6MB05RYPnEEE83crYiMsMbEuoBKLoRg+eVV0q5Vg4tq7e6yUr/J4yjCCZzCOQRwBXW4hSa0gMAjPMMrvHnKe/HevY9Fa8HLZ47hD7zPH/Khjrc=</latexit> 分割 分割 分割 A11 · B11 +A12 · B21 <latexit sha1_base64="2NuugJvo5CQalDlXxtZv4bIkBaY=">AAACFHicbZDLSsNAFIYn9VbjLerSzWARhEJJqqDLWjcuK9gLNCFMJpN26OTCzEQooQ/hxldx40IRty7c+TZO0oDa+sPAz3fO4cz5vYRRIU3zS6usrK6tb1Q39a3tnd09Y/+gJ+KUY9LFMYv5wEOCMBqRrqSSkUHCCQo9Rvre5Dqv9+8JFzSO7uQ0IU6IRhENKEZSIdeoX7mZZc1s7McStgsPbVuvw5w3f3hTcdeomQ2zEFw2VmlqoFTHNT5tP8ZpSCKJGRJiaJmJdDLEJcWMzHQ7FSRBeIJGZKhshEIinKw4agZPFPFhEHP1IgkL+nsiQ6EQ09BTnSGSY7FYy+F/tWEqg0sno1GSShLh+aIgZVDGME8I+pQTLNlUGYQ5VX+FeIw4wlLlqKsQrMWTl02v2bDOGs3b81qrXcZRBUfgGJwCC1yAFrgBHdAFGDyAJ/ACXrVH7Vl7097nrRWtnDkEf6R9fANgpZvi</latexit> 統治 A11 · B12 +A21 · B22 <latexit sha1_base64="j10lAKqCSykM3Zz/I9ux+hwKcHg=">AAACE3icbZDLSsNAFIYn9VbjLerSzWARRKEkUdBlrRuXFewFmhAmk0k7dHJhZiKUkHdw46u4caGIWzfufBunbRa19YeBn++cw5nz+ymjQprmj1ZZWV1b36hu6lvbO7t7xv5BRyQZx6SNE5bwno8EYTQmbUklI72UExT5jHT90e2k3n0kXNAkfpDjlLgRGsQ0pBhJhTzj7MbLLatwcJBI2FTeLqDj6OdQcXuO23bhGTWzbk4Fl41Vmhoo1fKMbydIcBaRWGKGhOhbZirdHHFJMSOF7mSCpAiP0ID0lY1RRISbT28q4IkiAQwTrl4s4ZTOT+QoEmIc+aozQnIoFmsT+F+tn8nw2s1pnGaSxHi2KMwYlAmcBAQDygmWbKwMwpyqv0I8RBxhqWLUVQjW4snLpmPXrYu6fX9ZazTLOKrgCByDU2CBK9AAd6AF2gCDJ/AC3sC79qy9ah/a56y1opUzh+CPtK9fAI6bug==</latexit> 統治 A21 · B11 +A22 · B21 <latexit sha1_base64="vjBP5uoZkqbFRiuJjMu8akRaZBU=">AAACE3icbVDLSsNAFJ3UV42vqEs3g0UQhZJEQZe1blxWsA9oQphMJu3QyYOZiVBC/8GNv+LGhSJu3bjzb5y0AbX1wMDhnHO5c4+fMiqkaX5plaXlldW16rq+sbm1vWPs7nVEknFM2jhhCe/5SBBGY9KWVDLSSzlBkc9I1x9dF373nnBBk/hOjlPiRmgQ05BiJJXkGSdXXm5bEwcHiYRNL7esCXQc/RQWuv2jq4xn1My6OQVcJFZJaqBEyzM+nSDBWURiiRkSom+ZqXRzxCXFjEx0JxMkRXiEBqSvaIwiItx8etMEHiklgGHC1YslnKq/J3IUCTGOfJWMkByKea8Q//P6mQwv3ZzGaSZJjGeLwoxBmcCiIBhQTrBkY0UQ5lT9FeIh4ghLVaOuSrDmT14kHbtundXt2/Nao1nWUQUH4BAcAwtcgAa4AS3QBhg8gCfwAl61R+1Ze9PeZ9GKVs7sgz/QPr4BAKebug==</latexit> 統治 A21 · B12 +A22 · B22 <latexit sha1_base64="nQkdyCkWWdIfMEfXAIHMln1L5rY=">AAACE3icbZDLSsNAFIYn9VbjLerSzWARRKEkUdBlrRuXFewFmhAmk0k7dHJhZiKU0Hdw46u4caGIWzfufBsnbUBt/WHg5zvncOb8fsqokKb5pVWWlldW16rr+sbm1vaOsbvXEUnGMWnjhCW85yNBGI1JW1LJSC/lBEU+I11/dF3Uu/eEC5rEd3KcEjdCg5iGFCOpkGecXHm5bU0cHCQSNr3csifQcfRTWHD7hyvvGTWzbk4FF41Vmhoo1fKMTydIcBaRWGKGhOhbZirdHHFJMSMT3ckESREeoQHpKxujiAg3n940gUeKBDBMuHqxhFP6eyJHkRDjyFedEZJDMV8r4H+1fibDSzencZpJEuPZojBjUCawCAgGlBMs2VgZhDlVf4V4iDjCUsWoqxCs+ZMXTceuW2d1+/a81miWcVTBATgEx8ACF6ABbkALtAEGD+AJvIBX7VF71t6091lrRStn9sEfaR/fA8mbvA==</latexit> 統治 結合 10
  11. /35 日本語版P64 再帰の「底」のケース ※簡易化のため は2のべき乗と想定 n <latexit sha1_base64="2QmInd+nHO60uhS7AYCvgpiU4a8=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlpuyXK27VnYOsEi8nFcjR6Je/eoOYpRFKwwTVuuu5ifEzqgxnAqelXqoxoWxMh9i1VNIItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2ooszYbEo2BG/55VXSrlW9i2qteVmp3+RxFOEETuEcPLiCOtxBA1rAAOEZXuHNeXRenHfnY9FacPKZY/gD5/MH2F+M9g==</latexit> 11

  12. /35 さっきのアルゴリズムの計算量を求める 行列の乗算にかかる時間を として漸化式で表してみる n ⇥ n <latexit sha1_base64="rU/xrREL14Gj7SkmwaoAogYXTA0=">AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Ae2oWy2m3bpZhN2J0IJ/RdePCji1X/jzX/jts1BWx8MPN6bYWZekEhh0HW/ncLa+sbmVnG7tLO7t39QPjxqmTjVjDdZLGPdCajhUijeRIGSdxLNaRRI3g7GtzO//cS1EbF6wEnC/YgOlQgFo2ilR0V6KCJuiOqXK27VnYOsEi8nFcjR6Je/eoOYpRFXyCQ1puu5CfoZ1SiY5NNSLzU8oWxMh7xrqaJ2jZ/NL56SM6sMSBhrWwrJXP09kdHImEkU2M6I4sgsezPxP6+bYnjtZ0IlKXLFFovCVBKMyex9MhCaM5QTSyjTwt5K2IhqytCGVLIheMsvr5JWrepdVGv3l5X6TR5HEU7gFM7Bgyuowx00oAkMFDzDK7w5xnlx3p2PRWvByWeO4Q+czx8D1JB8</latexit> T(n)

    <latexit sha1_base64="vM8hsiJM1mIid8s1EMCG6OMcapM=">AAAB63icbVBNSwMxEJ2tX7V+VT16CRahXspuFfRY9OKxQr+gXUo2zbahSXZJskJZ+he8eFDEq3/Im//GbLsHbX0w8Hhvhpl5QcyZNq777RQ2Nre2d4q7pb39g8Oj8vFJR0eJIrRNIh6pXoA15UzStmGG016sKBYBp91gep/53SeqNItky8xi6gs8lixkBJtMalXl5bBccWvuAmideDmpQI7msPw1GEUkEVQawrHWfc+NjZ9iZRjhdF4aJJrGmEzxmPYtlVhQ7aeLW+fowiojFEbKljRoof6eSLHQeiYC2ymwmehVLxP/8/qJCW/9lMk4MVSS5aIw4chEKHscjZiixPCZJZgoZm9FZIIVJsbGU7IheKsvr5NOveZd1eqP15XGXR5HEc7gHKrgwQ004AGa0AYCE3iGV3hzhPPivDsfy9aCk8+cwh84nz9C/Y25</latexit> -分割にかかる時間 ⇥(1) <latexit sha1_base64="mR4xVs/DXYAmOYoxJUW5uOvqr/U=">AAAB8HicbVA9TwJBEJ3DL8Qv1NJmIzHBhtyhiZZEG0tMADFwIXvLHGzYvbvs7pkQwq+wsdAYW3+Onf/GBa5Q8CWTvLw3k5l5QSK4Nq777eTW1jc2t/LbhZ3dvf2D4uFRS8epYthksYhVO6AaBY+wabgR2E4UUhkIfAhGtzP/4QmV5nHUMOMEfUkHEQ85o8ZKj93GEA0te+e9YsmtuHOQVeJlpAQZ6r3iV7cfs1RiZJigWnc8NzH+hCrDmcBpoZtqTCgb0QF2LI2oRO1P5gdPyZlV+iSMla3IkLn6e2JCpdZjGdhOSc1QL3sz8T+vk5rw2p/wKEkNRmyxKEwFMTGZfU/6XCEzYmwJZYrbWwkbUkWZsRkVbAje8surpFWteBeV6v1lqXaTxZGHEziFMnhwBTW4gzo0gYGEZ3iFN0c5L86787FozTnZzDH8gfP5A6+jj6w=</latexit> -再帰呼び出しにかかる時間 8T(n/2) <latexit sha1_base64="XaQSeogBBeZ02tdmu67A0KgEd+w=">AAAB7nicbVBNSwMxEJ3Ur1q/qh69BItQL3W3CvZY9OKxQr+gXUo2zbah2eySZIWy9Ed48aCIV3+PN/+NabsHbX0w8Hhvhpl5fiy4No7zjXIbm1vbO/ndwt7+weFR8fikraNEUdaikYhU1yeaCS5Zy3AjWDdWjIS+YB1/cj/3O09MaR7JppnGzAvJSPKAU2Ks1Kk1y/KqejkolpyKswBeJ25GSpChMSh+9YcRTUImDRVE657rxMZLiTKcCjYr9BPNYkInZMR6lkoSMu2li3Nn+MIqQxxEypY0eKH+nkhJqPU09G1nSMxYr3pz8T+vl5ig5qVcxolhki4XBYnAJsLz3/GQK0aNmFpCqOL2VkzHRBFqbEIFG4K7+vI6aVcr7nWl+nhTqt9lceThDM6hDC7cQh0eoAEtoDCBZ3iFNxSjF/SOPpatOZTNnMIfoM8fnWiOcA==</latexit> -結合にかかる時間 ⇥(n2) <latexit sha1_base64="3afWb1vL+AOktUowYQ1CxKg5hzM=">AAAB8nicbVBNSwMxEM36WetX1aOXYBHqpexWQY9FLx4r9Au2a8mm2TY0myzJrFCW/gwvHhTx6q/x5r8xbfegrQ8GHu/NMDMvTAQ34Lrfztr6xubWdmGnuLu3f3BYOjpuG5VqylpUCaW7ITFMcMlawEGwbqIZiUPBOuH4buZ3npg2XMkmTBIWxGQoecQpASv5veaIAanIx9pFv1R2q+4ceJV4OSmjHI1+6as3UDSNmQQqiDG+5yYQZEQDp4JNi73UsITQMRky31JJYmaCbH7yFJ9bZYAjpW1JwHP190RGYmMmcWg7YwIjs+zNxP88P4XoJsi4TFJgki4WRanAoPDsfzzgmlEQE0sI1dzeiumIaELBplS0IXjLL6+Sdq3qXVZrD1fl+m0eRwGdojNUQR66RnV0jxqohShS6Bm9ojcHnBfn3flYtK45+cwJ+gPn8wc3WpCN</latexit> T(n) = ( ⇥(1) n = 1 8T(n/2) + ⇥(n2) n > 1 <latexit sha1_base64="GR8m44C9HSDmcXqd/MYxaonOnuY=">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</latexit> -再帰の底のケース ⇥(1) <latexit sha1_base64="mR4xVs/DXYAmOYoxJUW5uOvqr/U=">AAAB8HicbVA9TwJBEJ3DL8Qv1NJmIzHBhtyhiZZEG0tMADFwIXvLHGzYvbvs7pkQwq+wsdAYW3+Onf/GBa5Q8CWTvLw3k5l5QSK4Nq777eTW1jc2t/LbhZ3dvf2D4uFRS8epYthksYhVO6AaBY+wabgR2E4UUhkIfAhGtzP/4QmV5nHUMOMEfUkHEQ85o8ZKj93GEA0te+e9YsmtuHOQVeJlpAQZ6r3iV7cfs1RiZJigWnc8NzH+hCrDmcBpoZtqTCgb0QF2LI2oRO1P5gdPyZlV+iSMla3IkLn6e2JCpdZjGdhOSc1QL3sz8T+vk5rw2p/wKEkNRmyxKEwFMTGZfU/6XCEzYmwJZYrbWwkbUkWZsRkVbAje8surpFWteBeV6v1lqXaTxZGHEziFMnhwBTW4gzo0gYGEZ3iFN0c5L86787FozTnZzDH8gfP5A6+jj6w=</latexit> 削れねぇぞおい. そこで-> ⇥(n3) <latexit sha1_base64="G8WRpL5exN/uz8BAetuwevHrbn8=">AAAB8nicbVBNSwMxEM3Wr1q/qh69BItQL2W3FfRY9OKxQr+gXUs2zbah2WRJZoWy9Gd48aCIV3+NN/+NabsHbX0w8Hhvhpl5QSy4Adf9dnIbm1vbO/ndwt7+weFR8fikbVSiKWtRJZTuBsQwwSVrAQfBurFmJAoE6wSTu7nfeWLacCWbMI2ZH5GR5CGnBKzU6zfHDEhZPtYuB8WSW3EXwOvEy0gJZWgMil/9oaJJxCRQQYzpeW4Mfko0cCrYrNBPDIsJnZAR61kqScSMny5OnuELqwxxqLQtCXih/p5ISWTMNApsZ0RgbFa9ufif10sgvPFTLuMEmKTLRWEiMCg8/x8PuWYUxNQSQjW3t2I6JppQsCkVbAje6svrpF2teLVK9eGqVL/N4sijM3SOyshD16iO7lEDtRBFCj2jV/TmgPPivDsfy9ack82coj9wPn8AON+Qjg==</latexit> のちのち登場するマスター法を用いると 計算量が であることがわかる 12
  13. /35 Strassenの方法 “決して自明ではない.(これは本書に現れる最も控え目な言明かもしれない.)” by CLRS Strassen, Volker, Gaussian Elimination is

    not Optimal, Numer. Math. 13, p. 354-356, 1969 13
  14. /35 1. さっきと同様に … に元の行列を分割 A11 <latexit sha1_base64="WtsRVvyqN1EY0jYrpA0wO4qTpYw=">AAAB7XicbVBNSwMxEJ31s9avqkcvwSJ4Kpsq6LHqxWMF+wHtUrJpto3NJkuSFcrS/+DFgyJe/T/e/Dem7R609cHA470ZZuaFieDG+v63t7K6tr6xWdgqbu/s7u2XDg6bRqWasgZVQul2SAwTXLKG5VawdqIZiUPBWuHoduq3npg2XMkHO05YEJOB5BGnxDqped3LMJ70SmW/4s+AlgnOSRly1Hulr25f0TRm0lJBjOlgP7FBRrTlVLBJsZsalhA6IgPWcVSSmJkgm107QadO6aNIaVfSopn6eyIjsTHjOHSdMbFDs+hNxf+8TmqjqyDjMkktk3S+KEoFsgpNX0d9rhm1YuwIoZq7WxEdEk2odQEVXQh48eVl0qxW8Hmlen9Rrt3kcRTgGE7gDDBcQg3uoA4NoPAIz/AKb57yXrx372PeuuLlM0fwB97nD+4JjrQ=</latexit> B22 <latexit

    sha1_base64="Tp6lmLSuM4gEcdNM4n4aPeq4+Fg=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRbBU9ldC3os9eKxgv2AdinZNNvGZpMlyQpl6X/w4kERr/4fb/4b03YP2vpg4PHeDDPzwoQzbVz32ylsbG5t7xR3S3v7B4dH5eOTtpapIrRFJJeqG2JNORO0ZZjhtJsoiuOQ0044uZ37nSeqNJPiwUwTGsR4JFjECDZWajcGme/PBuWKW3UXQOvEy0kFcjQH5a/+UJI0psIQjrXueW5iggwrwwins1I/1TTBZIJHtGepwDHVQba4doYurDJEkVS2hEEL9fdEhmOtp3FoO2NsxnrVm4v/eb3URDdBxkSSGirIclGUcmQkmr+OhkxRYvjUEkwUs7ciMsYKE2MDKtkQvNWX10nbr3pXVf++Vqk38jiKcAbncAkeXEMd7qAJLSDwCM/wCm+OdF6cd+dj2Vpw8plT+APn8wfynY63</latexit> 2. … を使って(加減算のみ)新しく10個の行列を作る A11 <latexit sha1_base64="WtsRVvyqN1EY0jYrpA0wO4qTpYw=">AAAB7XicbVBNSwMxEJ31s9avqkcvwSJ4Kpsq6LHqxWMF+wHtUrJpto3NJkuSFcrS/+DFgyJe/T/e/Dem7R609cHA470ZZuaFieDG+v63t7K6tr6xWdgqbu/s7u2XDg6bRqWasgZVQul2SAwTXLKG5VawdqIZiUPBWuHoduq3npg2XMkHO05YEJOB5BGnxDqped3LMJ70SmW/4s+AlgnOSRly1Hulr25f0TRm0lJBjOlgP7FBRrTlVLBJsZsalhA6IgPWcVSSmJkgm107QadO6aNIaVfSopn6eyIjsTHjOHSdMbFDs+hNxf+8TmqjqyDjMkktk3S+KEoFsgpNX0d9rhm1YuwIoZq7WxEdEk2odQEVXQh48eVl0qxW8Hmlen9Rrt3kcRTgGE7gDDBcQg3uoA4NoPAIz/AKb57yXrx372PeuuLlM0fwB97nD+4JjrQ=</latexit> B22 <latexit sha1_base64="Tp6lmLSuM4gEcdNM4n4aPeq4+Fg=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRbBU9ldC3os9eKxgv2AdinZNNvGZpMlyQpl6X/w4kERr/4fb/4b03YP2vpg4PHeDDPzwoQzbVz32ylsbG5t7xR3S3v7B4dH5eOTtpapIrRFJJeqG2JNORO0ZZjhtJsoiuOQ0044uZ37nSeqNJPiwUwTGsR4JFjECDZWajcGme/PBuWKW3UXQOvEy0kFcjQH5a/+UJI0psIQjrXueW5iggwrwwins1I/1TTBZIJHtGepwDHVQba4doYurDJEkVS2hEEL9fdEhmOtp3FoO2NsxnrVm4v/eb3URDdBxkSSGirIclGUcmQkmr+OhkxRYvjUEkwUs7ciMsYKE2MDKtkQvNWX10nbr3pXVf++Vqk38jiKcAbncAkeXEMd7qAJLSDwCM/wCm+OdF6cd+dj2Vpw8plT+APn8wfynY63</latexit> S1 = B12 B22 <latexit sha1_base64="TEQjHUdpzeSPoYWXZr64lccZb64=">AAAB/nicbZDLSsNAFIZPvNZ6i4orN4NFcGNJoqAbodSNy4r2Am0Ik+m0HTq5MDMRSgj4Km5cKOLW53Dn2zhps9DWHwY+/nMO58zvx5xJZVnfxtLyyuraemmjvLm1vbNr7u23ZJQIQpsk4pHo+FhSzkLaVExx2okFxYHPadsf3+T19iMVkkXhg5rE1A3wMGQDRrDSlmce3ns2ukZ1L7WdDJ3l4DiZZ1asqjUVWgS7gAoUanjmV68fkSSgoSIcS9m1rVi5KRaKEU6zci+RNMZkjIe0qzHEAZVuOj0/Qyfa6aNBJPQLFZq6vydSHEg5CXzdGWA1kvO13Pyv1k3U4MpNWRgnioZktmiQcKQilGeB+kxQovhEAyaC6VsRGWGBidKJlXUI9vyXF6HlVO3zqnN3UanVizhKcATHcAo2XEINbqEBTSCQwjO8wpvxZLwY78bHrHXJKGYO4I+Mzx+FaJNH</latexit> S2 = A11 + A12 <latexit sha1_base64="kQfTDG7+jZkRnJ6Qm+tVJoHUExg=">AAAB/nicbZDLSsNAFIZPvNZ6i4orN4NFEISSREE3QtWNy4r2Am0Ik+m0HTq5MDMRSgj4Km5cKOLW53Dn2zhNs9DWHwY+/nMO58zvx5xJZVnfxsLi0vLKammtvL6xubVt7uw2ZZQIQhsk4pFo+1hSzkLaUExx2o4FxYHPacsf3UzqrUcqJIvCBzWOqRvgQcj6jGClLc/cv/ccdImuvNS2M3SSg5N5ZsWqWrnQPNgFVKBQ3TO/ur2IJAENFeFYyo5txcpNsVCMcJqVu4mkMSYjPKAdjSEOqHTT/PwMHWmnh/qR0C9UKHd/T6Q4kHIc+LozwGooZ2sT879aJ1H9CzdlYZwoGpLpon7CkYrQJAvUY4ISxccaMBFM34rIEAtMlE6srEOwZ788D02nap9WnbuzSu26iKMEB3AIx2DDOdTgFurQAAIpPMMrvBlPxovxbnxMWxeMYmYP/sj4/AF9uZNC</latexit> S3 = A21 + A22 <latexit sha1_base64="zDtM0TbGxidxeXS1poJdYfnKV6c=">AAAB/nicbZDLSsNAFIYn9VbrLSqu3AwWQRBKkgq6EapuXFa0F2hDmEwn7dDJJMxMhBICvoobF4q49Tnc+TZO0yy09YeBj/+cwznz+zGjUlnWt1FaWl5ZXSuvVzY2t7Z3zN29towSgUkLRywSXR9JwignLUUVI91YEBT6jHT88c203nkkQtKIP6hJTNwQDTkNKEZKW555cO/V4SW88lLHzuBpDk7mmVWrZuWCi2AXUAWFmp751R9EOAkJV5ghKXu2FSs3RUJRzEhW6SeSxAiP0ZD0NHIUEumm+fkZPNbOAAaR0I8rmLu/J1IUSjkJfd0ZIjWS87Wp+V+tl6jgwk0pjxNFOJ4tChIGVQSnWcABFQQrNtGAsKD6VohHSCCsdGIVHYI9/+VFaDs1u15z7s6qjesijjI4BEfgBNjgHDTALWiCFsAgBc/gFbwZT8aL8W58zFpLRjGzD/7I+PwBgmSTRQ==</latexit> S4 = B21 B11 <latexit sha1_base64="nAbZAh8rJ5RA4MworYEGWIgLyi8=">AAAB/nicbZDLSgMxFIYz9VbrbVRcuQkWwY1lUgu6EUrduKxoL9AOQyZN29BMZkgyQhkGfBU3LhRx63O4823MtLPQ1h8CH/85h3Py+xFnSjvOt1VYWV1b3yhulra2d3b37P2DtgpjSWiLhDyUXR8rypmgLc00p91IUhz4nHb8yU1W7zxSqVgoHvQ0om6AR4INGcHaWJ59dO/V4DVseEkVpfA8A4RSzy47FWcmuAwohzLI1fTsr/4gJHFAhSYcK9VDTqTdBEvNCKdpqR8rGmEywSPaMyhwQJWbzM5P4alxBnAYSvOEhjP390SCA6WmgW86A6zHarGWmf/VerEeXrkJE1GsqSDzRcOYQx3CLAs4YJISzacGMJHM3ArJGEtMtEmsZEJAi19ehna1gi4q1btaud7I4yiCY3ACzgACl6AObkETtAABCXgGr+DNerJerHfrY95asPKZQ/BH1ucPhyCTSA==</latexit> S5 = A11 + A22 <latexit sha1_base64="j9LuFiwMdD9SkgTnUSCX/PhowjY=">AAAB/nicbZDLSsNAFIYn9VbrLSqu3AwWQRBKEhXdCFU3LivaC7QhTKaTduhkEmYmQgkBX8WNC0Xc+hzufBsnbRba+sPAx3/O4Zz5/ZhRqSzr2ygtLC4tr5RXK2vrG5tb5vZOS0aJwKSJIxaJjo8kYZSTpqKKkU4sCAp9Rtr+6Cavtx+JkDTiD2ocEzdEA04DipHSlmfu3Xtn8BJeealtZ/A4B8fJPLNq1ayJ4DzYBVRBoYZnfvX6EU5CwhVmSMqubcXKTZFQFDOSVXqJJDHCIzQgXY0chUS66eT8DB5qpw+DSOjHFZy4vydSFEo5Dn3dGSI1lLO13Pyv1k1UcOGmlMeJIhxPFwUJgyqCeRawTwXBio01ICyovhXiIRIIK51YRYdgz355HlpOzT6pOXen1fp1EUcZ7IMDcARscA7q4BY0QBNgkIJn8ArejCfjxXg3PqatJaOY2QV/ZHz+AIQBk0Y=</latexit> S6 = B11 + B22 <latexit sha1_base64="XZQUfN0BGg7GqL7rc6bKVAWXmcw=">AAAB/nicbZDLSsNAFIYn9VbrLSqu3AwWQRBKEkXdCKVuXFa0F2hDmEwn7dDJJMxMhBICvoobF4q49Tnc+TZO2iy09YeBj/+cwznz+zGjUlnWt1FaWl5ZXSuvVzY2t7Z3zN29towSgUkLRywSXR9JwignLUUVI91YEBT6jHT88U1e7zwSIWnEH9QkJm6IhpwGFCOlLc88uPcu4DVseKltZ/A0B8fJPLNq1ayp4CLYBVRBoaZnfvUHEU5CwhVmSMqebcXKTZFQFDOSVfqJJDHCYzQkPY0chUS66fT8DB5rZwCDSOjHFZy6vydSFEo5CX3dGSI1kvO13Pyv1ktUcOWmlMeJIhzPFgUJgyqCeRZwQAXBik00ICyovhXiERIIK51YRYdgz395EdpOzT6rOXfn1XqjiKMMDsEROAE2uAR1cAuaoAUwSMEzeAVvxpPxYrwbH7PWklHM7IM/Mj5/AIiyk0k=</latexit> S7 = A12 A22 <latexit sha1_base64="FJmPqr641/PqeOGkCsjXcSmn6OY=">AAAB/nicbZDLSsNAFIYn9VbrLSqu3AwWwY0liULdCFU3LivaC7QhTKaTduhkEmYmQgkBX8WNC0Xc+hzufBsnbRba+sPAx3/O4Zz5/ZhRqSzr2ygtLa+srpXXKxubW9s75u5eW0aJwKSFIxaJro8kYZSTlqKKkW4sCAp9Rjr++Cavdx6JkDTiD2oSEzdEQ04DipHSlmce3Ht1eAmvvNR2Mniag+Nknlm1atZUcBHsAqqgUNMzv/qDCCch4QozJGXPtmLlpkgoihnJKv1EkhjhMRqSnkaOQiLddHp+Bo+1M4BBJPTjCk7d3xMpCqWchL7uDJEayflabv5X6yUquHBTyuNEEY5ni4KEQRXBPAs4oIJgxSYaEBZU3wrxCAmElU6sokOw57+8CG2nZp/VnLvzauO6iKMMDsEROAE2qIMGuAVN0AIYpOAZvII348l4Md6Nj1lryShm9sEfGZ8/i9GTSw==</latexit> S8 = B21 + B22 <latexit sha1_base64="hy3O3mMBSE7kgjZGxdh5/TqwNXQ=">AAAB/nicbZDLSsNAFIYn9VbrLSqu3AwWQRBKEgW7EUrduKxoL9CGMJlO2qGTSZiZCCUEfBU3LhRx63O4822cpllo6w8DH/85h3Pm92NGpbKsb6O0srq2vlHerGxt7+zumfsHHRklApM2jlgkej6ShFFO2ooqRnqxICj0Gen6k5tZvftIhKQRf1DTmLghGnEaUIyUtjzz6N6rw2vY9FLHzuB5Dk7mmVWrZuWCy2AXUAWFWp75NRhGOAkJV5ghKfu2FSs3RUJRzEhWGSSSxAhP0Ij0NXIUEumm+fkZPNXOEAaR0I8rmLu/J1IUSjkNfd0ZIjWWi7WZ+V+tn6ig7qaUx4kiHM8XBQmDKoKzLOCQCoIVm2pAWFB9K8RjJBBWOrGKDsFe/PIydJyafVFz7i6rjWYRRxkcgxNwBmxwBRrgFrRAG2CQgmfwCt6MJ+PFeDc+5q0lo5g5BH9kfP4AjW2TTA==</latexit> S9 = A11 A21 <latexit sha1_base64="5DjcDQ82KwZdWk9Py4MDmmqG82g=">AAAB/nicbZDLSgMxFIYz9VbrbVRcuQkWwY1lUgV1IVTduKxoL9AOQybNtKGZzJBkhDIM+CpuXCji1udw59uYabvQ1h8CH/85h3Py+zFnSjvOt1VYWFxaXimultbWNza37O2dpooSSWiDRDySbR8rypmgDc00p+1YUhz6nLb84U1ebz1SqVgkHvQopm6I+4IFjGBtLM/eu/cu4CW88lKEMnicQxVlnl12Ks5YcB7QFMpgqrpnf3V7EUlCKjThWKkOcmLtplhqRjjNSt1E0RiTIe7TjkGBQ6rcdHx+Bg+N04NBJM0TGo7d3xMpDpUahb7pDLEeqNlabv5X6yQ6OHdTJuJEU0Emi4KEQx3BPAvYY5ISzUcGMJHM3ArJAEtMtEmsZEJAs1+eh2a1gk4q1bvTcu16GkcR7IMDcAQQOAM1cAvqoAEISMEzeAVv1pP1Yr1bH5PWgjWd2QV/ZH3+AIvqk0s=</latexit> S10 = B11 + B12 <latexit sha1_base64="Yf+gD4nz9Dtmda66+oPeRnUbmC0=">AAACAXicbZDLSsNAFIZP6q3WW9SN4GawCIJQkiroRih147KivUAbwmQ6bYdOLsxMhBLixldx40IRt76FO9/GaZqFtv4w8PGfczhzfi/iTCrL+jYKS8srq2vF9dLG5tb2jrm715JhLAhtkpCHouNhSTkLaFMxxWknEhT7Hqdtb3w9rbcfqJAsDO7VJKKOj4cBGzCClbZc8+DOTWwrRVeorsFO0WkG1dQ1y1bFyoQWwc6hDLkarvnV64ck9mmgCMdSdm0rUk6ChWKE07TUiyWNMBnjIe1qDLBPpZNkF6ToWDt9NAiFfoFCmft7IsG+lBPf050+ViM5X5ua/9W6sRpcOgkLoljRgMwWDWKOVIimcaA+E5QoPtGAiWD6r4iMsMBE6dBKOgR7/uRFaFUr9lmlentertXzOIpwCEdwAjZcQA1uoAFNIPAIz/AKb8aT8WK8Gx+z1oKRz+zDHxmfP8hslIk=</latexit> ⇥(n2) <latexit sha1_base64="3afWb1vL+AOktUowYQ1CxKg5hzM=">AAAB8nicbVBNSwMxEM36WetX1aOXYBHqpexWQY9FLx4r9Au2a8mm2TY0myzJrFCW/gwvHhTx6q/x5r8xbfegrQ8GHu/NMDMvTAQ34Lrfztr6xubWdmGnuLu3f3BYOjpuG5VqylpUCaW7ITFMcMlawEGwbqIZiUPBOuH4buZ3npg2XMkmTBIWxGQoecQpASv5veaIAanIx9pFv1R2q+4ceJV4OSmjHI1+6as3UDSNmQQqiDG+5yYQZEQDp4JNi73UsITQMRky31JJYmaCbH7yFJ9bZYAjpW1JwHP190RGYmMmcWg7YwIjs+zNxP88P4XoJsi4TFJgki4WRanAoPDsfzzgmlEQE0sI1dzeiumIaELBplS0IXjLL6+Sdq3qXVZrD1fl+m0eRwGdojNUQR66RnV0jxqohShS6Bm9ojcHnBfn3flYtK45+cwJ+gPn8wc3WpCN</latexit> オーダーは 3. 次の乗算を実行(再帰フェーズ) P1 = A11 · S1 <latexit sha1_base64="N/Ku60xTzeYmQN3XpNzpIsR267E=">AAAB/nicbVDLSgMxFM3UV62vUXHlJlgEV2VSBd0IVTcuK9oHtMOQyaRtaCYZkoxQhoK/4saFIm79Dnf+jWk7C209cOFwzr3ce0+YcKaN5307haXlldW14nppY3Nre8fd3WtqmSpCG0Ryqdoh1pQzQRuGGU7biaI4DjlthcObid96pEozKR7MKKF+jPuC9RjBxkqBe1APELyEV0GG0LhLImngfYACt+xVvCngIkE5KYMc9cD96kaSpDEVhnCsdQd5ifEzrAwjnI5L3VTTBJMh7tOOpQLHVPvZ9PwxPLZKBHtS2RIGTtXfExmOtR7Foe2MsRnoeW8i/ud1UtO78DMmktRQQWaLeimHRsJJFjBiihLDR5Zgopi9FZIBVpgYm1jJhoDmX14kzWoFnVaqd2fl2nUeRxEcgiNwAhA4BzVwC+qgAQjIwDN4BW/Ok/PivDsfs9aCk8/sgz9wPn8AdBCT4Q==</latexit> P2 = S2 · B22 <latexit sha1_base64="Zz7qZgACxw6FqCtBBzk+tl/IBA0=">AAACAHicbZDLSsNAFIYn9VbrLerChZvBIrgqSRR0I5S6cVnRXqANYTKZtkMnM2FmIpSQja/ixoUibn0Md76N0zYLbf1h4OM/53Dm/GHCqNKO822VVlbX1jfKm5Wt7Z3dPXv/oK1EKjFpYcGE7IZIEUY5aWmqGekmkqA4ZKQTjm+m9c4jkYoK/qAnCfFjNOR0QDHSxgrso2bgwWt4H2Re3seR0LBh0MsDu+rUnJngMrgFVEGhZmB/9SOB05hwjRlSquc6ifYzJDXFjOSVfqpIgvAYDUnPIEcxUX42OyCHp8aJ4EBI87iGM/f3RIZipSZxaDpjpEdqsTY1/6v1Uj248jPKk1QTjueLBimDWsBpGjCikmDNJgYQltT8FeIRkghrk1nFhOAunrwMba/mnte8u4tqvVHEUQbH4AScARdcgjq4BU3QAhjk4Bm8gjfryXqx3q2PeWvJKmYOwR9Znz9OVJTy</latexit> P3 = S3 · B11 <latexit sha1_base64="41NA/f+dap36gKZ1H7+8QFu6hyc=">AAACAHicbVBNS8NAEN3Ur1q/oh48eFksgqeStIJehFIvHivaVmhD2Gw27dLNbtjdCCXk4l/x4kERr/4Mb/4bt20O2vpg4PHeDDPzgoRRpR3n2yqtrK6tb5Q3K1vbO7t79v5BV4lUYtLBggn5ECBFGOWko6lm5CGRBMUBI71gfD31e49EKir4vZ4kxIvRkNOIYqSN5NtHbb8Br+CdnzXyAQ6Fhi0/c93ct6tOzZkBLhO3IFVQoO3bX4NQ4DQmXGOGlOq7TqK9DElNMSN5ZZAqkiA8RkPSN5SjmCgvmz2Qw1OjhDAS0hTXcKb+nshQrNQkDkxnjPRILXpT8T+vn+ro0ssoT1JNOJ4vilIGtYDTNGBIJcGaTQxBWFJzK8QjJBHWJrOKCcFdfHmZdOs1t1Gr355Xm60ijjI4BifgDLjgAjTBDWiDDsAgB8/gFbxZT9aL9W59zFtLVjFzCP7A+vwBTnKU8g==</latexit> P4 = A22 · S4 <latexit sha1_base64="iwuikfjV5P32cyxx7cFU62LiHSM=">AAACAHicbVBNS8NAEN3Ur1q/oh48eFksgqeS1IJehKoXjxVtLbQhbDabdulmN+xuhBJy8a948aCIV3+GN/+N2zYHbX0w8Hhvhpl5QcKo0o7zbZWWlldW18rrlY3Nre0de3evo0QqMWljwYTsBkgRRjlpa6oZ6SaSoDhg5CEYXU/8h0ciFRX8Xo8T4sVowGlEMdJG8u2Dlt+AF/DSz+r1vI9DoeGdnzVy3646NWcKuEjcglRBgZZvf/VDgdOYcI0ZUqrnOon2MiQ1xYzklX6qSILwCA1Iz1COYqK8bPpADo+NEsJISFNcw6n6eyJDsVLjODCdMdJDNe9NxP+8Xqqjcy+jPEk14Xi2KEoZ1AJO0oAhlQRrNjYEYUnNrRAPkURYm8wqJgR3/uVF0qnX3NNa/bZRbV4VcZTBITgCJ8AFZ6AJbkALtAEGOXgGr+DNerJerHfrY9ZasoqZffAH1ucPUGOU9Q==</latexit> P5 = S5 · S6 <latexit sha1_base64="KVRdTov5xwvqUIsm2YZJCOinWo8=">AAAB/3icbZDNSsNAFIUn9a/Wv6jgxs1gEVyVpFp1IxTduKxoa6ENYTKZtEMnmTAzEUrMwldx40IRt76GO9/GSZuFth4Y+Dj3Xu6d48WMSmVZ30ZpYXFpeaW8Wllb39jcMrd3OpInApM25oyLrockYTQibUUVI91YEBR6jNx7o6u8fv9AhKQ8ulPjmDghGkQ0oBgpbbnmXsttwAt466aNrI99rnI8zVyzatWsieA82AVUQaGWa371fY6TkEQKMyRlz7Zi5aRIKIoZySr9RJIY4REakJ7GCIVEOunk/gweaseHARf6RQpO3N8TKQqlHIee7gyRGsrZWm7+V+slKjh3UhrFiSIRni4KEgYVh3kY0KeCYMXGGhAWVN8K8RAJhJWOrKJDsGe/PA+des0+rtVvTqrNyyKOMtgHB+AI2OAMNME1aIE2wOARPINX8GY8GS/Gu/ExbS0Zxcwu+CPj8wcAEJTR</latexit> P6 = S7 · S8 <latexit sha1_base64="jP2eL+NPFQIoXiNJJsUEqVMiAhk=">AAAB/3icbZDNSsNAFIUn9a/Wv6jgxs1gEVyVpIrtRii6cVnRtkIbwmQyaYdOMmFmIpSYha/ixoUibn0Nd76NkzYLbT0w8HHuvdw7x4sZlcqyvo3S0vLK6lp5vbKxubW9Y+7udSVPBCYdzBkX9x6ShNGIdBRVjNzHgqDQY6Tnja/yeu+BCEl5dKcmMXFCNIxoQDFS2nLNg7Z7Di/grZs2sgH2ucqxmblm1apZU8FFsAuogkJt1/wa+BwnIYkUZkjKvm3FykmRUBQzklUGiSQxwmM0JH2NEQqJdNLp/Rk81o4PAy70ixScur8nUhRKOQk93RkiNZLztdz8r9ZPVNB0UhrFiSIRni0KEgYVh3kY0KeCYMUmGhAWVN8K8QgJhJWOrKJDsOe/vAjdes0+rdVvzqqtyyKOMjgER+AE2KABWuAatEEHYPAInsEreDOejBfj3fiYtZaMYmYf/JHx+QMH0ZTW</latexit> P7 = S9 · S10 <latexit sha1_base64="jR6YcqRgH0Y+XNvLVfr4yHK0xIA=">AAACAHicbZBPS8MwGMbT+W/Of1UPHrwEh+BptFOYHoShF48T3RxspaRptoWlSUlSYZRe/CpePCji1Y/hzW9juvWgmw8Efjzv+/LmfYKYUaUd59sqLS2vrK6V1ysbm1vbO/buXkeJRGLSxoIJ2Q2QIoxy0tZUM9KNJUFRwMhDML7O6w+PRCoq+L2exMSL0JDTAcVIG8u3D1p+A17COz+9yPo4FDpH18l8u+rUnKngIrgFVEGhlm9/9UOBk4hwjRlSquc6sfZSJDXFjGSVfqJIjPAYDUnPIEcRUV46PSCDx8YJ4UBI87iGU/f3RIoipSZRYDojpEdqvpab/9V6iR6ceynlcaIJx7NFg4RBLWCeBgypJFiziQGEJTV/hXiEJMLaZFYxIbjzJy9Cp15zT2v127Nq86qIowwOwRE4AS5ogCa4AS3QBhhk4Bm8gjfryXqx3q2PWWvJKmb2wR9Znz92zJUM</latexit> 7回の再帰 4. 次の計算で部分行列を得る C11 = P5 + P4 P2 + P6 <latexit sha1_base64="pImTiRKAbMnW/mkQ4WJWrScI9mo=">AAACB3icbVDLSgMxFL1TX7W+Rl0KEiyCIJaZWh8bodiNywr2Ae0wZNK0Dc08SDJCGbpz46+4caGIW3/BnX9jOp2FVg/kcjjnXm7u8SLOpLKsLyO3sLi0vJJfLaytb2xumds7TRnGgtAGCXko2h6WlLOANhRTnLYjQbHvcdryRrWp37qnQrIwuFPjiDo+HgSszwhWWnLN/Zqb2PYEXaG6e4aOda2gE13LKT93zaJVslKgv8TOSBEy1F3zs9sLSezTQBGOpezYVqScBAvFCKeTQjeWNMJkhAe0o2mAfSqdJL1jgg610kP9UOgXKJSqPycS7Es59j3d6WM1lPPeVPzP68Sqf+kkLIhiRQMyW9SPOVIhmoaCekxQovhYE0wE039FZIgFJkpHV9Ah2PMn/yXNcsk+LZVvK8XqdRZHHvbgAI7Ahguowg3UoQEEHuAJXuDVeDSejTfjfdaaM7KZXfgF4+MbV3qVJA==</latexit> C12 = P1 + P2 <latexit sha1_base64="6GsP5abxmO6x5wjFYfOIV0lhlcg=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDBZBEEoSBd0IxW5cVrAPaEOYTCft0MmDmYlYQn7FjQtF3Poj7vwbp20W2nrgXg7n3MvcOX7CmVSW9W2U1tY3NrfK25Wd3b39A/Ow2pFxKghtk5jHoudjSTmLaFsxxWkvERSHPqddf9Kc+d1HKiSLowc1Tagb4lHEAkaw0pJnVpteZjs5ukEtz0bnujueWbPq1hxoldgFqUGBlmd+DYYxSUMaKcKxlH3bSpSbYaEY4TSvDFJJE0wmeET7mkY4pNLN5rfn6FQrQxTEQlek0Fz9vZHhUMpp6OvJEKuxXPZm4n9eP1XBtZuxKEkVjcjioSDlSMVoFgQaMkGJ4lNNMBFM34rIGAtMlI6rokOwl7+8SjpO3b6oO/eXtcZtEUcZjuEEzsCGK2jAHbSgDQSe4Ble4c3IjRfj3fhYjJaMYucI/sD4/AFUk5IJ</latexit> C21 = P3 + P4 <latexit sha1_base64="e9RIFdplAiCrakdkBaZV7cG8Y50=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDBZBEErSFnQjFLtxWcE+oA1hMp20QycPZiZiCfkVNy4UceuPuPNvnLZZaOuBezmccy9z53gxZ1JZ1rdR2Njc2t4p7pb29g8Oj8zjcldGiSC0QyIeib6HJeUspB3FFKf9WFAceJz2vGlr7vceqZAsCh/ULKZOgMch8xnBSkuuWW65ac3O0A1qu3V0qXvDNStW1VoArRM7JxXI0XbNr+EoIklAQ0U4lnJgW7FyUiwUI5xmpWEiaYzJFI/pQNMQB1Q66eL2DJ1rZYT8SOgKFVqovzdSHEg5Czw9GWA1kaveXPzPGyTKv3ZSFsaJoiFZPuQnHKkIzYNAIyYoUXymCSaC6VsRmWCBidJxlXQI9uqX10m3VrXr1dp9o9K8zeMowimcwQXYcAVNuIM2dIDAEzzDK7wZmfFivBsfy9GCke+cwB8Ynz9asJIN</latexit> C22 = P5 + P1 P3 P7 <latexit sha1_base64="gPTLYkDxteldQjPz3ckpLa8JFeI=">AAACB3icbVDLSsNAFJ3UV62vqEtBBosgiCVJlboRit24rGAf0IYwmU7aoZNJmJkIJXTnxl9x40IRt/6CO//GaZqFVi/M4XDOvdy5x48ZlcqyvozC0vLK6lpxvbSxubW9Y+7utWWUCExaOGKR6PpIEkY5aSmqGOnGgqDQZ6Tjjxszv3NPhKQRv1OTmLghGnIaUIyUljzzsOGljjOFV7DpXcBTjTY801jNsOaZZatiZQX/EjsnZZBX0zM/+4MIJyHhCjMkZc+2YuWmSCiKGZmW+okkMcJjNCQ9TTkKiXTT7I4pPNbKAAaR0I8rmKk/J1IUSjkJfd0ZIjWSi95M/M/rJSq4dFPK40QRjueLgoRBFcFZKHBABcGKTTRBWFD9V4hHSCCsdHQlHYK9ePJf0nYqdrXi3J6X69d5HEVwAI7ACbBBDdTBDWiCFsDgATyBF/BqPBrPxpvxPm8tGPnMPvhVxsc3XCOVJw==</latexit> ⇥(n2) <latexit sha1_base64="3afWb1vL+AOktUowYQ1CxKg5hzM=">AAAB8nicbVBNSwMxEM36WetX1aOXYBHqpexWQY9FLx4r9Au2a8mm2TY0myzJrFCW/gwvHhTx6q/x5r8xbfegrQ8GHu/NMDMvTAQ34Lrfztr6xubWdmGnuLu3f3BYOjpuG5VqylpUCaW7ITFMcMlawEGwbqIZiUPBOuH4buZ3npg2XMkmTBIWxGQoecQpASv5veaIAanIx9pFv1R2q+4ceJV4OSmjHI1+6as3UDSNmQQqiDG+5yYQZEQDp4JNi73UsITQMRky31JJYmaCbH7yFJ9bZYAjpW1JwHP190RGYmMmcWg7YwIjs+zNxP88P4XoJsi4TFJgki4WRanAoPDsfzzgmlEQE0sI1dzeiumIaELBplS0IXjLL6+Sdq3qXVZrD1fl+m0eRwGdojNUQR66RnV0jxqohShS6Bm9ojcHnBfn3flYtK45+cwJ+gPn8wc3WpCN</latexit> オーダーは オーダーは ⇥(1) <latexit sha1_base64="mR4xVs/DXYAmOYoxJUW5uOvqr/U=">AAAB8HicbVA9TwJBEJ3DL8Qv1NJmIzHBhtyhiZZEG0tMADFwIXvLHGzYvbvs7pkQwq+wsdAYW3+Onf/GBa5Q8CWTvLw3k5l5QSK4Nq777eTW1jc2t/LbhZ3dvf2D4uFRS8epYthksYhVO6AaBY+wabgR2E4UUhkIfAhGtzP/4QmV5nHUMOMEfUkHEQ85o8ZKj93GEA0te+e9YsmtuHOQVeJlpAQZ6r3iV7cfs1RiZJigWnc8NzH+hCrDmcBpoZtqTCgb0QF2LI2oRO1P5gdPyZlV+iSMla3IkLn6e2JCpdZjGdhOSc1QL3sz8T+vk5rw2p/wKEkNRmyxKEwFMTGZfU/6XCEzYmwJZYrbWwkbUkWZsRkVbAje8surpFWteBeV6v1lqXaTxZGHEziFMnhwBTW4gzo0gYGEZ3iFN0c5L86787FozTnZzDH8gfP5A6+jj6w=</latexit> 14
  15. /35 正当性の証明 C12 = P1 + P2 <latexit sha1_base64="6GsP5abxmO6x5wjFYfOIV0lhlcg=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDBZBEEoSBd0IxW5cVrAPaEOYTCft0MmDmYlYQn7FjQtF3Poj7vwbp20W2nrgXg7n3MvcOX7CmVSW9W2U1tY3NrfK25Wd3b39A/Ow2pFxKghtk5jHoudjSTmLaFsxxWkvERSHPqddf9Kc+d1HKiSLowc1Tagb4lHEAkaw0pJnVpteZjs5ukEtz0bnujueWbPq1hxoldgFqUGBlmd+DYYxSUMaKcKxlH3bSpSbYaEY4TSvDFJJE0wmeET7mkY4pNLN5rfn6FQrQxTEQlek0Fz9vZHhUMpp6OvJEKuxXPZm4n9eP1XBtZuxKEkVjcjioSDlSMVoFgQaMkGJ4lNNMBFM34rIGAtMlI6rokOwl7+8SjpO3b6oO/eXtcZtEUcZjuEEzsCGK2jAHbSgDQSe4Ble4c3IjRfj3fhYjJaMYucI/sD4/AFUk5IJ</latexit> の場合

    A11 · B12 A11 · B22 + A11 · B22 + A12 · B22 <latexit sha1_base64="G+qIPTnRXSzwXq8iteRfiDawnRc=">AAACQXicbZBLSwMxFIUzPuv4qrp0EywWQSwzo6DLWjcuK9gHdIaSSdM2NPMguSOUoX/Njf/AnXs3LhRx68a0U8S2XggcvnPCTY4fC67Asl6MpeWV1bX13Ia5ubW9s5vf26+rKJGU1WgkItn0iWKCh6wGHARrxpKRwBes4Q9uxn7jgUnFo/AehjHzAtILeZdTAhq1883rdmrbI5d2IsAVrZ0RLp7hjOJf7Dgj03XN4imey2sDZ9CZjbfzBatkTQYvCnsqCmg61Xb+2e1ENAlYCFQQpVq2FYOXEgmcCqbXJ4rFhA5Ij7W0DEnAlJdOGhjhY006uBtJfULAE/r3RkoCpYaBr5MBgb6a98bwP6+VQPfKS3kYJ8BCmi3qJgJDhMd14g6XjIIYakGo5PqtmPaJJBR06aYuwZ7/8qKoOyX7vOTcXRTKlWkdOXSIjtAJstElKqNbVEU1RNEjekXv6MN4Mt6MT+Mriy4Z0zsHaGaM7x/BpKsO</latexit> A11 · B12 <latexit sha1_base64="lfXtMYapnUxevvQQIXGCxKgH+/A=">AAAB/XicbZDLSsNAFIYn9VbrLV52bgaL4KokVdBlrRuXFewF2hAmk2k7dDITZiZCDcFXceNCEbe+hzvfxkmbhbb+MPDxn3M4Z/4gZlRpx/m2Siura+sb5c3K1vbO7p69f9BRIpGYtLFgQvYCpAijnLQ11Yz0YklQFDDSDSY3eb37QKSigt/raUy8CI04HVKMtLF8++jaT103G+BQaNg0XM9gxberTs2ZCS6DW0AVFGr59tcgFDiJCNeYIaX6rhNrL0VSU8xIVhkkisQIT9CI9A1yFBHlpbPrM3hqnBAOhTSPazhzf0+kKFJqGgWmM0J6rBZruflfrZ/o4ZWXUh4nmnA8XzRMGNQC5lHAkEqCNZsaQFhScyvEYyQR1iawPAR38cvL0KnX3PNa/e6i2mgWcZTBMTgBZ8AFl6ABbkELtAEGj+AZvII368l6sd6tj3lrySpmDsEfWZ8/K26TvQ==</latexit> +A12 · B22 <latexit sha1_base64="XkwuXlc47m/Q5JnPc9HhkRS2oP4=">AAAB/nicbVDLSsNAFJ3UV62vqLhyM1gEQShJFHRZ68ZlBfuANoTJZNIOncyEmYlQQsBfceNCEbd+hzv/xqTNQlsPXDiccy/33uPHjCptWd9GZWV1bX2julnb2t7Z3TP3D7pKJBKTDhZMyL6PFGGUk46mmpF+LAmKfEZ6/uS28HuPRCoq+IOexsSN0IjTkGKkc8kzj85vvNR2siEOhIYtL3WcDNY8s241rBngMrFLUgcl2p75NQwETiLCNWZIqYFtxdpNkdQUM5LVhokiMcITNCKDnHIUEeWms/MzeJorAQyFzItrOFN/T6QoUmoa+XlnhPRYLXqF+J83SHR47aaUx4kmHM8XhQmDWsAiCxhQSbBm05wgLGl+K8RjJBHWeWJFCPbiy8uk6zTsi4Zzf1lvtso4quAYnIAzYIMr0AR3oA06AIMUPINX8GY8GS/Gu/Exb60Y5cwh+APj8weYOpP0</latexit> A11 <latexit sha1_base64="WtsRVvyqN1EY0jYrpA0wO4qTpYw=">AAAB7XicbVBNSwMxEJ31s9avqkcvwSJ4Kpsq6LHqxWMF+wHtUrJpto3NJkuSFcrS/+DFgyJe/T/e/Dem7R609cHA470ZZuaFieDG+v63t7K6tr6xWdgqbu/s7u2XDg6bRqWasgZVQul2SAwTXLKG5VawdqIZiUPBWuHoduq3npg2XMkHO05YEJOB5BGnxDqped3LMJ70SmW/4s+AlgnOSRly1Hulr25f0TRm0lJBjOlgP7FBRrTlVLBJsZsalhA6IgPWcVSSmJkgm107QadO6aNIaVfSopn6eyIjsTHjOHSdMbFDs+hNxf+8TmqjqyDjMkktk3S+KEoFsgpNX0d9rhm1YuwIoZq7WxEdEk2odQEVXQh48eVl0qxW8Hmlen9Rrt3kcRTgGE7gDDBcQg3uoA4NoPAIz/AKb57yXrx372PeuuLlM0fwB97nD+4JjrQ=</latexit> A12 <latexit sha1_base64="HvNqFpXTG6rWIkCLPWE0ieuTihk=">AAAB7XicbVBNSwMxEJ34WetX1aOXYBE8ld0q6LHqxWMF+wHtUrJpto3NJkuSFcrS/+DFgyJe/T/e/Dem7R609cHA470ZZuaFieDGet43WlldW9/YLGwVt3d29/ZLB4dNo1JNWYMqoXQ7JIYJLlnDcitYO9GMxKFgrXB0O/VbT0wbruSDHScsiMlA8ohTYp3UvO5lfnXSK5W9ijcDXiZ+TsqQo94rfXX7iqYxk5YKYkzH9xIbZERbTgWbFLupYQmhIzJgHUcliZkJstm1E3zqlD6OlHYlLZ6pvycyEhszjkPXGRM7NIveVPzP66Q2ugoyLpPUMknni6JUYKvw9HXc55pRK8aOEKq5uxXTIdGEWhdQ0YXgL768TJrVin9eqd5flGs3eRwFOIYTOAMfLqEGd1CHBlB4hGd4hTek0At6Rx/z1hWUzxzBH6DPH++OjrU=</latexit> A21 <latexit sha1_base64="H82LD9IAmpAW5rWkE2+2R7km4QU=">AAAB7XicbVBNSwMxEJ34WetX1aOXYBE8ld0q6LHqxWMF+wHtUrJpto3NJkuSFcrS/+DFgyJe/T/e/Dem7R609cHA470ZZuaFieDGet43WlldW9/YLGwVt3d29/ZLB4dNo1JNWYMqoXQ7JIYJLlnDcitYO9GMxKFgrXB0O/VbT0wbruSDHScsiMlA8ohTYp3UvO5lVX/SK5W9ijcDXiZ+TsqQo94rfXX7iqYxk5YKYkzH9xIbZERbTgWbFLupYQmhIzJgHUcliZkJstm1E3zqlD6OlHYlLZ6pvycyEhszjkPXGRM7NIveVPzP66Q2ugoyLpPUMknni6JUYKvw9HXc55pRK8aOEKq5uxXTIdGEWhdQ0YXgL768TJrVin9eqd5flGs3eRwFOIYTOAMfLqEGd1CHBlB4hGd4hTek0At6Rx/z1hWUzxzBH6DPH++PjrU=</latexit> A22 <latexit sha1_base64="gGMSnDhj1RvmVuBmOJuXydE//ow=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRbBU9ldC3qsevFYwX5Au5Rsmm1js8mSZIWy9D948aCIV/+PN/+NabsHbX0w8Hhvhpl5YcKZNq777RTW1jc2t4rbpZ3dvf2D8uFRS8tUEdokkkvVCbGmnAnaNMxw2kkUxXHIaTsc38789hNVmknxYCYJDWI8FCxiBBsrta77me9P++WKW3XnQKvEy0kFcjT65a/eQJI0psIQjrXuem5iggwrwwin01Iv1TTBZIyHtGupwDHVQTa/dorOrDJAkVS2hEFz9fdEhmOtJ3FoO2NsRnrZm4n/ed3URFdBxkSSGirIYlGUcmQkmr2OBkxRYvjEEkwUs7ciMsIKE2MDKtkQvOWXV0nLr3oXVf++Vqnf5HEU4QRO4Rw8uIQ63EEDmkDgEZ7hFd4c6bw4787HorXg5DPH8AfO5w/xFI62</latexit> B11 <latexit sha1_base64="8gCv0nL+iW+7P6Y4gWHTEJ+WgiI=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRbBU9lUQY+lXjxWsLXQLiWbZtvYbLIkWaEs/Q9ePCji1f/jzX9j2u5BWx8MPN6bYWZemAhurO9/e4W19Y3NreJ2aWd3b/+gfHjUNirVlLWoEkp3QmKY4JK1LLeCdRLNSBwK9hCOb2b+wxPThit5bycJC2IylDzilFgntRv9DONpv1zxq/4caJXgnFQgR7Nf/uoNFE1jJi0VxJgu9hMbZERbTgWblnqpYQmhYzJkXUcliZkJsvm1U3TmlAGKlHYlLZqrvycyEhsziUPXGRM7MsveTPzP66Y2ug4yLpPUMkkXi6JUIKvQ7HU04JpRKyaOEKq5uxXREdGEWhdQyYWAl19eJe1aFV9Ua3eXlXojj6MIJ3AK54DhCupwC01oAYVHeIZXePOU9+K9ex+L1oKXzxzDH3ifP++SjrU=</latexit> B12 <latexit sha1_base64="QuM51X7FDHygo6xFiwqXTVC2cok=">AAAB7XicbVBNSwMxEJ3Ur1q/qh69BIvgqexWQY+lXjxWsLXQLiWbZtvYbLIkWaEs/Q9ePCji1f/jzX9j2u5BWx8MPN6bYWZemAhurOd9o8La+sbmVnG7tLO7t39QPjxqG5VqylpUCaU7ITFMcMlallvBOolmJA4FewjHNzP/4Ylpw5W8t5OEBTEZSh5xSqyT2o1+5tem/XLFq3pz4FXi56QCOZr98ldvoGgaM2mpIMZ0fS+xQUa05VSwaamXGpYQOiZD1nVUkpiZIJtfO8VnThngSGlX0uK5+nsiI7Exkzh0nTGxI7PszcT/vG5qo+sg4zJJLZN0sShKBbYKz17HA64ZtWLiCKGau1sxHRFNqHUBlVwI/vLLq6Rdq/oX1drdZaXeyOMowgmcwjn4cAV1uIUmtIDCIzzDK7whhV7QO/pYtBZQPnMMf4A+fwDxF462</latexit> B21 <latexit sha1_base64="LRdxHHGIKQrElvWuyxMUGbU5xvU=">AAAB7XicbVBNSwMxEJ3Ur1q/qh69BIvgqexWQY+lXjxWsLXQLiWbZtvYbLIkWaEs/Q9ePCji1f/jzX9j2u5BWx8MPN6bYWZemAhurOd9o8La+sbmVnG7tLO7t39QPjxqG5VqylpUCaU7ITFMcMlallvBOolmJA4FewjHNzP/4Ylpw5W8t5OEBTEZSh5xSqyT2o1+VvOn/XLFq3pz4FXi56QCOZr98ldvoGgaM2mpIMZ0fS+xQUa05VSwaamXGpYQOiZD1nVUkpiZIJtfO8VnThngSGlX0uK5+nsiI7Exkzh0nTGxI7PszcT/vG5qo+sg4zJJLZN0sShKBbYKz17HA64ZtWLiCKGau1sxHRFNqHUBlVwI/vLLq6Rdq/oX1drdZaXeyOMowgmcwjn4cAV1uIUmtIDCIzzDK7whhV7QO/pYtBZQPnMMf4A+fwDxGI62</latexit> B22 <latexit sha1_base64="Tp6lmLSuM4gEcdNM4n4aPeq4+Fg=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRbBU9ldC3os9eKxgv2AdinZNNvGZpMlyQpl6X/w4kERr/4fb/4b03YP2vpg4PHeDDPzwoQzbVz32ylsbG5t7xR3S3v7B4dH5eOTtpapIrRFJJeqG2JNORO0ZZjhtJsoiuOQ0044uZ37nSeqNJPiwUwTGsR4JFjECDZWajcGme/PBuWKW3UXQOvEy0kFcjQH5a/+UJI0psIQjrXueW5iggwrwwins1I/1TTBZIJHtGepwDHVQba4doYurDJEkVS2hEEL9fdEhmOtp3FoO2NsxnrVm4v/eb3URDdBxkSSGirIclGUcmQkmr+OhkxRYvjUEkwUs7ciMsYKE2MDKtkQvNWX10nbr3pXVf++Vqk38jiKcAbncAkeXEMd7qAJLSDwCM/wCm+OdF6cd+dj2Vpw8plT+APn8wfynY63</latexit> C11 <latexit sha1_base64="GyVbAkGkl2JkUgwwoB3OD7RzOHs=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRbBU9lUQY/FXjxWsLXQLiWbZtvYbLIkWaEs/Q9ePCji1f/jzX9j2u5BWx8MPN6bYWZemAhurO9/e4W19Y3NreJ2aWd3b/+gfHjUNirVlLWoEkp3QmKY4JK1LLeCdRLNSBwK9hCOGzP/4Ylpw5W8t5OEBTEZSh5xSqyT2o1+hvG0X674VX8OtEpwTiqQo9kvf/UGiqYxk5YKYkwX+4kNMqItp4JNS73UsITQMRmyrqOSxMwE2fzaKTpzygBFSruSFs3V3xMZiY2ZxKHrjIkdmWVvJv7ndVMbXQcZl0lqmaSLRVEqkFVo9joacM2oFRNHCNXc3YroiGhCrQuo5ELAyy+vknatii+qtbvLSv0mj6MIJ3AK54DhCupwC01oAYVHeIZXePOU9+K9ex+L1oKXzxzDH3ifP/EbjrY=</latexit> C12 <latexit sha1_base64="PhDJdWnMIvK9Wdgl3YkVIwL1mMs=">AAAB7XicbVDLSgNBEOyNrxhfUY9eBoPgKexGQY/BXDxGMDGQLGF2MpuMmccyMyuEJf/gxYMiXv0fb/6Nk2QPmljQUFR1090VJZwZ6/vfXmFtfWNzq7hd2tnd2z8oHx61jUo1oS2iuNKdCBvKmaQtyyynnURTLCJOH6JxY+Y/PFFtmJL3dpLQUOChZDEj2Dqp3ehnQW3aL1f8qj8HWiVBTiqQo9kvf/UGiqSCSks4NqYb+IkNM6wtI5xOS73U0ASTMR7SrqMSC2rCbH7tFJ05ZYBipV1Ji+bq74kMC2MmInKdAtuRWfZm4n9eN7XxdZgxmaSWSrJYFKccWYVmr6MB05RYPnEEE83crYiMsMbEuoBKLoRg+eVV0q5Vg4tq7e6yUr/J4yjCCZzCOQRwBXW4hSa0gMAjPMMrvHnKe/HevY9Fa8HLZ47hD7zPH/Kgjrc=</latexit> C22 <latexit sha1_base64="cxrJJIC+l9L+ZIBe7sL1xZt+0Cw=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRbBU9ldC3os9uKxgv2AdinZNNvGZpMlyQpl6X/w4kERr/4fb/4b03YP2vpg4PHeDDPzwoQzbVz32ylsbG5t7xR3S3v7B4dH5eOTtpapIrRFJJeqG2JNORO0ZZjhtJsoiuOQ0044acz9zhNVmknxYKYJDWI8EixiBBsrtRuDzPdng3LFrboLoHXi5aQCOZqD8ld/KEkaU2EIx1r3PDcxQYaVYYTTWamfappgMsEj2rNU4JjqIFtcO0MXVhmiSCpbwqCF+nsiw7HW0zi0nTE2Y73qzcX/vF5qopsgYyJJDRVkuShKOTISzV9HQ6YoMXxqCSaK2VsRGWOFibEBlWwI3urL66TtV72rqn9fq9Rv8ziKcAbncAkeXEMd7qAJLSDwCM/wCm+OdF6cd+dj2Vpw8plT+APn8wf0Jo64</latexit> C21 <latexit sha1_base64="281MpEfJrh/xPJY7c3yuZWmIESY=">AAAB7XicbVDLSgNBEOyNrxhfUY9eBoPgKexGQY/BXDxGMDGQLGF2MpuMmccyMyuEJf/gxYMiXv0fb/6Nk2QPmljQUFR1090VJZwZ6/vfXmFtfWNzq7hd2tnd2z8oHx61jUo1oS2iuNKdCBvKmaQtyyynnURTLCJOH6JxY+Y/PFFtmJL3dpLQUOChZDEj2Dqp3ehntWDaL1f8qj8HWiVBTiqQo9kvf/UGiqSCSks4NqYb+IkNM6wtI5xOS73U0ASTMR7SrqMSC2rCbH7tFJ05ZYBipV1Ji+bq74kMC2MmInKdAtuRWfZm4n9eN7XxdZgxmaSWSrJYFKccWYVmr6MB05RYPnEEE83crYiMsMbEuoBKLoRg+eVV0q5Vg4tq7e6yUr/J4yjCCZzCOQRwBXW4hSa0gMAjPMMrvHnKe/HevY9Fa8HLZ47hD7zPH/Khjrc=</latexit> P1 = A11 · S1 <latexit sha1_base64="N/Ku60xTzeYmQN3XpNzpIsR267E=">AAAB/nicbVDLSgMxFM3UV62vUXHlJlgEV2VSBd0IVTcuK9oHtMOQyaRtaCYZkoxQhoK/4saFIm79Dnf+jWk7C209cOFwzr3ce0+YcKaN5307haXlldW14nppY3Nre8fd3WtqmSpCG0Ryqdoh1pQzQRuGGU7biaI4DjlthcObid96pEozKR7MKKF+jPuC9RjBxkqBe1APELyEV0GG0LhLImngfYACt+xVvCngIkE5KYMc9cD96kaSpDEVhnCsdQd5ifEzrAwjnI5L3VTTBJMh7tOOpQLHVPvZ9PwxPLZKBHtS2RIGTtXfExmOtR7Foe2MsRnoeW8i/ud1UtO78DMmktRQQWaLeimHRsJJFjBiihLDR5Zgopi9FZIBVpgYm1jJhoDmX14kzWoFnVaqd2fl2nUeRxEcgiNwAhA4BzVwC+qgAQjIwDN4BW/Ok/PivDsfs9aCk8/sgz9wPn8AdBCT4Q==</latexit> P2 = S2 · B22 <latexit sha1_base64="Zz7qZgACxw6FqCtBBzk+tl/IBA0=">AAACAHicbZDLSsNAFIYn9VbrLerChZvBIrgqSRR0I5S6cVnRXqANYTKZtkMnM2FmIpSQja/ixoUibn0Md76N0zYLbf1h4OM/53Dm/GHCqNKO822VVlbX1jfKm5Wt7Z3dPXv/oK1EKjFpYcGE7IZIEUY5aWmqGekmkqA4ZKQTjm+m9c4jkYoK/qAnCfFjNOR0QDHSxgrso2bgwWt4H2Re3seR0LBh0MsDu+rUnJngMrgFVEGhZmB/9SOB05hwjRlSquc6ifYzJDXFjOSVfqpIgvAYDUnPIEcxUX42OyCHp8aJ4EBI87iGM/f3RIZipSZxaDpjpEdqsTY1/6v1Uj248jPKk1QTjueLBimDWsBpGjCikmDNJgYQltT8FeIRkghrk1nFhOAunrwMba/mnte8u4tqvVHEUQbH4AScARdcgjq4BU3QAhjk4Bm8gjfryXqx3q2PeWvJKmYOwR9Znz9OVJTy</latexit> S1 = B12 B22 <latexit sha1_base64="TEQjHUdpzeSPoYWXZr64lccZb64=">AAAB/nicbZDLSsNAFIZPvNZ6i4orN4NFcGNJoqAbodSNy4r2Am0Ik+m0HTq5MDMRSgj4Km5cKOLW53Dn2zhps9DWHwY+/nMO58zvx5xJZVnfxtLyyuraemmjvLm1vbNr7u23ZJQIQpsk4pHo+FhSzkLaVExx2okFxYHPadsf3+T19iMVkkXhg5rE1A3wMGQDRrDSlmce3ns2ukZ1L7WdDJ3l4DiZZ1asqjUVWgS7gAoUanjmV68fkSSgoSIcS9m1rVi5KRaKEU6zci+RNMZkjIe0qzHEAZVuOj0/Qyfa6aNBJPQLFZq6vydSHEg5CXzdGWA1kvO13Pyv1k3U4MpNWRgnioZktmiQcKQilGeB+kxQovhEAyaC6VsRGWGBidKJlXUI9vyXF6HlVO3zqnN3UanVizhKcATHcAo2XEINbqEBTSCQwjO8wpvxZLwY78bHrHXJKGYO4I+Mzx+FaJNH</latexit> S2 = A11 + A12 <latexit sha1_base64="kQfTDG7+jZkRnJ6Qm+tVJoHUExg=">AAAB/nicbZDLSsNAFIZPvNZ6i4orN4NFEISSREE3QtWNy4r2Am0Ik+m0HTq5MDMRSgj4Km5cKOLW53Dn2zhNs9DWHwY+/nMO58zvx5xJZVnfxsLi0vLKammtvL6xubVt7uw2ZZQIQhsk4pFo+1hSzkLaUExx2o4FxYHPacsf3UzqrUcqJIvCBzWOqRvgQcj6jGClLc/cv/ccdImuvNS2M3SSg5N5ZsWqWrnQPNgFVKBQ3TO/ur2IJAENFeFYyo5txcpNsVCMcJqVu4mkMSYjPKAdjSEOqHTT/PwMHWmnh/qR0C9UKHd/T6Q4kHIc+LozwGooZ2sT879aJ1H9CzdlYZwoGpLpon7CkYrQJAvUY4ISxccaMBFM34rIEAtMlE6srEOwZ788D02nap9WnbuzSu26iKMEB3AIx2DDOdTgFurQAAIpPMMrvBlPxovxbnxMWxeMYmYP/sj4/AF9uZNC</latexit> 代入 C11, C21, C22 <latexit sha1_base64="wajoXUBzaVAwtXiqkpojphJeSxE=">AAAB/3icbVDLSsNAFL3xWesrKrhxM1gEF1KSKOiy2I3LCvYBbQiT6aQdOnkwMxFKzMJfceNCEbf+hjv/xmkbQVsPDPdwzr3cO8dPOJPKsr6MpeWV1bX10kZ5c2t7Z9fc22/JOBWENknMY9HxsaScRbSpmOK0kwiKQ5/Ttj+qT/z2PRWSxdGdGifUDfEgYgEjWGnJMw/rXmbb+RnS1fmpTu6ZFatqTYEWiV2QChRoeOZnrx+TNKSRIhxL2bWtRLkZFooRTvNyL5U0wWSEB7SraYRDKt1sen+OTrTSR0Es9IsUmqq/JzIcSjkOfd0ZYjWU895E/M/rpiq4cjMWJamiEZktClKOVIwmYaA+E5QoPtYEE8H0rYgMscBE6cjKOgR7/suLpOVU7fOqc3tRqV0XcZTgCI7hFGy4hBrcQAOaQOABnuAFXo1H49l4M95nrUtGMXMAf2B8fAPobpQa</latexit> も同じようにして検証できる. 15
  16. /35 Strassenの方法の計算量 行列の乗算にかかる時間を として漸化式で表してみる n ⇥ n <latexit sha1_base64="rU/xrREL14Gj7SkmwaoAogYXTA0=">AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Ae2oWy2m3bpZhN2J0IJ/RdePCji1X/jzX/jts1BWx8MPN6bYWZekEhh0HW/ncLa+sbmVnG7tLO7t39QPjxqmTjVjDdZLGPdCajhUijeRIGSdxLNaRRI3g7GtzO//cS1EbF6wEnC/YgOlQgFo2ilR0V6KCJuiOqXK27VnYOsEi8nFcjR6Je/eoOYpRFXyCQ1puu5CfoZ1SiY5NNSLzU8oWxMh7xrqaJ2jZ/NL56SM6sMSBhrWwrJXP09kdHImEkU2M6I4sgsezPxP6+bYnjtZ0IlKXLFFovCVBKMyex9MhCaM5QTSyjTwt5K2IhqytCGVLIheMsvr5JWrepdVGv3l5X6TR5HEU7gFM7Bgyuowx00oAkMFDzDK7w5xnlx3p2PRWvByWeO4Q+czx8D1JB8</latexit> T(n)

    <latexit sha1_base64="vM8hsiJM1mIid8s1EMCG6OMcapM=">AAAB63icbVBNSwMxEJ2tX7V+VT16CRahXspuFfRY9OKxQr+gXUo2zbahSXZJskJZ+he8eFDEq3/Im//GbLsHbX0w8Hhvhpl5QcyZNq777RQ2Nre2d4q7pb39g8Oj8vFJR0eJIrRNIh6pXoA15UzStmGG016sKBYBp91gep/53SeqNItky8xi6gs8lixkBJtMalXl5bBccWvuAmideDmpQI7msPw1GEUkEVQawrHWfc+NjZ9iZRjhdF4aJJrGmEzxmPYtlVhQ7aeLW+fowiojFEbKljRoof6eSLHQeiYC2ymwmehVLxP/8/qJCW/9lMk4MVSS5aIw4chEKHscjZiixPCZJZgoZm9FZIIVJsbGU7IheKsvr5NOveZd1eqP15XGXR5HEc7gHKrgwQ004AGa0AYCE3iGV3hzhPPivDsfy9aCk8+cwh84nz9C/Y25</latexit> -ステップ1 ⇥(1) <latexit sha1_base64="mR4xVs/DXYAmOYoxJUW5uOvqr/U=">AAAB8HicbVA9TwJBEJ3DL8Qv1NJmIzHBhtyhiZZEG0tMADFwIXvLHGzYvbvs7pkQwq+wsdAYW3+Onf/GBa5Q8CWTvLw3k5l5QSK4Nq777eTW1jc2t/LbhZ3dvf2D4uFRS8epYthksYhVO6AaBY+wabgR2E4UUhkIfAhGtzP/4QmV5nHUMOMEfUkHEQ85o8ZKj93GEA0te+e9YsmtuHOQVeJlpAQZ6r3iV7cfs1RiZJigWnc8NzH+hCrDmcBpoZtqTCgb0QF2LI2oRO1P5gdPyZlV+iSMla3IkLn6e2JCpdZjGdhOSc1QL3sz8T+vk5rw2p/wKEkNRmyxKEwFMTGZfU/6XCEzYmwJZYrbWwkbUkWZsRkVbAje8surpFWteBeV6v1lqXaTxZGHEziFMnhwBTW4gzo0gYGEZ3iFN0c5L86787FozTnZzDH8gfP5A6+jj6w=</latexit> -再帰の底のケース ⇥(1) <latexit sha1_base64="mR4xVs/DXYAmOYoxJUW5uOvqr/U=">AAAB8HicbVA9TwJBEJ3DL8Qv1NJmIzHBhtyhiZZEG0tMADFwIXvLHGzYvbvs7pkQwq+wsdAYW3+Onf/GBa5Q8CWTvLw3k5l5QSK4Nq777eTW1jc2t/LbhZ3dvf2D4uFRS8epYthksYhVO6AaBY+wabgR2E4UUhkIfAhGtzP/4QmV5nHUMOMEfUkHEQ85o8ZKj93GEA0te+e9YsmtuHOQVeJlpAQZ6r3iV7cfs1RiZJigWnc8NzH+hCrDmcBpoZtqTCgb0QF2LI2oRO1P5gdPyZlV+iSMla3IkLn6e2JCpdZjGdhOSc1QL3sz8T+vk5rw2p/wKEkNRmyxKEwFMTGZfU/6XCEzYmwJZYrbWwkbUkWZsRkVbAje8surpFWteBeV6v1lqXaTxZGHEziFMnhwBTW4gzo0gYGEZ3iFN0c5L86787FozTnZzDH8gfP5A6+jj6w=</latexit> 明らかにさっきよりよさそう -ステップ3 7T(n/2) <latexit sha1_base64="TZYGmTlaDXraUg1F2X/1ZUehcvU=">AAAB7nicbVBNSwMxEJ3Ur1q/qh69BItQL3W3CvVY9OKxQr+gXUo2zbah2eySZIWy9Ed48aCIV3+PN/+NabsHbX0w8Hhvhpl5fiy4No7zjXIbm1vbO/ndwt7+weFR8fikraNEUdaikYhU1yeaCS5Zy3AjWDdWjIS+YB1/cj/3O09MaR7JppnGzAvJSPKAU2Ks1Kk1y/KqejkolpyKswBeJ25GSpChMSh+9YcRTUImDRVE657rxMZLiTKcCjYr9BPNYkInZMR6lkoSMu2li3Nn+MIqQxxEypY0eKH+nkhJqPU09G1nSMxYr3pz8T+vl5jg1ku5jBPDJF0uChKBTYTnv+MhV4waMbWEUMXtrZiOiSLU2IQKNgR39eV10q5W3OtK9fGmVL/L4sjDGZxDGVyoQR0eoAEtoDCBZ3iFNxSjF/SOPpatOZTNnMIfoM8fm96Obw==</latexit> -ステップ2 ⇥(n2) <latexit sha1_base64="3afWb1vL+AOktUowYQ1CxKg5hzM=">AAAB8nicbVBNSwMxEM36WetX1aOXYBHqpexWQY9FLx4r9Au2a8mm2TY0myzJrFCW/gwvHhTx6q/x5r8xbfegrQ8GHu/NMDMvTAQ34Lrfztr6xubWdmGnuLu3f3BYOjpuG5VqylpUCaW7ITFMcMlawEGwbqIZiUPBOuH4buZ3npg2XMkmTBIWxGQoecQpASv5veaIAanIx9pFv1R2q+4ceJV4OSmjHI1+6as3UDSNmQQqiDG+5yYQZEQDp4JNi73UsITQMRky31JJYmaCbH7yFJ9bZYAjpW1JwHP190RGYmMmcWg7YwIjs+zNxP88P4XoJsi4TFJgki4WRanAoPDsfzzgmlEQE0sI1dzeiumIaELBplS0IXjLL6+Sdq3qXVZrD1fl+m0eRwGdojNUQR66RnV0jxqohShS6Bm9ojcHnBfn3flYtK45+cwJ+gPn8wc3WpCN</latexit> -ステップ4 ⇥(n2) <latexit sha1_base64="3afWb1vL+AOktUowYQ1CxKg5hzM=">AAAB8nicbVBNSwMxEM36WetX1aOXYBHqpexWQY9FLx4r9Au2a8mm2TY0myzJrFCW/gwvHhTx6q/x5r8xbfegrQ8GHu/NMDMvTAQ34Lrfztr6xubWdmGnuLu3f3BYOjpuG5VqylpUCaW7ITFMcMlawEGwbqIZiUPBOuH4buZ3npg2XMkmTBIWxGQoecQpASv5veaIAanIx9pFv1R2q+4ceJV4OSmjHI1+6as3UDSNmQQqiDG+5yYQZEQDp4JNi73UsITQMRky31JJYmaCbH7yFJ9bZYAjpW1JwHP190RGYmMmcWg7YwIjs+zNxP88P4XoJsi4TFJgki4WRanAoPDsfzzgmlEQE0sI1dzeiumIaELBplS0IXjLL6+Sdq3qXVZrD1fl+m0eRwGdojNUQR66RnV0jxqohShS6Bm9ojcHnBfn3flYtK45+cwJ+gPn8wc3WpCN</latexit> T(n) = ( ⇥(1) n = 1 7T(n/2) + ⇥(n2) n > 1 <latexit sha1_base64="4daNEoqGsJPYylXGJyS741YqCQA=">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</latexit> のちのち登場するマスター法を用いると 計算量が であることがわかる! ⇥(nlg 7) <latexit sha1_base64="Xvwa/sb11rTkkrzUHzV1K1rInkE=">AAAB+3icbVBNS8NAEJ3Ur1q/Yj16CRahXkpShXosevFYoV/QxLLZbtqlm03Y3Ygl5K948aCIV/+IN/+N2zYHbX0w8Hhvhpl5fsyoVLb9bRQ2Nre2d4q7pb39g8Mj87jclVEiMOngiEWi7yNJGOWko6hipB8LgkKfkZ4/vZ37vUciJI14W81i4oVozGlAMVJaGppltz0hClX5Q+qycdrIsouhWbFr9gLWOnFyUoEcraH55Y4inISEK8yQlAPHjpWXIqEoZiQruYkkMcJTNCYDTTkKifTSxe2Zda6VkRVEQhdX1kL9PZGiUMpZ6OvOEKmJXPXm4n/eIFHBtZdSHieKcLxcFCTMUpE1D8IaUUGwYjNNEBZU32rhCRIIKx1XSYfgrL68Trr1mnNZq99fVZo3eRxFOIUzqIIDDWjCHbSgAxie4Ble4c3IjBfj3fhYthaMfOYE/sD4/AGa75Qo</latexit> 16
  17. /35 似たようなアイデア A = (a + b)(c + d) =

    ac + ad + bc + bd B = ac C = bd <latexit sha1_base64="uwzbJRfZJPhtPg9NA+Ke4ABrqaQ=">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</latexit> (B C) + (A B C)i <latexit sha1_base64="lY1zKoenp8WqIYll0NKJMZ1TPv4=">AAAB+HicbVDLSgMxFL3js9ZHR126CRahIi0zVdBlbTcuK9gHtEPJpJk2NPMgyQh16Je4caGIWz/FnX9jpp2Fth643MM595Kb40acSWVZ38ba+sbm1nZuJ7+7t39QMA+P2jKMBaEtEvJQdF0sKWcBbSmmOO1GgmLf5bTjThqp33mkQrIweFDTiDo+HgXMYwQrLQ3MQqlebpxflG7LaUdsYBatijUHWiV2RoqQoTkwv/rDkMQ+DRThWMqebUXKSbBQjHA6y/djSSNMJnhEe5oG2KfSSeaHz9CZVobIC4WuQKG5+nsjwb6UU9/Vkz5WY7nspeJ/Xi9W3o2TsCCKFQ3I4iEv5kiFKE0BDZmgRPGpJpgIpm9FZIwFJkpnldch2MtfXiXtasW+rFTvr4q1ehZHDk7gFEpgwzXU4A6a0AICMTzDK7wZT8aL8W58LEbXjGznGP7A+PwBjIWQbQ==</latexit> 加減算で答えを得る まず,3回の乗算を先にやる 日本語版P69 17 a + bi <latexit sha1_base64="XJx5hn8e1xqZer/0VlkYvSV6BwI=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBZBEEpSBT0WvXisYD+gDWWz3bRLdzdhdyOU0L/gxYMiXv1D3vw3btIctPXBwOO9GWbmBTFn2rjut1NaW9/Y3CpvV3Z29/YPqodHHR0litA2iXikegHWlDNJ24YZTnuxolgEnHaD6V3md5+o0iySj2YWU1/gsWQhI9hkEr4I2LBac+tuDrRKvILUoEBrWP0ajCKSCCoN4VjrvufGxk+xMoxwOq8MEk1jTKZ4TPuWSiyo9tP81jk6s8oIhZGyJQ3K1d8TKRZaz0RgOwU2E73sZeJ/Xj8x4Y2fMhknhkqyWBQmHJkIZY+jEVOUGD6zBBPF7K2ITLDCxNh4KjYEb/nlVdJp1L3LeuPhqta8LeIowwmcwjl4cA1NuIcWtIHABJ7hFd4c4bw4787HorXkFDPH8AfO5w+qLo39</latexit> c + di <latexit sha1_base64="jx7hmu5VQuLI4YaH/smhw53BBeI=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBZBEEpSBT0WvXisYD+gDWWz2bRLdzdhdyOU0L/gxYMiXv1D3vw3btoctPXBwOO9GWbmBQln2rjut1NaW9/Y3CpvV3Z29/YPqodHHR2nitA2iXmsegHWlDNJ24YZTnuJolgEnHaDyV3ud5+o0iyWj2aaUF/gkWQRI9jkErkI2bBac+vuHGiVeAWpQYHWsPo1CGOSCioN4Vjrvucmxs+wMoxwOqsMUk0TTCZ4RPuWSiyo9rP5rTN0ZpUQRbGyJQ2aq78nMiy0norAdgpsxnrZy8X/vH5qohs/YzJJDZVksShKOTIxyh9HIVOUGD61BBPF7K2IjLHCxNh4KjYEb/nlVdJp1L3LeuPhqta8LeIowwmcwjl4cA1NuIcWtIHAGJ7hFd4c4bw4787HorXkFDPH8AfO5w+wRo4B</latexit> 3回の実数乗算を用いて複素数 と の積の計算ができることを示せ. ac bd <latexit sha1_base64="9pWPLasM7biK1KqQ+Xi1vioQz3s=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBiyWpgh6LXjxWMG2hDWWz2bRLdzdhdyOU0N/gxYMiXv1B3vw3btsctPXBwOO9GWbmhSln2rjut1NaW9/Y3CpvV3Z29/YPqodHbZ1kilCfJDxR3RBrypmkvmGG026qKBYhp51wfDfzO09UaZbIRzNJaSDwULKYEWys5GNyEUaDas2tu3OgVeIVpAYFWoPqVz9KSCaoNIRjrXuem5ogx8owwum00s80TTEZ4yHtWSqxoDrI58dO0ZlVIhQnypY0aK7+nsix0HoiQtspsBnpZW8m/uf1MhPfBDmTaWaoJItFccaRSdDscxQxRYnhE0swUczeisgIK0yMzadiQ/CWX14l7Ubdu6w3Hq5qzdsijjKcwCmcgwfX0IR7aIEPBBg8wyu8OdJ5cd6dj0VrySlmjuEPnM8fYjWOZw==</latexit> アルゴリズムは入力として を取り, 実数部 と虚数部 を別々に出力しなければならない. a, b, c, d <latexit sha1_base64="/0XJkWW6as7rG8doMBkOg3gNutA=">AAAB8XicbVBNSwMxEJ2tX7V+VT16CRbBQym7VdBj0YvHCvYD26Vks9k2NJssSVYoS/+FFw+KePXfePPfmLZ70NYHA4/3ZpiZFyScaeO6305hbX1jc6u4XdrZ3ds/KB8etbVMFaEtIrlU3QBrypmgLcMMp91EURwHnHaC8e3M7zxRpZkUD2aSUD/GQ8EiRrCx0iOuoqCKSBWFg3LFrblzoFXi5aQCOZqD8lc/lCSNqTCEY617npsYP8PKMMLptNRPNU0wGeMh7VkqcEy1n80vnqIzq4QoksqWMGiu/p7IcKz1JA5sZ4zNSC97M/E/r5ea6NrPmEhSQwVZLIpSjoxEs/dRyBQlhk8swUQxeysiI6wwMTakkg3BW355lbTrNe+iVr+/rDRu8jiKcAKncA4eXEED7qAJLSAg4Ble4c3Rzovz7nwsWgtOPnMMf+B8/gA36I9Q</latexit> ad + bc <latexit sha1_base64="k/sm0O6uixRdfw/78XLRPM6z7ZQ=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBZBEEpSBT0WvXisYD+gDWWz3bRLN5uwOxFK6I/w4kERr/4eb/4bt20O2vpg4PHeDDPzgkQKg6777RTW1jc2t4rbpZ3dvf2D8uFRy8SpZrzJYhnrTkANl0LxJgqUvJNoTqNA8nYwvpv57SeujYjVI04S7kd0qEQoGEUrtemAXJCA9csVt+rOQVaJl5MK5Gj0y1+9QczSiCtkkhrT9dwE/YxqFEzyaamXGp5QNqZD3rVU0YgbP5ufOyVnVhmQMNa2FJK5+nsio5ExkyiwnRHFkVn2ZuJ/XjfF8MbPhEpS5IotFoWpJBiT2e9kIDRnKCeWUKaFvZWwEdWUoU2oZEPwll9eJa1a1bus1h6uKvXbPI4inMApnIMH11CHe2hAExiM4Rle4c1JnBfn3flYtBacfOYY/sD5/AEMio65</latexit>
  18. /35 再帰がアルゴリズムの中に含まれてるとき 計算量出すの難しそう. ところで. 18

  19. /35 とにかく漸化式を出す. T(n) = ( ⇥(1) n = 1 8T(n/2)

    + ⇥(n2) n > 1 <latexit sha1_base64="GR8m44C9HSDmcXqd/MYxaonOnuY=">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</latexit> 単純な分割法を用いた行列積の求解の場合 解ければ万事OK c1n3  8(n/2)3 + ⇥(n2)  c2n3 <latexit sha1_base64="sqj27aWkT5I3X04oxWH8O41TyHY=">AAACGXicbVDLSgMxFM3UV62vqks3wSK0CHVmKthl0Y3LCn1BH0MmvW1DM5kxyQil9Dfc+CtuXCjiUlf+jeljoa0HAifnnEtyjx9xprRtf1uJtfWNza3kdmpnd2//IH14VFNhLClUachD2fCJAs4EVDXTHBqRBBL4HOr+8Gbq1x9AKhaKih5F0A5IX7Aeo0QbyUvb1HOw6BRwi8M9LmbFhZszt3PcqgxAk6zouLm5Rz13GvTSGTtvz4BXibMgGbRA2Ut/trohjQMQmnKiVNOxI90eE6kZ5TBJtWIFEaFD0oemoYIEoNrj2WYTfGaULu6F0hyh8Uz9PTEmgVKjwDfJgOiBWvam4n9eM9a9YnvMRBRrEHT+UC/mWId4WhPuMglU85EhhEpm/orpgEhCtSkzZUpwlldeJTU37xTy7t1lpnS9qCOJTtApyiIHXaESukVlVEUUPaJn9IrerCfrxXq3PubRhLWYOUZ/YH39ALcRnGU=</latexit> とりあえず と推測して代入 -> T(n) = ⇥(n3) <latexit sha1_base64="Did3UZygsJvbqqtgXH4Y1y1YpVs=">AAAB+3icbVBNS8NAEJ3Ur1q/Yj16WSxCeylJK+hFKHrxWKFf0May2W7bpZtN2N2IJfSvePGgiFf/iDf/jds2B219MPB4b4aZeX7EmdKO821lNja3tneyu7m9/YPDI/s431JhLAltkpCHsuNjRTkTtKmZ5rQTSYoDn9O2P7md++1HKhULRUNPI+oFeCTYkBGsjdS3842iKKFr1GuMqcZF8VAt9e2CU3YWQOvETUkBUtT79ldvEJI4oEITjpXquk6kvQRLzQins1wvVjTCZIJHtGuowAFVXrK4fYbOjTJAw1CaEhot1N8TCQ6Umga+6QywHqtVby7+53VjPbzyEiaiWFNBlouGMUc6RPMg0IBJSjSfGoKJZOZWRMZYYqJNXDkTgrv68jppVcputVy5vyjUbtI4snAKZ1AEFy6hBndQhyYQeIJneIU3a2a9WO/Wx7I1Y6UzJ/AH1ucPKP+SlQ==</latexit> 8(n/2)3 + ⇥(n2) <latexit sha1_base64="1EDQT2aw8qhorLeC/AJhdd+D6Kg=">AAAB/3icbVDLSsNAFJ3UV62vqODGzWARWoSapIJdFt24rNAXtGmZTCft0MkkzEyEErvwV9y4UMStv+HOv3HaZqGtBy4czrmXe+/xIkalsqxvI7O2vrG5ld3O7ezu7R+Yh0dNGcYCkwYOWSjaHpKEUU4aiipG2pEgKPAYaXnj25nfeiBC0pDX1SQiboCGnPoUI6WlvnlSKfBLp9grwwvYrY+IQgXec4p9M2+VrDngKrFTkgcpan3zqzsIcRwQrjBDUnZsK1JugoSimJFprhtLEiE8RkPS0ZSjgEg3md8/hedaGUA/FLq4gnP190SCAikngac7A6RGctmbif95nVj5FTehPIoV4XixyI8ZVCGchQEHVBCs2EQThAXVt0I8QgJhpSPL6RDs5ZdXSdMp2eWSc3+Vr96kcWTBKTgDBWCDa1AFd6AGGgCDR/AMXsGb8WS8GO/Gx6I1Y6Qzx+APjM8f9PuTgA==</latexit> 十分大きな のもとで正数 , が存在して c1 <latexit sha1_base64="bxLhQ885cIXIOA7SAduGcGiC4fY=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHM0nQj+hQ8pAzaqz0wPpev1xxq+4cZJV4OalAjka//NUbxCyNUBomqNZdz02Mn1FlOBM4LfVSjQllYzrErqWSRqj9bH7qlJxZZUDCWNmShszV3xMZjbSeRIHtjKgZ6WVvJv7ndVMTXvsZl0lqULLFojAVxMRk9jcZcIXMiIkllClubyVsRBVlxqZTsiF4yy+vklat6l1Ua/eXlfpNHkcRTuAUzsGDK6jDHTSgCQyG8Ayv8OYI58V5dz4WrQUnnzmGP3A+fwDs8Y2P</latexit> c2 <latexit sha1_base64="gxvawmigqvEzWrjqgYRso8ZdmLo=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHM0nQj+hQ8pAzaqz0wPq1frniVt05yCrxclKBHI1++as3iFkaoTRMUK27npsYP6PKcCZwWuqlGhPKxnSIXUsljVD72fzUKTmzyoCEsbIlDZmrvycyGmk9iQLbGVEz0sveTPzP66YmvPYzLpPUoGSLRWEqiInJ7G8y4AqZERNLKFPc3krYiCrKjE2nZEPwll9eJa1a1buo1u4vK/WbPI4inMApnIMHV1CHO2hAExgM4Rle4c0Rzovz7nwsWgtOPnMMf+B8/gDudY2Q</latexit> n <latexit sha1_base64="2QmInd+nHO60uhS7AYCvgpiU4a8=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlpuyXK27VnYOsEi8nFcjR6Je/eoOYpRFKwwTVuuu5ifEzqgxnAqelXqoxoWxMh9i1VNIItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2ooszYbEo2BG/55VXSrlW9i2qteVmp3+RxFOEETuEcPLiCOtxBA1rAAOEZXuHNeXRenHfnY9FacPKZY/gD5/MH2F+M9g==</latexit> 推測が合っている! 19
  20. /35 置換え法 1. 解の形を推定する. 2. 数学的帰納法を用いて定数を定め, 推定解がうまく働くことを示す. T(n)  2(cbn/2c

    lg(bn/2c)) + n  cn lg(n/2) + n = cn lg n cn lg 2 + n = cn lg n cn + n  cn lg n <latexit sha1_base64="G5XKM9LfnRjaJ2aVusSAvu7CFLk=">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</latexit> 例. T(n) = 2T(bn/2c) + n <latexit sha1_base64="a4CXFgOtz8PmVn5PrZCpKz22ilY=">AAACCXicbVDLSgMxFM3UV62vqks3wSK0CHVmFHQjFN24rNAXdIaSSTNtaCYZkoxQSrdu/BU3LhRx6x+4829Mp7PQ1rPJyTn3cu89Qcyo0rb9beVWVtfWN/Kbha3tnd294v5BS4lEYtLEggnZCZAijHLS1FQz0oklQVHASDsY3c789gORigre0OOY+BEacBpSjLSRekXYKPPKtWsej4VMCAk5PIOuJ9NP5dSUlOyqnQIuEycjJZCh3it+eX2Bk4hwjRlSquvYsfYnSGqKGZkWvESRGOERGpCuoRxFRPmT9JIpPDFKH4ZmjVBwDVP1d8cERUqNo8BURkgP1aI3E//zuokOr/wJ5XGiCcfzQWHCoBZwFgvsU0mwZmNDEJbU7ArxEEmEtQmvYEJwFk9eJi236pxX3fuLUu0miyMPjsAxKAMHXIIauAN10AQYPIJn8ArerCfrxXq3PualOSvrOQR/YH3+AL7Tl9g=</latexit> T(n) = O(n lg n) <latexit sha1_base64="xQ//vGOtWzkMI/f2Te9lhXGlWSM=">AAAB+HicbVBNS8NAEJ34WetHox69LBahvZSkCnoRil68WaFf0Iay2W7apZtN2N0INfSXePGgiFd/ijf/jds2B219MPB4b4aZeX7MmdKO822trW9sbm3ndvK7e/sHBfvwqKWiRBLaJBGPZMfHinImaFMzzWknlhSHPqdtf3w789uPVCoWiYaexNQL8VCwgBGsjdS3C42SKF/fl0SPD1MxLfftolNx5kCrxM1IETLU+/ZXbxCRJKRCE46V6rpOrL0US80Ip9N8L1E0xmSMh7RrqMAhVV46P3yKzowyQEEkTQmN5urviRSHSk1C33SGWI/UsjcT//O6iQ6uvJSJONFUkMWiIOFIR2iWAhowSYnmE0MwkczcisgIS0y0ySpvQnCXX14lrWrFPa9UHy6KtZssjhycwCmUwIVLqMEd1KEJBBJ4hld4s56sF+vd+li0rlnZzDH8gfX5A08Mkjg=</latexit> に対して と推定. bn/2c  n/2 <latexit sha1_base64="wBdP7txrpYMRtzV180ZSic7j8QE=">AAACBXicbZA9TwJBEIbn8AvxC7XUYiMxscI7NNGSaGOJiXwk3IXsLXuwYW/33N0zIYTGxr9iY6Extv4HO/+Ne0Ch4FRP3ncmM/OGCWfauO63k1taXlldy68XNja3tneKu3sNLVNFaJ1ILlUrxJpyJmjdMMNpK1EUxyGnzXBwnfnNB6o0k+LODBMaxLgnWMQINlbqFA99HnEpFRKnFeSrKfuc3mdCp1hyy+6k0CJ4MyjBrGqd4pfflSSNqTCEY63bnpuYYISVYYTTccFPNU0wGeAebVsUOKY6GE2+GKNjq3RRZPdHUhg0UX9PjHCs9TAObWeMTV/Pe5n4n9dOTXQZjJhIUkMFmS6KUo6MRFkkqMsUJYYPLWCimL0VkT5WmBgbXMGG4M2/vAiNStk7K1duz0vVq1kceTiAIzgBDy6gCjdQgzoQeIRneIU358l5cd6dj2lrzpnN7MOfcj5/AKFGl10=</latexit> より c 1 <latexit sha1_base64="/S6JPEF4gUnnuuXgFZAYuEyo5BE=">AAAB73icbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGC/YA2lM120i7dbNLdjVBC/4QXD4p49e9489+4bXPQ1gcDj/dmmJkXJIJr47rfztr6xubWdmGnuLu3f3BYOjpu6jhVDBssFrFqB1Sj4BIbhhuB7UQhjQKBrWB0N/NbT6g0j+WjmSToR3QgecgZNVZqM9Id4Jh4vVLZrbhzkFXi5aQMOeq90le3H7M0QmmYoFp3PDcxfkaV4UzgtNhNNSaUjegAO5ZKGqH2s/m9U3JulT4JY2VLGjJXf09kNNJ6EgW2M6JmqJe9mfif10lNeONnXCapQckWi8JUEBOT2fOkzxUyIyaWUKa4vZWwIVWUGRtR0YbgLb+8SprVindZqT5clWu3eRwFOIUzuAAPrqEG91CHBjAQ8Ayv8OaMnRfn3flYtK45+cwJ/IHz+QPqi487</latexit> と仮定 T(2) = 2T(b2/2c) + 2 = 2T(1) + 2 = 4 <latexit sha1_base64="TIZZWQz3qrDDpO7WumAckgm1uJE=">AAACG3icbZDLSsNAFIYn9VbrLerSzWARWoSaxIJuhKIblxV6gyaUyXTSDp1kwsxEKKHv4cZXceNCEVeCC9/GaVpBW89mPv7/HOac348Zlcqyvozcyura+kZ+s7C1vbO7Z+4ftCRPBCZNzBkXHR9JwmhEmooqRjqxICj0GWn7o5up374nQlIeNdQ4Jl6IBhENKEZKSz3TaZScMryC+nVZwDgX0DlzoCsyLsNT6Mxc+4erPbNoVays4DLYcyiCedV75ofb5zgJSaQwQ1J2bStWXoqEopiRScFNJIkRHqEB6WqMUEikl2a3TeCJVvow0HsFPFIwU39PpCiUchz6ujNEaigXvan4n9dNVHDppTSKE0UiPPsoSBhUHE6Dgn0qCFZsrAFhQfWuEA+RQFjpOAs6BHvx5GVoORX7vOLcVYu163kceXAEjkEJ2OAC1MAtqIMmwOABPIEX8Go8Gs/Gm/E+a80Z85lD8KeMz2+pq5rr</latexit> T(1) = 1 <latexit sha1_base64="hQnyleEopmMinXnvMtQERgxvPpo=">AAAB73icbVBNSwMxEJ3Ur1q/qh69BItQL2W3CnoRil48VugXtEvJptk2NJtdk6xQlv4JLx4U8erf8ea/MW33oK0PBh7vzTAzz48F18ZxvlFubX1jcyu/XdjZ3ds/KB4etXSUKMqaNBKR6vhEM8ElaxpuBOvEipHQF6ztj+9mfvuJKc0j2TCTmHkhGUoecEqMlTqNsnuOb7DbL5acijMHXiVuRkqQod4vfvUGEU1CJg0VROuu68TGS4kynAo2LfQSzWJCx2TIupZKEjLtpfN7p/jMKgMcRMqWNHiu/p5ISaj1JPRtZ0jMSC97M/E/r5uY4NpLuYwTwyRdLAoSgU2EZ8/jAVeMGjGxhFDF7a2Yjogi1NiICjYEd/nlVdKqVtyLSvXhslS7zeLIwwmcQhlcuIIa3EMdmkBBwDO8wht6RC/oHX0sWnMomzmGP0CfP4YZjlI=</latexit> T(3) = 2T(b3/2c) + 3 = 2T(2) + 3 = 5 <latexit sha1_base64="ogdvfavyvEVp2S2lRKQpegJrRD8=">AAACG3icbZDLSgMxFIYz9VbrbdSlm2ARWoQ6F0U3QtGNywq9QVtKJs20oZlkSDJCKX0PN76KGxeKuBJc+Dam0wpaPZt8/P855Jw/iBlV2nE+rczS8srqWnY9t7G5tb1j7+7VlUgkJjUsmJDNACnCKCc1TTUjzVgSFAWMNILh9dRv3BGpqOBVPYpJJ0J9TkOKkTZS1/aqBb8IL6F52yxkQkjon3iwLVMuwmPoz1zvm8+6dt4pOWnBv+DOIQ/mVena7+2ewElEuMYMKdVynVh3xkhqihmZ5NqJIjHCQ9QnLYMcRUR1xultE3hklB4MzV6h4Bqm6s+JMYqUGkWB6YyQHqhFbyr+57USHV50xpTHiSYczz4KEwa1gNOgYI9KgjUbGUBYUrMrxAMkEdYmzpwJwV08+S/UvZLrl7zb03z5ah5HFhyAQ1AALjgHZXADKqAGMLgHj+AZvFgP1pP1ar3NWjPWfGYf/Crr4wuzLprx</latexit> 境界条件のチェック 2 · 2 lg 2 = 4 <latexit sha1_base64="830PGGyN0TrgjykgLfDm4aJf8Hg=">AAAB+3icbVDLSsNAFJ34rPUV69LNYBFclSQWdCMU3bisYB/QhDKZTNqhk5kwMxFLyK+4caGIW3/EnX/jtM1CWw9cOJxzL/feE6aMKu0439ba+sbm1nZlp7q7t39waB/VukpkEpMOFkzIfogUYZSTjqaakX4qCUpCRnrh5Hbm9x6JVFTwBz1NSZCgEacxxUgbaWjXPB9HQkPPZ6PcK+A1bA7tutNw5oCrxC1JHZRoD+0vPxI4SwjXmCGlBq6T6iBHUlPMSFH1M0VShCdoRAaGcpQQFeTz2wt4ZpQIxkKa4hrO1d8TOUqUmiah6UyQHqtlbyb+5w0yHV8FOeVppgnHi0VxxqAWcBYEjKgkWLOpIQhLam6FeIwkwtrEVTUhuMsvr5Ku13AvGt59s966KeOogBNwCs6BCy5BC9yBNugADJ7AM3gFb1ZhvVjv1seidc0qZ47BH1ifP8XXkvc=</latexit> 2 · 3 lg 3 ⇡ 9.5098 <latexit sha1_base64="itlOI1bqMtNsoqwibfJ3c077Ino=">AAACBnicbVDLSsNAFJ3UV62vqEsRBovgKiStYrsrunFZwT6gCWUymbRDJ5kwMxFL6MqNv+LGhSJu/QZ3/o3TNgttPXDhcM693HuPnzAqlW1/G4WV1bX1jeJmaWt7Z3fP3D9oS54KTFqYMy66PpKE0Zi0FFWMdBNBUOQz0vFH11O/c0+EpDy+U+OEeBEaxDSkGCkt9c3jiosDrmDVZYOsOoEuShLBH2DdurDrtb5Zti17BrhMnJyUQY5m3/xyA47TiMQKMyRlz7ET5WVIKIoZmZTcVJIE4REakJ6mMYqI9LLZGxN4qpUAhlzoihWcqb8nMhRJOY583RkhNZSL3lT8z+ulKqx5GY2TVJEYzxeFKYOKw2kmMKCCYMXGmiAsqL4V4iESCCudXEmH4Cy+vEzaFcupWpXb83LjKo+jCI7ACTgDDrgEDXADmqAFMHgEz+AVvBlPxovxbnzMWwtGPnMI/sD4/AFsB5cp</latexit> c = 2 <latexit sha1_base64="+AwirKnTFIhmYbVv6fytSEfIZhY=">AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9CIUvXisaD+gDWWz3bRLN5uwOxFK6E/w4kERr/4ib/4bt20O2vpg4PHeDDPzgkQKg6777aysrq1vbBa2its7u3v7pYPDpolTzXiDxTLW7YAaLoXiDRQoeTvRnEaB5K1gdDv1W09cGxGrRxwn3I/oQIlQMIpWemDX1V6p7FbcGcgy8XJShhz1Xumr249ZGnGFTFJjOp6boJ9RjYJJPil2U8MTykZ0wDuWKhpx42ezUyfk1Cp9EsbalkIyU39PZDQyZhwFtjOiODSL3lT8z+ukGF75mVBJilyx+aIwlQRjMv2b9IXmDOXYEsq0sLcSNqSaMrTpFG0I3uLLy6RZrXjnler9Rbl2k8dRgGM4gTPw4BJqcAd1aACDATzDK7w50nlx3p2PeeuKk88cwR84nz+6y41u</latexit> ととると n 2 <latexit sha1_base64="s4zvsadSG/Do7s3CMVXM1H2YNcE=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Ae0oWy2k3bpZhN3N0IJ/RFePCji1d/jzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHssHM0nQj+hQ8pAzaqzUlr0hPpJav1xxq+4cZJV4OalAjka//NUbxCyNUBomqNZdz02Mn1FlOBM4LfVSjQllYzrErqWSRqj9bH7ulJxZZUDCWNmShszV3xMZjbSeRIHtjKgZ6WVvJv7ndVMTXvsZl0lqULLFojAVxMRk9jsZcIXMiIkllClubyVsRBVlxiZUsiF4yy+vklat6l1Ua/eXlfpNHkcRTuAUzsGDK6jDHTSgCQzG8Ayv8OYkzovz7nwsWgtOPnMMf+B8/gCmao8d</latexit> したがって で T(n)  cn lg n <latexit sha1_base64="OLTnSWDcTwsswpKAWpzb62cPBnE=">AAAB+nicbVBNS8NAEN3Ur1q/Uj16WSxCvZSkCnosevFYoV/QhrLZTtqlm03c3Sgl9qd48aCIV3+JN/+N2zYHbX0w8Hhvhpl5fsyZ0o7zbeXW1jc2t/LbhZ3dvf0Du3jYUlEiKTRpxCPZ8YkCzgQ0NdMcOrEEEvoc2v74Zua3H0AqFomGnsTghWQoWMAo0Ubq28VGWZz1ONxjKnp8mIpp3y45FWcOvErcjJRQhnrf/uoNIpqEIDTlRKmu68TaS4nUjHKYFnqJgpjQMRlC11BBQlBeOj99ik+NMsBBJE0Jjefq74mUhEpNQt90hkSP1LI3E//zuokOrryUiTjRIOhiUZBwrCM8ywEPmASq+cQQQiUzt2I6IpJQbdIqmBDc5ZdXSatacc8r1buLUu06iyOPjtEJKiMXXaIaukV11EQUPaJn9IrerCfrxXq3PhatOSubOUJ/YH3+AJHIk5A=</latexit> が成立 20
  21. /35 置換え法のTips T(n)  2(cbn/2c) + n  cn +

    n = O(n) <latexit sha1_base64="A6WPZfn9l946Uvt8DVTJrbue2Os=">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</latexit> ダメゼッタイ T(n) = O(n) <latexit sha1_base64="jT/aI2y3/PLd0AvL3nx+p63I9i4=">AAAB8nicbVDLSgMxFL1TX7W+qi7dBItQN2WmCroRim7cWaEvmA4lk2ba0EwyJBmhlH6GGxeKuPVr3Pk3pu0stPVAyOGce7n3njDhTBvX/XZya+sbm1v57cLO7t7+QfHwqKVlqghtEsml6oRYU84EbRpmOO0kiuI45LQdju5mfvuJKs2kaJhxQoMYDwSLGMHGSn6jLM7RDXqwX69YcivuHGiVeBkpQYZ6r/jV7UuSxlQYwrHWvucmJphgZRjhdFroppommIzwgPqWChxTHUzmK0/RmVX6KJLKPmHQXP3dMcGx1uM4tJUxNkO97M3E/zw/NdF1MGEiSQ0VZDEoSjkyEs3uR32mKDF8bAkmitldERlihYmxKRVsCN7yyaukVa14F5Xq42WpdpvFkYcTOIUyeHAFNbiHOjSBgIRneIU3xzgvzrvzsSjNOVnPMfyB8/kDqAePig==</latexit> さっきの例で と推定して 間違い! T(n)  cn <latexit sha1_base64="W8RhK0Ybrn2ujd0Rhh2l+OzXfj0=">AAAB83icbVDLSgNBEOyNrxhfUY9eBoMQL2E3CnoMevEYIYmBZAmzk95kyOzsOjMrhCW/4cWDIl79GW/+jZPHQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjlo5TxbDJYhGrdkA1Ci6xabgR2E4U0igQ+BCMbqf+wxMqzWPZMOME/YgOJA85o8ZK3UZZnpOuwEfCZK9YcivuDGSVeAtSggXqveJXtx+zNEJpmKBadzw3MX5GleFM4KTQTTUmlI3oADuWShqh9rPZzRNyZpU+CWNlSxoyU39PZDTSehwFtjOiZqiXvan4n9dJTXjtZ1wmqUHJ5ovCVBATk2kApM8VMiPGllCmuL2VsCFVlBkbU8GG4C2/vEpa1Yp3UaneX5ZqN4s48nACp1AGD66gBndQhyYwSOAZXuHNSZ0X5935mLfmnMXMMfyB8/kDjVKQuA==</latexit> の形まで もっていけなければNG 変数変換 T(n) = 2T(b p nc) + lg n <latexit sha1_base64="1+tHV0ipy653zGf8jdo0HcliaA0=">AAACD3icbVBLSwMxGMz6rPVV9eglWJQWoexWQS9C0YvHCn1BdynZNNuGZpM1yQpl6T/w4l/x4kERr169+W9Mt3vQ1oHAZGY+km/8iFGlbfvbWlpeWV1bz23kN7e2d3YLe/stJWKJSRMLJmTHR4owyklTU81IJ5IEhT4jbX90M/XbD0QqKnhDjyPihWjAaUAx0kbqFU4aJV6+qsJGyWUBE0K66l7qhE9cmV7Lpy4bQBMs2hU7BVwkTkaKIEO9V/hy+wLHIeEaM6RU17Ej7SVIaooZmeTdWJEI4REakK6hHIVEeUm6zwQeG6UPAyHN4Rqm6u+JBIVKjUPfJEOkh2rem4r/ed1YB5deQnkUa8Lx7KEgZlALOC0H9qkkWLOxIQhLav4K8RBJhLWpMG9KcOZXXiStasU5q1Tvzou166yOHDgER6AEHHABauAW1EETYPAInsEreLOerBfr3fqYRZesbOYA/IH1+QONAZvA</latexit> を解きたい. m = lg n <latexit sha1_base64="xCFCVxaeY2e1zAY/vknrJ/o02pQ=">AAAB8XicbVBNSwMxEJ34WetX1aOXYBE8ld0q6EUoevFYwX5gu5Rsmm1Dk+ySZIWy9F948aCIV/+NN/+NabsHbX0w8Hhvhpl5YSK4sZ73jVZW19Y3Ngtbxe2d3b390sFh08SppqxBYxHrdkgME1yxhuVWsHaiGZGhYK1wdDv1W09MGx6rBztOWCDJQPGIU2Kd9CjxNe6KQaYmvVLZq3gz4GXi56QMOeq90le3H9NUMmWpIMZ0fC+xQUa05VSwSbGbGpYQOiID1nFUEclMkM0unuBTp/RxFGtXyuKZ+nsiI9KYsQxdpyR2aBa9qfif10ltdBVkXCWpZYrOF0WpwDbG0/dxn2tGrRg7Qqjm7lZMh0QTal1IRReCv/jyMmlWK/55pXp/Ua7d5HEU4BhO4Ax8uIQa3EEdGkBBwTO8whsy6AW9o4956wrKZ47gD9DnD9gekGE=</latexit> T(2m) = 2T(2m/2) + m <latexit sha1_base64="hkN8v4Al//6An3sIDLClgJnB6LE=">AAACAXicbZDLSsNAFIYn9VbrLepGcDNYhBahJlHQjVB047JCb9DGMplO2qEzSZiZCCXUja/ixoUibn0Ld76NkzYLbf1h4OM/53Dm/F7EqFSW9W3klpZXVtfy64WNza3tHXN3rynDWGDSwCELRdtDkjAakIaiipF2JAjiHiMtb3ST1lsPREgaBnU1jojL0SCgPsVIaatnHtRLzj0vwyvopJTwU2dShieQ98yiVbGmgotgZ1AEmWo986vbD3HMSaAwQ1J2bCtSboKEopiRSaEbSxIhPEID0tEYIE6km0wvmMBj7fShHwr9AgWn7u+JBHEpx9zTnRypoZyvpeZ/tU6s/Es3oUEUKxLg2SI/ZlCFMI0D9qkgWLGxBoQF1X+FeIgEwkqHVtAh2PMnL0LTqdhnFefuvFi9zuLIg0NwBErABhegCm5BDTQABo/gGbyCN+PJeDHejY9Za87IZvbBHxmfP5JAk8M=</latexit> と置くと ※簡易化のため丸めの影響は無視 S(m) = 2S(m/2) + m <latexit sha1_base64="FOWYzkQrcsyo/C6+f3OiRBVikcM=">AAAB+XicbVDLSgMxFL1TX7W+Rl26CRahRagzo6AboejGZUX7gHYomTRtQ5OZIckUytA/ceNCEbf+iTv/xrSdhVYP3MvhnHvJzQlizpR2nC8rt7K6tr6R3yxsbe/s7tn7Bw0VJZLQOol4JFsBVpSzkNY105y2YkmxCDhtBqPbmd8cU6lYFD7qSUx9gQch6zOCtZG6tv1QEuVrz/Qzr4xOkejaRafizIH+EjcjRchQ69qfnV5EEkFDTThWqu06sfZTLDUjnE4LnUTRGJMRHtC2oSEWVPnp/PIpOjFKD/UjaSrUaK7+3EixUGoiAjMpsB6qZW8m/ue1E92/8lMWxommIVk81E840hGaxYB6TFKi+cQQTCQztyIyxBITbcIqmBDc5S//JQ2v4p5XvPuLYvUmiyMPR3AMJXDhEqpwBzWoA4ExPMELvFqp9Wy9We+L0ZyV7RzCL1gf37ASkRk=</latexit> と展開. S(m) = O(m lg m) <latexit sha1_base64="rj1r61iRcgOD+ghSjiLJbs4dZro=">AAAB+nicbVBNS8NAEJ3Ur1q/Uj16WSxCeylJFfQiFL14s6L9gDaUzXbbLt1Nwu5GKbE/xYsHRbz6S7z5b9y2OWjrg4HHezPMzPMjzpR2nG8rs7K6tr6R3cxtbe/s7tn5/YYKY0lonYQ8lC0fK8pZQOuaaU5bkaRY+Jw2/dHV1G8+UKlYGNzrcUQ9gQcB6zOCtZG6dv6uKEroAt0URYcPEjEpde2CU3ZmQMvETUkBUtS69lenF5JY0EATjpVqu06kvQRLzQink1wnVjTCZIQHtG1ogAVVXjI7fYKOjdJD/VCaCjSaqb8nEiyUGgvfdAqsh2rRm4r/ee1Y98+9hAVRrGlA5ov6MUc6RNMcUI9JSjQfG4KJZOZWRIZYYqJNWjkTgrv48jJpVMruSblye1qoXqZxZOEQjqAILpxBFa6hBnUg8AjP8Apv1pP1Yr1bH/PWjJXOHMAfWJ8/+TiSiA==</latexit> 実際 であるので T(n) = T(2m) = S(m) = O(m lg m) = O(lg n lg lg n) <latexit sha1_base64="SeZXzUypvZ1dwP4UDkDUJpecWsU=">AAACIXicbZDLTgIxFIY7eEO8jbp000hMhg2ZQRPZmBDduBMjtwSQdEqBhrYzaTsmZDKv4sZXceNCY9gZX8YOsFDwJG2//uectOf3Q0aVdt0vK7O2vrG5ld3O7ezu7R/Yh0cNFUQSkzoOWCBbPlKEUUHqmmpGWqEkiPuMNP3xTZpvPhGpaCBqehKSLkdDQQcUI22knl2uOaIAr2DNKT3yFB6c2XHn8A4bxjyZX1IWSbrPKSn07LxbdGcBV8FbQB4sotqzp51+gCNOhMYMKdX23FB3YyQ1xYwkuU6kSIjwGA1J26BAnKhuPJswgWdG6cNBIM0SGs7U3x0x4kpNuG8qOdIjtZxLxf9y7UgPyt2YijDSROD5Q4OIQR3A1C7Yp5JgzSYGEJbU/BXiEZIIa2NqzpjgLY+8Co1S0Tsvlu4v8pXrhR1ZcAJOgQM8cAkq4BZUQR1g8AxewTv4sF6sN+vTms5LM9ai5xj8Cev7B3+moCs=</latexit> と求まる. 21 S(m) = T(2m) <latexit sha1_base64="kYsi/Ki5fe+NNa5ViMLrzhlHqxs=">AAAB9HicbVBNS8NAEJ3Ur1q/qh69LBahvZSkCnoRil48VuwXtLFstpt26W4SdzeFEvo7vHhQxKs/xpv/xm2bg7Y+GHi8N8PMPC/iTGnb/rYya+sbm1vZ7dzO7t7+Qf7wqKnCWBLaICEPZdvDinIW0IZmmtN2JCkWHqctb3Q781tjKhULg7qeRNQVeBAwnxGsjeQ+FEUJXaN6sfIoSr18wS7bc6BV4qSkAClqvfxXtx+SWNBAE46V6jh2pN0ES80Ip9NcN1Y0wmSEB7RjaIAFVW4yP3qKzozSR34oTQUazdXfEwkWSk2EZzoF1kO17M3E/7xOrP0rN2FBFGsakMUiP+ZIh2iWAOozSYnmE0MwkczcisgQS0y0ySlnQnCWX14lzUrZOS9X7i8K1Zs0jiycwCkUwYFLqMId1KABBJ7gGV7hzRpbL9a79bFozVjpzDH8gfX5A9SykDA=</latexit> と置いて さっきの漸化式に似ていることに気づく.
  22. /35 でも, うまい推定解出すの運ゲーじゃん. 置換え法を使えば漸化式の解の正しさを 簡潔に証明できることは分かった. 再帰木を使えば良い 22

  23. /35 T(n) = ( ⇥(1) n = 1 8T(n/2) +

    ⇥(n2) n > 1 <latexit sha1_base64="GR8m44C9HSDmcXqd/MYxaonOnuY=">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</latexit> 再帰木の構築 c(n/2)2 <latexit sha1_base64="KDGWbid+M32IFDYr8EZ/KcklIpU=">AAAB73icbVBNS8NAEJ3Ur1q/qh69LBahXmoSBT0WvXisYD+gjWWz3bRLN5u4uxFK6J/w4kERr/4db/4bN20O2vpg4PHeDDPz/JgzpW372yqsrK6tbxQ3S1vbO7t75f2DlooSSWiTRDySHR8rypmgTc00p51YUhz6nLb98U3mt5+oVCwS93oSUy/EQ8ECRrA2UodUxZl7+uD2yxW7Zs+AlomTkwrkaPTLX71BRJKQCk04Vqrr2LH2Uiw1I5xOS71E0RiTMR7SrqECh1R56ezeKToxygAFkTQlNJqpvydSHCo1CX3TGWI9UoteJv7ndRMdXHkpE3GiqSDzRUHCkY5Q9jwaMEmJ5hNDMJHM3IrICEtMtImoZEJwFl9eJi235pzX3LuLSv06j6MIR3AMVXDgEupwCw1oAgEOz/AKb9aj9WK9Wx/z1oKVzxzCH1ifP2DijuE=</latexit> c(n/2)2 <latexit sha1_base64="KDGWbid+M32IFDYr8EZ/KcklIpU=">AAAB73icbVBNS8NAEJ3Ur1q/qh69LBahXmoSBT0WvXisYD+gjWWz3bRLN5u4uxFK6J/w4kERr/4db/4bN20O2vpg4PHeDDPz/JgzpW372yqsrK6tbxQ3S1vbO7t75f2DlooSSWiTRDySHR8rypmgTc00p51YUhz6nLb98U3mt5+oVCwS93oSUy/EQ8ECRrA2UodUxZl7+uD2yxW7Zs+AlomTkwrkaPTLX71BRJKQCk04Vqrr2LH2Uiw1I5xOS71E0RiTMR7SrqECh1R56ezeKToxygAFkTQlNJqpvydSHCo1CX3TGWI9UoteJv7ndRMdXHkpE3GiqSDzRUHCkY5Q9jwaMEmJ5hNDMJHM3IrICEtMtImoZEJwFl9eJi235pzX3LuLSv06j6MIR3AMVXDgEupwCw1oAgEOz/AKb9aj9WK9Wx/z1oKVzxzCH1ifP2DijuE=</latexit> c(n/2)2 <latexit sha1_base64="KDGWbid+M32IFDYr8EZ/KcklIpU=">AAAB73icbVBNS8NAEJ3Ur1q/qh69LBahXmoSBT0WvXisYD+gjWWz3bRLN5u4uxFK6J/w4kERr/4db/4bN20O2vpg4PHeDDPz/JgzpW372yqsrK6tbxQ3S1vbO7t75f2DlooSSWiTRDySHR8rypmgTc00p51YUhz6nLb98U3mt5+oVCwS93oSUy/EQ8ECRrA2UodUxZl7+uD2yxW7Zs+AlomTkwrkaPTLX71BRJKQCk04Vqrr2LH2Uiw1I5xOS71E0RiTMR7SrqECh1R56ezeKToxygAFkTQlNJqpvydSHCo1CX3TGWI9UoteJv7ndRMdXHkpE3GiqSDzRUHCkY5Q9jwaMEmJ5hNDMJHM3IrICEtMtImoZEJwFl9eJi235pzX3LuLSv06j6MIR3AMVXDgEupwCw1oAgEOz/AKb9aj9WK9Wx/z1oKVzxzCH1ifP2DijuE=</latexit> … 8個 64個 c(n/4)2 <latexit sha1_base64="T6qJtyReumukMUjj2bJzaaIeZeg=">AAAB73icbVBNSwMxEJ31s9avqkcvwSLUS92tBT0WvXisYD+gXUs2zbahSXZNskJZ+ie8eFDEq3/Hm//GtN2Dtj4YeLw3w8y8IOZMG9f9dlZW19Y3NnNb+e2d3b39wsFhU0eJIrRBIh6pdoA15UzShmGG03asKBYBp61gdDP1W09UaRbJezOOqS/wQLKQEWys1CYleV49e6j0CkW37M6AlomXkSJkqPcKX91+RBJBpSEca93x3Nj4KVaGEU4n+W6iaYzJCA9ox1KJBdV+Ort3gk6t0kdhpGxJg2bq74kUC63HIrCdApuhXvSm4n9eJzHhlZ8yGSeGSjJfFCYcmQhNn0d9pigxfGwJJorZWxEZYoWJsRHlbQje4svLpFkpexflyl21WLvO4sjBMZxACTy4hBrcQh0aQIDDM7zCm/PovDjvzse8dcXJZo7gD5zPH2PwjuM=</latexit> c(n/4)2 <latexit sha1_base64="T6qJtyReumukMUjj2bJzaaIeZeg=">AAAB73icbVBNSwMxEJ31s9avqkcvwSLUS92tBT0WvXisYD+gXUs2zbahSXZNskJZ+ie8eFDEq3/Hm//GtN2Dtj4YeLw3w8y8IOZMG9f9dlZW19Y3NnNb+e2d3b39wsFhU0eJIrRBIh6pdoA15UzShmGG03asKBYBp61gdDP1W09UaRbJezOOqS/wQLKQEWys1CYleV49e6j0CkW37M6AlomXkSJkqPcKX91+RBJBpSEca93x3Nj4KVaGEU4n+W6iaYzJCA9ox1KJBdV+Ort3gk6t0kdhpGxJg2bq74kUC63HIrCdApuhXvSm4n9eJzHhlZ8yGSeGSjJfFCYcmQhNn0d9pigxfGwJJorZWxEZYoWJsRHlbQje4svLpFkpexflyl21WLvO4sjBMZxACTy4hBrcQh0aQIDDM7zCm/PovDjvzse8dcXJZo7gD5zPH2PwjuM=</latexit> … … … T(1) <latexit sha1_base64="buKEJJi1AOp0SlRUCZR2gDaGe1Y=">AAAB63icbVBNSwMxEJ2tX7V+VT16CRahXspuFfRY9OKxQr+gXUo2zbahSXZJskJZ+he8eFDEq3/Im//GbLsHbX0w8Hhvhpl5QcyZNq777RQ2Nre2d4q7pb39g8Oj8vFJR0eJIrRNIh6pXoA15UzStmGG016sKBYBp91gep/53SeqNItky8xi6gs8lixkBJtMalW9y2G54tbcBdA68XJSgRzNYflrMIpIIqg0hGOt+54bGz/FyjDC6bw0SDSNMZniMe1bKrGg2k8Xt87RhVVGKIyULWnQQv09kWKh9UwEtlNgM9GrXib+5/UTE976KZNxYqgky0VhwpGJUPY4GjFFieEzSzBRzN6KyAQrTIyNp2RD8FZfXiedes27qtUfryuNuzyOIpzBOVTBgxtowAM0oQ0EJvAMr/DmCOfFeXc+lq0FJ585hT9wPn8A5j2NfA==</latexit> T(1) <latexit sha1_base64="buKEJJi1AOp0SlRUCZR2gDaGe1Y=">AAAB63icbVBNSwMxEJ2tX7V+VT16CRahXspuFfRY9OKxQr+gXUo2zbahSXZJskJZ+he8eFDEq3/Im//GbLsHbX0w8Hhvhpl5QcyZNq777RQ2Nre2d4q7pb39g8Oj8vFJR0eJIrRNIh6pXoA15UzStmGG016sKBYBp91gep/53SeqNItky8xi6gs8lixkBJtMalW9y2G54tbcBdA68XJSgRzNYflrMIpIIqg0hGOt+54bGz/FyjDC6bw0SDSNMZniMe1bKrGg2k8Xt87RhVVGKIyULWnQQv09kWKh9UwEtlNgM9GrXib+5/UTE976KZNxYqgky0VhwpGJUPY4GjFFieEzSzBRzN6KyAQrTIyNp2RD8FZfXiedes27qtUfryuNuzyOIpzBOVTBgxtowAM0oQ0EJvAMr/DmCOfFeXc+lq0FJ585hT9wPn8A5j2NfA==</latexit> T(1) <latexit sha1_base64="buKEJJi1AOp0SlRUCZR2gDaGe1Y=">AAAB63icbVBNSwMxEJ2tX7V+VT16CRahXspuFfRY9OKxQr+gXUo2zbahSXZJskJZ+he8eFDEq3/Im//GbLsHbX0w8Hhvhpl5QcyZNq777RQ2Nre2d4q7pb39g8Oj8vFJR0eJIrRNIh6pXoA15UzStmGG016sKBYBp91gep/53SeqNItky8xi6gs8lixkBJtMalW9y2G54tbcBdA68XJSgRzNYflrMIpIIqg0hGOt+54bGz/FyjDC6bw0SDSNMZniMe1bKrGg2k8Xt87RhVVGKIyULWnQQv09kWKh9UwEtlNgM9GrXib+5/UTE976KZNxYqgky0VhwpGJUPY4GjFFieEzSzBRzN6KyAQrTIyNp2RD8FZfXiedes27qtUfryuNuzyOIpzBOVTBgxtowAM0oQ0EJvAMr/DmCOfFeXc+lq0FJ585hT9wPn8A5j2NfA==</latexit> ※簡易化のため は8のべき乗と想定 n <latexit sha1_base64="2QmInd+nHO60uhS7AYCvgpiU4a8=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlpuyXK27VnYOsEi8nFcjR6Je/eoOYpRFKwwTVuuu5ifEzqgxnAqelXqoxoWxMh9i1VNIItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2ooszYbEo2BG/55VXSrlW9i2qteVmp3+RxFOEETuEcPLiCOtxBA1rAAOEZXuHNeXRenHfnY9FacPKZY/gD5/MH2F+M9g==</latexit> 木の高さ cn2 <latexit sha1_base64="HJGrz/aYh79gyTggVGXvrX09Jco=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Ae0sWy2m3bp7ibsboQS+he8eFDEq3/Im//GTZqDtj4YeLw3w8y8IOZMG9f9dkpr6xubW+Xtys7u3v5B9fCoo6NEEdomEY9UL8CaciZp2zDDaS9WFIuA024wvc387hNVmkXywcxi6gs8lixkBJtMIvKxMazW3LqbA60SryA1KNAaVr8Go4gkgkpDONa677mx8VOsDCOcziuDRNMYkyke076lEguq/TS/dY7OrDJCYaRsSYNy9fdEioXWMxHYToHNRC97mfif109MeO2nTMaJoZIsFoUJRyZC2eNoxBQlhs8swUQxeysiE6wwMTaeig3BW355lXQade+i3ri/rDVvijjKcAKncA4eXEET7qAFbSAwgWd4hTdHOC/Ou/OxaC05xcwx/IHz+QO53o4H</latexit> 8c(n/2)2 = 2cn2 <latexit sha1_base64="M5/sFW72hvDZ+SD/x6Rm/0N7zwQ=">AAAB+nicbVBNT8JAEJ36ifhV9OhlIzHBC7bVRC4mRC8eMZGPBArZLgts2G6b3a2GID/FiweN8eov8ea/cYEeFHzJJC/vzWRmXhBzprTjfFsrq2vrG5uZrez2zu7evp07qKkokYRWScQj2QiwopwJWtVMc9qIJcVhwGk9GN5M/foDlYpF4l6PYuqHuC9YjxGsjdSxcyVSEGfeadtDV8gjou117LxTdGZAy8RNSR5SVDr2V6sbkSSkQhOOlWq6Tqz9MZaaEU4n2VaiaIzJEPdp01CBQ6r88ez0CToxShf1ImlKaDRTf0+McajUKAxMZ4j1QC16U/E/r5noXskfMxEnmgoyX9RLONIRmuaAukxSovnIEEwkM7ciMsASE23SypoQ3MWXl0nNK7rnRe/uIl++TuPIwBEcQwFcuIQy3EIFqkDgEZ7hFd6sJ+vFerc+5q0rVjpzCH9gff4AtTuRtA==</latexit> 64c(n/4)2 = 4cn2 <latexit sha1_base64="qicqD/8CqIhjs0g0XmiRiWONqQE=">AAAB+3icbVDLSgNBEOz1GeNrjUcvg0GIl7i7LupFCHrxGME8IC9mJ5NkyOzsMjMrhpBf8eJBEa/+iDf/xkmyB00saCiquunuCmLOlHacb2tldW19YzOzld3e2d3btw9yVRUlktAKiXgk6wFWlDNBK5ppTuuxpDgMOK0Fw9upX3ukUrFIPOhRTFsh7gvWYwRrI3Xs3IVPCuLMP2176Br5RLS9jp13is4MaJm4KclDinLH/mp2I5KEVGjCsVIN14l1a4ylZoTTSbaZKBpjMsR92jBU4JCq1nh2+wSdGKWLepE0JTSaqb8nxjhUahQGpjPEeqAWvan4n9dIdO+qNWYiTjQVZL6ol3CkIzQNAnWZpETzkSGYSGZuRWSAJSbaxJU1IbiLLy+Tqld0z4vevZ8v3aRxZOAIjqEALlxCCe6gDBUg8ATP8Apv1sR6sd6tj3nripXOHMIfWJ8/L7mR9A==</latexit> コスト h = lg n <latexit sha1_base64="K5Tdf4fgMCRR5KDDec4xfOabsFU=">AAAB8XicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9CIUvXisYD+wDWWz3bRLN5uwOxFK6L/w4kERr/4bb/4bt20O2vpg4PHeDDPzgkQKg6777aysrq1vbBa2its7u3v7pYPDpolTzXiDxTLW7YAaLoXiDRQoeTvRnEaB5K1gdDv1W09cGxGrBxwn3I/oQIlQMIpWehySa9KVg0xNeqWyW3FnIMvEy0kZctR7pa9uP2ZpxBUySY3peG6CfkY1Cib5pNhNDU8oG9EB71iqaMSNn80unpBTq/RJGGtbCslM/T2R0ciYcRTYzoji0Cx6U/E/r5NieOVnQiUpcsXmi8JUEozJ9H3SF5ozlGNLKNPC3krYkGrK0IZUtCF4iy8vk2a14p1XqvcX5dpNHkcBjuEEzsCDS6jBHdShAQwUPMMrvDnGeXHenY9564qTzxzBHzifP9BdkFw=</latexit> 1個 個 8h <latexit sha1_base64="bc4l3VpRiR8rg8Y3pwrlrNlMHD0=">AAAB6nicbVDLTgJBEOzFF+IL9ehlIjHxRHbRRI5ELx4xyiOBlcwOvTBhdnYzM2tCCJ/gxYPGePWLvPk3DrAHBSvppFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLRzcxvPaHSPJYPZpygH9GB5CFn1Fjpvvo47BVLbtmdg6wSLyMlyFDvFb+6/ZilEUrDBNW647mJ8SdUGc4ETgvdVGNC2YgOsGOppBFqfzI/dUrOrNInYaxsSUPm6u+JCY20HkeB7YyoGeplbyb+53VSE1b9CZdJalCyxaIwFcTEZPY36XOFzIixJZQpbm8lbEgVZcamU7AheMsvr5JmpexdlCt3l6XadRZHHk7gFM7BgyuowS3UoQEMBvAMr/DmCOfFeXc+Fq05J5s5hj9wPn8A/UaNmg==</latexit> ⇥(n3) <latexit sha1_base64="G8WRpL5exN/uz8BAetuwevHrbn8=">AAAB8nicbVBNSwMxEM3Wr1q/qh69BItQL2W3FfRY9OKxQr+gXUs2zbah2WRJZoWy9Gd48aCIV3+NN/+NabsHbX0w8Hhvhpl5QSy4Adf9dnIbm1vbO/ndwt7+weFR8fikbVSiKWtRJZTuBsQwwSVrAQfBurFmJAoE6wSTu7nfeWLacCWbMI2ZH5GR5CGnBKzU6zfHDEhZPtYuB8WSW3EXwOvEy0gJZWgMil/9oaJJxCRQQYzpeW4Mfko0cCrYrNBPDIsJnZAR61kqScSMny5OnuELqwxxqLQtCXih/p5ISWTMNApsZ0RgbFa9ufif10sgvPFTLuMEmKTLRWEiMCg8/x8PuWYUxNQSQjW3t2I6JppQsCkVbAje6svrpF2teLVK9eGqVL/N4sijM3SOyshD16iO7lEDtRBFCj2jV/TmgPPivDsfy9ack82coj9wPn8AON+Qjg==</latexit> 8h <latexit sha1_base64="bc4l3VpRiR8rg8Y3pwrlrNlMHD0=">AAAB6nicbVDLTgJBEOzFF+IL9ehlIjHxRHbRRI5ELx4xyiOBlcwOvTBhdnYzM2tCCJ/gxYPGePWLvPk3DrAHBSvppFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLRzcxvPaHSPJYPZpygH9GB5CFn1Fjpvvo47BVLbtmdg6wSLyMlyFDvFb+6/ZilEUrDBNW647mJ8SdUGc4ETgvdVGNC2YgOsGOppBFqfzI/dUrOrNInYaxsSUPm6u+JCY20HkeB7YyoGeplbyb+53VSE1b9CZdJalCyxaIwFcTEZPY36XOFzIixJZQpbm8lbEgVZcamU7AheMsvr5JmpexdlCt3l6XadRZHHk7gFM7BgyuowS3UoQEMBvAMr/DmCOfFeXc+Fq05J5s5hj9wPn8A/UaNmg==</latexit> ボトムの要素数: 8lg n = (23)ln n = (2lg n)3 = n3 <latexit sha1_base64="8Svmj+p7SdePFUTthaqIDzxYafs=">AAACH3icbZDLSsNAFIYnXmu9RV26GSxCuylJK9qNUHTjsoK9QJuWyXTSDp1MwsxEKCFv4sZXceNCEXHXt3HSZlFbDwz8fP85nDm/GzIqlWXNjI3Nre2d3dxefv/g8OjYPDltySASmDRxwALRcZEkjHLSVFQx0gkFQb7LSNud3Kd++5kISQP+pKYhcXw04tSjGCmNBuZ1rR/32CjmSQJvYbHSj6tJKUV8CWUNpdTUiPerA7Ngla15wXVhZ6IAsmoMzJ/eMMCRT7jCDEnZta1QOTESimJGknwvkiREeIJGpKslRz6RTjy/L4GXmgyhFwj9uIJzujwRI1/Kqe/qTh+psVz1Uvif142UV3NiysNIEY4Xi7yIQRXANCw4pIJgxaZaICyo/ivEYyQQVjrSvA7BXj15XbQqZbtarjxeFep3WRw5cA4uQBHY4AbUwQNogCbA4AW8gQ/wabwa78aX8b1o3TCymTPwp4zZL3KzoVk=</latexit> 部分問題の サイズは1/2 c(n2) <latexit sha1_base64="MWun7Fy0INQnzZjP0n0C7r39z3c=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRahXspuK+ix6MVjBfsB7VqyabaNzSZLkhXK0v/gxYMiXv0/3vw3pu0etPXBwOO9GWbmBTFn2rjut5NbW9/Y3MpvF3Z29/YPiodHLS0TRWiTSC5VJ8CaciZo0zDDaSdWFEcBp+1gfDPz209UaSbFvZnE1I/wULCQEWys1CJl8VA97xdLbsWdA60SLyMlyNDoF796A0mSiApDONa667mx8VOsDCOcTgu9RNMYkzEe0q6lAkdU++n82ik6s8oAhVLZEgbN1d8TKY60nkSB7YywGellbyb+53UTE175KRNxYqggi0VhwpGRaPY6GjBFieETSzBRzN6KyAgrTIwNqGBD8JZfXiWtasWrVap3F6X6dRZHHk7gFMrgwSXU4RYa0AQCj/AMr/DmSOfFeXc+Fq05J5s5hj9wPn8AgOuObA==</latexit> オリジナルの呼出: 23
  24. /35 T(n) = ( ⇥(1) n = 1 8T(n/2) +

    ⇥(n2) n > 1 <latexit sha1_base64="GR8m44C9HSDmcXqd/MYxaonOnuY=">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</latexit> 再帰木の構築 T(n) = cn2 + 2cn2 + 4cn2 + . . . + 2h 1cn2 + ⇥(n3) = cn2 h 1 X k=0 2k + ⇥(n3) = cn2 2h 1 + ⇥(n3) = cn2(n 1) + ⇥(n3) = cn3 cn2 + ⇥(n3) = ⇥ n3 <latexit sha1_base64="6WLvbXtmjebXqpUwGDfWIUUc6ek=">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</latexit> h = lg n <latexit sha1_base64="K5Tdf4fgMCRR5KDDec4xfOabsFU=">AAAB8XicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9CIUvXisYD+wDWWz3bRLN5uwOxFK6L/w4kERr/4bb/4bt20O2vpg4PHeDDPzgkQKg6777aysrq1vbBa2its7u3v7pYPDpolTzXiDxTLW7YAaLoXiDRQoeTvRnEaB5K1gdDv1W09cGxGrBxwn3I/oQIlQMIpWehySa9KVg0xNeqWyW3FnIMvEy0kZctR7pa9uP2ZpxBUySY3peG6CfkY1Cib5pNhNDU8oG9EB71iqaMSNn80unpBTq/RJGGtbCslM/T2R0ciYcRTYzoji0Cx6U/E/r5NieOVnQiUpcsXmi8JUEozJ9H3SF5ozlGNLKNPC3krYkGrK0IZUtCF4iy8vk2a14p1XqvcX5dpNHkcBjuEEzsCDS6jBHdShAQwUPMMrvDnGeXHenY9564qTzxzBHzifP9BdkFw=</latexit> より 推測ができた. あとは置換え法で証明すれば良い(済). 24
  25. /35 分割統治の計算量を求めるプロセス 一般化できそう 25

  26. /35 マスター定理 a 1 <latexit sha1_base64="e23j/eQrjEgtifTnZjpOaYePq68=">AAAB73icbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGC/YA2lM120i7dbNLdjVBC/4QXD4p49e9489+4bXPQ1gcDj/dmmJkXJIJr47rfztr6xubWdmGnuLu3f3BYOjpu6jhVDBssFrFqB1Sj4BIbhhuB7UQhjQKBrWB0N/NbT6g0j+WjmSToR3QgecgZNVZqU9Id4Jh4vVLZrbhzkFXi5aQMOeq90le3H7M0QmmYoFp3PDcxfkaV4UzgtNhNNSaUjegAO5ZKGqH2s/m9U3JulT4JY2VLGjJXf09kNNJ6EgW2M6JmqJe9mfif10lNeONnXCapQckWi8JUEBOT2fOkzxUyIyaWUKa4vZWwIVWUGRtR0YbgLb+8SprVindZqT5clWu3eRwFOIUzuAAPrqEG91CHBjAQ8Ayv8OaMnRfn3flYtK45+cwJ/IHz+QPndY85</latexit> f(n) <latexit sha1_base64="pG5iFsNQm8e14VQk0dg/VxXte+4=">AAAB63icbVBNSwMxEJ2tX7V+VT16CRahXspuK+ix6MVjBfsB7VKyabYNTbJLkhXK0r/gxYMiXv1D3vw3Zts9aOuDgcd7M8zMC2LOtHHdb6ewsbm1vVPcLe3tHxwelY9POjpKFKFtEvFI9QKsKWeStg0znPZiRbEIOO0G07vM7z5RpVkkH80spr7AY8lCRrDJpLAqL4fliltzF0DrxMtJBXK0huWvwSgiiaDSEI617ntubPwUK8MIp/PSINE0xmSKx7RvqcSCaj9d3DpHF1YZoTBStqRBC/X3RIqF1jMR2E6BzUSvepn4n9dPTHjjp0zGiaGSLBeFCUcmQtnjaMQUJYbPLMFEMXsrIhOsMDE2npINwVt9eZ106jWvUas/XFWat3kcRTiDc6iCB9fQhHtoQRsITOAZXuHNEc6L8+58LFsLTj5zCn/gfP4AXnuNyw==</latexit> T(n)

    <latexit sha1_base64="vM8hsiJM1mIid8s1EMCG6OMcapM=">AAAB63icbVBNSwMxEJ2tX7V+VT16CRahXspuFfRY9OKxQr+gXUo2zbahSXZJskJZ+he8eFDEq3/Im//GbLsHbX0w8Hhvhpl5QcyZNq777RQ2Nre2d4q7pb39g8Oj8vFJR0eJIrRNIh6pXoA15UzStmGG016sKBYBp91gep/53SeqNItky8xi6gs8lixkBJtMalXl5bBccWvuAmideDmpQI7msPw1GEUkEVQawrHWfc+NjZ9iZRjhdF4aJJrGmEzxmPYtlVhQ7aeLW+fowiojFEbKljRoof6eSLHQeiYC2ymwmehVLxP/8/qJCW/9lMk4MVSS5aIw4chEKHscjZiixPCZJZgoZm9FZIIVJsbGU7IheKsvr5NOveZd1eqP15XGXR5HEc7gHKrgwQ004AGa0AYCE3iGV3hzhPPivDsfy9aCk8+cwh84nz9C/Y25</latexit> T(n) = aT(n/b) + f(n) <latexit sha1_base64="ohCH7vdM3XKzxE6JUirDO0Tn1ro=">AAAB/nicbVDLSgMxFL1TX7W+RsWVm2ARWoQ6UwXdCEU3Liv0BW0pmTTThmYyQ5IRylDwV9y4UMSt3+HOvzFtZ6GtBy73cM695OZ4EWdKO863lVlZXVvfyG7mtrZ3dvfs/YOGCmNJaJ2EPJQtDyvKmaB1zTSnrUhSHHicNr3R3dRvPlKpWChqehzRboAHgvmMYG2knn1UK4giukHY9HOviM6Qb4SenXdKzgxombgpyUOKas/+6vRDEgdUaMKxUm3XiXQ3wVIzwukk14kVjTAZ4QFtGypwQFU3mZ0/QadG6SM/lKaERjP190aCA6XGgWcmA6yHatGbiv957Vj7192EiSjWVJD5Q37MkQ7RNAvUZ5ISzceGYCKZuRWRIZaYaJNYzoTgLn55mTTKJfeiVH64zFdu0ziycAwnUAAXrqAC91CFOhBI4Ble4c16sl6sd+tjPpqx0p1D+APr8weOKZKm</latexit> と を定数, を関数とする. 非負整数上の関数 を漸化式 n/b <latexit sha1_base64="rSqccx9IdhZwyOJi1mpvSHd4EhM=">AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4qkkV9Fj04rGi/YA2lM120y7dbMLuRCihP8GLB0W8+ou8+W/ctjlo64OBx3szzMwLEikMuu63s7K6tr6xWdgqbu/s7u2XDg6bJk414w0Wy1i3A2q4FIo3UKDk7URzGgWSt4LR7dRvPXFtRKwecZxwP6IDJULBKFrpQZ0HvVLZrbgzkGXi5aQMOeq90le3H7M04gqZpMZ0PDdBP6MaBZN8UuymhieUjeiAdyxVNOLGz2anTsipVfokjLUthWSm/p7IaGTMOApsZ0RxaBa9qfif10kxvPYzoZIUuWLzRWEqCcZk+jfpC80ZyrEllGlhbyVsSDVlaNMp2hC8xZeXSbNa8S4q1fvLcu0mj6MAx3ACZ+DBFdTgDurQAAYDeIZXeHOk8+K8Ox/z1hUnnzmCP3A+fwD/B42b</latexit> bn/bc <latexit sha1_base64="JcRhJvBRulJbnEAjxVmR6m43Pr4=">AAAB/HicbZA7T8MwFIVvyquUV6Aji0WFxFSSggRjBQtjkehDaqrKcZ3WqmNHtoNUVeWvsDCAECs/hI1/g5tmgJY7fTrnXvn4hAln2njet1NYW9/Y3Cpul3Z29/YP3MOjlpapIrRJJJeqE2JNORO0aZjhtJMoiuOQ03Y4vp377UeqNJPiwUwS2ovxULCIEWys1HfLAY+4lAqJ8xAFKuO+W/GqXjZoFfwcKpBPo+9+BQNJ0pgKQzjWuut7ielNsTKMcDorBammCSZjPKRdiwLHVPemWfgZOrXKAEU2QiSFQZn6+2KKY60ncWg3Y2xGetmbi/953dRE170pE0lqqCCLh6KUIyPRvAk0YIoSwycWMFHMZkVkhBUmxvZVsiX4y19ehVat6l9Ua/eXlfpNXkcRjuEEzsCHK6jDHTSgCQQm8Ayv8OY8OS/Ou/OxWC04+U0Z/ozz+QNE+5SG</latexit> dn/be <latexit sha1_base64="/Z2tUkIRTe0iwjiNDsUS9Cn97s8=">AAAB+nicbZDLSsNAFIZPvNZ6S3XpZrAIrmpSBV0W3bisYC/QhDKZTtqhk0mYmSgl9lHcuFDErU/izrdx0mahrT8MfPznHM6ZP0g4U9pxvq2V1bX1jc3SVnl7Z3dv364ctFWcSkJbJOax7AZYUc4EbWmmOe0mkuIo4LQTjG/yeueBSsVica8nCfUjPBQsZARrY/XtiscJZRyJswB5Mse+XXVqzkxoGdwCqlCo2be/vEFM0ogKTThWquc6ifYzLDUjnE7LXqpogskYD2nPoMARVX42O32KTowzQGEszRMazdzfExmOlJpEgemMsB6pxVpu/lfrpTq88jMmklRTQeaLwpQjHaM8BzRgkhLNJwYwkczcisgIS0y0SatsQnAXv7wM7XrNPa/V7y6qjesijhIcwTGcgguX0IBbaEILCDzCM7zCm/VkvVjv1se8dcUqZg7hj6zPH1hdk2g=</latexit> によって定義する. ここで, は または を意味するものと解釈する. b > 1 <latexit sha1_base64="Oaj7SatYDW4Wm2nNKQXdZTfaWvE=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoCcpevFYwbSFNpTNdtIu3WzC7kYopb/BiwdFvPqDvPlv3LY5aOuDgcd7M8zMC1PBtXHdb6ewtr6xuVXcLu3s7u0flA+PmjrJFEOfJSJR7ZBqFFyib7gR2E4V0jgU2ApHdzO/9YRK80Q+mnGKQUwHkkecUWMlPyQ3xOuVK27VnYOsEi8nFcjR6JW/uv2EZTFKwwTVuuO5qQkmVBnOBE5L3UxjStmIDrBjqaQx6mAyP3ZKzqzSJ1GibElD5urviQmNtR7Hoe2MqRnqZW8m/ud1MhNdBxMu08ygZItFUSaIScjsc9LnCpkRY0soU9zeStiQKsqMzadkQ/CWX14lzVrVu6jWHi4r9ds8jiKcwCmcgwdXUId7aIAPDDg8wyu8OdJ5cd6dj0VrwclnjuEPnM8fZamNwQ==</latexit> このとき, は漸近的に次の限界を持つ. T(n) <latexit sha1_base64="vM8hsiJM1mIid8s1EMCG6OMcapM=">AAAB63icbVBNSwMxEJ2tX7V+VT16CRahXspuFfRY9OKxQr+gXUo2zbahSXZJskJZ+he8eFDEq3/Im//GbLsHbX0w8Hhvhpl5QcyZNq777RQ2Nre2d4q7pb39g8Oj8vFJR0eJIrRNIh6pXoA15UzStmGG016sKBYBp91gep/53SeqNItky8xi6gs8lixkBJtMalXl5bBccWvuAmideDmpQI7msPw1GEUkEVQawrHWfc+NjZ9iZRjhdF4aJJrGmEzxmPYtlVhQ7aeLW+fowiojFEbKljRoof6eSLHQeiYC2ymwmehVLxP/8/qJCW/9lMk4MVSS5aIw4chEKHscjZiixPCZJZgoZm9FZIIVJsbGU7IheKsvr5NOveZd1eqP15XGXR5HEc7gHKrgwQ004AGa0AYCE3iGV3hzhPPivDsfy9aCk8+cwh84nz9C/Y25</latexit> 1. ある定数 に対して ならば, である. ✏ > 0 <latexit sha1_base64="iQxt5eBAWyi4CnJXD1FTsFM/HhA=">AAAB83icbVBNSwMxEJ2tX7V+VT16CRbBU9mtgp6k6MVjBfsB3aVk02wbmk1CkhVK6d/w4kERr/4Zb/4b03YP2vpg4PHeDDPzYsWZsb7/7RXW1jc2t4rbpZ3dvf2D8uFRy8hME9okkkvdibGhnAnatMxy2lGa4jTmtB2P7mZ++4lqw6R4tGNFoxQPBEsYwdZJYUiVYVwKdIP8XrniV/050CoJclKBHI1e+SvsS5KlVFjCsTHdwFc2mmBtGeF0WgozQxUmIzygXUcFTqmJJvObp+jMKX2USO1KWDRXf09McGrMOI1dZ4rt0Cx7M/E/r5vZ5DqaMKEySwVZLEoyjqxEswBQn2lKLB87golm7lZEhlhjYl1MJRdCsPzyKmnVqsFFtfZwWanf5nEU4QRO4RwCuII63EMDmkBAwTO8wpuXeS/eu/exaC14+cwx/IH3+QP4DpD6</latexit> f(n) = O(n logb a ✏) <latexit sha1_base64="/Co+zHNANF4vde42xchIx70RmKc=">AAACCXicbVC7SgNBFJ2NrxhfUUubwSAkhWE3CtoIQRs7I5gHZNdldjKbDJmdWWZmhbBsa+Ov2FgoYusf2Pk3Th6FJh64cDjnXu69J4gZVdq2v63c0vLK6lp+vbCxubW9U9zdaymRSEyaWDAhOwFShFFOmppqRjqxJCgKGGkHw6ux334gUlHB7/QoJl6E+pyGFCNtJL8IwzKvwAt4U+b3qctE3w9SlB27JFaUCZ5V/GLJrtoTwEXizEgJzNDwi19uT+AkIlxjhpTqOnasvRRJTTEjWcFNFIkRHqI+6RrKUUSUl04+yeCRUXowFNIU13Ci/p5IUaTUKApMZ4T0QM17Y/E/r5vo8NxLKY8TTTieLgoTBrWA41hgj0qCNRsZgrCk5laIB0girE14BROCM//yImnVqs5JtXZ7WqpfzuLIgwNwCMrAAWegDq5BAzQBBo/gGbyCN+vJerHerY9pa86azeyDP7A+fwDWiJkw</latexit> T(n) = ⇥(nlogb a) <latexit sha1_base64="NXmsTiZOS/6q9hWSxepYpDy9Ol0=">AAACBXicbVDLSsNAFJ3UV62vqEtdDBah3ZSkCroRim5cVugLmhgm00k7dDIJMxOhhGzc+CtuXCji1n9w5984bbPQ1gMXDufcy733+DGjUlnWt1FYWV1b3yhulra2d3b3zP2DjowSgUkbRywSPR9JwignbUUVI71YEBT6jHT98c3U7z4QIWnEW2oSEzdEQ04DipHSkmcetyq8Cq+g0xoRhSr8PnVYNPT8FGVZ1TPLVs2aAS4TOydlkKPpmV/OIMJJSLjCDEnZt61YuSkSimJGspKTSBIjPEZD0teUo5BIN519kcFTrQxgEAldXMGZ+nsiRaGUk9DXnSFSI7noTcX/vH6igks3pTxOFOF4vihIGFQRnEYCB1QQrNhEE4QF1bdCPEICYaWDK+kQ7MWXl0mnXrPPavW783LjOo+jCI7ACagAG1yABrgFTdAGGDyCZ/AK3own48V4Nz7mrQUjnzkEf2B8/gC2gJd2</latexit> f(n) = ⇥(nlogb a) <latexit sha1_base64="wxEUjKX9v7jbdUtsLF9regJ8Cl0=">AAACBXicbVDLSsNAFJ3UV62vqEtdDBah3ZSkCroRim5cVugLmhgm00k7dDIJMxOhhGzc+CtuXCji1n9w5984bbPQ1gMXDufcy733+DGjUlnWt1FYWV1b3yhulra2d3b3zP2DjowSgUkbRywSPR9JwignbUUVI71YEBT6jHT98c3U7z4QIWnEW2oSEzdEQ04DipHSkmceBxVehVfQaY2IQhV+nzosGnp+irKs6pllq2bNAJeJnZMyyNH0zC9nEOEkJFxhhqTs21as3BQJRTEjWclJJIkRHqMh6WvKUUikm86+yOCpVgYwiIQuruBM/T2RolDKSejrzhCpkVz0puJ/Xj9RwaWbUh4ninA8XxQkDKoITiOBAyoIVmyiCcKC6lshHiGBsNLBlXQI9uLLy6RTr9lntfrdeblxncdRBEfgBFSADS5AA9yCJmgDDB7BM3gFb8aT8WK8Gx/z1oKRzxyCPzA+fwDTrpeI</latexit> T(n) = ⇥(nlogb a lg n) <latexit sha1_base64="0prIF2fK13+RYI2+hJ5S8C3FtR0=">AAACC3icbVDLSsNAFJ3UV62vqEs3Q4vQbkpSBd0IRTcuK/QFTQyT6aQdOpmEmYlQQvZu/BU3LhRx6w+482+ctllo64ELh3Pu5d57/JhRqSzr2yisrW9sbhW3Szu7e/sH5uFRV0aJwKSDIxaJvo8kYZSTjqKKkX4sCAp9Rnr+5Gbm9x6IkDTibTWNiRuiEacBxUhpyTPL7SqvwSvotMdEoSq/Tx0WjTw/RVnmsFHKs5pnVqy6NQdcJXZOKiBHyzO/nGGEk5BwhRmScmBbsXJTJBTFjGQlJ5EkRniCRmSgKUchkW46/yWDp1oZwiASuriCc/X3RIpCKaehrztDpMZy2ZuJ/3mDRAWXbkp5nCjC8WJRkDCoIjgLBg6pIFixqSYIC6pvhXiMBMJKx1fSIdjLL6+SbqNun9Ubd+eV5nUeRxGcgDKoAhtcgCa4BS3QARg8gmfwCt6MJ+PFeDc+Fq0FI585Bn9gfP4AweGaRw==</latexit> 2. ならば, である. f(n) = ⌦(nlogb a+✏) <latexit sha1_base64="EHlFmqSFcIuKRsfUNaViZF2E21Y=">AAACDnicbVA9SwNBEN3zM8avU0ubxRBIEMKdCtoIQRs7I5gYyMWwt5lLFvd2j909IRz5BTb+FRsLRWyt7fw3bj4KjT4YeLw3w8y8MOFMG8/7cubmFxaXlnMr+dW19Y1Nd2u7oWWqKNSp5FI1Q6KBMwF1wwyHZqKAxCGHm/DufOTf3IPSTIprM0igHZOeYBGjxFip4xajkijjUxxcxtAjJXGbBVz2OmFGhvsBJJpxKYbljlvwKt4Y+C/xp6SApqh13M+gK2kagzCUE61bvpeYdkaUYZTDMB+kGhJC70gPWpYKEoNuZ+N3hrholS6OpLIlDB6rPycyEms9iEPbGRPT17PeSPzPa6UmOmlnTCSpAUEni6KUYyPxKBvcZQqo4QNLCFXM3oppnyhCjU0wb0PwZ1/+SxoHFf+wcnB1VKieTePIoV20h0rIR8eoii5QDdURRQ/oCb2gV+fReXbenPdJ65wzndlBv+B8fAPDk5tW</latexit> c < 1 <latexit sha1_base64="WfaFCDew75a8b7g+U5VJ8Sh5PJI=">AAAB7HicbVA9SwNBEJ2LXzF+RS1tFoNgFe6ioIVF0MYygpcEkiPsbeaSJXt7x+6eEEJ+g42FIrb+IDv/jZvkCk18MPB4b4aZeWEquDau++0U1tY3NreK26Wd3b39g/LhUVMnmWLos0Qkqh1SjYJL9A03AtupQhqHAlvh6G7mt55QaZ7IRzNOMYjpQPKIM2qs5DNyQ7xeueJW3TnIKvFyUoEcjV75q9tPWBajNExQrTuem5pgQpXhTOC01M00ppSN6AA7lkoaow4m82On5MwqfRIlypY0ZK7+npjQWOtxHNrOmJqhXvZm4n9eJzPRdTDhMs0MSrZYFGWCmITMPid9rpAZMbaEMsXtrYQNqaLM2HxKNgRv+eVV0qxVvYtq7eGyUr/N4yjCCZzCOXhwBXW4hwb4wIDDM7zCmyOdF+fd+Vi0Fpx85hj+wPn8AWQljcA=</latexit> af(n/b)  cf(n) <latexit sha1_base64="4j+T1jhEsTzYQ7+9RfN+cZD7oE8=">AAAB+3icbVDLTgIxFO3gC/E14tJNIzGBDc6giS6JblxiIo8EJqRT7kBDpzO2HSOZ8CtuXGiMW3/EnX9jgVkoeJKbnJ5zb3rv8WPOlHacbyu3tr6xuZXfLuzs7u0f2IfFlooSSaFJIx7Jjk8UcCagqZnm0IklkNDn0PbHNzO//QhSsUjc60kMXkiGggWMEm2kvl0kQVmc+RXc4/CAqXlU+nbJqTpz4FXiZqSEMjT69ldvENEkBKEpJ0p1XSfWXkqkZpTDtNBLFMSEjskQuoYKEoLy0vnuU3xqlAEOImlKaDxXf0+kJFRqEvqmMyR6pJa9mfif1010cOWlTMSJBkEXHwUJxzrCsyDwgEmgmk8MIVQysyumIyIJ1SauggnBXT55lbRqVfe8Wru7KNWvszjy6BidoDJy0SWqo1vUQE1E0RN6Rq/ozZpaL9a79bFozVnZzBH6A+vzB59SkuA=</latexit> T(n) = ⇥(f(n)) <latexit sha1_base64="w2Wkwq9Zk6Rm7qw8MFnERSx3Lww=">AAAB/HicbVDLSgMxFM3UV62v0S7dBIvQbspMFXQjFN24rNAXtEPJpHfa0ExmSDLCUOqvuHGhiFs/xJ1/Y9rOQlsPXDg5515y7/FjzpR2nG8rt7G5tb2T3y3s7R8cHtnHJ20VJZJCi0Y8kl2fKOBMQEszzaEbSyChz6HjT+7mfucRpGKRaOo0Bi8kI8ECRok20sAuNsuigm9wvzkGTcqBeVUGdsmpOgvgdeJmpIQyNAb2V38Y0SQEoSknSvVcJ9belEjNKIdZoZ8oiAmdkBH0DBUkBOVNF8vP8LlRhjiIpCmh8UL9PTEloVJp6JvOkOixWvXm4n9eL9HBtTdlIk40CLr8KEg41hGeJ4GHTALVPDWEUMnMrpiOiSRUm7wKJgR39eR10q5V3Ytq7eGyVL/N4sijU3SGyshFV6iO7lEDtRBFKXpGr+jNerJerHfrY9mas7KZIvoD6/MHjKKSxQ==</latexit> ✏ > 0 <latexit sha1_base64="iQxt5eBAWyi4CnJXD1FTsFM/HhA=">AAAB83icbVBNSwMxEJ2tX7V+VT16CRbBU9mtgp6k6MVjBfsB3aVk02wbmk1CkhVK6d/w4kERr/4Zb/4b03YP2vpg4PHeDDPzYsWZsb7/7RXW1jc2t4rbpZ3dvf2D8uFRy8hME9okkkvdibGhnAnatMxy2lGa4jTmtB2P7mZ++4lqw6R4tGNFoxQPBEsYwdZJYUiVYVwKdIP8XrniV/050CoJclKBHI1e+SvsS5KlVFjCsTHdwFc2mmBtGeF0WgozQxUmIzygXUcFTqmJJvObp+jMKX2USO1KWDRXf09McGrMOI1dZ4rt0Cx7M/E/r5vZ5DqaMKEySwVZLEoyjqxEswBQn2lKLB87golm7lZEhlhjYl1MJRdCsPzyKmnVqsFFtfZwWanf5nEU4QRO4RwCuII63EMDmkBAwTO8wpuXeS/eu/exaC14+cwx/IH3+QP4DpD6</latexit> 3. ある定数 に対して であり, しかもある定数 と 十分大きな全ての に対して ならば である. n <latexit sha1_base64="2QmInd+nHO60uhS7AYCvgpiU4a8=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlpuyXK27VnYOsEi8nFcjR6Je/eoOYpRFKwwTVuuu5ifEzqgxnAqelXqoxoWxMh9i1VNIItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2ooszYbEo2BG/55VXSrlW9i2qteVmp3+RxFOEETuEcPLiCOtxBA1rAAOEZXuHNeXRenHfnY9FacPKZY/gD5/MH2F+M9g==</latexit> ※全てのケースをカバーできているわけではない. 他にも細かい条件あり 26
  27. /35 とりあえず当てはめてみる T(n) = 8T(n/2) + ⇥ n2 <latexit sha1_base64="jDlukj/haOrDtcfSLetZLQACrNI=">AAACEnicbVA9SwNBEN2L3/ErammzGIQEId5FQRtBtLFUyIeQi2FvM5cs7u0du3NCOPIbbPwrNhaK2FrZ+W/cxBRqfDDM470ZducFiRQGXffTyc3Mzs0vLC7ll1dW19YLG5sNE6eaQ53HMtbXATMghYI6CpRwnWhgUSChGdyej/zmHWgjYlXDQQLtiPWUCAVnaKVOoVwrqTI9oce2031aLe/5tT4g8yWEWFI3WXXoa9HrY7lTKLoVdww6TbwJKZIJLjuFD78b8zQChVwyY1qem2A7YxoFlzDM+6mBhPFb1oOWpYpFYNrZ+KQh3bVKl4axtqWQjtWfGxmLjBlEgZ2MGPbNX28k/ue1UgyP25lQSYqg+PdDYSopxnSUD+0KDRzlwBLGtbB/pbzPNONoU8zbELy/J0+TRrXiHVSqV4fF07NJHItkm+yQEvHIETklF+SS1Akn9+SRPJMX58F5cl6dt+/RnDPZ2SK/4Lx/Aa0xmwo=</latexit>

    a <latexit sha1_base64="wqZLPcGml9Og5FDdUdFUNWEF4FE=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlJu2XK27VnYOsEi8nFcjR6Je/eoOYpRFKwwTVuuu5ifEzqgxnAqelXqoxoWxMh9i1VNIItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2ooszYbEo2BG/55VXSrlW9i2qteVmp3+RxFOEETuEcPLiCOtxBA1rAAOEZXuHNeXRenHfnY9FacPKZY/gD5/MHxKuM6Q==</latexit> b <latexit sha1_base64="YJjhR7RY5hyNtVLBH/MerrmOQ7I=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlZtAvV9yqOwdZJV5OKpCj0S9/9QYxSyOUhgmqdddzE+NnVBnOBE5LvVRjQtmYDrFrqaQRaj+bHzolZ1YZkDBWtqQhc/X3REYjrSdRYDsjakZ62ZuJ/3nd1ITXfsZlkhqUbLEoTAUxMZl9TQZcITNiYgllittbCRtRRZmx2ZRsCN7yy6ukXat6F9Va87JSv8njKMIJnMI5eHAFdbiDBrSAAcIzvMKb8+i8OO/Ox6K14OQzx/AHzucPxi+M6g==</latexit> O(n logb a ✏) = O(n 2) = f(n) <latexit sha1_base64="SYnr/i+TAO/XXf9QYuiW0q03MK4=">AAACFHicbZDLSgMxFIYz9VbrrerSTbAUWsQyUwXdCEU37qxgL9AZSybNtKGZZEgyQhn6EG58FTcuFHHrwp1vY6btQlsPBD7+P4dzzu9HjCpt299WZml5ZXUtu57b2Nza3snv7jWViCUmDSyYkG0fKcIoJw1NNSPtSBIU+oy0/OFV6rceiFRU8Ds9iogXoj6nAcVIG6mbP7op8fvEZaIPu4k/hujYJZGiTPBxGV7A1K2mEJR4uZsv2BV7UnARnBkUwKzq3fyX2xM4DgnXmCGlOo4daS9BUlPMyDjnxopECA9Rn3QMchQS5SWTo8awaJQeDIQ0j2s4UX93JChUahSanYsh0gM176Xif14n1sG5l1AexZpwPB0UxAxqAdOEYI9KgjUbGUBYUrMrxAMkEdYmx5wJwZk/eRGa1YpzUqnenhZql7M4suAAHIIScMAZqIFrUAcNgMEjeAav4M16sl6sd+tj+jVjzXr2wZ+yPn8AmXub+Q==</latexit> ✏ = 1 <latexit sha1_base64="V69ZFXDYFrQbzLFBt0au3hKZ6Hw=">AAAB83icbVBNSwMxEM3Wr1q/qh69BIvgqexWQS9C0YvHCvYDukvJprNtaDYJSVYopX/DiwdFvPpnvPlvTNs9aOuDgcd7M8zMixVnxvr+t1dYW9/Y3Cpul3Z29/YPyodHLSMzTaFJJZe6ExMDnAloWmY5dJQGksYc2vHobua3n0AbJsWjHSuIUjIQLGGUWCeFISjDuBT4Bge9csWv+nPgVRLkpIJyNHrlr7AvaZaCsJQTY7qBr2w0IdoyymFaCjMDitARGUDXUUFSMNFkfvMUnzmljxOpXQmL5+rviQlJjRmnsetMiR2aZW8m/ud1M5tcRxMmVGZB0MWiJOPYSjwLAPeZBmr52BFCNXO3YjokmlDrYiq5EILll1dJq1YNLqq1h8tK/TaPo4hO0Ck6RwG6QnV0jxqoiShS6Bm9ojcv8168d+9j0Vrw8plj9Afe5w/4DJD6</latexit> logb a = log2 8 = 3 <latexit sha1_base64="IDj9pqK+dgx+9nd1x5i/OZb69lc=">AAACAnicbZDLSsNAFIYnXmu9RV2Jm8EiuCpJK9iNUHTjsoK9QBvCZDpph05mwsxEKKG48VXcuFDErU/hzrdxkmahrT8MfPznHM6cP4gZVdpxvq2V1bX1jc3SVnl7Z3dv3z447CiRSEzaWDAhewFShFFO2ppqRnqxJCgKGOkGk5us3n0gUlHB7/U0Jl6ERpyGFCNtLN8+HjAx8oMUzeAVzLmWNjKu+3bFqTq54DK4BVRAoZZvfw2GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifYMcRUR5aX7CDJ4ZZwhDIc3jGubu74kURUpNo8B0RkiP1WItM/+r9RMdNryU8jjRhOP5ojBhUAuY5QGHVBKs2dQAwpKav0I8RhJhbVIrmxDcxZOXoVOruvVq7e6i0rwu4iiBE3AKzoELLkET3IIWaAMMHsEzeAVv1pP1Yr1bH/PWFauYOQJ/ZH3+ABcQle0=</latexit> であり と置くと 1. ある定数 に対して ならば, である. ✏ > 0 <latexit sha1_base64="iQxt5eBAWyi4CnJXD1FTsFM/HhA=">AAAB83icbVBNSwMxEJ2tX7V+VT16CRbBU9mtgp6k6MVjBfsB3aVk02wbmk1CkhVK6d/w4kERr/4Zb/4b03YP2vpg4PHeDDPzYsWZsb7/7RXW1jc2t4rbpZ3dvf2D8uFRy8hME9okkkvdibGhnAnatMxy2lGa4jTmtB2P7mZ++4lqw6R4tGNFoxQPBEsYwdZJYUiVYVwKdIP8XrniV/050CoJclKBHI1e+SvsS5KlVFjCsTHdwFc2mmBtGeF0WgozQxUmIzygXUcFTqmJJvObp+jMKX2USO1KWDRXf09McGrMOI1dZ4rt0Cx7M/E/r5vZ5DqaMKEySwVZLEoyjqxEswBQn2lKLB87golm7lZEhlhjYl1MJRdCsPzyKmnVqsFFtfZwWanf5nEU4QRO4RwCuII63EMDmkBAwTO8wpuXeS/eu/exaC14+cwx/IH3+QP4DpD6</latexit> f(n) = O(n logb a ✏) <latexit sha1_base64="/Co+zHNANF4vde42xchIx70RmKc=">AAACCXicbVC7SgNBFJ2NrxhfUUubwSAkhWE3CtoIQRs7I5gHZNdldjKbDJmdWWZmhbBsa+Ov2FgoYusf2Pk3Th6FJh64cDjnXu69J4gZVdq2v63c0vLK6lp+vbCxubW9U9zdaymRSEyaWDAhOwFShFFOmppqRjqxJCgKGGkHw6ux334gUlHB7/QoJl6E+pyGFCNtJL8IwzKvwAt4U+b3qctE3w9SlB27JFaUCZ5V/GLJrtoTwEXizEgJzNDwi19uT+AkIlxjhpTqOnasvRRJTTEjWcFNFIkRHqI+6RrKUUSUl04+yeCRUXowFNIU13Ci/p5IUaTUKApMZ4T0QM17Y/E/r5vo8NxLKY8TTTieLgoTBrWA41hgj0qCNRsZgrCk5laIB0girE14BROCM//yImnVqs5JtXZ7WqpfzuLIgwNwCMrAAWegDq5BAzQBBo/gGbyCN+vJerHerY9pa86azeyDP7A+fwDWiJkw</latexit> T(n) = ⇥(nlogb a) <latexit sha1_base64="NXmsTiZOS/6q9hWSxepYpDy9Ol0=">AAACBXicbVDLSsNAFJ3UV62vqEtdDBah3ZSkCroRim5cVugLmhgm00k7dDIJMxOhhGzc+CtuXCji1n9w5984bbPQ1gMXDufcy733+DGjUlnWt1FYWV1b3yhulra2d3b3zP2DjowSgUkbRywSPR9JwignbUUVI71YEBT6jHT98c3U7z4QIWnEW2oSEzdEQ04DipHSkmcetyq8Cq+g0xoRhSr8PnVYNPT8FGVZ1TPLVs2aAS4TOydlkKPpmV/OIMJJSLjCDEnZt61YuSkSimJGspKTSBIjPEZD0teUo5BIN519kcFTrQxgEAldXMGZ+nsiRaGUk9DXnSFSI7noTcX/vH6igks3pTxOFOF4vihIGFQRnEYCB1QQrNhEE4QF1bdCPEICYaWDK+kQ7MWXl0mnXrPPavW783LjOo+jCI7ACagAG1yABrgFTdAGGDyCZ/AK3own48V4Nz7mrQUjnzkEf2B8/gC2gJd2</latexit> マスター定理 f(n) <latexit sha1_base64="pG5iFsNQm8e14VQk0dg/VxXte+4=">AAAB63icbVBNSwMxEJ2tX7V+VT16CRahXspuK+ix6MVjBfsB7VKyabYNTbJLkhXK0r/gxYMiXv1D3vw3Zts9aOuDgcd7M8zMC2LOtHHdb6ewsbm1vVPcLe3tHxwelY9POjpKFKFtEvFI9QKsKWeStg0znPZiRbEIOO0G07vM7z5RpVkkH80spr7AY8lCRrDJpLAqL4fliltzF0DrxMtJBXK0huWvwSgiiaDSEI617ntubPwUK8MIp/PSINE0xmSKx7RvqcSCaj9d3DpHF1YZoTBStqRBC/X3RIqF1jMR2E6BzUSvepn4n9dPTHjjp0zGiaGSLBeFCUcmQtnjaMQUJYbPLMFEMXsrIhOsMDE2npINwVt9eZ106jWvUas/XFWat3kcRTiDc6iCB9fQhHtoQRsITOAZXuHNEc6L8+58LFsLTj5zCn/gfP4AXnuNyw==</latexit> 単純な分割法を用いた行列積の求解の場合, 計算量は ⇥(n3) <latexit sha1_base64="G8WRpL5exN/uz8BAetuwevHrbn8=">AAAB8nicbVBNSwMxEM3Wr1q/qh69BItQL2W3FfRY9OKxQr+gXUs2zbah2WRJZoWy9Gd48aCIV3+NN/+NabsHbX0w8Hhvhpl5QSy4Adf9dnIbm1vbO/ndwt7+weFR8fikbVSiKWtRJZTuBsQwwSVrAQfBurFmJAoE6wSTu7nfeWLacCWbMI2ZH5GR5CGnBKzU6zfHDEhZPtYuB8WSW3EXwOvEy0gJZWgMil/9oaJJxCRQQYzpeW4Mfko0cCrYrNBPDIsJnZAR61kqScSMny5OnuELqwxxqLQtCXih/p5ISWTMNApsZ0RgbFa9ufif10sgvPFTLuMEmKTLRWEiMCg8/x8PuWYUxNQSQjW3t2I6JppQsCkVbAje6svrpF2teLVK9eGqVL/N4sijM3SOyshD16iO7lEDtRBFCj2jV/TmgPPivDsfy9ack82coj9wPn8AON+Qjg==</latexit> T(n) = 7T(n/2) + ⇥ n2 <latexit sha1_base64="/rB1TSg8k2JoMxnPx1rU1KIHEGU=">AAACEnicbVA9SwNBEN2L3/ErammzGIQEId5FQRtBtLFUyIeQi2FvM5cs7u0du3NCOPIbbPwrNhaK2FrZ+W/cxBRqfDDM470ZducFiRQGXffTyc3Mzs0vLC7ll1dW19YLG5sNE6eaQ53HMtbXATMghYI6CpRwnWhgUSChGdyej/zmHWgjYlXDQQLtiPWUCAVnaKVOoVwrqTI9oUe2031aLe/5tT4g8yWEWFI3WXXoa9HrY7lTKLoVdww6TbwJKZIJLjuFD78b8zQChVwyY1qem2A7YxoFlzDM+6mBhPFb1oOWpYpFYNrZ+KQh3bVKl4axtqWQjtWfGxmLjBlEgZ2MGPbNX28k/ue1UgyP25lQSYqg+PdDYSopxnSUD+0KDRzlwBLGtbB/pbzPNONoU8zbELy/J0+TRrXiHVSqV4fF07NJHItkm+yQEvHIETklF+SS1Akn9+SRPJMX58F5cl6dt+/RnDPZ2SK/4Lx/AauMmwk=</latexit> 練習. 紙と鉛筆を用意してStrassenの方法 の場合を 確かめてください. ケース 1-3 のどれに該当する? 27
  28. /35 練習問題(CLRS本文より) T(n) = T(2n/3) + 1 <latexit sha1_base64="uOzRyMKvp9IwOqAGJVoNx/hrpss=">AAAB+3icbVBNS8NAEJ3Ur1q/Yj16WSxCi1CTVtCLUPTisUK/oA1ls920SzebsLsRS+lf8eJBEa/+EW/+G7dtDlp9MPB4b4aZeX7MmdKO82Vl1tY3Nrey27md3b39A/sw31JRIgltkohHsuNjRTkTtKmZ5rQTS4pDn9O2P76d++0HKhWLRENPYuqFeChYwAjWRurb+UZRlNA1ahQr4rxaQmfI7dsFp+wsgP4SNyUFSFHv25+9QUSSkApNOFaq6zqx9qZYakY4neV6iaIxJmM8pF1DBQ6p8qaL22fo1CgDFETSlNBoof6cmOJQqUnom84Q65Fa9ebif1430cGVN2UiTjQVZLkoSDjSEZoHgQZMUqL5xBBMJDO3IjLCEhNt4sqZENzVl/+SVqXsVsuV+4tC7SaNIwvHcAJFcOESanAHdWgCgUd4ghd4tWbWs/VmvS9bM1Y6cwS/YH18Aw21kTY=</latexit> T(n)

    = ⇥(lg n) <latexit sha1_base64="VfRrw4tcbJM2X3Ibz+W2HhZOvEg=">AAAB/nicbVBNS8NAEJ3Ur1q/ouLJy2IR2ktJqqAXoejFY4V+QRPKZrttl242YXcjlFDwr3jxoIhXf4c3/43bNgdtfTDweG+GmXlBzJnSjvNt5dbWNza38tuFnd29/QP78KilokQS2iQRj2QnwIpyJmhTM81pJ5YUhwGn7WB8N/Pbj1QqFomGnsTUD/FQsAEjWBupZ580SqKMbpDXGFGNSx4fpmJa7tlFp+LMgVaJm5EiZKj37C+vH5EkpEITjpXquk6s/RRLzQin04KXKBpjMsZD2jVU4JAqP52fP0XnRumjQSRNCY3m6u+JFIdKTcLAdIZYj9SyNxP/87qJHlz7KRNxoqkgi0WDhCMdoVkWqM8kJZpPDMFEMnMrIiMsMdEmsYIJwV1+eZW0qhX3olJ9uCzWbrM48nAKZ1ACF66gBvdQhyYQSOEZXuHNerJerHfrY9Gas7KZY/gD6/MHEoiUSQ==</latexit> T(n) = 3T(n/4) + n lg n <latexit sha1_base64="u5ytlS0HEPcL9U6CP7zItrP24Fg=">AAACAXicbVDLSgMxFL1TX7W+Rt0IboJFaBHqTFvQjVB047JCX9AOJZOmbWgmMyQZoQx146+4caGIW//CnX9j+lho9cDlHs65l+QeP+JMacf5slIrq2vrG+nNzNb2zu6evX/QUGEsCa2TkIey5WNFORO0rpnmtBVJigOf06Y/upn6zXsqFQtFTY8j6gV4IFifEayN1LWPajmRR1eoZPp5OY/OkOjwQSImXTvrFJwZ0F/iLkgWFqh27c9OLyRxQIUmHCvVdp1IewmWmhFOJ5lOrGiEyQgPaNtQgQOqvGR2wQSdGqWH+qE0JTSaqT83EhwoNQ58MxlgPVTL3lT8z2vHun/pJUxEsaaCzB/qxxzpEE3jQD0mKdF8bAgmkpm/IjLEEhNtQsuYENzlk/+SRrHglgrFu3K2cr2IIw3HcAI5cOECKnALVagDgQd4ghd4tR6tZ+vNep+PpqzFziH8gvXxDVTElEY=</latexit> ケース2 ケース3 時間あったら確かめてみて. T(n) = ⇥(n lg n) <latexit sha1_base64="n/meLb/IGhtQh9iLQ2nVZrlTM4w=">AAAB/3icbVBNS8NAEN3Ur1q/ooIXL4tFaC8lqYJehKIXjxX6ITShbLbbdulmE3YnQok9+Fe8eFDEq3/Dm//GbZuDtj4YeLw3w8y8IBZcg+N8W7mV1bX1jfxmYWt7Z3fP3j9o6ShRlDVpJCJ1HxDNBJesCRwEu48VI2EgWDsY3Uz99gNTmkeyAeOY+SEZSN7nlICRuvZRoyTL+Ap7jSEDUpKeGKRyUu7aRafizICXiZuRIspQ79pfXi+iScgkUEG07rhODH5KFHAq2KTgJZrFhI7IgHUMlSRk2k9n90/wqVF6uB8pUxLwTP09kZJQ63EYmM6QwFAvelPxP6+TQP/ST7mME2CSzhf1E4EhwtMwcI8rRkGMDSFUcXMrpkOiCAUTWcGE4C6+vExa1Yp7VqnenRdr11kceXSMTlAJuegC1dAtqqMmougRPaNX9GY9WS/Wu/Uxb81Z2cwh+gPr8wfld5TB</latexit> 28
  29. /35 マスター定理の証明の概要 ※簡易化のため は のべき乗と仮定. 正確な議論は4.6.2を参照(難しくはない) n <latexit sha1_base64="2QmInd+nHO60uhS7AYCvgpiU4a8=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlpuyXK27VnYOsEi8nFcjR6Je/eoOYpRFKwwTVuuu5ifEzqgxnAqelXqoxoWxMh9i1VNIItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2ooszYbEo2BG/55VXSrlW9i2qteVmp3+RxFOEETuEcPLiCOtxBA1rAAOEZXuHNeXRenHfnY9FacPKZY/gD5/MH2F+M9g==</latexit> b

    <latexit sha1_base64="YJjhR7RY5hyNtVLBH/MerrmOQ7I=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlZtAvV9yqOwdZJV5OKpCj0S9/9QYxSyOUhgmqdddzE+NnVBnOBE5LvVRjQtmYDrFrqaQRaj+bHzolZ1YZkDBWtqQhc/X3REYjrSdRYDsjakZ62ZuJ/3nd1ITXfsZlkhqUbLEoTAUxMZl9TQZcITNiYgllittbCRtRRZmx2ZRsCN7yy6ukXat6F9Va87JSv8njKMIJnMI5eHAFdbiDBrSAAcIzvMKb8+i8OO/Ox6K14OQzx/AHzucPxi+M6g==</latexit> 補題4.2 T(n) = ( ⇥(1) n = 1 aT(n/b) + f(n) n = bi <latexit sha1_base64="0zNGyDJ/Y9Gc7UcOWqfemC5un8A=">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</latexit> T(n) = ⇥(nlogb a) + logb n 1 X j=0 ajf(n/bj) <latexit sha1_base64="fiHrfLLobOXSGIgTc0/a3LzWxyc=">AAACNXicbZBNSyNBEIZ7/NjVqLvRPXopDELCsnEmu6AXQdyLBw8KSRQycajp9CSd9PQM3T1CGOZPefF/eNo97GFFvPoX7HwIrm5Bw8P7VlFdb5gKro3r/nYWFpeWP3xcWS2trW98+lze3GrrJFOUtWgiEnUZomaCS9Yy3Ah2mSqGcSjYRTj6OfEvrpnSPJFNM05ZN8a+5BGnaKwUlE+bVVmDQ/CbA2YQqvIq90XSD0LAogZfwddZHOTDQ7d4MXJZwDfwCsCrfFhAVJWwB+GEa0G54tbdacF78OZQIfM6C8p3fi+hWcykoQK17nhuaro5KsOpYEXJzzRLkY6wzzoWJcZMd/Pp1QXsWqUHUaLskwam6uuJHGOtx3FoO2M0A/3Wm4j/8zqZiQ66OZdpZpiks0VRJsAkMIkQelwxasTYAlLF7V+BDlAhNTbokg3Be3vye2g36t73euP8R+XoeB7HCtkmO6RKPLJPjsgJOSMtQskN+UX+knvn1vnjPDiPs9YFZz7zhfxTztMzNQOouA==</latexit> であるとき 補題4.3 g(n) = logb n 1 X j=0 ajf(n/bj) <latexit sha1_base64="qL53cSu9dmByO8Al1Gk78N909xo=">AAACF3icbVDLSsNAFJ3UV62vqEs3g0VoF9akCropFN24rGAf0LRhMp20004mYWYilJC/cOOvuHGhiFvd+TdOHwttPXDhcM693HuPFzEqlWV9G5mV1bX1jexmbmt7Z3fP3D9oyDAWmNRxyELR8pAkjHJSV1Qx0ooEQYHHSNMb3Uz85gMRkob8Xo0j0glQn1OfYqS05JqlfoEXYQU6Mg7cZFix0m7isLDveglP4Sm0U4i6Q+gX+JnXHRZdM2+VrCngMrHnJA/mqLnml9MLcRwQrjBDUrZtK1KdBAlFMSNpzokliRAeoT5pa8pRQGQnmf6VwhOt9KAfCl1cwan6eyJBgZTjwNOdAVIDuehNxP+8dqz8q05CeRQrwvFskR8zqEI4CQn2qCBYsbEmCAuqb4V4gATCSkeZ0yHYiy8vk0a5ZJ+XyncX+er1PI4sOALHoABscAmq4BbUQB1g8AiewSt4M56MF+Pd+Ji1Zoz5zCH4A+PzB3rmnZ0=</latexit> の漸近的な限界は以下. f(n) = O(n logb a ✏) <latexit sha1_base64="/Co+zHNANF4vde42xchIx70RmKc=">AAACCXicbVC7SgNBFJ2NrxhfUUubwSAkhWE3CtoIQRs7I5gHZNdldjKbDJmdWWZmhbBsa+Ov2FgoYusf2Pk3Th6FJh64cDjnXu69J4gZVdq2v63c0vLK6lp+vbCxubW9U9zdaymRSEyaWDAhOwFShFFOmppqRjqxJCgKGGkHw6ux334gUlHB7/QoJl6E+pyGFCNtJL8IwzKvwAt4U+b3qctE3w9SlB27JFaUCZ5V/GLJrtoTwEXizEgJzNDwi19uT+AkIlxjhpTqOnasvRRJTTEjWcFNFIkRHqI+6RrKUUSUl04+yeCRUXowFNIU13Ci/p5IUaTUKApMZ4T0QM17Y/E/r5vo8NxLKY8TTTieLgoTBrWA41hgj0qCNRsZgrCk5laIB0girE14BROCM//yImnVqs5JtXZ7WqpfzuLIgwNwCMrAAWegDq5BAzQBBo/gGbyCN+vJerHerY9pa86azeyDP7A+fwDWiJkw</latexit> 1. ならば g(n) = O(n logb a) <latexit sha1_base64="zToNuAikPGXJ+PDkWb+jZaUn3fs=">AAACAHicbVBNS8NAEN3Ur1q/oh48eFksQnspSRX0IhS9eLOCbYU2hs120y7d7IbdjVBCLv4VLx4U8erP8Oa/cdvmoK0PBh7vzTAzL4gZVdpxvq3C0vLK6lpxvbSxubW9Y+/utZVIJCYtLJiQ9wFShFFOWppqRu5jSVAUMNIJRlcTv/NIpKKC3+lxTLwIDTgNKUbaSL59MKjwKryANxX+kPaYGPhBirKs6ttlp+ZMAReJm5MyyNH07a9eX+AkIlxjhpTquk6svRRJTTEjWamXKBIjPEID0jWUo4goL50+kMFjo/RhKKQpruFU/T2RokipcRSYzgjpoZr3JuJ/XjfR4bmXUh4nmnA8WxQmDGoBJ2nAPpUEazY2BGFJza0QD5FEWJvMSiYEd/7lRdKu19yTWv32tNy4zOMogkNwBCrABWegAa5BE7QABhl4Bq/gzXqyXqx362PWWrDymX3wB9bnD+KYlVQ=</latexit> 2. ならば f(n) = ⇥ nlogb a <latexit sha1_base64="Ks1AlFFCTG7WiuRanz+qDw6sENc=">AAACEHicbVA9SwNBEN3zM8avU0ubxSDGJtxFQRshaGMZIVEhF8PeZi5Z3Ns7dueEcOQn2PhXbCwUsbW089+4+Sj8ejDweG+GmXlhKoVBz/t0Zmbn5hcWC0vF5ZXVtXV3Y/PSJJnm0OSJTPR1yAxIoaCJAiVcpxpYHEq4Cm/PRv7VHWgjEtXAQQrtmPWUiARnaKWOuxeV1f5J0OgDskBChGV1kwcy6dFOHg4pGwZa9Pq433FLXsUbg/4l/pSUyBT1jvsRdBOexaCQS2ZMy/dSbOdMo+AShsUgM5Ayfst60LJUsRhMOx8/NKS7VunSKNG2FNKx+n0iZ7Exg9geuBsz7Jvf3kj8z2tlGB23c6HSDEHxyaIokxQTOkqHdoUGjnJgCeNa2Fsp7zPNONoMizYE//fLf8llteIfVKoXh6Xa6TSOAtkmO6RMfHJEauSc1EmTcHJPHskzeXEenCfn1XmbtM4405kt8gPO+xfRe5x3</latexit> g(n) = ⇥(nlogb a lg n) <latexit sha1_base64="BLkJ0TYEYNeTtHA0a2yDIQ46FUY=">AAACDHicbVDLSgMxFM34rPVVdekmWIR2U2aqoBuh6MZlhb6gM5ZMmpmGZpIhyQhlmA9w46+4caGIWz/AnX9j2s5CWw8EDuecy809fsyo0rb9ba2srq1vbBa2its7u3v7pYPDjhKJxKSNBROy5yNFGOWkralmpBdLgiKfka4/vpn63QciFRW8pScx8SIUchpQjLSRBqVyWOFVeAXd1ohoVOH3qctEOPBTlGXQZWHKs6pJ2TV7BrhMnJyUQY7moPTlDgVOIsI1ZkipvmPH2kuR1BQzkhXdRJEY4TEKSd9QjiKivHR2TAZPjTKEgZDmcQ1n6u+JFEVKTSLfJCOkR2rRm4r/ef1EB5deSnmcaMLxfFGQMKgFnDYDh1QSrNnEEIQlNX+FeIQkwtr0VzQlOIsnL5NOveac1ep35+XGdV5HARyDE1ABDrgADXALmqANMHgEz+AVvFlP1ov1bn3MoytWPnME/sD6/AFBbZqE</latexit> 3. ある定数 について ならば af(n/b)  cf(n) <latexit sha1_base64="89Dtktvg5iBhTl/9HDfEOUNoOOM=">AAAB/3icbVDLSgMxFL1TX7W+RgU3boJFaDd1pgq6LLpxWcE+oB1KJs20wUxmTDJCqV34K25cKOLW33Dn35hpZ6GtBy6cnHMvuff4MWdKO863lVtaXlldy68XNja3tnfs3b2mihJJaINEPJJtHyvKmaANzTSn7VhSHPqctvy7q9RvPVCpWCRu9SimXogHggWMYG2knn2AUVAS6AT5ZdTl9B6R9F3u2UWn4kyBFombkSJkqPfsr24/IklIhSYcK9VxnVh7Yyw1I5xOCt1E0RiTOzygHUMFDqnyxtP9J+jYKH0URNKU0Giq/p4Y41CpUeibzhDroZr3UvE/r5Po4MIbMxEnmgoy+yhIONIRSsNAfSYp0XxkCCaSmV0RGWKJiTaRFUwI7vzJi6RZrbinlerNWbF2mcWRh0M4ghK4cA41uIY6NIDAIzzDK7xZT9aL9W59zFpzVjazD39gff4ABHKTiA==</latexit> c < 1 <latexit sha1_base64="WfaFCDew75a8b7g+U5VJ8Sh5PJI=">AAAB7HicbVA9SwNBEJ2LXzF+RS1tFoNgFe6ioIVF0MYygpcEkiPsbeaSJXt7x+6eEEJ+g42FIrb+IDv/jZvkCk18MPB4b4aZeWEquDau++0U1tY3NreK26Wd3b39g/LhUVMnmWLos0Qkqh1SjYJL9A03AtupQhqHAlvh6G7mt55QaZ7IRzNOMYjpQPKIM2qs5DNyQ7xeueJW3TnIKvFyUoEcjV75q9tPWBajNExQrTuem5pgQpXhTOC01M00ppSN6AA7lkoaow4m82On5MwqfRIlypY0ZK7+npjQWOtxHNrOmJqhXvZm4n9eJzPRdTDhMs0MSrZYFGWCmITMPid9rpAZMbaEMsXtrYQNqaLM2HxKNgRv+eVV0qxVvYtq7eGyUr/N4yjCCZzCOXhwBXW4hwb4wIDDM7zCmyOdF+fd+Vi0Fpx85hj+wPn8AWQljcA=</latexit> g(n) = ⇥(f(n)) <latexit sha1_base64="wh23oR1Z63dPu8Io4X6DagjulUE=">AAAB/HicbVDLSgMxFL1TX7W+Rrt0EyxCuykzVdCNUHTjskJf0JaSSTNtaCYzJBlhGOqvuHGhiFs/xJ1/Y9rOQlsPXDg5515y7/EizpR2nG8rt7G5tb2T3y3s7R8cHtnHJ20VxpLQFgl5KLseVpQzQVuaaU67kaQ48DjteNO7ud95pFKxUDR1EtFBgMeC+YxgbaShXRyXRQXdoH5zQjUu++ZVGdolp+osgNaJm5ESZGgM7a/+KCRxQIUmHCvVc51ID1IsNSOczgr9WNEIkyke056hAgdUDdLF8jN0bpQR8kNpSmi0UH9PpDhQKgk80xlgPVGr3lz8z+vF2r8epExEsaaCLD/yY450iOZJoBGTlGieGIKJZGZXRCZYYqJNXgUTgrt68jpp16ruRbX2cFmq32Zx5OEUzqAMLlxBHe6hAS0gkMAzvMKb9WS9WO/Wx7I1Z2UzRfgD6/MHqsSS2A==</latexit> 細かい条件は マスター定理のとこ見て. 補題4.3を補題4.2に代入してマスター定理を得る. 29
  30. /35 補題4.2の証明 T(1) <latexit sha1_base64="buKEJJi1AOp0SlRUCZR2gDaGe1Y=">AAAB63icbVBNSwMxEJ2tX7V+VT16CRahXspuFfRY9OKxQr+gXUo2zbahSXZJskJZ+he8eFDEq3/Im//GbLsHbX0w8Hhvhpl5QcyZNq777RQ2Nre2d4q7pb39g8Oj8vFJR0eJIrRNIh6pXoA15UzStmGG016sKBYBp91gep/53SeqNItky8xi6gs8lixkBJtMalW9y2G54tbcBdA68XJSgRzNYflrMIpIIqg0hGOt+54bGz/FyjDC6bw0SDSNMZniMe1bKrGg2k8Xt87RhVVGKIyULWnQQv09kWKh9UwEtlNgM9GrXib+5/UTE976KZNxYqgky0VhwpGJUPY4GjFFieEzSzBRzN6KyAQrTIyNp2RD8FZfXiedes27qtUfryuNuzyOIpzBOVTBgxtowAM0oQ0EJvAMr/DmCOfFeXc+lq0FJ585hT9wPn8A5j2NfA==</latexit> T(1) <latexit sha1_base64="buKEJJi1AOp0SlRUCZR2gDaGe1Y=">AAAB63icbVBNSwMxEJ2tX7V+VT16CRahXspuFfRY9OKxQr+gXUo2zbahSXZJskJZ+he8eFDEq3/Im//GbLsHbX0w8Hhvhpl5QcyZNq777RQ2Nre2d4q7pb39g8Oj8vFJR0eJIrRNIh6pXoA15UzStmGG016sKBYBp91gep/53SeqNItky8xi6gs8lixkBJtMalW9y2G54tbcBdA68XJSgRzNYflrMIpIIqg0hGOt+54bGz/FyjDC6bw0SDSNMZniMe1bKrGg2k8Xt87RhVVGKIyULWnQQv09kWKh9UwEtlNgM9GrXib+5/UTE976KZNxYqgky0VhwpGJUPY4GjFFieEzSzBRzN6KyAQrTIyNp2RD8FZfXiedes27qtUfryuNuzyOIpzBOVTBgxtowAM0oQ0EJvAMr/DmCOfFeXc+lq0FJ585hT9wPn8A5j2NfA==</latexit> T(1) <latexit

    sha1_base64="buKEJJi1AOp0SlRUCZR2gDaGe1Y=">AAAB63icbVBNSwMxEJ2tX7V+VT16CRahXspuFfRY9OKxQr+gXUo2zbahSXZJskJZ+he8eFDEq3/Im//GbLsHbX0w8Hhvhpl5QcyZNq777RQ2Nre2d4q7pb39g8Oj8vFJR0eJIrRNIh6pXoA15UzStmGG016sKBYBp91gep/53SeqNItky8xi6gs8lixkBJtMalW9y2G54tbcBdA68XJSgRzNYflrMIpIIqg0hGOt+54bGz/FyjDC6bw0SDSNMZniMe1bKrGg2k8Xt87RhVVGKIyULWnQQv09kWKh9UwEtlNgM9GrXib+5/UTE976KZNxYqgky0VhwpGJUPY4GjFFieEzSzBRzN6KyAQrTIyNp2RD8FZfXiedes27qtUfryuNuzyOIpzBOVTBgxtowAM0oQ0EJvAMr/DmCOfFeXc+lq0FJ585hT9wPn8A5j2NfA==</latexit> コスト T(n) = ( ⇥(1) n = 1 aT(n/b) + f(n) n = bi <latexit sha1_base64="0zNGyDJ/Y9Gc7UcOWqfemC5un8A=">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</latexit> 1個 … f(n/b) <latexit sha1_base64="bbxsYw45pXiImHHRW9xdIiRyfAc=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRahXupuK+ix6MVjBfsB7VKyabaNzSZLkhXK0v/gxYMiXv0/3vw3pu0etPXBwOO9GWbmBTFn2rjut5NbW9/Y3MpvF3Z29/YPiodHLS0TRWiTSC5VJ8CaciZo0zDDaSdWFEcBp+1gfDvz209UaSbFg5nE1I/wULCQEWys1ArL4iI47xdLbsWdA60SLyMlyNDoF796A0mSiApDONa667mx8VOsDCOcTgu9RNMYkzEe0q6lAkdU++n82ik6s8oAhVLZEgbN1d8TKY60nkSB7YywGellbyb+53UTE177KRNxYqggi0VhwpGRaPY6GjBFieETSzBRzN6KyAgrTIwNqGBD8JZfXiWtasWrVar3l6X6TRZHHk7gFMrgwRXU4Q4a0AQCj/AMr/DmSOfFeXc+Fq05J5s5hj9wPn8AhtyOcA==</latexit> f(n/b) <latexit sha1_base64="bbxsYw45pXiImHHRW9xdIiRyfAc=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRahXupuK+ix6MVjBfsB7VKyabaNzSZLkhXK0v/gxYMiXv0/3vw3pu0etPXBwOO9GWbmBTFn2rjut5NbW9/Y3MpvF3Z29/YPiodHLS0TRWiTSC5VJ8CaciZo0zDDaSdWFEcBp+1gfDvz209UaSbFg5nE1I/wULCQEWys1ArL4iI47xdLbsWdA60SLyMlyNDoF796A0mSiApDONa667mx8VOsDCOcTgu9RNMYkzEe0q6lAkdU++n82ik6s8oAhVLZEgbN1d8TKY60nkSB7YywGellbyb+53UTE177KRNxYqggi0VhwpGRaPY6GjBFieETSzBRzN6KyAgrTIwNqGBD8JZfXiWtasWrVar3l6X6TRZHHk7gFMrgwRXU4Q4a0AQCj/AMr/DmSOfFeXc+Fq05J5s5hj9wPn8AhtyOcA==</latexit> f(n/b) <latexit sha1_base64="bbxsYw45pXiImHHRW9xdIiRyfAc=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRahXupuK+ix6MVjBfsB7VKyabaNzSZLkhXK0v/gxYMiXv0/3vw3pu0etPXBwOO9GWbmBTFn2rjut5NbW9/Y3MpvF3Z29/YPiodHLS0TRWiTSC5VJ8CaciZo0zDDaSdWFEcBp+1gfDvz209UaSbFg5nE1I/wULCQEWys1ArL4iI47xdLbsWdA60SLyMlyNDoF796A0mSiApDONa667mx8VOsDCOcTgu9RNMYkzEe0q6lAkdU++n82ik6s8oAhVLZEgbN1d8TKY60nkSB7YywGellbyb+53UTE177KRNxYqggi0VhwpGRaPY6GjBFieETSzBRzN6KyAgrTIwNqGBD8JZfXiWtasWrVar3l6X6TRZHHk7gFMrgwRXU4Q4a0AQCj/AMr/DmSOfFeXc+Fq05J5s5hj9wPn8AhtyOcA==</latexit> … … … f(n/b2) <latexit sha1_base64="ppBU1ZgGPryECax/amE/X0pnNm8=">AAAB73icbVBNSwMxEJ2tX7V+VT16CRahXupuFfRY9OKxgv2Adi3ZNNuGZpM1yQpl6Z/w4kERr/4db/4b03YP2vpg4PHeDDPzgpgzbVz328mtrK6tb+Q3C1vbO7t7xf2DppaJIrRBJJeqHWBNORO0YZjhtB0riqOA01Ywupn6rSeqNJPi3oxj6kd4IFjICDZWaodlcRY8VE97xZJbcWdAy8TLSAky1HvFr25fkiSiwhCOte54bmz8FCvDCKeTQjfRNMZkhAe0Y6nAEdV+Ort3gk6s0kehVLaEQTP190SKI63HUWA7I2yGetGbiv95ncSEV37KRJwYKsh8UZhwZCSaPo/6TFFi+NgSTBSztyIyxAoTYyMq2BC8xZeXSbNa8c4r1buLUu06iyMPR3AMZfDgEmpwC3VoAAEOz/AKb86j8+K8Ox/z1pyTzRzCHzifP68RjxQ=</latexit> f(n/b2) <latexit sha1_base64="ppBU1ZgGPryECax/amE/X0pnNm8=">AAAB73icbVBNSwMxEJ2tX7V+VT16CRahXupuFfRY9OKxgv2Adi3ZNNuGZpM1yQpl6Z/w4kERr/4db/4b03YP2vpg4PHeDDPzgpgzbVz328mtrK6tb+Q3C1vbO7t7xf2DppaJIrRBJJeqHWBNORO0YZjhtB0riqOA01Ywupn6rSeqNJPi3oxj6kd4IFjICDZWaodlcRY8VE97xZJbcWdAy8TLSAky1HvFr25fkiSiwhCOte54bmz8FCvDCKeTQjfRNMZkhAe0Y6nAEdV+Ort3gk6s0kehVLaEQTP190SKI63HUWA7I2yGetGbiv95ncSEV37KRJwYKsh8UZhwZCSaPo/6TFFi+NgSTBSztyIyxAoTYyMq2BC8xZeXSbNa8c4r1buLUu06iyMPR3AMZfDgEmpwC3VoAAEOz/AKb86j8+K8Ox/z1pyTzRzCHzifP68RjxQ=</latexit> 木の高さ h = logb n <latexit sha1_base64="ciU+qBghhDXhIk4bcM14FEWbq6Q=">AAAB8nicbVBNS8NAEJ3Ur1q/qh69BIvgqSRV0ItQ9OKxgv2ANJTNdtsu3eyG3YlQQn+GFw+KePXXePPfuG1z0NYHA4/3ZpiZFyWCG/S8b6ewtr6xuVXcLu3s7u0flA+PWkalmrImVULpTkQME1yyJnIUrJNoRuJIsHY0vpv57SemDVfyEScJC2MylHzAKUErBaObrlDDXpTJaa9c8areHO4q8XNSgRyNXvmr21c0jZlEKogxge8lGGZEI6eCTUvd1LCE0DEZssBSSWJmwmx+8tQ9s0rfHShtS6I7V39PZCQ2ZhJHtjMmODLL3kz8zwtSHFyHGZdJikzSxaJBKlxU7ux/t881oygmlhCqub3VpSOiCUWbUsmG4C+/vEpatap/Ua09XFbqt3kcRTiBUzgHH66gDvfQgCZQUPAMr/DmoPPivDsfi9aCk88cwx84nz9otZFW</latexit> ボトムの要素数: ah = alogb n = nlogb a <latexit sha1_base64="bD/rYKrYCbA69BHcIvBtbzaWw0o=">AAACDHicbVDLSgMxFM3UV62vqks3wSK4KjNV0I1QdOOygm2FdlrupJk2NJMMSUYow3yAG3/FjQtF3PoB7vwb0weirQcC555zLzf3BDFn2rjul5NbWl5ZXcuvFzY2t7Z3irt7DS0TRWidSC7VXQCaciZo3TDD6V2sKEQBp81geDX2m/dUaSbFrRnF1I+gL1jICBgrdYsl6AzwBYZO2uay3w1SkWW2Fj81ZJntcsvuBHiReDNSQjPUusXPdk+SJKLCEA5atzw3Nn4KyjDCaVZoJ5rGQIbQpy1LBURU++nkmAwfWaWHQ6nsEwZP1N8TKURaj6LAdkZgBnreG4v/ea3EhOd+ykScGCrIdFGYcGwkHieDe0xRYvjIEiCK2b9iMgAFxNj8CjYEb/7kRdKolL2TcuXmtFS9nMWRRwfoEB0jD52hKrpGNVRHBD2gJ/SCXp1H59l5c96nrTlnNrOP/sD5+AZE2psm</latexit> 部分問題の サイズは1/b <latexit sha1_base64="5g7wtq/dPK+ZVAv1fP0i/m6uf3s=">AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4qkkV9Fj04rGi/YA2lM120y7dbMLuRCihP8GLB0W8+ou8+W/ctjlo64OBx3szzMwLEikMuu63s7K6tr6xWdgqbu/s7u2XDg6bJk414w0Wy1i3A2q4FIo3UKDk7URzGgWSt4LR7dRvPXFtRKwecZxwP6IDJULBKFrpwTsPeqWyW3FnIMvEy0kZctR7pa9uP2ZpxBUySY3peG6CfkY1Cib5pNhNDU8oG9EB71iqaMSNn81OnZBTq/RJGGtbCslM/T2R0ciYcRTYzoji0Cx6U/E/r5NieO1nQiUpcsXmi8JUEozJ9G/SF5ozlGNLKNPC3krYkGrK0KZTtCF4iy8vk2a14l1UqveX5dpNHkcBjuEEzsCDK6jBHdShAQwG8Ayv8OZI58V5dz7mrStOPnMEf+B8/gCiGY1e</latexit> f(n) <latexit sha1_base64="pG5iFsNQm8e14VQk0dg/VxXte+4=">AAAB63icbVBNSwMxEJ2tX7V+VT16CRahXspuK+ix6MVjBfsB7VKyabYNTbJLkhXK0r/gxYMiXv1D3vw3Zts9aOuDgcd7M8zMC2LOtHHdb6ewsbm1vVPcLe3tHxwelY9POjpKFKFtEvFI9QKsKWeStg0znPZiRbEIOO0G07vM7z5RpVkkH80spr7AY8lCRrDJpLAqL4fliltzF0DrxMtJBXK0huWvwSgiiaDSEI617ntubPwUK8MIp/PSINE0xmSKx7RvqcSCaj9d3DpHF1YZoTBStqRBC/X3RIqF1jMR2E6BzUSvepn4n9dPTHjjp0zGiaGSLBeFCUcmQtnjaMQUJYbPLMFEMXsrIhOsMDE2npINwVt9eZ106jWvUas/XFWat3kcRTiDc6iCB9fQhHtoQRsITOAZXuHNEc6L8+58LFsLTj5zCn/gfP4AXnuNyw==</latexit> 個 a <latexit sha1_base64="wqZLPcGml9Og5FDdUdFUNWEF4FE=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlJu2XK27VnYOsEi8nFcjR6Je/eoOYpRFKwwTVuuu5ifEzqgxnAqelXqoxoWxMh9i1VNIItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2ooszYbEo2BG/55VXSrlW9i2qteVmp3+RxFOEETuEcPLiCOtxBA1rAAOEZXuHNeXRenHfnY9FacPKZY/gD5/MHxKuM6Q==</latexit> af(n/b) <latexit sha1_base64="kTATK3zvM93kIbGSwS49oz8ctmk=">AAAB7nicbVDLSgNBEOyNrxhfUY9eBoMQL3E3CnoMevEYwTwgWULvZDYZMju7zMwKIeQjvHhQxKvf482/cZLsQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TRVmDxiJW7QA1E1yyhuFGsHaiGEaBYK1gdDfzW09MaR7LRzNOmB/hQPKQUzRWamFYlhfBea9YcivuHGSVeBkpQYZ6r/jV7cc0jZg0VKDWHc9NjD9BZTgVbFroppolSEc4YB1LJUZM+5P5uVNyZpU+CWNlSxoyV39PTDDSehwFtjNCM9TL3kz8z+ukJrzxJ1wmqWGSLhaFqSAmJrPfSZ8rRo0YW4JUcXsroUNUSI1NqGBD8JZfXiXNasW7rFQfrkq12yyOPJzAKZTBg2uowT3UoQEURvAMr/DmJM6L8+58LFpzTjZzDH/gfP4AQSOO2w==</latexit> 個 a2 <latexit sha1_base64="wBSr/2JlGBldLsQ19aBVcDVq8o4=">AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGi/YA2ls120y7dbMLuRCihP8GLB0W8+ou8+W/ctjlo64OBx3szzMwLEikMuu63s7K6tr6xWdgqbu/s7u2XDg6bJk414w0Wy1i3A2q4FIo3UKDk7URzGgWSt4LRzdRvPXFtRKwecJxwP6IDJULBKFrpnj5We6WyW3FnIMvEy0kZctR7pa9uP2ZpxBUySY3peG6CfkY1Cib5pNhNDU8oG9EB71iqaMSNn81OnZBTq/RJGGtbCslM/T2R0ciYcRTYzoji0Cx6U/E/r5NieOVnQiUpcsXmi8JUEozJ9G/SF5ozlGNLKNPC3krYkGrK0KZTtCF4iy8vk2a14p1XqncX5dp1HkcBjuEEzsCDS6jBLdShAQwG8Ayv8OZI58V5dz7mrStOPnMEf+B8/gDp5I2N</latexit> a2f(n/b2) <latexit sha1_base64="YbTHF0IRTs3WYDtihtzjQCG+CdI=">AAAB9XicbVBNT8JAEJ3iF+IX6tHLRmKCF2yLiR6JXjxiIh8JFLJdtrBhu212txrS8D+8eNAYr/4Xb/4bF+hBwZdM8vLeTGbm+TFnStv2t5VbW9/Y3MpvF3Z29/YPiodHTRUlktAGiXgk2z5WlDNBG5ppTtuxpDj0OW3549uZ33qkUrFIPOhJTL0QDwULGMHaSD3cc1FQFugC+T33vF8s2RV7DrRKnIyUIEO9X/zqDiKShFRowrFSHceOtZdiqRnhdFroJorGmIzxkHYMFTikykvnV0/RmVEGKIikKaHRXP09keJQqUnom84Q65Fa9mbif14n0cG1lzIRJ5oKslgUJBzpCM0iQAMmKdF8YggmkplbERlhiYk2QRVMCM7yy6uk6VacasW9vyzVbrI48nACp1AGB66gBndQhwYQkPAMr/BmPVkv1rv1sWjNWdnMMfyB9fkDm+6QoQ==</latexit> 個 nlogb a <latexit sha1_base64="46nMdh0D1BIunbdV2fMTilJNlLs=">AAAB9HicbVBNSwMxEJ2tX7V+VT16CRbBU9mtgh6LXjxWsB/QriWbZtvQbLIm2UJZ9nd48aCIV3+MN/+NabsHbX0w8Hhvhpl5QcyZNq777RTW1jc2t4rbpZ3dvf2D8uFRS8tEEdokkkvVCbCmnAnaNMxw2okVxVHAaTsY38789oQqzaR4MNOY+hEeChYygo2VfPGY9rgc9oMUZ1m/XHGr7hxolXg5qUCORr/81RtIkkRUGMKx1l3PjY2fYmUY4TQr9RJNY0zGeEi7lgocUe2n86MzdGaVAQqlsiUMmqu/J1IcaT2NAtsZYTPSy95M/M/rJia89lMm4sRQQRaLwoQjI9EsATRgihLDp5Zgopi9FZERVpgYm1PJhuAtv7xKWrWqd1Gt3V9W6jd5HEU4gVM4Bw+uoA530IAmEHiCZ3iFN2fivDjvzseiteDkM8fwB87nD12Dknw=</latexit> ⇥(nlogb a) <latexit sha1_base64="XStyKTxb25kT0njUXYz9UqdDyLg=">AAAB/nicbVDLSsNAFJ34rPUVFVduBotQNyWpgi6LblxW6AuaGCbTSTt0MhNmJkIJAX/FjQtF3Pod7vwbp20W2nrgwuGce7n3njBhVGnH+bZWVtfWNzZLW+Xtnd29ffvgsKNEKjFpY8GE7IVIEUY5aWuqGeklkqA4ZKQbjm+nfveRSEUFb+lJQvwYDTmNKEbaSIF97LVGRKMqf8g8JoZBmKE8Pw/silNzZoDLxC1IBRRoBvaXNxA4jQnXmCGl+q6TaD9DUlPMSF72UkUShMdoSPqGchQT5Wez83N4ZpQBjIQ0xTWcqb8nMhQrNYlD0xkjPVKL3lT8z+unOrr2M8qTVBOO54uilEEt4DQLOKCSYM0mhiAsqbkV4hGSCGuTWNmE4C6+vEw69Zp7UavfX1YaN0UcJXACTkEVuOAKNMAdaII2wCADz+AVvFlP1ov1bn3MW1esYuYI/IH1+QMtfJWg</latexit> T(n) = ⇥ nlogb a + logb n 1 X j=0 ajf n/bj <latexit sha1_base64="ZoQUxMNb6FuoTu2I/Zx/dE6Mpos=">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</latexit> 足合わせて を得る. f(n) <latexit sha1_base64="pG5iFsNQm8e14VQk0dg/VxXte+4=">AAAB63icbVBNSwMxEJ2tX7V+VT16CRahXspuK+ix6MVjBfsB7VKyabYNTbJLkhXK0r/gxYMiXv1D3vw3Zts9aOuDgcd7M8zMC2LOtHHdb6ewsbm1vVPcLe3tHxwelY9POjpKFKFtEvFI9QKsKWeStg0znPZiRbEIOO0G07vM7z5RpVkkH80spr7AY8lCRrDJpLAqL4fliltzF0DrxMtJBXK0huWvwSgiiaDSEI617ntubPwUK8MIp/PSINE0xmSKx7RvqcSCaj9d3DpHF1YZoTBStqRBC/X3RIqF1jMR2E6BzUSvepn4n9dPTHjjp0zGiaGSLBeFCUcmQtnjaMQUJYbPLMFEMXsrIhOsMDE2npINwVt9eZ106jWvUas/XFWat3kcRTiDc6iCB9fQhHtoQRsITOAZXuHNEc6L8+58LFsLTj5zCn/gfP4AXnuNyw==</latexit> オリジナルの呼出: 30
  31. /35 補題4.3の証明1 g(n) = logb n 1 X j=0 ajf(n/bj)

    <latexit sha1_base64="qL53cSu9dmByO8Al1Gk78N909xo=">AAACF3icbVDLSsNAFJ3UV62vqEs3g0VoF9akCropFN24rGAf0LRhMp20004mYWYilJC/cOOvuHGhiFvd+TdOHwttPXDhcM693HuPFzEqlWV9G5mV1bX1jexmbmt7Z3fP3D9oyDAWmNRxyELR8pAkjHJSV1Qx0ooEQYHHSNMb3Uz85gMRkob8Xo0j0glQn1OfYqS05JqlfoEXYQU6Mg7cZFix0m7isLDveglP4Sm0U4i6Q+gX+JnXHRZdM2+VrCngMrHnJA/mqLnml9MLcRwQrjBDUrZtK1KdBAlFMSNpzokliRAeoT5pa8pRQGQnmf6VwhOt9KAfCl1cwan6eyJBgZTjwNOdAVIDuehNxP+8dqz8q05CeRQrwvFskR8zqEI4CQn2qCBYsbEmCAuqb4V4gATCSkeZ0yHYiy8vk0a5ZJ+XyncX+er1PI4sOALHoABscAmq4BbUQB1g8AiewSt4M56MF+Pd+Ji1Zoz5zCH4A+PzB3rmnZ0=</latexit> の漸近的な限界を求めたい. f(n) = O(n logb a ✏) <latexit sha1_base64="/Co+zHNANF4vde42xchIx70RmKc=">AAACCXicbVC7SgNBFJ2NrxhfUUubwSAkhWE3CtoIQRs7I5gHZNdldjKbDJmdWWZmhbBsa+Ov2FgoYusf2Pk3Th6FJh64cDjnXu69J4gZVdq2v63c0vLK6lp+vbCxubW9U9zdaymRSEyaWDAhOwFShFFOmppqRjqxJCgKGGkHw6ux334gUlHB7/QoJl6E+pyGFCNtJL8IwzKvwAt4U+b3qctE3w9SlB27JFaUCZ5V/GLJrtoTwEXizEgJzNDwi19uT+AkIlxjhpTqOnasvRRJTTEjWcFNFIkRHqI+6RrKUUSUl04+yeCRUXowFNIU13Ci/p5IUaTUKApMZ4T0QM17Y/E/r5vo8NxLKY8TTTieLgoTBrWA41hgj0qCNRsZgrCk5laIB0girE14BROCM//yImnVqs5JtXZ7WqpfzuLIgwNwCMrAAWegDq5BAzQBBo/gGbyCN+vJerHerY9pa86azeyDP7A+fwDWiJkw</latexit> 1. ならば g(n) = O(n logb a) <latexit sha1_base64="zToNuAikPGXJ+PDkWb+jZaUn3fs=">AAACAHicbVBNS8NAEN3Ur1q/oh48eFksQnspSRX0IhS9eLOCbYU2hs120y7d7IbdjVBCLv4VLx4U8erP8Oa/cdvmoK0PBh7vzTAzL4gZVdpxvq3C0vLK6lpxvbSxubW9Y+/utZVIJCYtLJiQ9wFShFFOWppqRu5jSVAUMNIJRlcTv/NIpKKC3+lxTLwIDTgNKUbaSL59MKjwKryANxX+kPaYGPhBirKs6ttlp+ZMAReJm5MyyNH07a9eX+AkIlxjhpTquk6svRRJTTEjWamXKBIjPEID0jWUo4goL50+kMFjo/RhKKQpruFU/T2RokipcRSYzgjpoZr3JuJ/XjfR4bmXUh4nmnA8WxQmDGoBJ2nAPpUEazY2BGFJza0QD5FEWJvMSiYEd/7lRdKu19yTWv32tNy4zOMogkNwBCrABWegAa5BE7QABhl4Bq/gzXqyXqx362PWWrDymX3wB9bnD+KYlVQ=</latexit> g(n) = O 0 @ logb n 1 X j=0 a j ⇣ n bj ⌘logb a ✏ 1 A <latexit sha1_base64="VbqBCmzXHh7gsXwYnF0tTEq4mZY=">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</latexit> f(n) <latexit sha1_base64="pG5iFsNQm8e14VQk0dg/VxXte+4=">AAAB63icbVBNSwMxEJ2tX7V+VT16CRahXspuK+ix6MVjBfsB7VKyabYNTbJLkhXK0r/gxYMiXv1D3vw3Zts9aOuDgcd7M8zMC2LOtHHdb6ewsbm1vVPcLe3tHxwelY9POjpKFKFtEvFI9QKsKWeStg0znPZiRbEIOO0G07vM7z5RpVkkH80spr7AY8lCRrDJpLAqL4fliltzF0DrxMtJBXK0huWvwSgiiaDSEI617ntubPwUK8MIp/PSINE0xmSKx7RvqcSCaj9d3DpHF1YZoTBStqRBC/X3RIqF1jMR2E6BzUSvepn4n9dPTHjjp0zGiaGSLBeFCUcmQtnjaMQUJYbPLMFEMXsrIhOsMDE2npINwVt9eZ106jWvUas/XFWat3kcRTiDc6iCB9fQhHtoQRsITOAZXuHNEc6L8+58LFsLTj5zCn/gfP4AXnuNyw==</latexit> g(n) <latexit sha1_base64="H/IG5bwopmw0jogcQhVX+ha6RXU=">AAAB63icbVBNSwMxEJ2tX7V+VT16CRahXspuK+ix6MVjBfsB7VKyabYNTbJLkhXK0r/gxYMiXv1D3vw3Zts9aOuDgcd7M8zMC2LOtHHdb6ewsbm1vVPcLe3tHxwelY9POjpKFKFtEvFI9QKsKWeStg0znPZiRbEIOO0G07vM7z5RpVkkH80spr7AY8lCRrDJpHFVXg7LFbfmLoDWiZeTCuRoDctfg1FEEkGlIRxr3ffc2PgpVoYRTuelQaJpjMkUj2nfUokF1X66uHWOLqwyQmGkbEmDFurviRQLrWcisJ0Cm4le9TLxP6+fmPDGT5mME0MlWS4KE45MhLLH0YgpSgyfWYKJYvZWRCZYYWJsPCUbgrf68jrp1Gteo1Z/uKo0b/M4inAG51AFD66hCffQgjYQmMAzvMKbI5wX5935WLYWnHzmFP7A+fwBYAKNzA==</latexit> まず を に代入 logb n 1 X j=0 a j ⇣ n bj ⌘logb a ✏ = n logb a ✏ logb n 1 X j=0 ✓ ab✏ blogb a ◆j = n logb a ✏ logb n 1 X j=0 (b ✏)j = n logb a ✏ ✓ b✏ logb n 1 b✏ 1 ◆ = n logb a ✏ ✓ n✏ 1 b✏ 1 ◆ = n logb a ✏ O (n ✏) = O n logb a <latexit sha1_base64="aNxrwGffN9JVggjb12IJGMBFDcg=">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</latexit> 中身を評価 もとに戻してdone 31
  32. /35 補題4.3の証明2 g(n) = logb n 1 X j=0 ajf(n/bj)

    <latexit sha1_base64="qL53cSu9dmByO8Al1Gk78N909xo=">AAACF3icbVDLSsNAFJ3UV62vqEs3g0VoF9akCropFN24rGAf0LRhMp20004mYWYilJC/cOOvuHGhiFvd+TdOHwttPXDhcM693HuPFzEqlWV9G5mV1bX1jexmbmt7Z3fP3D9oyDAWmNRxyELR8pAkjHJSV1Qx0ooEQYHHSNMb3Uz85gMRkob8Xo0j0glQn1OfYqS05JqlfoEXYQU6Mg7cZFix0m7isLDveglP4Sm0U4i6Q+gX+JnXHRZdM2+VrCngMrHnJA/mqLnml9MLcRwQrjBDUrZtK1KdBAlFMSNpzokliRAeoT5pa8pRQGQnmf6VwhOt9KAfCl1cwan6eyJBgZTjwNOdAVIDuehNxP+8dqz8q05CeRQrwvFskR8zqEI4CQn2qCBYsbEmCAuqb4V4gATCSkeZ0yHYiy8vk0a5ZJ+XyncX+er1PI4sOALHoABscAmq4BbUQB1g8AiewSt4M56MF+Pd+Ji1Zoz5zCH4A+PzB3rmnZ0=</latexit> の漸近的な限界を求めたい. 2. ならば f(n) = ⇥ nlogb a <latexit sha1_base64="Ks1AlFFCTG7WiuRanz+qDw6sENc=">AAACEHicbVA9SwNBEN3zM8avU0ubxSDGJtxFQRshaGMZIVEhF8PeZi5Z3Ns7dueEcOQn2PhXbCwUsbW089+4+Sj8ejDweG+GmXlhKoVBz/t0Zmbn5hcWC0vF5ZXVtXV3Y/PSJJnm0OSJTPR1yAxIoaCJAiVcpxpYHEq4Cm/PRv7VHWgjEtXAQQrtmPWUiARnaKWOuxeV1f5J0OgDskBChGV1kwcy6dFOHg4pGwZa9Pq433FLXsUbg/4l/pSUyBT1jvsRdBOexaCQS2ZMy/dSbOdMo+AShsUgM5Ayfst60LJUsRhMOx8/NKS7VunSKNG2FNKx+n0iZ7Exg9geuBsz7Jvf3kj8z2tlGB23c6HSDEHxyaIokxQTOkqHdoUGjnJgCeNa2Fsp7zPNONoMizYE//fLf8llteIfVKoXh6Xa6TSOAtkmO6RMfHJEauSc1EmTcHJPHskzeXEenCfn1XmbtM4405kt8gPO+xfRe5x3</latexit> g(n) = ⇥(nlogb a lg n) <latexit sha1_base64="BLkJ0TYEYNeTtHA0a2yDIQ46FUY=">AAACDHicbVDLSgMxFM34rPVVdekmWIR2U2aqoBuh6MZlhb6gM5ZMmpmGZpIhyQhlmA9w46+4caGIWz/AnX9j2s5CWw8EDuecy809fsyo0rb9ba2srq1vbBa2its7u3v7pYPDjhKJxKSNBROy5yNFGOWkralmpBdLgiKfka4/vpn63QciFRW8pScx8SIUchpQjLSRBqVyWOFVeAXd1ohoVOH3qctEOPBTlGXQZWHKs6pJ2TV7BrhMnJyUQY7moPTlDgVOIsI1ZkipvmPH2kuR1BQzkhXdRJEY4TEKSd9QjiKivHR2TAZPjTKEgZDmcQ1n6u+JFEVKTSLfJCOkR2rRm4r/ef1EB5deSnmcaMLxfFGQMKgFnDYDh1QSrNnEEIQlNX+FeIQkwtr0VzQlOIsnL5NOveac1ep35+XGdV5HARyDE1ABDrgADXALmqANMHgEz+AVvFlP1ov1bn3MoytWPnME/sD6/AFBbZqE</latexit> f(n) <latexit sha1_base64="pG5iFsNQm8e14VQk0dg/VxXte+4=">AAAB63icbVBNSwMxEJ2tX7V+VT16CRahXspuK+ix6MVjBfsB7VKyabYNTbJLkhXK0r/gxYMiXv1D3vw3Zts9aOuDgcd7M8zMC2LOtHHdb6ewsbm1vVPcLe3tHxwelY9POjpKFKFtEvFI9QKsKWeStg0znPZiRbEIOO0G07vM7z5RpVkkH80spr7AY8lCRrDJpLAqL4fliltzF0DrxMtJBXK0huWvwSgiiaDSEI617ntubPwUK8MIp/PSINE0xmSKx7RvqcSCaj9d3DpHF1YZoTBStqRBC/X3RIqF1jMR2E6BzUSvepn4n9dPTHjjp0zGiaGSLBeFCUcmQtnjaMQUJYbPLMFEMXsrIhOsMDE2npINwVt9eZ106jWvUas/XFWat3kcRTiDc6iCB9fQhHtoQRsITOAZXuHNEc6L8+58LFsLTj5zCn/gfP4AXnuNyw==</latexit> g(n) <latexit sha1_base64="H/IG5bwopmw0jogcQhVX+ha6RXU=">AAAB63icbVBNSwMxEJ2tX7V+VT16CRahXspuK+ix6MVjBfsB7VKyabYNTbJLkhXK0r/gxYMiXv1D3vw3Zts9aOuDgcd7M8zMC2LOtHHdb6ewsbm1vVPcLe3tHxwelY9POjpKFKFtEvFI9QKsKWeStg0znPZiRbEIOO0G07vM7z5RpVkkH80spr7AY8lCRrDJpHFVXg7LFbfmLoDWiZeTCuRoDctfg1FEEkGlIRxr3ffc2PgpVoYRTuelQaJpjMkUj2nfUokF1X66uHWOLqwyQmGkbEmDFurviRQLrWcisJ0Cm4le9TLxP6+fmPDGT5mME0MlWS4KE45MhLLH0YgpSgyfWYKJYvZWRCZYYWJsPCUbgrf68jrp1Gteo1Z/uKo0b/M4inAG51AFD66hCffQgjYQmMAzvMKbI5wX5935WLYWnHzmFP7A+fwBYAKNzA==</latexit> まず を に代入 g(n) = ⇥ 0 @ logb n 1 X j=0 aj ⇣ n bj ⌘logb a 1 A <latexit sha1_base64="QBLS8MVoVNa+EdN3xkMW9vtKSSM=">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</latexit> logb n 1 X j=0 aj ⇣ n bj ⌘logb a = nlogb a logb n 1 X j=0 ⇣ a blogb a ⌘j = nlogb a logb n 1 X j=0 1 = nlogb a logb n <latexit sha1_base64="ecHlZmsiitUNdZaqrsWwW5L6WLE=">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</latexit> 中身を評価 もとに戻してdone 32
  33. /35 補題4.3の証明3 g(n) = logb n 1 X j=0 ajf(n/bj)

    <latexit sha1_base64="qL53cSu9dmByO8Al1Gk78N909xo=">AAACF3icbVDLSsNAFJ3UV62vqEs3g0VoF9akCropFN24rGAf0LRhMp20004mYWYilJC/cOOvuHGhiFvd+TdOHwttPXDhcM693HuPFzEqlWV9G5mV1bX1jexmbmt7Z3fP3D9oyDAWmNRxyELR8pAkjHJSV1Qx0ooEQYHHSNMb3Uz85gMRkob8Xo0j0glQn1OfYqS05JqlfoEXYQU6Mg7cZFix0m7isLDveglP4Sm0U4i6Q+gX+JnXHRZdM2+VrCngMrHnJA/mqLnml9MLcRwQrjBDUrZtK1KdBAlFMSNpzokliRAeoT5pa8pRQGQnmf6VwhOt9KAfCl1cwan6eyJBgZTjwNOdAVIDuehNxP+8dqz8q05CeRQrwvFskR8zqEI4CQn2qCBYsbEmCAuqb4V4gATCSkeZ0yHYiy8vk0a5ZJ+XyncX+er1PI4sOALHoABscAmq4BbUQB1g8AiewSt4M56MF+Pd+Ji1Zoz5zCH4A+PzB3rmnZ0=</latexit> の漸近的な限界を求めたい. 3. ある定数 について ならば af(n/b)  cf(n) <latexit sha1_base64="89Dtktvg5iBhTl/9HDfEOUNoOOM=">AAAB/3icbVDLSgMxFL1TX7W+RgU3boJFaDd1pgq6LLpxWcE+oB1KJs20wUxmTDJCqV34K25cKOLW33Dn35hpZ6GtBy6cnHMvuff4MWdKO863lVtaXlldy68XNja3tnfs3b2mihJJaINEPJJtHyvKmaANzTSn7VhSHPqctvy7q9RvPVCpWCRu9SimXogHggWMYG2knn2AUVAS6AT5ZdTl9B6R9F3u2UWn4kyBFombkSJkqPfsr24/IklIhSYcK9VxnVh7Yyw1I5xOCt1E0RiTOzygHUMFDqnyxtP9J+jYKH0URNKU0Giq/p4Y41CpUeibzhDroZr3UvE/r5Po4MIbMxEnmgoy+yhIONIRSsNAfSYp0XxkCCaSmV0RGWKJiTaRFUwI7vzJi6RZrbinlerNWbF2mcWRh0M4ghK4cA41uIY6NIDAIzzDK7xZT9aL9W59zFpzVjazD39gff4ABHKTiA==</latexit> c < 1 <latexit sha1_base64="WfaFCDew75a8b7g+U5VJ8Sh5PJI=">AAAB7HicbVA9SwNBEJ2LXzF+RS1tFoNgFe6ioIVF0MYygpcEkiPsbeaSJXt7x+6eEEJ+g42FIrb+IDv/jZvkCk18MPB4b4aZeWEquDau++0U1tY3NreK26Wd3b39g/LhUVMnmWLos0Qkqh1SjYJL9A03AtupQhqHAlvh6G7mt55QaZ7IRzNOMYjpQPKIM2qs5DNyQ7xeueJW3TnIKvFyUoEcjV75q9tPWBajNExQrTuem5pgQpXhTOC01M00ppSN6AA7lkoaow4m82On5MwqfRIlypY0ZK7+npjQWOtxHNrOmJqhXvZm4n9eJzPRdTDhMs0MSrZYFGWCmITMPid9rpAZMbaEMsXtrYQNqaLM2HxKNgRv+eVV0qxVvYtq7eGyUr/N4yjCCZzCOXhwBXW4hwb4wIDDM7zCmyOdF+fd+Vi0Fpx85hj+wPn8AWQljcA=</latexit> g(n) = ⇥(f(n)) <latexit sha1_base64="wh23oR1Z63dPu8Io4X6DagjulUE=">AAAB/HicbVDLSgMxFL1TX7W+Rrt0EyxCuykzVdCNUHTjskJf0JaSSTNtaCYzJBlhGOqvuHGhiFs/xJ1/Y9rOQlsPXDg5515y7/EizpR2nG8rt7G5tb2T3y3s7R8cHtnHJ20VxpLQFgl5KLseVpQzQVuaaU67kaQ48DjteNO7ud95pFKxUDR1EtFBgMeC+YxgbaShXRyXRQXdoH5zQjUu++ZVGdolp+osgNaJm5ESZGgM7a/+KCRxQIUmHCvVc51ID1IsNSOczgr9WNEIkyke056hAgdUDdLF8jN0bpQR8kNpSmi0UH9PpDhQKgk80xlgPVGr3lz8z+vF2r8epExEsaaCLD/yY450iOZJoBGTlGieGIKJZGZXRCZYYqJNXgUTgrt68jpp16ruRbX2cFmq32Zx5OEUzqAMLlxBHe6hAS0gkMAzvMKb9WS9WO/Wx7I1Z2UzRfgD6/MHqsSS2A==</latexit> f(n) <latexit sha1_base64="pG5iFsNQm8e14VQk0dg/VxXte+4=">AAAB63icbVBNSwMxEJ2tX7V+VT16CRahXspuK+ix6MVjBfsB7VKyabYNTbJLkhXK0r/gxYMiXv1D3vw3Zts9aOuDgcd7M8zMC2LOtHHdb6ewsbm1vVPcLe3tHxwelY9POjpKFKFtEvFI9QKsKWeStg0znPZiRbEIOO0G07vM7z5RpVkkH80spr7AY8lCRrDJpLAqL4fliltzF0DrxMtJBXK0huWvwSgiiaDSEI617ntubPwUK8MIp/PSINE0xmSKx7RvqcSCaj9d3DpHF1YZoTBStqRBC/X3RIqF1jMR2E6BzUSvepn4n9dPTHjjp0zGiaGSLBeFCUcmQtnjaMQUJYbPLMFEMXsrIhOsMDE2npINwVt9eZ106jWvUas/XFWat3kcRTiDc6iCB9fQhHtoQRsITOAZXuHNEc6L8+58LFsLTj5zCn/gfP4AXnuNyw==</latexit> g(n) <latexit sha1_base64="H/IG5bwopmw0jogcQhVX+ha6RXU=">AAAB63icbVBNSwMxEJ2tX7V+VT16CRahXspuK+ix6MVjBfsB7VKyabYNTbJLkhXK0r/gxYMiXv1D3vw3Zts9aOuDgcd7M8zMC2LOtHHdb6ewsbm1vVPcLe3tHxwelY9POjpKFKFtEvFI9QKsKWeStg0znPZiRbEIOO0G07vM7z5RpVkkH80spr7AY8lCRrDJpHFVXg7LFbfmLoDWiZeTCuRoDctfg1FEEkGlIRxr3ffc2PgpVoYRTuelQaJpjMkUj2nfUokF1X66uHWOLqwyQmGkbEmDFurviRQLrWcisJ0Cm4le9TLxP6+fmPDGT5mME0MlWS4KE45MhLLH0YgpSgyfWYKJYvZWRCZYYWJsPCUbgrf68jrp1Gteo1Z/uKo0b/M4inAG51AFD66hCffQgjYQmMAzvMKbI5wX5935WLYWnHzmFP7A+fwBYAKNzA==</latexit> まず の定義に が含まれているので自明に g(n) = ⌦(f(n)) <latexit sha1_base64="dMvcn6XqAASrVRMs4U/RRebc4Cg=">AAAB+nicbZDNSgMxFIUz9a/Wv6ku3QSL0G7KTBV0IxTduLOCrYV2KJn0zjQ0kxmSjFJqH8WNC0Xc+iTufBvTdhbaeiDwce693JvjJ5wp7TjfVm5ldW19I79Z2Nre2d2zi/stFaeSQpPGPJZtnyjgTEBTM82hnUggkc/h3h9eTev3DyAVi8WdHiXgRSQULGCUaGP17GJYFpWL7k0EISkHhis9u+RUnZnwMrgZlFCmRs/+6vZjmkYgNOVEqY7rJNobE6kZ5TApdFMFCaFDEkLHoCARKG88O32Cj43Tx0EszRMaz9zfE2MSKTWKfNMZET1Qi7Wp+V+tk+rg3BszkaQaBJ0vClKOdYynOeA+k0A1HxkgVDJzK6YDIgnVJq2CCcFd/PIytGpV96Rauz0t1S+zOPLoEB2hMnLRGaqja9RATUTRI3pGr+jNerJerHfrY96as7KZA/RH1ucP5ZCSdw==</latexit> g(n) <latexit sha1_base64="H/IG5bwopmw0jogcQhVX+ha6RXU=">AAAB63icbVBNSwMxEJ2tX7V+VT16CRahXspuK+ix6MVjBfsB7VKyabYNTbJLkhXK0r/gxYMiXv1D3vw3Zts9aOuDgcd7M8zMC2LOtHHdb6ewsbm1vVPcLe3tHxwelY9POjpKFKFtEvFI9QKsKWeStg0znPZiRbEIOO0G07vM7z5RpVkkH80spr7AY8lCRrDJpHFVXg7LFbfmLoDWiZeTCuRoDctfg1FEEkGlIRxr3ffc2PgpVoYRTuelQaJpjMkUj2nfUokF1X66uHWOLqwyQmGkbEmDFurviRQLrWcisJ0Cm4le9TLxP6+fmPDGT5mME0MlWS4KE45MhLLH0YgpSgyfWYKJYvZWRCZYYWJsPCUbgrf68jrp1Gteo1Z/uKo0b/M4inAG51AFD66hCffQgjYQmMAzvMKbI5wX5935WLYWnHzmFP7A+fwBYAKNzA==</latexit> 次に を直接評価する. g(n) = logb n 1 X j=0 a j f n/b j  logb n 1 X j=0 c j f(n) + O(1)  f(n) 1 X j=0 c j + O(1) = f(n) ✓ 1 1 c ◆ + O(1) = O(f(n)) <latexit sha1_base64="HalHR2XQAYXSjD0npdueAWgeKwI=">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</latexit> af(n/b)  cf(n) <latexit sha1_base64="89Dtktvg5iBhTl/9HDfEOUNoOOM=">AAAB/3icbVDLSgMxFL1TX7W+RgU3boJFaDd1pgq6LLpxWcE+oB1KJs20wUxmTDJCqV34K25cKOLW33Dn35hpZ6GtBy6cnHMvuff4MWdKO863lVtaXlldy68XNja3tnfs3b2mihJJaINEPJJtHyvKmaANzTSn7VhSHPqctvy7q9RvPVCpWCRu9SimXogHggWMYG2knn2AUVAS6AT5ZdTl9B6R9F3u2UWn4kyBFombkSJkqPfsr24/IklIhSYcK9VxnVh7Yyw1I5xOCt1E0RiTOzygHUMFDqnyxtP9J+jYKH0URNKU0Giq/p4Y41CpUeibzhDroZr3UvE/r5Po4MIbMxEnmgoy+yhIONIRSsNAfSYp0XxkCCaSmV0RGWKJiTaRFUwI7vzJi6RZrbinlerNWbF2mcWRh0M4ghK4cA41uIY6NIDAIzzDK7xZT9aL9W59zFpzVjazD39gff4ABHKTiA==</latexit> より c < 1 <latexit sha1_base64="WfaFCDew75a8b7g+U5VJ8Sh5PJI=">AAAB7HicbVA9SwNBEJ2LXzF+RS1tFoNgFe6ioIVF0MYygpcEkiPsbeaSJXt7x+6eEEJ+g42FIrb+IDv/jZvkCk18MPB4b4aZeWEquDau++0U1tY3NreK26Wd3b39g/LhUVMnmWLos0Qkqh1SjYJL9A03AtupQhqHAlvh6G7mt55QaZ7IRzNOMYjpQPKIM2qs5DNyQ7xeueJW3TnIKvFyUoEcjV75q9tPWBajNExQrTuem5pgQpXhTOC01M00ppSN6AA7lkoaow4m82On5MwqfRIlypY0ZK7+npjQWOtxHNrOmJqhXvZm4n9eJzPRdTDhMs0MSrZYFGWCmITMPid9rpAZMbaEMsXtrYQNqaLM2HxKNgRv+eVV0qxVvYtq7eGyUr/N4yjCCZzCOXhwBXW4hwb4wIDDM7zCmyOdF+fd+Vi0Fpx85hj+wPn8AWQljcA=</latexit> より n <latexit sha1_base64="2QmInd+nHO60uhS7AYCvgpiU4a8=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlpuyXK27VnYOsEi8nFcjR6Je/eoOYpRFKwwTVuuu5ifEzqgxnAqelXqoxoWxMh9i1VNIItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2ooszYbEo2BG/55VXSrlW9i2qteVmp3+RxFOEETuEcPLiCOtxBA1rAAOEZXuHNeXRenHfnY9FacPKZY/gD5/MH2F+M9g==</latexit> 十分大きな の仮定から 漏れた寄与 挟めたのでdone. 33
  34. /35 マスター定理の見方 a 1 <latexit sha1_base64="e23j/eQrjEgtifTnZjpOaYePq68=">AAAB73icbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGC/YA2lM120i7dbNLdjVBC/4QXD4p49e9489+4bXPQ1gcDj/dmmJkXJIJr47rfztr6xubWdmGnuLu3f3BYOjpu6jhVDBssFrFqB1Sj4BIbhhuB7UQhjQKBrWB0N/NbT6g0j+WjmSToR3QgecgZNVZqU9Id4Jh4vVLZrbhzkFXi5aQMOeq90le3H7M0QmmYoFp3PDcxfkaV4UzgtNhNNSaUjegAO5ZKGqH2s/m9U3JulT4JY2VLGjJXf09kNNJ6EgW2M6JmqJe9mfif10lNeONnXCapQckWi8JUEBOT2fOkzxUyIyaWUKa4vZWwIVWUGRtR0YbgLb+8SprVindZqT5clWu3eRwFOIUzuAAPrqEG91CHBjAQ8Ayv8OaMnRfn3flYtK45+cwJ/IHz+QPndY85</latexit> f(n) <latexit sha1_base64="pG5iFsNQm8e14VQk0dg/VxXte+4=">AAAB63icbVBNSwMxEJ2tX7V+VT16CRahXspuK+ix6MVjBfsB7VKyabYNTbJLkhXK0r/gxYMiXv1D3vw3Zts9aOuDgcd7M8zMC2LOtHHdb6ewsbm1vVPcLe3tHxwelY9POjpKFKFtEvFI9QKsKWeStg0znPZiRbEIOO0G07vM7z5RpVkkH80spr7AY8lCRrDJpLAqL4fliltzF0DrxMtJBXK0huWvwSgiiaDSEI617ntubPwUK8MIp/PSINE0xmSKx7RvqcSCaj9d3DpHF1YZoTBStqRBC/X3RIqF1jMR2E6BzUSvepn4n9dPTHjjp0zGiaGSLBeFCUcmQtnjaMQUJYbPLMFEMXsrIhOsMDE2npINwVt9eZ106jWvUas/XFWat3kcRTiDc6iCB9fQhHtoQRsITOAZXuHNEc6L8+58LFsLTj5zCn/gfP4AXnuNyw==</latexit> T(n)

    <latexit sha1_base64="vM8hsiJM1mIid8s1EMCG6OMcapM=">AAAB63icbVBNSwMxEJ2tX7V+VT16CRahXspuFfRY9OKxQr+gXUo2zbahSXZJskJZ+he8eFDEq3/Im//GbLsHbX0w8Hhvhpl5QcyZNq777RQ2Nre2d4q7pb39g8Oj8vFJR0eJIrRNIh6pXoA15UzStmGG016sKBYBp91gep/53SeqNItky8xi6gs8lixkBJtMalXl5bBccWvuAmideDmpQI7msPw1GEUkEVQawrHWfc+NjZ9iZRjhdF4aJJrGmEzxmPYtlVhQ7aeLW+fowiojFEbKljRoof6eSLHQeiYC2ymwmehVLxP/8/qJCW/9lMk4MVSS5aIw4chEKHscjZiixPCZJZgoZm9FZIIVJsbGU7IheKsvr5NOveZd1eqP15XGXR5HEc7gHKrgwQ004AGa0AYCE3iGV3hzhPPivDsfy9aCk8+cwh84nz9C/Y25</latexit> T(n) = aT(n/b) + f(n) <latexit sha1_base64="ohCH7vdM3XKzxE6JUirDO0Tn1ro=">AAAB/nicbVDLSgMxFL1TX7W+RsWVm2ARWoQ6UwXdCEU3Liv0BW0pmTTThmYyQ5IRylDwV9y4UMSt3+HOvzFtZ6GtBy73cM695OZ4EWdKO863lVlZXVvfyG7mtrZ3dvfs/YOGCmNJaJ2EPJQtDyvKmaB1zTSnrUhSHHicNr3R3dRvPlKpWChqehzRboAHgvmMYG2knn1UK4giukHY9HOviM6Qb4SenXdKzgxombgpyUOKas/+6vRDEgdUaMKxUm3XiXQ3wVIzwukk14kVjTAZ4QFtGypwQFU3mZ0/QadG6SM/lKaERjP190aCA6XGgWcmA6yHatGbiv957Vj7192EiSjWVJD5Q37MkQ7RNAvUZ5ISzceGYCKZuRWRIZaYaJNYzoTgLn55mTTKJfeiVH64zFdu0ziycAwnUAAXrqAC91CFOhBI4Ble4c16sl6sd+tjPpqx0p1D+APr8weOKZKm</latexit> と を定数, を関数とする. 非負整数上の関数 を漸化式 n/b <latexit sha1_base64="rSqccx9IdhZwyOJi1mpvSHd4EhM=">AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4qkkV9Fj04rGi/YA2lM120y7dbMLuRCihP8GLB0W8+ou8+W/ctjlo64OBx3szzMwLEikMuu63s7K6tr6xWdgqbu/s7u2XDg6bJk414w0Wy1i3A2q4FIo3UKDk7URzGgWSt4LR7dRvPXFtRKwecZxwP6IDJULBKFrpQZ0HvVLZrbgzkGXi5aQMOeq90le3H7M04gqZpMZ0PDdBP6MaBZN8UuymhieUjeiAdyxVNOLGz2anTsipVfokjLUthWSm/p7IaGTMOApsZ0RxaBa9qfif10kxvPYzoZIUuWLzRWEqCcZk+jfpC80ZyrEllGlhbyVsSDVlaNMp2hC8xZeXSbNa8S4q1fvLcu0mj6MAx3ACZ+DBFdTgDurQAAYDeIZXeHOk8+K8Ox/z1hUnnzmCP3A+fwD/B42b</latexit> bn/bc <latexit sha1_base64="JcRhJvBRulJbnEAjxVmR6m43Pr4=">AAAB/HicbZA7T8MwFIVvyquUV6Aji0WFxFSSggRjBQtjkehDaqrKcZ3WqmNHtoNUVeWvsDCAECs/hI1/g5tmgJY7fTrnXvn4hAln2njet1NYW9/Y3Cpul3Z29/YP3MOjlpapIrRJJJeqE2JNORO0aZjhtJMoiuOQ03Y4vp377UeqNJPiwUwS2ovxULCIEWys1HfLAY+4lAqJ8xAFKuO+W/GqXjZoFfwcKpBPo+9+BQNJ0pgKQzjWuut7ielNsTKMcDorBammCSZjPKRdiwLHVPemWfgZOrXKAEU2QiSFQZn6+2KKY60ncWg3Y2xGetmbi/953dRE170pE0lqqCCLh6KUIyPRvAk0YIoSwycWMFHMZkVkhBUmxvZVsiX4y19ehVat6l9Ua/eXlfpNXkcRjuEEzsCHK6jDHTSgCQQm8Ayv8OY8OS/Ou/OxWC04+U0Z/ozz+QNE+5SG</latexit> dn/be <latexit sha1_base64="/Z2tUkIRTe0iwjiNDsUS9Cn97s8=">AAAB+nicbZDLSsNAFIZPvNZ6S3XpZrAIrmpSBV0W3bisYC/QhDKZTtqhk0mYmSgl9lHcuFDErU/izrdx0mahrT8MfPznHM6ZP0g4U9pxvq2V1bX1jc3SVnl7Z3dv364ctFWcSkJbJOax7AZYUc4EbWmmOe0mkuIo4LQTjG/yeueBSsVica8nCfUjPBQsZARrY/XtiscJZRyJswB5Mse+XXVqzkxoGdwCqlCo2be/vEFM0ogKTThWquc6ifYzLDUjnE7LXqpogskYD2nPoMARVX42O32KTowzQGEszRMazdzfExmOlJpEgemMsB6pxVpu/lfrpTq88jMmklRTQeaLwpQjHaM8BzRgkhLNJwYwkczcisgIS0y0SatsQnAXv7wM7XrNPa/V7y6qjesijhIcwTGcgguX0IBbaEILCDzCM7zCm/VkvVjv1se8dcUqZg7hj6zPH1hdk2g=</latexit> によって定義する. ここで, は または を意味するものと解釈する. b > 1 <latexit sha1_base64="Oaj7SatYDW4Wm2nNKQXdZTfaWvE=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoCcpevFYwbSFNpTNdtIu3WzC7kYopb/BiwdFvPqDvPlv3LY5aOuDgcd7M8zMC1PBtXHdb6ewtr6xuVXcLu3s7u0flA+PmjrJFEOfJSJR7ZBqFFyib7gR2E4V0jgU2ApHdzO/9YRK80Q+mnGKQUwHkkecUWMlPyQ3xOuVK27VnYOsEi8nFcjR6JW/uv2EZTFKwwTVuuO5qQkmVBnOBE5L3UxjStmIDrBjqaQx6mAyP3ZKzqzSJ1GibElD5urviQmNtR7Hoe2MqRnqZW8m/ud1MhNdBxMu08ygZItFUSaIScjsc9LnCpkRY0soU9zeStiQKsqMzadkQ/CWX14lzVrVu6jWHi4r9ds8jiKcwCmcgwdXUId7aIAPDDg8wyu8OdJ5cd6dj0VrwclnjuEPnM8fZamNwQ==</latexit> このとき, は漸近的に次の限界を持つ. T(n) <latexit sha1_base64="vM8hsiJM1mIid8s1EMCG6OMcapM=">AAAB63icbVBNSwMxEJ2tX7V+VT16CRahXspuFfRY9OKxQr+gXUo2zbahSXZJskJZ+he8eFDEq3/Im//GbLsHbX0w8Hhvhpl5QcyZNq777RQ2Nre2d4q7pb39g8Oj8vFJR0eJIrRNIh6pXoA15UzStmGG016sKBYBp91gep/53SeqNItky8xi6gs8lixkBJtMalXl5bBccWvuAmideDmpQI7msPw1GEUkEVQawrHWfc+NjZ9iZRjhdF4aJJrGmEzxmPYtlVhQ7aeLW+fowiojFEbKljRoof6eSLHQeiYC2ymwmehVLxP/8/qJCW/9lMk4MVSS5aIw4chEKHscjZiixPCZJZgoZm9FZIIVJsbGU7IheKsvr5NOveZd1eqP15XGXR5HEc7gHKrgwQ004AGa0AYCE3iGV3hzhPPivDsfy9aCk8+cwh84nz9C/Y25</latexit> 1. ある定数 に対して ならば, である. ✏ > 0 <latexit sha1_base64="iQxt5eBAWyi4CnJXD1FTsFM/HhA=">AAAB83icbVBNSwMxEJ2tX7V+VT16CRbBU9mtgp6k6MVjBfsB3aVk02wbmk1CkhVK6d/w4kERr/4Zb/4b03YP2vpg4PHeDDPzYsWZsb7/7RXW1jc2t4rbpZ3dvf2D8uFRy8hME9okkkvdibGhnAnatMxy2lGa4jTmtB2P7mZ++4lqw6R4tGNFoxQPBEsYwdZJYUiVYVwKdIP8XrniV/050CoJclKBHI1e+SvsS5KlVFjCsTHdwFc2mmBtGeF0WgozQxUmIzygXUcFTqmJJvObp+jMKX2USO1KWDRXf09McGrMOI1dZ4rt0Cx7M/E/r5vZ5DqaMKEySwVZLEoyjqxEswBQn2lKLB87golm7lZEhlhjYl1MJRdCsPzyKmnVqsFFtfZwWanf5nEU4QRO4RwCuII63EMDmkBAwTO8wpuXeS/eu/exaC14+cwx/IH3+QP4DpD6</latexit> f(n) = O(n logb a ✏) <latexit sha1_base64="/Co+zHNANF4vde42xchIx70RmKc=">AAACCXicbVC7SgNBFJ2NrxhfUUubwSAkhWE3CtoIQRs7I5gHZNdldjKbDJmdWWZmhbBsa+Ov2FgoYusf2Pk3Th6FJh64cDjnXu69J4gZVdq2v63c0vLK6lp+vbCxubW9U9zdaymRSEyaWDAhOwFShFFOmppqRjqxJCgKGGkHw6ux334gUlHB7/QoJl6E+pyGFCNtJL8IwzKvwAt4U+b3qctE3w9SlB27JFaUCZ5V/GLJrtoTwEXizEgJzNDwi19uT+AkIlxjhpTqOnasvRRJTTEjWcFNFIkRHqI+6RrKUUSUl04+yeCRUXowFNIU13Ci/p5IUaTUKApMZ4T0QM17Y/E/r5vo8NxLKY8TTTieLgoTBrWA41hgj0qCNRsZgrCk5laIB0girE14BROCM//yImnVqs5JtXZ7WqpfzuLIgwNwCMrAAWegDq5BAzQBBo/gGbyCN+vJerHerY9pa86azeyDP7A+fwDWiJkw</latexit> T(n) = ⇥(nlogb a) <latexit sha1_base64="NXmsTiZOS/6q9hWSxepYpDy9Ol0=">AAACBXicbVDLSsNAFJ3UV62vqEtdDBah3ZSkCroRim5cVugLmhgm00k7dDIJMxOhhGzc+CtuXCji1n9w5984bbPQ1gMXDufcy733+DGjUlnWt1FYWV1b3yhulra2d3b3zP2DjowSgUkbRywSPR9JwignbUUVI71YEBT6jHT98c3U7z4QIWnEW2oSEzdEQ04DipHSkmcetyq8Cq+g0xoRhSr8PnVYNPT8FGVZ1TPLVs2aAS4TOydlkKPpmV/OIMJJSLjCDEnZt61YuSkSimJGspKTSBIjPEZD0teUo5BIN519kcFTrQxgEAldXMGZ+nsiRaGUk9DXnSFSI7noTcX/vH6igks3pTxOFOF4vihIGFQRnEYCB1QQrNhEE4QF1bdCPEICYaWDK+kQ7MWXl0mnXrPPavW783LjOo+jCI7ACagAG1yABrgFTdAGGDyCZ/AK3own48V4Nz7mrQUjnzkEf2B8/gC2gJd2</latexit> f(n) = ⇥(nlogb a) <latexit sha1_base64="wxEUjKX9v7jbdUtsLF9regJ8Cl0=">AAACBXicbVDLSsNAFJ3UV62vqEtdDBah3ZSkCroRim5cVugLmhgm00k7dDIJMxOhhGzc+CtuXCji1n9w5984bbPQ1gMXDufcy733+DGjUlnWt1FYWV1b3yhulra2d3b3zP2DjowSgUkbRywSPR9JwignbUUVI71YEBT6jHT98c3U7z4QIWnEW2oSEzdEQ04DipHSkmceBxVehVfQaY2IQhV+nzosGnp+irKs6pllq2bNAJeJnZMyyNH0zC9nEOEkJFxhhqTs21as3BQJRTEjWclJJIkRHqMh6WvKUUikm86+yOCpVgYwiIQuruBM/T2RolDKSejrzhCpkVz0puJ/Xj9RwaWbUh4ninA8XxQkDKoITiOBAyoIVmyiCcKC6lshHiGBsNLBlXQI9uLLy6RTr9lntfrdeblxncdRBEfgBFSADS5AA9yCJmgDDB7BM3gFb8aT8WK8Gx/z1oKRzxyCPzA+fwDTrpeI</latexit> T(n) = ⇥(nlogb a lg n) <latexit sha1_base64="0prIF2fK13+RYI2+hJ5S8C3FtR0=">AAACC3icbVDLSsNAFJ3UV62vqEs3Q4vQbkpSBd0IRTcuK/QFTQyT6aQdOpmEmYlQQvZu/BU3LhRx6w+482+ctllo64ELh3Pu5d57/JhRqSzr2yisrW9sbhW3Szu7e/sH5uFRV0aJwKSDIxaJvo8kYZSTjqKKkX4sCAp9Rnr+5Gbm9x6IkDTibTWNiRuiEacBxUhpyTPL7SqvwSvotMdEoSq/Tx0WjTw/RVnmsFHKs5pnVqy6NQdcJXZOKiBHyzO/nGGEk5BwhRmScmBbsXJTJBTFjGQlJ5EkRniCRmSgKUchkW46/yWDp1oZwiASuriCc/X3RIpCKaehrztDpMZy2ZuJ/3mDRAWXbkp5nCjC8WJRkDCoIjgLBg6pIFixqSYIC6pvhXiMBMJKx1fSIdjLL6+SbqNun9Ubd+eV5nUeRxGcgDKoAhtcgCa4BS3QARg8gmfwCt6MJ+PFeDc+Fq0FI585Bn9gfP4AweGaRw==</latexit> 2. ならば, である. f(n) = ⌦(nlogb a+✏) <latexit sha1_base64="EHlFmqSFcIuKRsfUNaViZF2E21Y=">AAACDnicbVA9SwNBEN3zM8avU0ubxRBIEMKdCtoIQRs7I5gYyMWwt5lLFvd2j909IRz5BTb+FRsLRWyt7fw3bj4KjT4YeLw3w8y8MOFMG8/7cubmFxaXlnMr+dW19Y1Nd2u7oWWqKNSp5FI1Q6KBMwF1wwyHZqKAxCGHm/DufOTf3IPSTIprM0igHZOeYBGjxFip4xajkijjUxxcxtAjJXGbBVz2OmFGhvsBJJpxKYbljlvwKt4Y+C/xp6SApqh13M+gK2kagzCUE61bvpeYdkaUYZTDMB+kGhJC70gPWpYKEoNuZ+N3hrholS6OpLIlDB6rPycyEms9iEPbGRPT17PeSPzPa6UmOmlnTCSpAUEni6KUYyPxKBvcZQqo4QNLCFXM3oppnyhCjU0wb0PwZ1/+SxoHFf+wcnB1VKieTePIoV20h0rIR8eoii5QDdURRQ/oCb2gV+fReXbenPdJ65wzndlBv+B8fAPDk5tW</latexit> c < 1 <latexit sha1_base64="WfaFCDew75a8b7g+U5VJ8Sh5PJI=">AAAB7HicbVA9SwNBEJ2LXzF+RS1tFoNgFe6ioIVF0MYygpcEkiPsbeaSJXt7x+6eEEJ+g42FIrb+IDv/jZvkCk18MPB4b4aZeWEquDau++0U1tY3NreK26Wd3b39g/LhUVMnmWLos0Qkqh1SjYJL9A03AtupQhqHAlvh6G7mt55QaZ7IRzNOMYjpQPKIM2qs5DNyQ7xeueJW3TnIKvFyUoEcjV75q9tPWBajNExQrTuem5pgQpXhTOC01M00ppSN6AA7lkoaow4m82On5MwqfRIlypY0ZK7+npjQWOtxHNrOmJqhXvZm4n9eJzPRdTDhMs0MSrZYFGWCmITMPid9rpAZMbaEMsXtrYQNqaLM2HxKNgRv+eVV0qxVvYtq7eGyUr/N4yjCCZzCOXhwBXW4hwb4wIDDM7zCmyOdF+fd+Vi0Fpx85hj+wPn8AWQljcA=</latexit> af(n/b)  cf(n) <latexit sha1_base64="4j+T1jhEsTzYQ7+9RfN+cZD7oE8=">AAAB+3icbVDLTgIxFO3gC/E14tJNIzGBDc6giS6JblxiIo8EJqRT7kBDpzO2HSOZ8CtuXGiMW3/EnX9jgVkoeJKbnJ5zb3rv8WPOlHacbyu3tr6xuZXfLuzs7u0f2IfFlooSSaFJIx7Jjk8UcCagqZnm0IklkNDn0PbHNzO//QhSsUjc60kMXkiGggWMEm2kvl0kQVmc+RXc4/CAqXlU+nbJqTpz4FXiZqSEMjT69ldvENEkBKEpJ0p1XSfWXkqkZpTDtNBLFMSEjskQuoYKEoLy0vnuU3xqlAEOImlKaDxXf0+kJFRqEvqmMyR6pJa9mfif1010cOWlTMSJBkEXHwUJxzrCsyDwgEmgmk8MIVQysyumIyIJ1SauggnBXT55lbRqVfe8Wru7KNWvszjy6BidoDJy0SWqo1vUQE1E0RN6Rq/ozZpaL9a79bFozVnZzBH6A+vzB59SkuA=</latexit> T(n) = ⇥(f(n)) <latexit sha1_base64="w2Wkwq9Zk6Rm7qw8MFnERSx3Lww=">AAAB/HicbVDLSgMxFM3UV62v0S7dBIvQbspMFXQjFN24rNAXtEPJpHfa0ExmSDLCUOqvuHGhiFs/xJ1/Y9rOQlsPXDg5515y7/FjzpR2nG8rt7G5tb2T3y3s7R8cHtnHJ20VJZJCi0Y8kl2fKOBMQEszzaEbSyChz6HjT+7mfucRpGKRaOo0Bi8kI8ECRok20sAuNsuigm9wvzkGTcqBeVUGdsmpOgvgdeJmpIQyNAb2V38Y0SQEoSknSvVcJ9belEjNKIdZoZ8oiAmdkBH0DBUkBOVNF8vP8LlRhjiIpCmh8UL9PTEloVJp6JvOkOixWvXm4n9eL9HBtTdlIk40CLr8KEg41hGeJ4GHTALVPDWEUMnMrpiOiSRUm7wKJgR39eR10q5V3Ytq7eGyVL/N4sijU3SGyshFV6iO7lEDtRBFKXpGr+jNerJerHfrY9mas7KZIvoD6/MHjKKSxQ==</latexit> ✏ > 0 <latexit sha1_base64="iQxt5eBAWyi4CnJXD1FTsFM/HhA=">AAAB83icbVBNSwMxEJ2tX7V+VT16CRbBU9mtgp6k6MVjBfsB3aVk02wbmk1CkhVK6d/w4kERr/4Zb/4b03YP2vpg4PHeDDPzYsWZsb7/7RXW1jc2t4rbpZ3dvf2D8uFRy8hME9okkkvdibGhnAnatMxy2lGa4jTmtB2P7mZ++4lqw6R4tGNFoxQPBEsYwdZJYUiVYVwKdIP8XrniV/050CoJclKBHI1e+SvsS5KlVFjCsTHdwFc2mmBtGeF0WgozQxUmIzygXUcFTqmJJvObp+jMKX2USO1KWDRXf09McGrMOI1dZ4rt0Cx7M/E/r5vZ5DqaMKEySwVZLEoyjqxEswBQn2lKLB87golm7lZEhlhjYl1MJRdCsPzyKmnVqsFFtfZwWanf5nEU4QRO4RwCuII63EMDmkBAwTO8wpuXeS/eu/exaC14+cwx/IH3+QP4DpD6</latexit> 3. ある定数 に対して であり, しかもある定数 と 十分大きな全ての に対して ならば である. n <latexit sha1_base64="2QmInd+nHO60uhS7AYCvgpiU4a8=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlpuyXK27VnYOsEi8nFcjR6Je/eoOYpRFKwwTVuuu5ifEzqgxnAqelXqoxoWxMh9i1VNIItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2ooszYbEo2BG/55VXSrlW9i2qteVmp3+RxFOEETuEcPLiCOtxBA1rAAOEZXuHNeXRenHfnY9FacPKZY/gD5/MH2F+M9g==</latexit> 再帰の底の寄与が大きい場合 節点の寄与が大きい場合 底と節点の寄与が同じくらいのとき T(n) = ⇥(nlogb a) + logb n 1 X j=0 ajf(n/bj) <latexit sha1_base64="fiHrfLLobOXSGIgTc0/a3LzWxyc=">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</latexit> 再帰の底の寄与 節点の寄与 34
  35. /35 まとめ 大きい問題を分割, 部分問題を再帰的に解いて得られた解を結合して 元の問題の答えを得る. 分割統治法 計算量の求め方 まず漸化式で表す. 次にマスター定理を適用(可能な場合). ※実用上はStrassenのアルゴリズムが採用されることは少ないらしい.

    詳しくはCLRSの文献ノート(P92)参照. 35