{"id":478,"date":"2021-04-14T20:00:15","date_gmt":"2021-04-14T12:00:15","guid":{"rendered":"http:\/\/120.55.184.7\/?p=478"},"modified":"2025-04-15T18:16:01","modified_gmt":"2025-04-15T10:16:01","slug":"%e7%ba%bf%e6%ae%b5%e6%a0%91%ef%bc%9a%e9%ab%98%e6%95%88%e5%a4%84%e7%90%86%e5%8c%ba%e9%97%b4%e6%9f%a5%e8%af%a2%e4%b8%8e%e6%9b%b4%e6%96%b0%e7%9a%84%e6%95%b0%e6%8d%ae%e7%bb%93%e6%9e%84","status":"publish","type":"post","link":"https:\/\/beijian99.top\/?p=478","title":{"rendered":"\u7ebf\u6bb5\u6811\uff1a\u9ad8\u6548\u5904\u7406\u533a\u95f4\u67e5\u8be2\u4e0e\u66f4\u65b0\u7684\u6570\u636e\u7ed3\u6784"},"content":{"rendered":"\n<h1 class=\"wp-block-heading\">\u4ec0\u4e48\u662f\u7ebf\u6bb5\u6811\uff1f<\/h1>\n\n\n\n<p>\u7ebf\u6bb5\u6811\u662f\u4e00\u79cd\u4e8c\u53c9\u6811\u6570\u636e\u7ed3\u6784\uff0c\u7528\u4e8e\u5b58\u50a8\u533a\u95f4\u6216\u7ebf\u6bb5\u7684\u4fe1\u606f\u3002\u5b83\u80fd\u591f\u5728O(log n)\u65f6\u95f4\u590d\u6742\u5ea6\u5185\u5b8c\u6210\u533a\u95f4\u67e5\u8be2\u548c\u5355\u70b9\/\u533a\u95f4\u66f4\u65b0\u64cd\u4f5c\uff0c\u5982\u533a\u95f4\u548c\u3001\u533a\u95f4\u6700\u5c0f\u503c\/\u6700\u5927\u503c\u3001\u533a\u95f4\u4e58\u79ef\u7b49\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u57fa\u672c\u7279\u6027<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u5b8c\u5168\u4e8c\u53c9\u6811<\/strong>\uff1a\u7ebf\u6bb5\u6811\u901a\u5e38\u5b9e\u73b0\u4e3a\u5b8c\u5168\u4e8c\u53c9\u6811<\/li>\n\n\n\n<li><strong>\u5b58\u50a8\u533a\u95f4\u4fe1\u606f<\/strong>\uff1a\u6bcf\u4e2a\u8282\u70b9\u5b58\u50a8\u4e00\u4e2a\u533a\u95f4\u7684\u67d0\u79cd\u805a\u5408\u4fe1\u606f<\/li>\n\n\n\n<li><strong>\u9ad8\u6548\u64cd\u4f5c<\/strong>\uff1a\u67e5\u8be2\u548c\u66f4\u65b0\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3aO(log n)<\/li>\n\n\n\n<li><strong>\u7a7a\u95f4\u590d\u6742\u5ea6<\/strong>\uff1aO(n)\uff0c\u9700\u89812n-1\u4e2a\u8282\u70b9\u5b58\u50a8n\u4e2a\u5143\u7d20\u7684\u6570\u7ec4<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">\u7ebf\u6bb5\u6811\u7684\u7ed3\u6784<\/h2>\n\n\n\n<p>\u7ebf\u6bb5\u6811\u7684\u6bcf\u4e2a\u8282\u70b9\u4ee3\u8868\u4e00\u4e2a\u533a\u95f4[l, r]\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u53f6\u5b50\u8282\u70b9<\/strong>\uff1a\u4ee3\u8868\u5355\u4e2a\u5143\u7d20\u7684\u533a\u95f4[i, i]<\/li>\n\n\n\n<li><strong>\u5185\u90e8\u8282\u70b9<\/strong>\uff1a\u4ee3\u8868\u5408\u5e76\u7684\u5b50\u533a\u95f4\uff0c\u901a\u5e38\u4e3a\u5de6\u53f3\u5b50\u8282\u70b9\u7684\u533a\u95f4\u5408\u5e76<\/li>\n<\/ul>\n\n\n<div class=\"wp-block-syntaxhighlighter-code \"><pre class=\"brush: plain; title: ; notranslate\" title=\"\">\n\u793a\u4f8b\uff1a\u6570\u7ec4&#x5B;1, 3, 5, 7, 9, 11]\u7684\u7ebf\u6bb5\u6811(\u533a\u95f4\u548c)\n\n            &#x5B;0,5](36)\n           \/        \\\n      &#x5B;0,2](9)      &#x5B;3,5](27)\n      \/     \\        \/     \\\n &#x5B;0,1](4) &#x5B;2,2](5) &#x5B;3,4](16) &#x5B;5,5](11)\n  \/   \\             \/   \\\n&#x5B;0,0](1) &#x5B;1,1](3) &#x5B;3,3](7) &#x5B;4,4](9)\n<\/pre><\/div>\n\n\n<h2 class=\"wp-block-heading\">\u7ebf\u6bb5\u6811\u7684\u6784\u5efa<\/h2>\n\n\n\n<p>\u6784\u5efa\u7ebf\u6bb5\u6811\u662f\u4e00\u4e2a\u81ea\u5e95\u5411\u4e0a\u7684\u8fc7\u7a0b\uff1a<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\u4ece\u53f6\u5b50\u8282\u70b9\u5f00\u59cb\uff0c\u6bcf\u4e2a\u53f6\u5b50\u8282\u70b9\u5bf9\u5e94\u6570\u7ec4\u4e2d\u7684\u4e00\u4e2a\u5143\u7d20<\/li>\n\n\n\n<li>\u9012\u5f52\u5730\u6784\u5efa\u7236\u8282\u70b9\uff0c\u5408\u5e76\u5b50\u8282\u70b9\u7684\u4fe1\u606f<\/li>\n<\/ol>\n\n\n<div class=\"wp-block-syntaxhighlighter-code \"><pre class=\"brush: plain; title: ; notranslate\" title=\"\">\ndef build(node, l, r, arr, tree):\n    if l == r:\n        tree&#x5B;node] = arr&#x5B;l]\n        return\n    mid = (l + r) \/\/ 2\n    build(2*node+1, l, mid, arr, tree)\n    build(2*node+2, mid+1, r, arr, tree)\n    tree&#x5B;node] = tree&#x5B;2*node+1] + tree&#x5B;2*node+2]  # \u4ee5\u533a\u95f4\u548c\u4e3a\u4f8b\n<\/pre><\/div>\n\n\n<h2 class=\"wp-block-heading\">\u7ebf\u6bb5\u6811\u7684\u67e5\u8be2<\/h2>\n\n\n\n<p>\u67e5\u8be2\u533a\u95f4[q_l, q_r]\u7684\u4fe1\u606f\uff1a<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\u5982\u679c\u5f53\u524d\u8282\u70b9\u533a\u95f4\u5b8c\u5168\u5305\u542b\u5728\u67e5\u8be2\u533a\u95f4\u5185\uff0c\u76f4\u63a5\u8fd4\u56de\u8282\u70b9\u503c<\/li>\n\n\n\n<li>\u5982\u679c\u4e0e\u67e5\u8be2\u533a\u95f4\u65e0\u4ea4\u96c6\uff0c\u8fd4\u56de\u4e0d\u5f71\u54cd\u7ed3\u679c\u7684\u503c\uff08\u5982\u6c42\u548c\u76840\uff0c\u6700\u5c0f\u503c\u7684\u221e\uff09<\/li>\n\n\n\n<li>\u5426\u5219\u9012\u5f52\u67e5\u8be2\u5de6\u53f3\u5b50\u6811\u5e76\u5408\u5e76\u7ed3\u679c<\/li>\n<\/ol>\n\n\n<div class=\"wp-block-syntaxhighlighter-code \"><pre class=\"brush: plain; title: ; notranslate\" title=\"\">\ndef query(node, l, r, q_l, q_r, tree):\n    if r &amp;lt; q_l or l &gt; q_r:  # \u65e0\u4ea4\u96c6\n        return 0\n    if q_l &amp;lt;= l and r &amp;lt;= q_r:  # \u5b8c\u5168\u5305\u542b\n        return tree&#x5B;node]\n    mid = (l + r) \/\/ 2\n    left = query(2*node+1, l, mid, q_l, q_r, tree)\n    right = query(2*node+2, mid+1, r, q_l, q_r, tree)\n    return left + right\n<\/pre><\/div>\n\n\n<h2 class=\"wp-block-heading\">\u7ebf\u6bb5\u6811\u7684\u66f4\u65b0<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">\u5355\u70b9\u66f4\u65b0<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\u627e\u5230\u5bf9\u5e94\u7684\u53f6\u5b50\u8282\u70b9<\/li>\n\n\n\n<li>\u66f4\u65b0\u53f6\u5b50\u8282\u70b9\u7684\u503c<\/li>\n\n\n\n<li>\u9012\u5f52\u5411\u4e0a\u66f4\u65b0\u6240\u6709\u53d7\u5f71\u54cd\u7684\u7236\u8282\u70b9<\/li>\n<\/ol>\n\n\n<div class=\"wp-block-syntaxhighlighter-code \"><pre class=\"brush: plain; title: ; notranslate\" title=\"\">\ndef update(node, l, r, idx, val, tree):\n    if l == r:\n        tree&#x5B;node] = val\n        return\n    mid = (l + r) \/\/ 2\n    if idx &amp;lt;= mid:\n        update(2*node+1, l, mid, idx, val, tree)\n    else:\n        update(2*node+2, mid+1, r, idx, val, tree)\n    tree&#x5B;node] = tree&#x5B;2*node+1] + tree&#x5B;2*node+2]\n<\/pre><\/div>\n\n\n<h3 class=\"wp-block-heading\">\u533a\u95f4\u66f4\u65b0\uff08\u5ef6\u8fdf\u4f20\u64ad\uff09<\/h3>\n\n\n\n<p>\u5bf9\u4e8e\u533a\u95f4\u66f4\u65b0\u64cd\u4f5c\uff0c\u53ef\u4ee5\u4f7f\u7528\u5ef6\u8fdf\u4f20\u64ad\uff08Lazy Propagation\uff09\u6280\u672f\u6765\u4f18\u5316\uff1a<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\u6807\u8bb0\u9700\u8981\u66f4\u65b0\u4f46\u5c1a\u672a\u5b9e\u9645\u6267\u884c\u7684\u533a\u95f4<\/li>\n\n\n\n<li>\u5728\u67e5\u8be2\u6216\u8fdb\u4e00\u6b65\u66f4\u65b0\u65f6\u4f20\u64ad\u8fd9\u4e9b\u6807\u8bb0<\/li>\n<\/ol>\n\n\n\n<h2 class=\"wp-block-heading\">\u7ebf\u6bb5\u6811\u7684\u5e94\u7528<\/h2>\n\n\n\n<p>\u7ebf\u6bb5\u6811\u53ef\u4ee5\u89e3\u51b3\u591a\u79cd\u533a\u95f4\u64cd\u4f5c\u95ee\u9898\uff1a<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>\u533a\u95f4\u6c42\u548c<\/strong>\uff1a\u67e5\u8be2\u4efb\u610f\u533a\u95f4\u7684\u5143\u7d20\u548c<\/li>\n\n\n\n<li><strong>\u533a\u95f4\u6700\u503c<\/strong>\uff1a\u67e5\u8be2\u533a\u95f4\u6700\u5c0f\u503c\/\u6700\u5927\u503c<\/li>\n\n\n\n<li><strong>\u533a\u95f4\u7edf\u8ba1<\/strong>\uff1a\u7edf\u8ba1\u6ee1\u8db3\u7279\u5b9a\u6761\u4ef6\u7684\u5143\u7d20\u6570\u91cf<\/li>\n\n\n\n<li><strong>\u533a\u95f4\u8986\u76d6<\/strong>\uff1a\u6279\u91cf\u4fee\u6539\u533a\u95f4\u5185\u7684\u5143\u7d20\u503c<\/li>\n\n\n\n<li><strong>\u9006\u5e8f\u5bf9\u8ba1\u6570<\/strong>\uff1a\u7edf\u8ba1\u6570\u7ec4\u4e2d\u7684\u9006\u5e8f\u5bf9\u6570\u91cf<\/li>\n\n\n\n<li><strong>\u626b\u63cf\u7ebf\u7b97\u6cd5<\/strong>\uff1a\u7528\u4e8e\u8ba1\u7b97\u51e0\u4f55\u4e2d\u7684\u77e9\u5f62\u9762\u79ef\u5e76\u7b49\u95ee\u9898<\/li>\n<\/ol>\n\n\n\n<h2 class=\"wp-block-heading\">\u7ebf\u6bb5\u6811\u7684\u53d8\u79cd<\/h2>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>zkw\u7ebf\u6bb5\u6811<\/strong>\uff1a\u975e\u9012\u5f52\u5b9e\u73b0\uff0c\u6548\u7387\u66f4\u9ad8<\/li>\n\n\n\n<li><strong>\u52a8\u6001\u5f00\u70b9\u7ebf\u6bb5\u6811<\/strong>\uff1a\u8282\u7701\u7a7a\u95f4\uff0c\u9002\u7528\u4e8e\u7a00\u758f\u6570\u636e<\/li>\n\n\n\n<li><strong>\u4e8c\u7ef4\u7ebf\u6bb5\u6811<\/strong>\uff1a\u5904\u7406\u4e8c\u7ef4\u5e73\u9762\u4e0a\u7684\u533a\u95f4\u67e5\u8be2<\/li>\n\n\n\n<li><strong>\u6301\u4e45\u5316\u7ebf\u6bb5\u6811<\/strong>\uff1a\u652f\u6301\u5386\u53f2\u7248\u672c\u67e5\u8be2<\/li>\n<\/ol>\n\n\n\n<h2 class=\"wp-block-heading\">\u4ee3\u7801\u793a\u4f8b\uff08Python\u5b9e\u73b0\u533a\u95f4\u548c\u7ebf\u6bb5\u6811\uff09<\/h2>\n\n\n<div class=\"wp-block-syntaxhighlighter-code \"><pre class=\"brush: python; gutter: true; title: ; notranslate\" title=\"\">\nclass SegmentTree:\n    def __init__(self, data):\n        self.n = len(data)\n        self.size = 1\n        while self.size &lt; self.n:\n            self.size &lt;&lt;= 1\n        self.tree = &#x5B;0] * (2 * self.size)\n        # \u521d\u59cb\u5316\u53f6\u5b50\u8282\u70b9\n        for i in range(self.n):\n            self.tree&#x5B;self.size + i] = data&#x5B;i]\n        # \u6784\u5efa\u5185\u90e8\u8282\u70b9\n        for i in range(self.size - 1, 0, -1):\n            self.tree&#x5B;i] = self.tree&#x5B;2 * i] + self.tree&#x5B;2 * i + 1]\n\n    def update(self, index, value):\n        pos = self.size + index\n        self.tree&#x5B;pos] = value\n        pos &gt;&gt;= 1\n        while pos &gt;= 1:\n            new_val = self.tree&#x5B;2 * pos] + self.tree&#x5B;2 * pos + 1]\n            if self.tree&#x5B;pos] == new_val:\n                break\n            self.tree&#x5B;pos] = new_val\n            pos &gt;&gt;= 1\n\n    def query(self, l, r):\n        res = 0\n        l += self.size\n        r += self.size\n        while l &lt;= r:\n            if l % 2 == 1:\n                res += self.tree&#x5B;l]\n                l += 1\n            if r % 2 == 0:\n                res += self.tree&#x5B;r]\n                r -= 1\n            l &gt;&gt;= 1\n            r &gt;&gt;= 1\n        return res\n<\/pre><\/div>\n\n\n<h2 class=\"wp-block-heading\"><strong>\u4ee3\u7801\u793a\u4f8b\uff08Golang\u5b9e\u73b0\u533a\u95f4\u548c\u7ebf\u6bb5\u6811\uff09<\/strong><\/h2>\n\n\n<div class=\"wp-block-syntaxhighlighter-code \"><pre class=\"brush: go; gutter: true; title: ; notranslate\" title=\"\">\npackage main\n\nimport (\n\t&quot;fmt&quot;\n)\n\ntype SegmentTree struct {\n\tdata &#x5B;]int\n\ttree &#x5B;]int\n\tsize int\n}\n\n\/\/ NewSegmentTree \u521b\u5efa\u7ebf\u6bb5\u6811\nfunc NewSegmentTree(data &#x5B;]int) *SegmentTree {\n\tn := len(data)\n\tst := &amp;SegmentTree{\n\t\tdata: data,\n\t\tsize: 1,\n\t}\n\t\n\t\/\/ \u8ba1\u7b97\u7ebf\u6bb5\u6811\u5927\u5c0f(\u6700\u63a5\u8fd1\u4e14\u5927\u4e8e\u7b49\u4e8en\u76842\u7684\u5e42\u6b21)\n\tfor st.size &lt; n {\n\t\tst.size &lt;&lt;= 1\n\t}\n\t\n\t\/\/ \u521d\u59cb\u5316\u7ebf\u6bb5\u6811\n\tst.tree = make(&#x5B;]int, 2*st.size)\n\t\n\t\/\/ \u586b\u5145\u53f6\u5b50\u8282\u70b9\n\tfor i := 0; i &lt; n; i++ {\n\t\tst.tree&#x5B;st.size+i] = data&#x5B;i]\n\t}\n\t\n\t\/\/ \u6784\u5efa\u5185\u90e8\u8282\u70b9\n\tfor i := st.size - 1; i &gt; 0; i-- {\n\t\tst.tree&#x5B;i] = st.tree&#x5B;2*i] + st.tree&#x5B;2*i+1]\n\t}\n\t\n\treturn st\n}\n\n\/\/ Update \u66f4\u65b0\u6307\u5b9a\u4f4d\u7f6e\u7684\u503c\nfunc (st *SegmentTree) Update(index int, value int) {\n\tpos := st.size + index\n\tst.tree&#x5B;pos] = value\n\t\n\t\/\/ \u5411\u4e0a\u66f4\u65b0\u7236\u8282\u70b9\n\tfor pos &gt; 1 {\n\t\tpos &gt;&gt;= 1\n\t\tnewVal := st.tree&#x5B;2*pos] + st.tree&#x5B;2*pos+1]\n\t\tif st.tree&#x5B;pos] == newVal {\n\t\t\tbreak \/\/ \u5982\u679c\u6ca1\u6709\u53d8\u5316\uff0c\u53ef\u4ee5\u63d0\u524d\u7ec8\u6b62\n\t\t}\n\t\tst.tree&#x5B;pos] = newVal\n\t}\n}\n\n\/\/ Query \u67e5\u8be2\u533a\u95f4&#x5B;l, r]\u7684\u548c\nfunc (st *SegmentTree) Query(l, r int) int {\n\tres := 0\n\tl += st.size\n\tr += st.size\n\t\n\tfor l &lt;= r {\n\t\t\/\/ \u5982\u679cl\u662f\u53f3\u5b50\u8282\u70b9\uff0c\u5355\u72ec\u5904\u7406\n\t\tif l%2 == 1 {\n\t\t\tres += st.tree&#x5B;l]\n\t\t\tl++\n\t\t}\n\t\t\n\t\t\/\/ \u5982\u679cr\u662f\u5de6\u5b50\u8282\u70b9\uff0c\u5355\u72ec\u5904\u7406\n\t\tif r%2 == 0 {\n\t\t\tres += st.tree&#x5B;r]\n\t\t\tr--\n\t\t}\n\t\t\n\t\t\/\/ \u79fb\u52a8\u5230\u7236\u8282\u70b9\n\t\tl &gt;&gt;= 1\n\t\tr &gt;&gt;= 1\n\t}\n\t\n\treturn res\n}\n\nfunc main() {\n\tdata := &#x5B;]int{1, 3, 5, 7, 9, 11}\n\tst := NewSegmentTree(data)\n\t\n\tfmt.Println(&quot;\u521d\u59cb\u7ebf\u6bb5\u6811:&quot;)\n\tfmt.Println(st.tree)\n\t\n\tfmt.Println(&quot;\\n\u67e5\u8be2\u533a\u95f4&#x5B;1,4]\u7684\u548c:&quot;, st.Query(1, 4))\n\t\n\tfmt.Println(&quot;\\n\u66f4\u65b0\u7d22\u5f152\u7684\u503c\u4e3a10&quot;)\n\tst.Update(2, 10)\n\tfmt.Println(&quot;\u66f4\u65b0\u540e\u7684\u7ebf\u6bb5\u6811:&quot;, st.tree)\n\t\n\tfmt.Println(&quot;\\n\u67e5\u8be2\u533a\u95f4&#x5B;0,5]\u7684\u548c:&quot;, st.Query(0, 5))\n\tfmt.Println(&quot;\u67e5\u8be2\u533a\u95f4&#x5B;2,3]\u7684\u548c:&quot;, st.Query(2, 3))\n}\n<\/pre><\/div>\n\n\n<h2 class=\"wp-block-heading\">\u603b\u7ed3<\/h2>\n\n\n\n<p>\u7ebf\u6bb5\u6811\u662f\u4e00\u79cd\u529f\u80fd\u5f3a\u5927\u4e14\u7075\u6d3b\u7684\u6570\u636e\u7ed3\u6784\uff0c\u7279\u522b\u9002\u5408\u5904\u7406\u5404\u79cd\u533a\u95f4\u67e5\u8be2\u548c\u66f4\u65b0\u95ee\u9898\u3002\u867d\u7136\u5b9e\u73b0\u8d77\u6765\u6bd4\u7b80\u5355\u7684\u524d\u7f00\u6570\u7ec4\u6216\u5dee\u5206\u6570\u7ec4\u66f4\u590d\u6742\uff0c\u4f46\u5728\u9700\u8981\u9891\u7e41\u6df7\u5408\u67e5\u8be2\u548c\u66f4\u65b0\u7684\u573a\u666f\u4e0b\uff0c\u7ebf\u6bb5\u6811\u7684\u6027\u80fd\u4f18\u52bf\u975e\u5e38\u660e\u663e\u3002<\/p>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u4ec0\u4e48\u662f\u7ebf\u6bb5\u6811\uff1f 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[&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":476,"comment_status":"open","ping_status":"open","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 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