{"id":927,"date":"2021-04-30T17:23:46","date_gmt":"2021-04-30T09:23:46","guid":{"rendered":"https:\/\/beijian99.top\/?p=927"},"modified":"2025-05-06T16:14:54","modified_gmt":"2025-05-06T08:14:54","slug":"%e6%a0%91%e7%8a%b6%e6%95%b0%e7%bb%84%ef%bc%88fenwick-tree%ef%bc%89","status":"publish","type":"post","link":"https:\/\/beijian99.top\/?p=927","title":{"rendered":"\u6811\u72b6\u6570\u7ec4\uff08Fenwick Tree\uff09"},"content":{"rendered":"\n

\u6811\u72b6\u6570\u7ec4\u662f\u4e00\u79cd\u9ad8\u6548\u7ef4\u62a4\u524d\u7f00\u548c\u7684\u6570\u636e\u7ed3\u6784\uff0c\u652f\u6301\u5355\u70b9\u66f4\u65b0\u548c\u524d\u7f00\u67e5\u8be2\u64cd\u4f5c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a O(logn)\u3002\u5176\u6838\u5fc3\u601d\u60f3\u662f\u901a\u8fc7\u4e8c\u8fdb\u5236\u5206\u89e3\u5c06\u524d\u7f00\u548c\u62c6\u5206\u4e3a\u82e5\u5e72\u5b50\u6570\u7ec4\u7684\u548c\u3002<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\n\n\n\n

\u200b\u200b\u6838\u5fc3\u601d\u60f3\u200b<\/strong>\u200b<\/p>\n\n\n\n

    \n
  1. \u4e8c\u8fdb\u5236\u7d22\u5f15\uff1a
    \u2022 \u6bcf\u4e2a\u6574\u6570 i \u53ef\u4ee5\u8868\u793a\u4e3a\u4e8c\u8fdb\u5236\u5f62\u5f0f\uff08\u5982 6=1102\u200b\uff09\u3002 \u2022 \u5b9a\u4e49 lowbit(i)<\/code> \u4e3a i \u7684\u4e8c\u8fdb\u5236\u8868\u793a\u4e2d\u6700\u4f4e\u4f4d\u7684 1<\/code> \u6240\u5bf9\u5e94\u7684\u503c\uff08\u5982 lowbit(6) = 2<\/code>\uff09\u3002 \u2022 lowbit(i)<\/code> \u7684\u8ba1\u7b97\u65b9\u5f0f\uff1ai & -i<\/code>\uff08\u5229\u7528\u8865\u7801\u6027\u8d28\uff09\u3002<\/li>\n\n\n\n
  2. \u6811\u72b6\u7ed3\u6784\uff1a
    \u2022 \u6570\u7ec4 tree<\/code> \u4e2d\uff0ctree[i]<\/code> \u5b58\u50a8\u7684\u662f\u539f\u6570\u7ec4 a<\/code> \u4e2d\u533a\u95f4 [i - lowbit(i) + 1, i]<\/code> \u7684\u548c\u3002 \u2022 \u901a\u8fc7 lowbit<\/code> \u8df3\u8dc3\u5f0f\u8bbf\u95ee\uff0c\u5c06\u524d\u7f00\u548c\u5206\u89e3\u4e3a O(logn) \u4e2a tree<\/code> \u8282\u70b9\u7684\u548c\u3002<\/li>\n<\/ol>\n\n\n\n
    \n\n\n\n

    \u200b\u200b\u64cd\u4f5c\u5b9e\u73b0\u200b<\/strong>\u200b<\/p>\n\n\n\n

      \n
    1. \u5355\u70b9\u66f4\u65b0\uff08update\uff09\uff1a
      \u2022 \u4fee\u6539 a[x]<\/code> \u540e\uff0c\u9700\u8981\u66f4\u65b0\u6240\u6709\u5305\u542b a[x]<\/code> \u7684 tree<\/code> \u8282\u70b9\u3002 \u2022 \u4ece x<\/code> \u5f00\u59cb\uff0c\u6bcf\u6b21\u5411\u4e0a\u8df3 lowbit(x)<\/code>\uff0c\u76f4\u5230\u8d85\u51fa\u6570\u7ec4\u8303\u56f4\u3002<\/li>\n\n\n\n
    2. \u524d\u7f00\u67e5\u8be2\uff08query\uff09\uff1a
      \u2022 \u67e5\u8be2\u524d\u7f00\u548c sum[1..x]<\/code> \u65f6\uff0c\u4ece x<\/code> \u5f00\u59cb\uff0c\u6bcf\u6b21\u7d2f\u52a0 tree[x]<\/code>\uff0c\u7136\u540e\u5411\u4e0a\u8df3 x -= lowbit(x)<\/code>\uff0c\u76f4\u5230 x = 0<\/code>\u3002<\/li>\n<\/ol>\n\n\n\n
      \n\n\n\n

      \u200b\u200b\u6b63\u786e\u6027\u8bc1\u660e\u200b<\/strong>\u200b<\/p>\n\n\n\n

        \n
      1. \u66f4\u65b0\u8986\u76d6\u6027\uff1a
        \u2022 \u6bcf\u6b21\u66f4\u65b0 x<\/code> \u65f6\uff0cx + lowbit(x)<\/code> \u4f1a\u8df3\u5230\u4e0b\u4e00\u4e2a\u8986\u76d6 a[x]<\/code> \u7684\u7236\u8282\u70b9\u3002\u4f8b\u5982\uff1a \u25e6 \u66f4\u65b0 x=1<\/code>\uff08lowbit(1)=1<\/code>\uff09\u4f1a\u5f71\u54cd 1, 2, 4, 8,...<\/code>\u3002 \u25e6 \u66f4\u65b0 x=6<\/code>\uff08lowbit(6)=2<\/code>\uff09\u4f1a\u5f71\u54cd 6, 8,...<\/code>\u3002 \u2022 \u901a\u8fc7\u4e8c\u8fdb\u5236\u4f4d\u7684\u8fdb\u4f4d\uff0c\u786e\u4fdd\u6240\u6709\u76f8\u5173\u533a\u95f4\u88ab\u66f4\u65b0\u3002<\/li>\n\n\n\n
      2. \u67e5\u8be2\u5b8c\u6574\u6027\uff1a
        \u2022 \u524d\u7f00\u548c sum[1..x]<\/code> \u88ab\u5206\u89e3\u4e3a tree[x] + tree[x - lowbit(x)] + ...<\/code>\u3002 \u2022 \u6bcf\u6b21 x -= lowbit(x)<\/code> \u4f1a\u8df3\u5230\u4e0b\u4e00\u4e2a\u4e0d\u91cd\u53e0\u7684\u533a\u95f4\uff0c\u786e\u4fdd\u6240\u6709\u5b50\u533a\u95f4\u88ab\u7d2f\u52a0\u3002<\/li>\n<\/ol>\n\n\n\n
        \n\n\n\n

        \u200b\u200b\u65f6\u95f4\u590d\u6742\u5ea6\u8bc1\u660e\u200b<\/strong>\u200b
        \u2022 \u5355\u70b9\u66f4\u65b0\uff1a\u6bcf\u6b21\u8df3\u8dc3 lowbit(x)<\/code>\uff0c\u76f8\u5f53\u4e8e\u5728\u4e8c\u8fdb\u5236\u8868\u793a\u4e2d\u4ece\u53f3\u5411\u5de6\u79fb\u52a8 1<\/code> \u7684\u4f4d\u7f6e\uff0c\u6700\u591a O(logn) \u6b21\u3002<\/p>\n\n\n\n

        \u2022 \u524d\u7f00\u67e5\u8be2\uff1a\u540c\u7406\uff0c\u6bcf\u6b21\u8df3\u8dc3 lowbit(x)<\/code>\uff0c\u6700\u591a O(logn) \u6b21\u3002<\/p>\n\n\n\n

        \u2022 \u521d\u59cb\u5316\uff1a\u901a\u8fc7 n \u6b21\u66f4\u65b0\u64cd\u4f5c\u6784\u5efa\uff0c\u590d\u6742\u5ea6 O(nlogn)\u3002<\/p>\n\n\n\n

        \n\n\n\n

        \u200b\u200bC++ \u5b9e\u73b0\u200b<\/strong><\/p>\n\n\n\n

        \u200b<\/p>\n\n\n

        \n#include <bits\/stdc++.h>\nusing namespace std;\n\nconst int MAXN = 5e5 + 5;\nint a[MAXN], c[MAXN];\nint n, m;\nint lowbit(int x)\n{\n    return x & -x;\n}\n\nvoid update(int pos, int value)\n{\n    while (pos <= n)\n    {\n        c[pos] += value;\n        pos += lowbit(pos);\n    }\n}\n\nint getPreSum(int right)\n{\n    int sum = 0;\n    while (right > 0)\n    {\n        sum += c[right];\n        right -= lowbit(right);\n    }\n    return sum;\n}\n\nint getRangeSum(int left, int right)\n{\n    return getPreSum(right) - getPreSum(left - 1);\n}\n\nint main()\n{\n\n    scanf("%d%d", &n, &m);\n    for (int i = 1; i <= n; i++)\n    {\n        scanf("%d", &a[i]);\n        update(i, a[i]);\n    }\n\n    int op, l, r;\n    while (m--)\n    {\n        scanf("%d%d%d", &op, &l, &r);\n        switch (op)\n        {\n        case 1:\n            update(l, r);\n            break;\n        case 2:\n            printf("%d\\n", getRangeSum(l, r));\n            break;\n        default:\n            break;\n        }\n    }\n    return 0;\n}\n<\/pre><\/div>\n\n\n
        \u200b<\/p>\n\n\n\n
        \n\n\n\n
        \u200b\u200b\u5e94\u7528\u573a\u666f\u200b<\/strong>\u200b
        \u2022 \u52a8\u6001\u7ef4\u62a4\u524d\u7f00\u548c\uff08\u5982\u9891\u7e41\u66f4\u65b0\u7684\u6570\u7ec4\uff09\u3002<\/p>\n\n\n\n
        \u2022 \u9006\u5e8f\u5bf9\u8ba1\u6570\uff08\u79bb\u6563\u5316\u540e\u63d2\u5165\u7edf\u8ba1\uff09\u3002<\/p>\n\n\n\n
        \u2022 \u66ff\u4ee3\u7ebf\u6bb5\u6811\uff08\u5f53\u4ec5\u9700\u524d\u7f00\u64cd\u4f5c\u65f6\u66f4\u9ad8\u6548\uff09\u3002<\/p>\n\n\n\n
        \u6811\u72b6\u6570\u7ec4\u901a\u8fc7\u5de7\u5999\u7684\u4e8c\u8fdb\u5236\u8bbe\u8ba1\uff0c\u4ee5\u6781\u4f4e\u7684\u4ee3\u7801\u590d\u6742\u5ea6\u5b9e\u73b0\u4e86\u9ad8\u6548\u7684\u524d\u7f00\u64cd\u4f5c\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"
        \u6811\u72b6\u6570\u7ec4\u662f\u4e00\u79cd\u9ad8\u6548\u7ef4\u62a4\u524d\u7f00\u548c\u7684\u6570\u636e\u7ed3\u6784\uff0c\u652f\u6301\u5355\u70b9\u66f4\u65b0\u548c\u524d\u7f00\u67e5\u8be2\u64cd\u4f5c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a O(logn)\u3002\u5176\u6838\u5fc3\u601d\u60f3 […]<\/p>\n","protected":false},"author":1,"featured_media":935,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[4],"tags":[],"class_list":["post-927","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algorithm"],"_links":{"self":[{"href":"https:\/\/beijian99.top\/index.php?rest_route=\/wp\/v2\/posts\/927","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/beijian99.top\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/beijian99.top\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/beijian99.top\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/beijian99.top\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=927"}],"version-history":[{"count":3,"href":"https:\/\/beijian99.top\/index.php?rest_route=\/wp\/v2\/posts\/927\/revisions"}],"predecessor-version":[{"id":939,"href":"https:\/\/beijian99.top\/index.php?rest_route=\/wp\/v2\/posts\/927\/revisions\/939"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/beijian99.top\/index.php?rest_route=\/wp\/v2\/media\/935"}],"wp:attachment":[{"href":"https:\/\/beijian99.top\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=927"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/beijian99.top\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=927"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/beijian99.top\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=927"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}