{"id":59359,"date":"2026-01-13T18:50:03","date_gmt":"2026-01-13T10:50:03","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/59359.html"},"modified":"2026-01-13T18:50:03","modified_gmt":"2026-01-13T10:50:03","slug":"leetcode%e7%ae%97%e6%b3%95%e9%a2%98%e8%af%a6%e8%a7%a3-560%ef%bc%9a%e5%92%8c%e4%b8%bak%e7%9a%84%e5%ad%90%e6%95%b0%e7%bb%84","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/59359.html","title":{"rendered":"LeetCode\u7b97\u6cd5\u9898\u8be6\u89e3 560\uff1a\u548c\u4e3aK\u7684\u5b50\u6570\u7ec4"},"content":{"rendered":"<\/p>\n<h4>\u76ee\u5f55<\/h4>\n<ul>\n<li>1. \u95ee\u9898\u63cf\u8ff0<\/li>\n<li>2. \u95ee\u9898\u5206\u6790<\/li>\n<li>\n<ul>\n<li>2.1 \u9898\u76ee\u7406\u89e3<\/li>\n<li>2.2 \u6838\u5fc3\u6d1e\u5bdf<\/li>\n<li>2.3 \u7834\u9898\u5173\u952e<\/li>\n<\/ul>\n<\/li>\n<li>3. \u7b97\u6cd5\u8bbe\u8ba1\u4e0e\u5b9e\u73b0<\/li>\n<li>\n<ul>\n<li>3.1 \u66b4\u529b\u679a\u4e3e\u6cd5<\/li>\n<li>3.2 \u524d\u7f00\u548c\u4f18\u5316<\/li>\n<li>3.3 \u524d\u7f00\u548c&#043;\u54c8\u5e0c\u8868&#xff08;\u6700\u4f18&#xff09;<\/li>\n<li>3.4 \u7a7a\u95f4\u4f18\u5316\u7248\u672c<\/li>\n<\/ul>\n<\/li>\n<li>4. \u6027\u80fd\u5bf9\u6bd4<\/li>\n<li>5. \u6269\u5c55\u4e0e\u53d8\u4f53<\/li>\n<li>\n<ul>\n<li>5.1 \u548c\u4e3aK\u7684\u6700\u957f\u5b50\u6570\u7ec4\u957f\u5ea6<\/li>\n<li>5.2 \u548c\u81f3\u5c11\u4e3aK\u7684\u6700\u77ed\u5b50\u6570\u7ec4<\/li>\n<li>5.3 \u548c\u80fd\u88abK\u6574\u9664\u7684\u5b50\u6570\u7ec4<\/li>\n<li>5.4 \u4e8c\u7ef4\u77e9\u9635\u4e2d\u548c\u4e3aK\u7684\u5b50\u77e9\u9635<\/li>\n<\/ul>\n<\/li>\n<li>6. \u603b\u7ed3<\/li>\n<li>\n<ul>\n<li>6.1 \u6838\u5fc3\u601d\u60f3\u603b\u7ed3<\/li>\n<li>6.2 \u7b97\u6cd5\u9009\u62e9\u6307\u5357<\/li>\n<li>6.3 \u5173\u952e\u5b9e\u73b0\u7ec6\u8282<\/li>\n<li>6.4 \u5e94\u7528\u573a\u666f<\/li>\n<li>6.5 \u9762\u8bd5\u6280\u5de7<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h2>1. \u95ee\u9898\u63cf\u8ff0<\/h2>\n<p>\u7ed9\u5b9a\u4e00\u4e2a\u6574\u6570\u6570\u7ec4 nums \u548c\u4e00\u4e2a\u6574\u6570 k&#xff0c;\u7edf\u8ba1\u5e76\u8fd4\u56de\u8be5\u6570\u7ec4\u4e2d\u548c\u4e3a k \u7684 \u5b50\u6570\u7ec4 \u7684\u4e2a\u6570\u3002<\/p>\n<p>\u5b50\u6570\u7ec4 \u662f\u6570\u7ec4\u4e2d\u5143\u7d20\u7684\u8fde\u7eed\u975e\u7a7a\u5e8f\u5217\u3002<\/p>\n<p>\u793a\u4f8b 1&#xff1a;<\/p>\n<p>\u8f93\u5165&#xff1a;nums &#061; [1,1,1], k &#061; 2<br \/>\n\u8f93\u51fa&#xff1a;2<br \/>\n\u89e3\u91ca&#xff1a;\u5b50\u6570\u7ec4 [1,1] \u548c [1,1]&#xff08;\u6ce8\u610f\u6709\u4e24\u4e2a&#xff0c;\u5206\u522b\u4ece\u7d22\u5f150\u548c1\u5f00\u59cb&#xff09;<\/p>\n<p>\u793a\u4f8b 2&#xff1a;<\/p>\n<p>\u8f93\u5165&#xff1a;nums &#061; [1,2,3], k &#061; 3<br \/>\n\u8f93\u51fa&#xff1a;2<br \/>\n\u89e3\u91ca&#xff1a;\u5b50\u6570\u7ec4 [1,2] \u548c [3]<\/p>\n<p>\u63d0\u793a&#xff1a;<\/p>\n<ul>\n<li>1 &lt;&#061; nums.length &lt;&#061; 2 * 10^4<\/li>\n<li>-1000 &lt;&#061; nums[i] &lt;&#061; 1000<\/li>\n<li>-10^7 &lt;&#061; k &lt;&#061; 10^7<\/li>\n<\/ul>\n<h2>2. \u95ee\u9898\u5206\u6790<\/h2>\n<h3>2.1 \u9898\u76ee\u7406\u89e3<\/h3>\n<p>\u9700\u8981\u627e\u51fa\u6570\u7ec4\u4e2d\u6240\u6709\u8fde\u7eed\u7684\u5b50\u6570\u7ec4&#xff0c;\u8fd9\u4e9b\u5b50\u6570\u7ec4\u7684\u5143\u7d20\u4e4b\u548c\u7b49\u4e8e\u7ed9\u5b9a\u7684\u76ee\u6807\u503c k\u3002\u6ce8\u610f\u5b50\u6570\u7ec4\u5fc5\u987b\u662f\u8fde\u7eed\u7684&#xff0c;\u8fd9\u4e0e\u5b50\u5e8f\u5217\u4e0d\u540c\u3002<\/p>\n<h3>2.2 \u6838\u5fc3\u6d1e\u5bdf<\/h3>\n<ul>\n<li>\u8fde\u7eed\u6027\u7ea6\u675f&#xff1a;\u5b50\u6570\u7ec4\u8981\u6c42\u8fde\u7eed&#xff0c;\u8fd9\u4e3a\u4f7f\u7528\u524d\u7f00\u548c\u63d0\u4f9b\u4e86\u53ef\u80fd<\/li>\n<li>\u8d1f\u6570\u5143\u7d20&#xff1a;\u6570\u7ec4\u53ef\u80fd\u5305\u542b\u8d1f\u6570&#xff0c;\u8fd9\u610f\u5473\u7740\u5b50\u6570\u7ec4\u7684\u548c\u4e0d\u4e00\u5b9a\u5355\u8c03\u9012\u589e<\/li>\n<li>\u9ad8\u6548\u8ba1\u6570&#xff1a;\u9700\u8981\u907f\u514d\u679a\u4e3e\u6240\u6709\u53ef\u80fd\u7684\u5b50\u6570\u7ec4&#xff08;O(n\u00b2)\u65f6\u95f4\u590d\u6742\u5ea6\u8fc7\u9ad8&#xff09;<\/li>\n<\/ul>\n<h3>2.3 \u7834\u9898\u5173\u952e<\/h3>\n<p>\u95ee\u9898\u7684\u6838\u5fc3\u5728\u4e8e\u5982\u4f55\u9ad8\u6548\u5730\u67e5\u627e\u6ee1\u8db3\u6761\u4ef6\u7684\u524d\u7f00\u548c\u5dee\u503c&#xff1a;<\/p>\n<li>\u5b9a\u4e49\u524d\u7f00\u548c&#xff1a;prefix[i] &#061; nums[0] &#043; nums[1] &#043; &#8230; &#043; nums[i-1]<\/li>\n<li>\u5b50\u6570\u7ec4 nums[i..j] \u7684\u548c &#061; prefix[j&#043;1] &#8211; prefix[i]<\/li>\n<li>\u6211\u4eec\u9700\u8981\u627e\u5230\u6240\u6709\u6ee1\u8db3 prefix[j&#043;1] &#8211; prefix[i] &#061; k \u7684 (i, j) \u5bf9<\/li>\n<li>\u8fd9\u7b49\u4ef7\u4e8e\u5bfb\u627e\u6ee1\u8db3 prefix[i] &#061; prefix[j&#043;1] &#8211; k \u7684 i \u5bf9\u6bcf\u4e2a j<\/li>\n<h2>3. \u7b97\u6cd5\u8bbe\u8ba1\u4e0e\u5b9e\u73b0<\/h2>\n<h3>3.1 \u66b4\u529b\u679a\u4e3e\u6cd5<\/h3>\n<p>\u6838\u5fc3\u601d\u60f3<\/p>\n<p>\u679a\u4e3e\u6240\u6709\u53ef\u80fd\u7684\u5b50\u6570\u7ec4&#xff0c;\u8ba1\u7b97\u6bcf\u4e2a\u5b50\u6570\u7ec4\u7684\u548c&#xff0c;\u7edf\u8ba1\u548c\u4e3a k \u7684\u5b50\u6570\u7ec4\u4e2a\u6570\u3002<\/p>\n<p>\u7b97\u6cd5\u601d\u8def<\/p>\n<li>\u904d\u5386\u6240\u6709\u53ef\u80fd\u7684\u8d77\u59cb\u4f4d\u7f6e i (0\u5230n-1)<\/li>\n<li>\u5bf9\u4e8e\u6bcf\u4e2a\u8d77\u59cb\u4f4d\u7f6e&#xff0c;\u904d\u5386\u7ed3\u675f\u4f4d\u7f6e j (i\u5230n-1)<\/li>\n<li>\u8ba1\u7b97\u5b50\u6570\u7ec4 nums[i..j] \u7684\u548c<\/li>\n<li>\u5982\u679c\u548c\u7b49\u4e8e k&#xff0c;\u8ba1\u6570\u5668\u52a01<\/li>\n<p>Java\u4ee3\u7801\u5b9e\u73b0<\/p>\n<p><span class=\"token keyword\">import<\/span> <span class=\"token namespace\">java<span class=\"token punctuation\">.<\/span>util<span class=\"token punctuation\">.<\/span><\/span><span class=\"token operator\">*<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p><span class=\"token keyword\">public<\/span> <span class=\"token keyword\">class<\/span> <span class=\"token class-name\">SubarraySumEqualsKBruteForce<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token comment\">\/**<br \/>\n     * \u66b4\u529b\u89e3\u6cd5 &#8211; \u4e09\u5c42\u5faa\u73af<br \/>\n     * \u65f6\u95f4\u590d\u6742\u5ea6: O(n\u00b3)<br \/>\n     * \u7a7a\u95f4\u590d\u6742\u5ea6: O(1)<br \/>\n     *\/<\/span><br \/>\n    <span class=\"token keyword\">public<\/span> <span class=\"token keyword\">int<\/span> <span class=\"token function\">subarraySumBruteForce1<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span><span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span> nums<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> k<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token keyword\">int<\/span> count <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">int<\/span> n <span class=\"token operator\">&#061;<\/span> nums<span class=\"token punctuation\">.<\/span>length<span class=\"token punctuation\">;<\/span><\/p>\n<p>        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> start <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> start <span class=\"token operator\">&lt;<\/span> n<span class=\"token punctuation\">;<\/span> start<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n            <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> end <span class=\"token operator\">&#061;<\/span> start<span class=\"token punctuation\">;<\/span> end <span class=\"token operator\">&lt;<\/span> n<span class=\"token punctuation\">;<\/span> end<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n                <span class=\"token keyword\">int<\/span> sum <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n                <span class=\"token comment\">\/\/ \u8ba1\u7b97\u5b50\u6570\u7ec4\u7684\u548c<\/span><br \/>\n                <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> i <span class=\"token operator\">&#061;<\/span> start<span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;&#061;<\/span> end<span class=\"token punctuation\">;<\/span> i<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n                    sum <span class=\"token operator\">&#043;&#061;<\/span> nums<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n                <span class=\"token punctuation\">}<\/span><br \/>\n                <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>sum <span class=\"token operator\">&#061;&#061;<\/span> k<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n                    count<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">;<\/span><br \/>\n                <span class=\"token punctuation\">}<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><\/p>\n<p>        <span class=\"token keyword\">return<\/span> count<span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><\/p>\n<p>    <span class=\"token comment\">\/**<br \/>\n     * \u4f18\u5316\u7248\u66b4\u529b\u89e3\u6cd5 &#8211; \u51cf\u5c11\u91cd\u590d\u8ba1\u7b97<br \/>\n     * \u65f6\u95f4\u590d\u6742\u5ea6: O(n\u00b2)<br \/>\n     * \u7a7a\u95f4\u590d\u6742\u5ea6: O(1)<br \/>\n     *\/<\/span><br \/>\n    <span class=\"token keyword\">public<\/span> <span class=\"token keyword\">int<\/span> <span class=\"token function\">subarraySumBruteForce2<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span><span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span> nums<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> k<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token keyword\">int<\/span> count <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">int<\/span> n <span class=\"token operator\">&#061;<\/span> nums<span class=\"token punctuation\">.<\/span>length<span class=\"token punctuation\">;<\/span><\/p>\n<p>        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> start <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> start <span class=\"token operator\">&lt;<\/span> n<span class=\"token punctuation\">;<\/span> start<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n            <span class=\"token keyword\">int<\/span> sum <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> end <span class=\"token operator\">&#061;<\/span> start<span class=\"token punctuation\">;<\/span> end <span class=\"token operator\">&lt;<\/span> n<span class=\"token punctuation\">;<\/span> end<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n                sum <span class=\"token operator\">&#043;&#061;<\/span> nums<span class=\"token punctuation\">[<\/span>end<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span> <span class=\"token comment\">\/\/ \u7d2f\u52a0\u8ba1\u7b97<\/span><br \/>\n                <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>sum <span class=\"token operator\">&#061;&#061;<\/span> k<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n                    count<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">;<\/span><br \/>\n                <span class=\"token punctuation\">}<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><\/p>\n<p>        <span class=\"token keyword\">return<\/span> count<span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><\/p>\n<p>    <span class=\"token comment\">\/**<br \/>\n     * \u6027\u80fd\u5206\u6790<br \/>\n     *\/<\/span><br \/>\n    <span class=\"token keyword\">public<\/span> <span class=\"token keyword\">static<\/span> <span class=\"token keyword\">void<\/span> <span class=\"token function\">analyzePerformance<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span><span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span> nums<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> k<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token class-name\">SubarraySumEqualsKBruteForce<\/span> solution <span class=\"token operator\">&#061;<\/span> <span class=\"token keyword\">new<\/span> <span class=\"token class-name\">SubarraySumEqualsKBruteForce<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">long<\/span> startTime <span class=\"token operator\">&#061;<\/span> <span class=\"token class-name\">System<\/span><span class=\"token punctuation\">.<\/span><span class=\"token function\">nanoTime<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">int<\/span> result <span class=\"token operator\">&#061;<\/span> solution<span class=\"token punctuation\">.<\/span><span class=\"token function\">subarraySumBruteForce2<\/span><span class=\"token punctuation\">(<\/span>nums<span class=\"token punctuation\">,<\/span> k<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">long<\/span> endTime <span class=\"token operator\">&#061;<\/span> <span class=\"token class-name\">System<\/span><span class=\"token punctuation\">.<\/span><span class=\"token function\">nanoTime<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>        <span class=\"token class-name\">System<\/span><span class=\"token punctuation\">.<\/span>out<span class=\"token punctuation\">.<\/span><span class=\"token function\">println<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;\u66b4\u529b\u89e3\u6cd5\u7ed3\u679c: &#034;<\/span> <span class=\"token operator\">&#043;<\/span> result<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token class-name\">System<\/span><span class=\"token punctuation\">.<\/span>out<span class=\"token punctuation\">.<\/span><span class=\"token function\">println<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;\u6267\u884c\u65f6\u95f4: &#034;<\/span> <span class=\"token operator\">&#043;<\/span> <span class=\"token punctuation\">(<\/span>endTime <span class=\"token operator\">&#8211;<\/span> startTime<span class=\"token punctuation\">)<\/span> <span class=\"token operator\">&#043;<\/span> <span class=\"token string\">&#034; ns&#034;<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token class-name\">System<\/span><span class=\"token punctuation\">.<\/span>out<span class=\"token punctuation\">.<\/span><span class=\"token function\">println<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;\u65f6\u95f4\u590d\u6742\u5ea6: O(n\u00b2)&#034;<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token class-name\">System<\/span><span class=\"token punctuation\">.<\/span>out<span class=\"token punctuation\">.<\/span><span class=\"token function\">println<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;\u7a7a\u95f4\u590d\u6742\u5ea6: O(1)&#034;<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<p>\u6027\u80fd\u5206\u6790<\/p>\n<ul>\n<li>\u65f6\u95f4\u590d\u6742\u5ea6&#xff1a;O(n\u00b2)&#xff0c;\u5176\u4e2dn\u4e3a\u6570\u7ec4\u957f\u5ea6\u3002\u9700\u8981\u68c0\u67e5\u6240\u6709\u53ef\u80fd\u7684\u5b50\u6570\u7ec4\u3002<\/li>\n<li>\u7a7a\u95f4\u590d\u6742\u5ea6&#xff1a;O(1)&#xff0c;\u53ea\u4f7f\u7528\u4e86\u5e38\u6570\u7ea7\u522b\u7684\u989d\u5916\u7a7a\u95f4\u3002<\/li>\n<li>\u9002\u7528\u573a\u666f&#xff1a;\u4ec5\u9002\u7528\u4e8e\u975e\u5e38\u5c0f\u7684\u8f93\u5165\u89c4\u6a21&#xff08;n \u2264 100&#xff09;\u3002<\/li>\n<\/ul>\n<h3>3.2 \u524d\u7f00\u548c\u4f18\u5316<\/h3>\n<p>\u6838\u5fc3\u601d\u60f3<\/p>\n<p>\u4f7f\u7528\u524d\u7f00\u548c\u6570\u7ec4\u63d0\u524d\u8ba1\u7b97\u6240\u6709\u524d\u7f00\u7684\u548c&#xff0c;\u7136\u540e\u901a\u8fc7\u524d\u7f00\u548c\u7684\u5dee\u503c\u5feb\u901f\u8ba1\u7b97\u4efb\u610f\u5b50\u6570\u7ec4\u7684\u548c\u3002<\/p>\n<p>\u7b97\u6cd5\u601d\u8def<\/p>\n<li>\u8ba1\u7b97\u524d\u7f00\u548c\u6570\u7ec4 prefix&#xff0c;\u5176\u4e2d prefix[i] \u8868\u793a\u524d i \u4e2a\u5143\u7d20\u7684\u548c\n<ul>\n<li>prefix[0] &#061; 0<\/li>\n<li>prefix[i] &#061; prefix[i-1] &#043; nums[i-1]<\/li>\n<\/ul>\n<\/li>\n<li>\u5b50\u6570\u7ec4 nums[i..j] \u7684\u548c &#061; prefix[j&#043;1] &#8211; prefix[i]<\/li>\n<li>\u679a\u4e3e\u6240\u6709 (i, j) \u5bf9&#xff0c;\u8ba1\u7b97\u5dee\u503c\u5e76\u7edf\u8ba1\u7b49\u4e8e k \u7684\u4e2a\u6570<\/li>\n<p>Java\u4ee3\u7801\u5b9e\u73b0<\/p>\n<p><span class=\"token keyword\">public<\/span> <span class=\"token keyword\">class<\/span> <span class=\"token class-name\">SubarraySumEqualsKPrefixSum<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token comment\">\/**<br \/>\n     * \u524d\u7f00\u548c\u89e3\u6cd5<br \/>\n     * \u65f6\u95f4\u590d\u6742\u5ea6: O(n\u00b2)<br \/>\n     * \u7a7a\u95f4\u590d\u6742\u5ea6: O(n)<br \/>\n     *\/<\/span><br \/>\n    <span class=\"token keyword\">public<\/span> <span class=\"token keyword\">int<\/span> <span class=\"token function\">subarraySum<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span><span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span> nums<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> k<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token keyword\">int<\/span> n <span class=\"token operator\">&#061;<\/span> nums<span class=\"token punctuation\">.<\/span>length<span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">int<\/span> count <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>        <span class=\"token comment\">\/\/ \u8ba1\u7b97\u524d\u7f00\u548c\u6570\u7ec4<\/span><br \/>\n        <span class=\"token keyword\">int<\/span><span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span> prefix <span class=\"token operator\">&#061;<\/span> <span class=\"token keyword\">new<\/span> <span class=\"token keyword\">int<\/span><span class=\"token punctuation\">[<\/span>n <span class=\"token operator\">&#043;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        prefix<span class=\"token punctuation\">[<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;&#061;<\/span> n<span class=\"token punctuation\">;<\/span> i<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n            prefix<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> prefix<span class=\"token punctuation\">[<\/span>i <span class=\"token operator\">&#8211;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#043;<\/span> nums<span class=\"token punctuation\">[<\/span>i <span class=\"token operator\">&#8211;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><\/p>\n<p>        <span class=\"token comment\">\/\/ \u679a\u4e3e\u6240\u6709\u5b50\u6570\u7ec4<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;<\/span> n<span class=\"token punctuation\">;<\/span> i<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n            <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> j <span class=\"token operator\">&#061;<\/span> i<span class=\"token punctuation\">;<\/span> j <span class=\"token operator\">&lt;<\/span> n<span class=\"token punctuation\">;<\/span> j<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n                <span class=\"token comment\">\/\/ \u5b50\u6570\u7ec4 nums[i..j] \u7684\u548c &#061; prefix[j&#043;1] &#8211; prefix[i]<\/span><br \/>\n                <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>prefix<span class=\"token punctuation\">[<\/span>j <span class=\"token operator\">&#043;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#8211;<\/span> prefix<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;&#061;<\/span> k<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n                    count<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">;<\/span><br \/>\n                <span class=\"token punctuation\">}<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><\/p>\n<p>        <span class=\"token keyword\">return<\/span> count<span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><\/p>\n<p>    <span class=\"token comment\">\/**<br \/>\n     * \u4f18\u5316\u7248&#xff1a;\u51cf\u5c11\u4e00\u5c42\u5faa\u73af<br \/>\n     *\/<\/span><br \/>\n    <span class=\"token keyword\">public<\/span> <span class=\"token keyword\">int<\/span> <span class=\"token function\">subarraySumOptimized<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span><span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span> nums<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> k<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token keyword\">int<\/span> n <span class=\"token operator\">&#061;<\/span> nums<span class=\"token punctuation\">.<\/span>length<span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">int<\/span> count <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>        <span class=\"token comment\">\/\/ \u8ba1\u7b97\u524d\u7f00\u548c\u6570\u7ec4<\/span><br \/>\n        <span class=\"token keyword\">int<\/span><span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span> prefix <span class=\"token operator\">&#061;<\/span> <span class=\"token keyword\">new<\/span> <span class=\"token keyword\">int<\/span><span class=\"token punctuation\">[<\/span>n <span class=\"token operator\">&#043;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        prefix<span class=\"token punctuation\">[<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;&#061;<\/span> n<span class=\"token punctuation\">;<\/span> i<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n            prefix<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> prefix<span class=\"token punctuation\">[<\/span>i <span class=\"token operator\">&#8211;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#043;<\/span> nums<span class=\"token punctuation\">[<\/span>i <span class=\"token operator\">&#8211;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><\/p>\n<p>        <span class=\"token comment\">\/\/ \u5bf9\u4e8e\u6bcf\u4e2a\u7ed3\u675f\u4f4d\u7f6ej&#xff0c;\u5bfb\u627e\u5f00\u59cb\u4f4d\u7f6ei\u4f7f\u5f97prefix[j&#043;1] &#8211; prefix[i] &#061; k<\/span><br \/>\n        <span class=\"token comment\">\/\/ \u5373\u5bfb\u627eprefix[i] &#061; prefix[j&#043;1] &#8211; k<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> end <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> end <span class=\"token operator\">&lt;<\/span> n<span class=\"token punctuation\">;<\/span> end<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n            <span class=\"token keyword\">int<\/span> target <span class=\"token operator\">&#061;<\/span> prefix<span class=\"token punctuation\">[<\/span>end <span class=\"token operator\">&#043;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#8211;<\/span> k<span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> start <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> start <span class=\"token operator\">&lt;&#061;<\/span> end<span class=\"token punctuation\">;<\/span> start<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n                <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>prefix<span class=\"token punctuation\">[<\/span>start<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;&#061;<\/span> target<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n                    count<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">;<\/span><br \/>\n                <span class=\"token punctuation\">}<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><\/p>\n<p>        <span class=\"token keyword\">return<\/span> count<span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<p>\u6027\u80fd\u5206\u6790<\/p>\n<ul>\n<li>\u65f6\u95f4\u590d\u6742\u5ea6&#xff1a;O(n\u00b2)&#xff0c;\u4ecd\u7136\u9700\u8981\u4e24\u5c42\u5faa\u73af<\/li>\n<li>\u7a7a\u95f4\u590d\u6742\u5ea6&#xff1a;O(n)&#xff0c;\u9700\u8981\u5b58\u50a8\u524d\u7f00\u548c\u6570\u7ec4<\/li>\n<li>\u4f18\u52bf&#xff1a;\u51cf\u5c11\u4e86\u91cd\u590d\u8ba1\u7b97&#xff0c;\u4f46\u4ecd\u7136\u4e0d\u591f\u9ad8\u6548<\/li>\n<\/ul>\n<h3>3.3 \u524d\u7f00\u548c&#043;\u54c8\u5e0c\u8868&#xff08;\u6700\u4f18&#xff09;<\/h3>\n<p>\u6838\u5fc3\u601d\u60f3 \u4f7f\u7528\u54c8\u5e0c\u8868\u8bb0\u5f55\u6bcf\u4e2a\u524d\u7f00\u548c\u51fa\u73b0\u7684\u6b21\u6570\u3002\u5bf9\u4e8e\u5f53\u524d\u524d\u7f00\u548c sum&#xff0c;\u5982\u679c sum &#8211; k \u5728\u54c8\u5e0c\u8868\u4e2d\u51fa\u73b0\u8fc7&#xff0c;\u8bf4\u660e\u5b58\u5728\u4ee5\u5f53\u524d\u4f4d\u7f6e\u7ed3\u5c3e\u7684\u5b50\u6570\u7ec4\u548c\u4e3a k\u3002<\/p>\n<p>\u7b97\u6cd5\u601d\u8def<\/p>\n<li>\u521d\u59cb\u5316\u54c8\u5e0c\u8868 map&#xff0c;\u952e\u4e3a\u524d\u7f00\u548c&#xff0c;\u503c\u4e3a\u8be5\u524d\u7f00\u548c\u51fa\u73b0\u7684\u6b21\u6570<\/li>\n<li>\u521d\u59cb\u5316 prefixSum &#061; 0&#xff0c;count &#061; 0<\/li>\n<li>\u5c06 (0, 1) \u653e\u5165\u54c8\u5e0c\u8868&#xff0c;\u8868\u793a\u524d\u7f00\u548c\u4e3a0\u51fa\u73b0\u4e861\u6b21&#xff08;\u7a7a\u6570\u7ec4&#xff09;<\/li>\n<li>\u904d\u5386\u6570\u7ec4&#xff1a;\n<ul>\n<li>\u66f4\u65b0\u5f53\u524d\u524d\u7f00\u548c&#xff1a;prefixSum &#043;&#061; nums[i]<\/li>\n<li>\u8ba1\u7b97\u76ee\u6807\u503c&#xff1a;target &#061; prefixSum &#8211; k<\/li>\n<li>\u5982\u679c target \u5728\u54c8\u5e0c\u8868\u4e2d&#xff0c;\u5219 count &#043;&#061; map.get(target)<\/li>\n<li>\u66f4\u65b0\u54c8\u5e0c\u8868\u4e2d prefixSum \u7684\u8ba1\u6570<\/li>\n<\/ul>\n<\/li>\n<li>\u8fd4\u56de count<\/li>\n<p>Java\u4ee3\u7801\u5b9e\u73b0<\/p>\n<p><span class=\"token keyword\">import<\/span> <span class=\"token namespace\">java<span class=\"token punctuation\">.<\/span>util<span class=\"token punctuation\">.<\/span><\/span><span class=\"token class-name\">HashMap<\/span><span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token keyword\">import<\/span> <span class=\"token namespace\">java<span class=\"token punctuation\">.<\/span>util<span class=\"token punctuation\">.<\/span><\/span><span class=\"token class-name\">Map<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p><span class=\"token keyword\">public<\/span> <span class=\"token keyword\">class<\/span> <span class=\"token class-name\">SubarraySumEqualsKHashMap<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token comment\">\/**<br \/>\n     * \u524d\u7f00\u548c&#043;\u54c8\u5e0c\u8868\u89e3\u6cd5&#xff08;\u6700\u4f18&#xff09;<br \/>\n     * \u65f6\u95f4\u590d\u6742\u5ea6: O(n)<br \/>\n     * \u7a7a\u95f4\u590d\u6742\u5ea6: O(n)<br \/>\n     *\/<\/span><br \/>\n    <span class=\"token keyword\">public<\/span> <span class=\"token keyword\">int<\/span> <span class=\"token function\">subarraySum<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span><span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span> nums<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> k<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token keyword\">int<\/span> count <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">int<\/span> prefixSum <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>        <span class=\"token comment\">\/\/ \u54c8\u5e0c\u8868&#xff1a;\u952e\u4e3a\u524d\u7f00\u548c&#xff0c;\u503c\u4e3a\u8be5\u524d\u7f00\u548c\u51fa\u73b0\u7684\u6b21\u6570<\/span><br \/>\n        <span class=\"token class-name\">Map<\/span><span class=\"token generics\"><span class=\"token punctuation\">&lt;<\/span><span class=\"token class-name\">Integer<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token class-name\">Integer<\/span><span class=\"token punctuation\">&gt;<\/span><\/span> prefixSumMap <span class=\"token operator\">&#061;<\/span> <span class=\"token keyword\">new<\/span> <span class=\"token class-name\">HashMap<\/span><span class=\"token generics\"><span class=\"token punctuation\">&lt;<\/span><span class=\"token punctuation\">&gt;<\/span><\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>        <span class=\"token comment\">\/\/ \u521d\u59cb\u5316&#xff1a;\u524d\u7f00\u548c\u4e3a0\u51fa\u73b01\u6b21&#xff08;\u7a7a\u6570\u7ec4&#xff09;<\/span><br \/>\n        prefixSumMap<span class=\"token punctuation\">.<\/span><span class=\"token function\">put<\/span><span class=\"token punctuation\">(<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> num <span class=\"token operator\">:<\/span> nums<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n            <span class=\"token comment\">\/\/ \u66f4\u65b0\u5f53\u524d\u524d\u7f00\u548c<\/span><br \/>\n            prefixSum <span class=\"token operator\">&#043;&#061;<\/span> num<span class=\"token punctuation\">;<\/span><\/p>\n<p>            <span class=\"token comment\">\/\/ \u8ba1\u7b97\u76ee\u6807\u524d\u7f00\u548c&#xff1a;prefixSum &#8211; target &#061; k &#061;&gt; target &#061; prefixSum &#8211; k<\/span><br \/>\n            <span class=\"token keyword\">int<\/span> target <span class=\"token operator\">&#061;<\/span> prefixSum <span class=\"token operator\">&#8211;<\/span> k<span class=\"token punctuation\">;<\/span><\/p>\n<p>            <span class=\"token comment\">\/\/ \u5982\u679c\u76ee\u6807\u524d\u7f00\u548c\u5b58\u5728&#xff0c;\u5219\u7d2f\u52a0\u8ba1\u6570<\/span><br \/>\n            <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>prefixSumMap<span class=\"token punctuation\">.<\/span><span class=\"token function\">containsKey<\/span><span class=\"token punctuation\">(<\/span>target<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n                count <span class=\"token operator\">&#043;&#061;<\/span> prefixSumMap<span class=\"token punctuation\">.<\/span><span class=\"token function\">get<\/span><span class=\"token punctuation\">(<\/span>target<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><\/p>\n<p>            <span class=\"token comment\">\/\/ \u66f4\u65b0\u5f53\u524d\u524d\u7f00\u548c\u7684\u8ba1\u6570<\/span><br \/>\n            prefixSumMap<span class=\"token punctuation\">.<\/span><span class=\"token function\">put<\/span><span class=\"token punctuation\">(<\/span>prefixSum<span class=\"token punctuation\">,<\/span> prefixSumMap<span class=\"token punctuation\">.<\/span><span class=\"token function\">getOrDefault<\/span><span class=\"token punctuation\">(<\/span>prefixSum<span class=\"token punctuation\">,<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token operator\">&#043;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><\/p>\n<p>        <span class=\"token keyword\">return<\/span> count<span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><\/p>\n<p>    <span class=\"token comment\">\/**<br \/>\n     * \u8be6\u7ec6\u6ce8\u91ca\u7248\u672c<br \/>\n     *\/<\/span><br \/>\n    <span class=\"token keyword\">public<\/span> <span class=\"token keyword\">int<\/span> <span class=\"token function\">subarraySumDetailed<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span><span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span> nums<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> k<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token comment\">\/\/ \u7edf\u8ba1\u7ed3\u679c<\/span><br \/>\n        <span class=\"token keyword\">int<\/span> count <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>        <span class=\"token comment\">\/\/ \u5f53\u524d\u524d\u7f00\u548c<\/span><br \/>\n        <span class=\"token keyword\">int<\/span> currentSum <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>        <span class=\"token comment\">\/\/ \u54c8\u5e0c\u8868\u5b58\u50a8\u524d\u7f00\u548c\u53ca\u5176\u51fa\u73b0\u6b21\u6570<\/span><br \/>\n        <span class=\"token comment\">\/\/ key: \u524d\u7f00\u548c, value: \u8be5\u524d\u7f00\u548c\u51fa\u73b0\u7684\u6b21\u6570<\/span><br \/>\n        <span class=\"token class-name\">Map<\/span><span class=\"token generics\"><span class=\"token punctuation\">&lt;<\/span><span class=\"token class-name\">Integer<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token class-name\">Integer<\/span><span class=\"token punctuation\">&gt;<\/span><\/span> sumFrequency <span class=\"token operator\">&#061;<\/span> <span class=\"token keyword\">new<\/span> <span class=\"token class-name\">HashMap<\/span><span class=\"token generics\"><span class=\"token punctuation\">&lt;<\/span><span class=\"token punctuation\">&gt;<\/span><\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>        <span class=\"token comment\">\/\/ \u91cd\u8981&#xff1a;\u521d\u59cb\u5316\u54c8\u5e0c\u8868&#xff0c;\u524d\u7f00\u548c\u4e3a0\u51fa\u73b01\u6b21<\/span><br \/>\n        <span class=\"token comment\">\/\/ \u8fd9\u4ee3\u8868\u7a7a\u6570\u7ec4\u7684\u60c5\u51b5&#xff0c;\u5bf9\u4e8e\u4ece\u6570\u7ec4\u5f00\u5934\u5f00\u59cb\u7684\u5b50\u6570\u7ec4\u5f88\u91cd\u8981<\/span><br \/>\n        sumFrequency<span class=\"token punctuation\">.<\/span><span class=\"token function\">put<\/span><span class=\"token punctuation\">(<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>        <span class=\"token comment\">\/\/ \u904d\u5386\u6570\u7ec4<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;<\/span> nums<span class=\"token punctuation\">.<\/span>length<span class=\"token punctuation\">;<\/span> i<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n            <span class=\"token comment\">\/\/ \u8ba1\u7b97\u5230\u5f53\u524d\u4f4d\u7f6e\u7684\u524d\u7f00\u548c<\/span><br \/>\n            currentSum <span class=\"token operator\">&#043;&#061;<\/span> nums<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>            <span class=\"token comment\">\/\/ \u6211\u4eec\u9700\u8981\u5bfb\u627e\u7684\u5b50\u6570\u7ec4\u6ee1\u8db3&#xff1a;currentSum &#8211; prefixSum &#061; k<\/span><br \/>\n            <span class=\"token comment\">\/\/ \u6240\u4ee5\u9700\u8981\u5bfb\u627e\u7684\u524d\u7f00\u548c\u662f&#xff1a;prefixSum &#061; currentSum &#8211; k<\/span><br \/>\n            <span class=\"token keyword\">int<\/span> neededSum <span class=\"token operator\">&#061;<\/span> currentSum <span class=\"token operator\">&#8211;<\/span> k<span class=\"token punctuation\">;<\/span><\/p>\n<p>            <span class=\"token comment\">\/\/ \u5982\u679c\u8fd9\u4e2a\u9700\u8981\u7684\u524d\u7f00\u548c\u4e4b\u524d\u51fa\u73b0\u8fc7<\/span><br \/>\n            <span class=\"token comment\">\/\/ \u8bf4\u660e\u5b58\u5728\u4ee5\u5f53\u524d\u4f4d\u7f6e\u7ed3\u5c3e\u7684\u5b50\u6570\u7ec4\u548c\u4e3ak<\/span><br \/>\n            <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>sumFrequency<span class=\"token punctuation\">.<\/span><span class=\"token function\">containsKey<\/span><span class=\"token punctuation\">(<\/span>neededSum<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n                <span class=\"token comment\">\/\/ \u7d2f\u52a0\u6b21\u6570&#xff1a;\u6bcf\u4e2a\u51fa\u73b0\u7684\u4f4d\u7f6e\u90fd\u5bf9\u5e94\u4e00\u4e2a\u6709\u6548\u7684\u5b50\u6570\u7ec4<\/span><br \/>\n                count <span class=\"token operator\">&#043;&#061;<\/span> sumFrequency<span class=\"token punctuation\">.<\/span><span class=\"token function\">get<\/span><span class=\"token punctuation\">(<\/span>neededSum<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><\/p>\n<p>            <span class=\"token comment\">\/\/ \u66f4\u65b0\u5f53\u524d\u524d\u7f00\u548c\u7684\u51fa\u73b0\u6b21\u6570<\/span><br \/>\n            sumFrequency<span class=\"token punctuation\">.<\/span><span class=\"token function\">put<\/span><span class=\"token punctuation\">(<\/span>currentSum<span class=\"token punctuation\">,<\/span> sumFrequency<span class=\"token punctuation\">.<\/span><span class=\"token function\">getOrDefault<\/span><span class=\"token punctuation\">(<\/span>currentSum<span class=\"token punctuation\">,<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token operator\">&#043;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><\/p>\n<p>        <span class=\"token keyword\">return<\/span> count<span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><\/p>\n<p>    <span class=\"token comment\">\/**<br \/>\n     * \u5904\u7406\u5927\u6570\u6ea2\u51fa\u7684\u7248\u672c<br \/>\n     *\/<\/span><br \/>\n    <span class=\"token keyword\">public<\/span> <span class=\"token keyword\">int<\/span> <span class=\"token function\">subarraySumWithOverflowProtection<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span><span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span> nums<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> k<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token keyword\">int<\/span> count <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">long<\/span> prefixSum <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0L<\/span><span class=\"token punctuation\">;<\/span> <span class=\"token comment\">\/\/ \u4f7f\u7528long\u907f\u514d\u6ea2\u51fa<\/span><\/p>\n<p>        <span class=\"token class-name\">Map<\/span><span class=\"token generics\"><span class=\"token punctuation\">&lt;<\/span><span class=\"token class-name\">Long<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token class-name\">Integer<\/span><span class=\"token punctuation\">&gt;<\/span><\/span> map <span class=\"token operator\">&#061;<\/span> <span class=\"token keyword\">new<\/span> <span class=\"token class-name\">HashMap<\/span><span class=\"token generics\"><span class=\"token punctuation\">&lt;<\/span><span class=\"token punctuation\">&gt;<\/span><\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        map<span class=\"token punctuation\">.<\/span><span class=\"token function\">put<\/span><span class=\"token punctuation\">(<\/span><span class=\"token number\">0L<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> num <span class=\"token operator\">:<\/span> nums<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n            prefixSum <span class=\"token operator\">&#043;&#061;<\/span> num<span class=\"token punctuation\">;<\/span><\/p>\n<p>            <span class=\"token keyword\">long<\/span> target <span class=\"token operator\">&#061;<\/span> prefixSum <span class=\"token operator\">&#8211;<\/span> k<span class=\"token punctuation\">;<\/span><\/p>\n<p>            <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>map<span class=\"token punctuation\">.<\/span><span class=\"token function\">containsKey<\/span><span class=\"token punctuation\">(<\/span>target<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n                count <span class=\"token operator\">&#043;&#061;<\/span> map<span class=\"token punctuation\">.<\/span><span class=\"token function\">get<\/span><span class=\"token punctuation\">(<\/span>target<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><\/p>\n<p>            map<span class=\"token punctuation\">.<\/span><span class=\"token function\">put<\/span><span class=\"token punctuation\">(<\/span>prefixSum<span class=\"token punctuation\">,<\/span> map<span class=\"token punctuation\">.<\/span><span class=\"token function\">getOrDefault<\/span><span class=\"token punctuation\">(<\/span>prefixSum<span class=\"token punctuation\">,<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token operator\">&#043;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><\/p>\n<p>        <span class=\"token keyword\">return<\/span> count<span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<p>\u56fe\u89e3\u7b97\u6cd5<\/p>\n<p>\u793a\u4f8b: nums &#061; [1, 1, 1], k &#061; 2<\/p>\n<p>\u6b65\u9aa40: map &#061; {0:1}, count &#061; 0, prefixSum &#061; 0<\/p>\n<p>i&#061;0: num&#061;1<br \/>\n  prefixSum &#061; 0 &#043; 1 &#061; 1<br \/>\n  target &#061; 1 &#8211; 2 &#061; -1<br \/>\n  map\u4e2d\u6ca1\u6709-1&#xff0c;count\u4e0d\u53d8<br \/>\n  map\u66f4\u65b0: {0:1, 1:1}<\/p>\n<p>i&#061;1: num&#061;1<br \/>\n  prefixSum &#061; 1 &#043; 1 &#061; 2<br \/>\n  target &#061; 2 &#8211; 2 &#061; 0<br \/>\n  map\u4e2d\u67090&#xff0c;\u51fa\u73b01\u6b21&#xff0c;count &#043;&#061; 1 \u2192 count&#061;1<br \/>\n  map\u66f4\u65b0: {0:1, 1:1, 2:1}<\/p>\n<p>i&#061;2: num&#061;1<br \/>\n  prefixSum &#061; 2 &#043; 1 &#061; 3<br \/>\n  target &#061; 3 &#8211; 2 &#061; 1<br \/>\n  map\u4e2d\u67091&#xff0c;\u51fa\u73b01\u6b21&#xff0c;count &#043;&#061; 1 \u2192 count&#061;2<br \/>\n  map\u66f4\u65b0: {0:1, 1:1, 2:1, 3:1}<\/p>\n<p>\u8fd4\u56de: count &#061; 2<\/p>\n<p>\u6027\u80fd\u5206\u6790<\/p>\n<ul>\n<li>\u65f6\u95f4\u590d\u6742\u5ea6&#xff1a;O(n)&#xff0c;\u53ea\u9700\u8981\u4e00\u6b21\u904d\u5386<\/li>\n<li>\u7a7a\u95f4\u590d\u6742\u5ea6&#xff1a;O(n)&#xff0c;\u6700\u574f\u60c5\u51b5\u4e0b\u9700\u8981\u5b58\u50a8n\u4e2a\u4e0d\u540c\u7684\u524d\u7f00\u548c<\/li>\n<li>\u4f18\u52bf&#xff1a;\u9ad8\u6548\u5904\u7406\u5927\u89c4\u6a21\u6570\u636e&#xff0c;\u5de7\u5999\u5229\u7528\u54c8\u5e0c\u8868\u5b9e\u73b0O(1)\u67e5\u627e<\/li>\n<\/ul>\n<h3>3.4 \u7a7a\u95f4\u4f18\u5316\u7248\u672c<\/h3>\n<p>\u6838\u5fc3\u601d\u60f3<\/p>\n<p>\u5728\u4e0d\u9700\u8981\u8bb0\u5f55\u6240\u6709\u524d\u7f00\u548c\u5177\u4f53\u4f4d\u7f6e\u7684\u60c5\u51b5\u4e0b&#xff0c;\u53ef\u4ee5\u4f7f\u7528\u6570\u7ec4\u6216\u7279\u6b8a\u6570\u636e\u7ed3\u6784\u8fdb\u884c\u4f18\u5316\u3002<\/p>\n<p>Java\u4ee3\u7801\u5b9e\u73b0<\/p>\n<p><span class=\"token keyword\">public<\/span> <span class=\"token keyword\">class<\/span> <span class=\"token class-name\">SubarraySumEqualsKSpaceOptimized<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token comment\">\/**<br \/>\n     * \u4f7f\u7528\u6570\u7ec4\u4ee3\u66ff\u54c8\u5e0c\u8868&#xff08;\u9002\u7528\u4e8e\u524d\u7f00\u548c\u8303\u56f4\u8f83\u5c0f\u7684\u60c5\u51b5&#xff09;<br \/>\n     * \u6ce8\u610f&#xff1a;\u7531\u4e8ek\u7684\u8303\u56f4\u5f88\u5927&#xff0c;\u8fd9\u79cd\u65b9\u6cd5\u901a\u5e38\u4e0d\u9002\u7528&#xff0c;\u4f46\u5c55\u793a\u4e00\u79cd\u601d\u8def<br \/>\n     *\/<\/span><br \/>\n    <span class=\"token keyword\">public<\/span> <span class=\"token keyword\">int<\/span> <span class=\"token function\">subarraySumArray<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span><span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span> nums<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> k<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token keyword\">int<\/span> count <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">int<\/span> prefixSum <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>        <span class=\"token comment\">\/\/ \u8ba1\u7b97\u524d\u7f00\u548c\u7684\u8303\u56f4<\/span><br \/>\n        <span class=\"token keyword\">int<\/span> minSum <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span> maxSum <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">int<\/span> current <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> num <span class=\"token operator\">:<\/span> nums<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n            current <span class=\"token operator\">&#043;&#061;<\/span> num<span class=\"token punctuation\">;<\/span><br \/>\n            minSum <span class=\"token operator\">&#061;<\/span> <span class=\"token class-name\">Math<\/span><span class=\"token punctuation\">.<\/span><span class=\"token function\">min<\/span><span class=\"token punctuation\">(<\/span>minSum<span class=\"token punctuation\">,<\/span> current<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            maxSum <span class=\"token operator\">&#061;<\/span> <span class=\"token class-name\">Math<\/span><span class=\"token punctuation\">.<\/span><span class=\"token function\">max<\/span><span class=\"token punctuation\">(<\/span>maxSum<span class=\"token punctuation\">,<\/span> current<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><\/p>\n<p>        <span class=\"token comment\">\/\/ \u521b\u5efa\u6570\u7ec4&#xff0c;\u6ce8\u610f\u5904\u7406\u8d1f\u6570\u504f\u79fb<\/span><br \/>\n        <span class=\"token keyword\">int<\/span> offset <span class=\"token operator\">&#061;<\/span> <span class=\"token operator\">&#8211;<\/span>minSum<span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">int<\/span><span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span> sumCount <span class=\"token operator\">&#061;<\/span> <span class=\"token keyword\">new<\/span> <span class=\"token keyword\">int<\/span><span class=\"token punctuation\">[<\/span>maxSum <span class=\"token operator\">&#8211;<\/span> minSum <span class=\"token operator\">&#043;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>        <span class=\"token comment\">\/\/ \u521d\u59cb\u5316&#xff1a;\u524d\u7f00\u548c\u4e3a0\u51fa\u73b01\u6b21<\/span><br \/>\n        sumCount<span class=\"token punctuation\">[<\/span><span class=\"token number\">0<\/span> <span class=\"token operator\">&#043;<\/span> offset<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> num <span class=\"token operator\">:<\/span> nums<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n            prefixSum <span class=\"token operator\">&#043;&#061;<\/span> num<span class=\"token punctuation\">;<\/span><\/p>\n<p>            <span class=\"token keyword\">int<\/span> target <span class=\"token operator\">&#061;<\/span> prefixSum <span class=\"token operator\">&#8211;<\/span> k<span class=\"token punctuation\">;<\/span><\/p>\n<p>            <span class=\"token comment\">\/\/ \u68c0\u67e5\u76ee\u6807\u503c\u662f\u5426\u5728\u6709\u6548\u8303\u56f4\u5185<\/span><br \/>\n            <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>target <span class=\"token operator\">&gt;&#061;<\/span> minSum <span class=\"token operator\">&amp;&amp;<\/span> target <span class=\"token operator\">&lt;&#061;<\/span> maxSum<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n                count <span class=\"token operator\">&#043;&#061;<\/span> sumCount<span class=\"token punctuation\">[<\/span>target <span class=\"token operator\">&#043;<\/span> offset<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><\/p>\n<p>            <span class=\"token comment\">\/\/ \u66f4\u65b0\u5f53\u524d\u524d\u7f00\u548c\u7684\u8ba1\u6570<\/span><br \/>\n            sumCount<span class=\"token punctuation\">[<\/span>prefixSum <span class=\"token operator\">&#043;<\/span> offset<span class=\"token punctuation\">]<\/span><span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><\/p>\n<p>        <span class=\"token keyword\">return<\/span> count<span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><\/p>\n<p>    <span class=\"token comment\">\/**<br \/>\n     * \u6d41\u5f0f\u5904\u7406\u7248\u672c&#xff08;\u9002\u7528\u4e8e\u6570\u636e\u6d41\u573a\u666f&#xff09;<br \/>\n     *\/<\/span><br \/>\n    <span class=\"token keyword\">public<\/span> <span class=\"token keyword\">static<\/span> <span class=\"token keyword\">class<\/span> <span class=\"token class-name\">StreamProcessor<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token keyword\">private<\/span> <span class=\"token class-name\">Map<\/span><span class=\"token generics\"><span class=\"token punctuation\">&lt;<\/span><span class=\"token class-name\">Integer<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token class-name\">Integer<\/span><span class=\"token punctuation\">&gt;<\/span><\/span> prefixSumMap<span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">private<\/span> <span class=\"token keyword\">int<\/span> prefixSum<span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">private<\/span> <span class=\"token keyword\">int<\/span> totalCount<span class=\"token punctuation\">;<\/span><\/p>\n<p>        <span class=\"token keyword\">public<\/span> <span class=\"token class-name\">StreamProcessor<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n            <span class=\"token keyword\">this<\/span><span class=\"token punctuation\">.<\/span>prefixSumMap <span class=\"token operator\">&#061;<\/span> <span class=\"token keyword\">new<\/span> <span class=\"token class-name\">HashMap<\/span><span class=\"token generics\"><span class=\"token punctuation\">&lt;<\/span><span class=\"token punctuation\">&gt;<\/span><\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token keyword\">this<\/span><span class=\"token punctuation\">.<\/span>prefixSum <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token keyword\">this<\/span><span class=\"token punctuation\">.<\/span>totalCount <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token keyword\">this<\/span><span class=\"token punctuation\">.<\/span>prefixSumMap<span class=\"token punctuation\">.<\/span><span class=\"token function\">put<\/span><span class=\"token punctuation\">(<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span> <span class=\"token comment\">\/\/ \u521d\u59cb\u5316\u7a7a\u6570\u7ec4<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><\/p>\n<p>        <span class=\"token comment\">\/**<br \/>\n         * \u5904\u7406\u4e0b\u4e00\u4e2a\u6570\u5b57<br \/>\n         * &#064;param num \u4e0b\u4e00\u4e2a\u6570\u5b57<br \/>\n         * &#064;param k \u76ee\u6807\u503c<br \/>\n         * &#064;return \u5230\u5f53\u524d\u4f4d\u7f6e\u4e3a\u6b62&#xff0c;\u548c\u4e3ak\u7684\u5b50\u6570\u7ec4\u4e2a\u6570<br \/>\n         *\/<\/span><br \/>\n        <span class=\"token keyword\">public<\/span> <span class=\"token keyword\">int<\/span> <span class=\"token function\">processNext<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> num<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> k<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n            prefixSum <span class=\"token operator\">&#043;&#061;<\/span> num<span class=\"token punctuation\">;<\/span><\/p>\n<p>            <span class=\"token keyword\">int<\/span> target <span class=\"token operator\">&#061;<\/span> prefixSum <span class=\"token operator\">&#8211;<\/span> k<span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>prefixSumMap<span class=\"token punctuation\">.<\/span><span class=\"token function\">containsKey<\/span><span class=\"token punctuation\">(<\/span>target<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n                totalCount <span class=\"token operator\">&#043;&#061;<\/span> prefixSumMap<span class=\"token punctuation\">.<\/span><span class=\"token function\">get<\/span><span class=\"token punctuation\">(<\/span>target<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><\/p>\n<p>            prefixSumMap<span class=\"token punctuation\">.<\/span><span class=\"token function\">put<\/span><span class=\"token punctuation\">(<\/span>prefixSum<span class=\"token punctuation\">,<\/span> prefixSumMap<span class=\"token punctuation\">.<\/span><span class=\"token function\">getOrDefault<\/span><span class=\"token punctuation\">(<\/span>prefixSum<span class=\"token punctuation\">,<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token operator\">&#043;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>            <span class=\"token keyword\">return<\/span> totalCount<span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><\/p>\n<p>        <span class=\"token comment\">\/**<br \/>\n         * \u83b7\u53d6\u5f53\u524d\u603b\u6570<br \/>\n         *\/<\/span><br \/>\n        <span class=\"token keyword\">public<\/span> <span class=\"token keyword\">int<\/span> <span class=\"token function\">getTotalCount<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n            <span class=\"token keyword\">return<\/span> totalCount<span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><\/p>\n<p>        <span class=\"token comment\">\/**<br \/>\n         * \u91cd\u7f6e\u5904\u7406\u5668<br \/>\n         *\/<\/span><br \/>\n        <span class=\"token keyword\">public<\/span> <span class=\"token keyword\">void<\/span> <span class=\"token function\">reset<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n            <span class=\"token keyword\">this<\/span><span class=\"token punctuation\">.<\/span>prefixSumMap<span class=\"token punctuation\">.<\/span><span class=\"token function\">clear<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token keyword\">this<\/span><span class=\"token punctuation\">.<\/span>prefixSum <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token keyword\">this<\/span><span class=\"token punctuation\">.<\/span>totalCount <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token keyword\">this<\/span><span class=\"token punctuation\">.<\/span>prefixSumMap<span class=\"token punctuation\">.<\/span><span class=\"token function\">put<\/span><span class=\"token punctuation\">(<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<h2>4. \u6027\u80fd\u5bf9\u6bd4<\/h2>\n<table>\n<tr>\u7b97\u6cd5\u65f6\u95f4\u590d\u6742\u5ea6\u7a7a\u95f4\u590d\u6742\u5ea6\u4f18\u52bf\u52a3\u52bf<\/tr>\n<tbody>\n<tr>\n<td>\u66b4\u529b\u679a\u4e3e&#xff08;\u57fa\u7840&#xff09;<\/td>\n<td>O(n\u00b3)<\/td>\n<td>O(1)<\/td>\n<td>\u5b9e\u73b0\u7b80\u5355<\/td>\n<td>\u6548\u7387\u6781\u4f4e<\/td>\n<\/tr>\n<tr>\n<td>\u66b4\u529b\u679a\u4e3e&#xff08;\u4f18\u5316&#xff09;<\/td>\n<td>O(n\u00b2)<\/td>\n<td>O(1)<\/td>\n<td>\u65e0\u9700\u989d\u5916\u7a7a\u95f4<\/td>\n<td>\u4ec5\u9002\u7528\u4e8e\u5c0f\u6570\u636e<\/td>\n<\/tr>\n<tr>\n<td>\u524d\u7f00\u548c\u6570\u7ec4<\/td>\n<td>O(n\u00b2)<\/td>\n<td>O(n)<\/td>\n<td>\u51cf\u5c11\u91cd\u590d\u8ba1\u7b97<\/td>\n<td>\u4ecd\u7136\u9700\u8981\u4e24\u5c42\u5faa\u73af<\/td>\n<\/tr>\n<tr>\n<td>\u524d\u7f00\u548c&#043;\u54c8\u5e0c\u8868<\/td>\n<td>O(n)<\/td>\n<td>O(n)<\/td>\n<td>\u6548\u7387\u6700\u9ad8<\/td>\n<td>\u9700\u8981\u989d\u5916\u7a7a\u95f4<\/td>\n<\/tr>\n<tr>\n<td>\u6d41\u5f0f\u5904\u7406<\/td>\n<td>O(1)\u6bcf\u6b21<\/td>\n<td>O(n)<\/td>\n<td>\u9002\u5408\u6570\u636e\u6d41<\/td>\n<td>\u9700\u8981\u7ef4\u62a4\u72b6\u6001<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u6027\u80fd\u6d4b\u8bd5\u7ed3\u679c&#xff08;\u6570\u7ec4\u957f\u5ea6&#061;10000&#xff09;&#xff1a;<\/p>\n<ul>\n<li>\u66b4\u529b\u679a\u4e3e&#xff08;\u4f18\u5316&#xff09;&#xff1a;~500 ms<\/li>\n<li>\u524d\u7f00\u548c\u6570\u7ec4&#xff1a;~200 ms<\/li>\n<li>\u524d\u7f00\u548c&#043;\u54c8\u5e0c\u8868&#xff1a;~10 ms<\/li>\n<\/ul>\n<p>\u5185\u5b58\u5360\u7528\u5206\u6790&#xff1a;<\/p>\n<ul>\n<li>\u54c8\u5e0c\u8868\u89e3\u6cd5&#xff1a;\u6700\u574f\u60c5\u51b5\u5b58\u50a8n\u4e2a\u952e\u503c\u5bf9&#xff0c;\u7ea6\u51e0\u5341KB<\/li>\n<li>\u524d\u7f00\u548c\u6570\u7ec4&#xff1a;\u5b58\u50a8n&#043;1\u4e2a\u6574\u6570&#xff0c;\u7ea640KB&#xff08;n&#061;10000&#xff09;<\/li>\n<\/ul>\n<h2>5. \u6269\u5c55\u4e0e\u53d8\u4f53<\/h2>\n<h3>5.1 \u548c\u4e3aK\u7684\u6700\u957f\u5b50\u6570\u7ec4\u957f\u5ea6<\/h3>\n<p><span class=\"token keyword\">public<\/span> <span class=\"token keyword\">class<\/span> <span class=\"token class-name\">LongestSubarraySumEqualsK<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token comment\">\/**<br \/>\n     * \u6c42\u548c\u4e3aK\u7684\u6700\u957f\u5b50\u6570\u7ec4\u957f\u5ea6<br \/>\n     * \u65f6\u95f4\u590d\u6742\u5ea6: O(n)<br \/>\n     * \u7a7a\u95f4\u590d\u6742\u5ea6: O(n)<br \/>\n     *\/<\/span><br \/>\n    <span class=\"token keyword\">public<\/span> <span class=\"token keyword\">int<\/span> <span class=\"token function\">longestSubarraySumK<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span><span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span> nums<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> k<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token comment\">\/\/ \u54c8\u5e0c\u8868&#xff1a;\u952e\u4e3a\u524d\u7f00\u548c&#xff0c;\u503c\u4e3a\u8be5\u524d\u7f00\u548c\u7b2c\u4e00\u6b21\u51fa\u73b0\u7684\u4f4d\u7f6e<\/span><br \/>\n        <span class=\"token class-name\">Map<\/span><span class=\"token generics\"><span class=\"token punctuation\">&lt;<\/span><span class=\"token class-name\">Integer<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token class-name\">Integer<\/span><span class=\"token punctuation\">&gt;<\/span><\/span> prefixSumMap <span class=\"token operator\">&#061;<\/span> <span class=\"token keyword\">new<\/span> <span class=\"token class-name\">HashMap<\/span><span class=\"token generics\"><span class=\"token punctuation\">&lt;<\/span><span class=\"token punctuation\">&gt;<\/span><\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        prefixSumMap<span class=\"token punctuation\">.<\/span><span class=\"token function\">put<\/span><span class=\"token punctuation\">(<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token operator\">&#8211;<\/span><span class=\"token number\">1<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span> <span class=\"token comment\">\/\/ \u524d\u7f00\u548c\u4e3a0\u51fa\u73b0\u5728\u7d22\u5f15-1\u5904&#xff08;\u7a7a\u6570\u7ec4&#xff09;<\/span><\/p>\n<p>        <span class=\"token keyword\">int<\/span> maxLength <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">int<\/span> prefixSum <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;<\/span> nums<span class=\"token punctuation\">.<\/span>length<span class=\"token punctuation\">;<\/span> i<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n            prefixSum <span class=\"token operator\">&#043;&#061;<\/span> nums<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>            <span class=\"token comment\">\/\/ \u5982\u679c\u5b58\u5728prefixSum &#8211; k&#xff0c;\u8ba1\u7b97\u5b50\u6570\u7ec4\u957f\u5ea6<\/span><br \/>\n            <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>prefixSumMap<span class=\"token punctuation\">.<\/span><span class=\"token function\">containsKey<\/span><span class=\"token punctuation\">(<\/span>prefixSum <span class=\"token operator\">&#8211;<\/span> k<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n                <span class=\"token keyword\">int<\/span> startIndex <span class=\"token operator\">&#061;<\/span> prefixSumMap<span class=\"token punctuation\">.<\/span><span class=\"token function\">get<\/span><span class=\"token punctuation\">(<\/span>prefixSum <span class=\"token operator\">&#8211;<\/span> k<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n                maxLength <span class=\"token operator\">&#061;<\/span> <span class=\"token class-name\">Math<\/span><span class=\"token punctuation\">.<\/span><span class=\"token function\">max<\/span><span class=\"token punctuation\">(<\/span>maxLength<span class=\"token punctuation\">,<\/span> i <span class=\"token operator\">&#8211;<\/span> startIndex<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><\/p>\n<p>            <span class=\"token comment\">\/\/ \u53ea\u8bb0\u5f55\u524d\u7f00\u548c\u7b2c\u4e00\u6b21\u51fa\u73b0\u7684\u4f4d\u7f6e&#xff0c;\u4ee5\u5f97\u5230\u6700\u957f\u5b50\u6570\u7ec4<\/span><br \/>\n            <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token operator\">!<\/span>prefixSumMap<span class=\"token punctuation\">.<\/span><span class=\"token function\">containsKey<\/span><span class=\"token punctuation\">(<\/span>prefixSum<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n                prefixSumMap<span class=\"token punctuation\">.<\/span><span class=\"token function\">put<\/span><span class=\"token punctuation\">(<\/span>prefixSum<span class=\"token punctuation\">,<\/span> i<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><\/p>\n<p>        <span class=\"token keyword\">return<\/span> maxLength<span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><\/p>\n<p>    <span class=\"token comment\">\/**<br \/>\n     * \u5904\u7406\u5305\u542b\u8d1f\u6570\u7684\u60c5\u51b5<br \/>\n     *\/<\/span><br \/>\n    <span class=\"token keyword\">public<\/span> <span class=\"token keyword\">int<\/span> <span class=\"token function\">longestSubarraySumKWithNegatives<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span><span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span> nums<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> k<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token comment\">\/\/ \u4f7f\u7528TreeMap\u53ef\u4ee5\u627e\u5230\u6240\u6709\u5c0f\u4e8e\u7b49\u4e8e\u67d0\u4e2a\u503c\u7684\u524d\u7f00\u548c<\/span><br \/>\n        <span class=\"token comment\">\/\/ \u4f46\u65f6\u95f4\u590d\u6742\u5ea6\u4f1a\u53d8\u4e3aO(n log n)<\/span><br \/>\n        <span class=\"token keyword\">return<\/span> <span class=\"token function\">longestSubarraySumK<\/span><span class=\"token punctuation\">(<\/span>nums<span class=\"token punctuation\">,<\/span> k<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span> <span class=\"token comment\">\/\/ \u539f\u65b9\u6cd5\u5df2\u7ecf\u652f\u6301\u8d1f\u6570<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<h3>5.2 \u548c\u81f3\u5c11\u4e3aK\u7684\u6700\u77ed\u5b50\u6570\u7ec4<\/h3>\n<p><span class=\"token keyword\">import<\/span> <span class=\"token namespace\">java<span class=\"token punctuation\">.<\/span>util<span class=\"token punctuation\">.<\/span><\/span><span class=\"token class-name\">Deque<\/span><span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token keyword\">import<\/span> <span class=\"token namespace\">java<span class=\"token punctuation\">.<\/span>util<span class=\"token punctuation\">.<\/span><\/span><span class=\"token class-name\">LinkedList<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p><span class=\"token keyword\">public<\/span> <span class=\"token keyword\">class<\/span> <span class=\"token class-name\">ShortestSubarraySumAtLeastK<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token comment\">\/**<br \/>\n     * \u6c42\u548c\u81f3\u5c11\u4e3aK\u7684\u6700\u77ed\u5b50\u6570\u7ec4\u957f\u5ea6<br \/>\n     * \u4f7f\u7528\u5355\u8c03\u961f\u5217\u4f18\u5316<br \/>\n     * \u65f6\u95f4\u590d\u6742\u5ea6: O(n)<br \/>\n     * \u7a7a\u95f4\u590d\u6742\u5ea6: O(n)<br \/>\n     *\/<\/span><br \/>\n    <span class=\"token keyword\">public<\/span> <span class=\"token keyword\">int<\/span> <span class=\"token function\">shortestSubarray<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span><span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span> nums<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> k<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token keyword\">int<\/span> n <span class=\"token operator\">&#061;<\/span> nums<span class=\"token punctuation\">.<\/span>length<span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">long<\/span><span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span> prefix <span class=\"token operator\">&#061;<\/span> <span class=\"token keyword\">new<\/span> <span class=\"token keyword\">long<\/span><span class=\"token punctuation\">[<\/span>n <span class=\"token operator\">&#043;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>        <span class=\"token comment\">\/\/ \u8ba1\u7b97\u524d\u7f00\u548c<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;<\/span> n<span class=\"token punctuation\">;<\/span> i<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n            prefix<span class=\"token punctuation\">[<\/span>i <span class=\"token operator\">&#043;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> prefix<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#043;<\/span> nums<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><\/p>\n<p>        <span class=\"token keyword\">int<\/span> minLength <span class=\"token operator\">&#061;<\/span> <span class=\"token class-name\">Integer<\/span><span class=\"token punctuation\">.<\/span>MAX_VALUE<span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token class-name\">Deque<\/span><span class=\"token generics\"><span class=\"token punctuation\">&lt;<\/span><span class=\"token class-name\">Integer<\/span><span class=\"token punctuation\">&gt;<\/span><\/span> deque <span class=\"token operator\">&#061;<\/span> <span class=\"token keyword\">new<\/span> <span class=\"token class-name\">LinkedList<\/span><span class=\"token generics\"><span class=\"token punctuation\">&lt;<\/span><span class=\"token punctuation\">&gt;<\/span><\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;&#061;<\/span> n<span class=\"token punctuation\">;<\/span> i<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n            <span class=\"token comment\">\/\/ \u7ef4\u62a4\u5355\u8c03\u9012\u589e\u961f\u5217<\/span><br \/>\n            <span class=\"token keyword\">while<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token operator\">!<\/span>deque<span class=\"token punctuation\">.<\/span><span class=\"token function\">isEmpty<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token operator\">&amp;&amp;<\/span> prefix<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&lt;&#061;<\/span> prefix<span class=\"token punctuation\">[<\/span>deque<span class=\"token punctuation\">.<\/span><span class=\"token function\">getLast<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n                deque<span class=\"token punctuation\">.<\/span><span class=\"token function\">removeLast<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><\/p>\n<p>            <span class=\"token comment\">\/\/ \u68c0\u67e5\u961f\u9996\u5143\u7d20\u662f\u5426\u6ee1\u8db3\u6761\u4ef6<\/span><br \/>\n            <span class=\"token keyword\">while<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token operator\">!<\/span>deque<span class=\"token punctuation\">.<\/span><span class=\"token function\">isEmpty<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token operator\">&amp;&amp;<\/span> prefix<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#8211;<\/span> prefix<span class=\"token punctuation\">[<\/span>deque<span class=\"token punctuation\">.<\/span><span class=\"token function\">getFirst<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&gt;&#061;<\/span> k<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n                minLength <span class=\"token operator\">&#061;<\/span> <span class=\"token class-name\">Math<\/span><span class=\"token punctuation\">.<\/span><span class=\"token function\">min<\/span><span class=\"token punctuation\">(<\/span>minLength<span class=\"token punctuation\">,<\/span> i <span class=\"token operator\">&#8211;<\/span> deque<span class=\"token punctuation\">.<\/span><span class=\"token function\">removeFirst<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><\/p>\n<p>            deque<span class=\"token punctuation\">.<\/span><span class=\"token function\">addLast<\/span><span class=\"token punctuation\">(<\/span>i<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><\/p>\n<p>        <span class=\"token keyword\">return<\/span> minLength <span class=\"token operator\">&#061;&#061;<\/span> <span class=\"token class-name\">Integer<\/span><span class=\"token punctuation\">.<\/span>MAX_VALUE <span class=\"token operator\">?<\/span> <span class=\"token operator\">&#8211;<\/span><span class=\"token number\">1<\/span> <span class=\"token operator\">:<\/span> minLength<span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<h3>5.3 \u548c\u80fd\u88abK\u6574\u9664\u7684\u5b50\u6570\u7ec4<\/h3>\n<p><span class=\"token keyword\">import<\/span> <span class=\"token namespace\">java<span class=\"token punctuation\">.<\/span>util<span class=\"token punctuation\">.<\/span><\/span><span class=\"token class-name\">HashMap<\/span><span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token keyword\">import<\/span> <span class=\"token namespace\">java<span class=\"token punctuation\">.<\/span>util<span class=\"token punctuation\">.<\/span><\/span><span class=\"token class-name\">Map<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p><span class=\"token keyword\">public<\/span> <span class=\"token keyword\">class<\/span> <span class=\"token class-name\">SubarraySumsDivisibleByK<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token comment\">\/**<br \/>\n     * \u7edf\u8ba1\u548c\u80fd\u88abK\u6574\u9664\u7684\u5b50\u6570\u7ec4\u4e2a\u6570<br \/>\n     * \u65f6\u95f4\u590d\u6742\u5ea6: O(n)<br \/>\n     * \u7a7a\u95f4\u590d\u6742\u5ea6: O(n)<br \/>\n     *\/<\/span><br \/>\n    <span class=\"token keyword\">public<\/span> <span class=\"token keyword\">int<\/span> <span class=\"token function\">subarraysDivByK<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span><span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span> nums<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> k<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token comment\">\/\/ \u54c8\u5e0c\u8868&#xff1a;\u952e\u4e3a\u524d\u7f00\u548c\u6a21K\u7684\u4f59\u6570&#xff0c;\u503c\u4e3a\u8be5\u4f59\u6570\u51fa\u73b0\u7684\u6b21\u6570<\/span><br \/>\n        <span class=\"token class-name\">Map<\/span><span class=\"token generics\"><span class=\"token punctuation\">&lt;<\/span><span class=\"token class-name\">Integer<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token class-name\">Integer<\/span><span class=\"token punctuation\">&gt;<\/span><\/span> remainderMap <span class=\"token operator\">&#061;<\/span> <span class=\"token keyword\">new<\/span> <span class=\"token class-name\">HashMap<\/span><span class=\"token generics\"><span class=\"token punctuation\">&lt;<\/span><span class=\"token punctuation\">&gt;<\/span><\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        remainderMap<span class=\"token punctuation\">.<\/span><span class=\"token function\">put<\/span><span class=\"token punctuation\">(<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span> <span class=\"token comment\">\/\/ \u4f59\u6570\u4e3a0\u51fa\u73b01\u6b21&#xff08;\u7a7a\u6570\u7ec4&#xff09;<\/span><\/p>\n<p>        <span class=\"token keyword\">int<\/span> count <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">int<\/span> prefixSum <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> num <span class=\"token operator\">:<\/span> nums<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n            prefixSum <span class=\"token operator\">&#043;&#061;<\/span> num<span class=\"token punctuation\">;<\/span><\/p>\n<p>            <span class=\"token comment\">\/\/ \u8ba1\u7b97\u4f59\u6570&#xff0c;\u786e\u4fdd\u4f59\u6570\u4e3a\u6b63\u6570<\/span><br \/>\n            <span class=\"token keyword\">int<\/span> remainder <span class=\"token operator\">&#061;<\/span> prefixSum <span class=\"token operator\">%<\/span> k<span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>remainder <span class=\"token operator\">&lt;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n                remainder <span class=\"token operator\">&#043;&#061;<\/span> k<span class=\"token punctuation\">;<\/span> <span class=\"token comment\">\/\/ \u786e\u4fdd\u4f59\u6570\u5728[0, k-1]\u8303\u56f4\u5185<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><\/p>\n<p>            <span class=\"token comment\">\/\/ \u5982\u679c\u76f8\u540c\u7684\u4f59\u6570\u4e4b\u524d\u51fa\u73b0\u8fc7&#xff0c;\u8bf4\u660e\u5b58\u5728\u5b50\u6570\u7ec4\u548c\u80fd\u88abk\u6574\u9664<\/span><br \/>\n            <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>remainderMap<span class=\"token punctuation\">.<\/span><span class=\"token function\">containsKey<\/span><span class=\"token punctuation\">(<\/span>remainder<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n                count <span class=\"token operator\">&#043;&#061;<\/span> remainderMap<span class=\"token punctuation\">.<\/span><span class=\"token function\">get<\/span><span class=\"token punctuation\">(<\/span>remainder<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><\/p>\n<p>            <span class=\"token comment\">\/\/ \u66f4\u65b0\u4f59\u6570\u7684\u8ba1\u6570<\/span><br \/>\n            remainderMap<span class=\"token punctuation\">.<\/span><span class=\"token function\">put<\/span><span class=\"token punctuation\">(<\/span>remainder<span class=\"token punctuation\">,<\/span> remainderMap<span class=\"token punctuation\">.<\/span><span class=\"token function\">getOrDefault<\/span><span class=\"token punctuation\">(<\/span>remainder<span class=\"token punctuation\">,<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token operator\">&#043;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><\/p>\n<p>        <span class=\"token keyword\">return<\/span> count<span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><\/p>\n<p>    <span class=\"token comment\">\/**<br \/>\n     * \u4f7f\u7528\u6570\u7ec4\u66ff\u4ee3\u54c8\u5e0c\u8868&#xff08;\u66f4\u9ad8\u6548&#xff09;<br \/>\n     *\/<\/span><br \/>\n    <span class=\"token keyword\">public<\/span> <span class=\"token keyword\">int<\/span> <span class=\"token function\">subarraysDivByKArray<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span><span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span> nums<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> k<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token comment\">\/\/ \u4f59\u6570\u8303\u56f4\u662f[0, k-1]<\/span><br \/>\n        <span class=\"token keyword\">int<\/span><span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span> remainderCount <span class=\"token operator\">&#061;<\/span> <span class=\"token keyword\">new<\/span> <span class=\"token keyword\">int<\/span><span class=\"token punctuation\">[<\/span>k<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        remainderCount<span class=\"token punctuation\">[<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span> <span class=\"token comment\">\/\/ \u4f59\u6570\u4e3a0\u51fa\u73b01\u6b21<\/span><\/p>\n<p>        <span class=\"token keyword\">int<\/span> count <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">int<\/span> prefixSum <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> num <span class=\"token operator\">:<\/span> nums<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n            prefixSum <span class=\"token operator\">&#043;&#061;<\/span> num<span class=\"token punctuation\">;<\/span><\/p>\n<p>            <span class=\"token comment\">\/\/ \u8ba1\u7b97\u4f59\u6570&#xff0c;\u786e\u4fdd\u4e3a\u6b63<\/span><br \/>\n            <span class=\"token keyword\">int<\/span> remainder <span class=\"token operator\">&#061;<\/span> prefixSum <span class=\"token operator\">%<\/span> k<span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>remainder <span class=\"token operator\">&lt;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n                remainder <span class=\"token operator\">&#043;&#061;<\/span> k<span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><\/p>\n<p>            <span class=\"token comment\">\/\/ \u76f8\u540c\u4f59\u6570\u7684\u524d\u7f00\u548c\u4e4b\u5dee\u80fd\u88abk\u6574\u9664<\/span><br \/>\n            count <span class=\"token operator\">&#043;&#061;<\/span> remainderCount<span class=\"token punctuation\">[<\/span>remainder<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>            <span class=\"token comment\">\/\/ \u66f4\u65b0\u4f59\u6570\u8ba1\u6570<\/span><br \/>\n            remainderCount<span class=\"token punctuation\">[<\/span>remainder<span class=\"token punctuation\">]<\/span><span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><\/p>\n<p>        <span class=\"token keyword\">return<\/span> count<span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<h3>5.4 \u4e8c\u7ef4\u77e9\u9635\u4e2d\u548c\u4e3aK\u7684\u5b50\u77e9\u9635<\/h3>\n<p><span class=\"token keyword\">import<\/span> <span class=\"token namespace\">java<span class=\"token punctuation\">.<\/span>util<span class=\"token punctuation\">.<\/span><\/span><span class=\"token class-name\">HashMap<\/span><span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token keyword\">import<\/span> <span class=\"token namespace\">java<span class=\"token punctuation\">.<\/span>util<span class=\"token punctuation\">.<\/span><\/span><span class=\"token class-name\">Map<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p><span class=\"token keyword\">public<\/span> <span class=\"token keyword\">class<\/span> <span class=\"token class-name\">SubmatrixSumEqualsK<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token comment\">\/**<br \/>\n     * \u4e8c\u7ef4\u77e9\u9635\u4e2d\u548c\u4e3aK\u7684\u5b50\u77e9\u9635\u4e2a\u6570<br \/>\n     * \u65f6\u95f4\u590d\u6742\u5ea6: O(m\u00b2 * n) \u6216 O(n\u00b2 * m)<br \/>\n     * \u7a7a\u95f4\u590d\u6742\u5ea6: O(n) \u6216 O(m)<br \/>\n     *\/<\/span><br \/>\n    <span class=\"token keyword\">public<\/span> <span class=\"token keyword\">int<\/span> <span class=\"token function\">numSubmatrixSumTarget<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span><span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span> matrix<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> target<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token keyword\">int<\/span> m <span class=\"token operator\">&#061;<\/span> matrix<span class=\"token punctuation\">.<\/span>length<span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">int<\/span> n <span class=\"token operator\">&#061;<\/span> matrix<span class=\"token punctuation\">[<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">.<\/span>length<span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">int<\/span> count <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>        <span class=\"token comment\">\/\/ \u679a\u4e3e\u77e9\u9635\u7684\u4e0a\u8fb9\u754c<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> top <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> top <span class=\"token operator\">&lt;<\/span> m<span class=\"token punctuation\">;<\/span> top<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n            <span class=\"token comment\">\/\/ \u538b\u7f29\u6570\u7ec4&#xff1a;\u5c06\u591a\u884c\u538b\u7f29\u4e3a\u4e00\u884c<\/span><br \/>\n            <span class=\"token keyword\">int<\/span><span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span> colSum <span class=\"token operator\">&#061;<\/span> <span class=\"token keyword\">new<\/span> <span class=\"token keyword\">int<\/span><span class=\"token punctuation\">[<\/span>n<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>            <span class=\"token comment\">\/\/ \u679a\u4e3e\u77e9\u9635\u7684\u4e0b\u8fb9\u754c<\/span><br \/>\n            <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> bottom <span class=\"token operator\">&#061;<\/span> top<span class=\"token punctuation\">;<\/span> bottom <span class=\"token operator\">&lt;<\/span> m<span class=\"token punctuation\">;<\/span> bottom<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n                <span class=\"token comment\">\/\/ \u66f4\u65b0\u5217\u548c<\/span><br \/>\n                <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> col <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> col <span class=\"token operator\">&lt;<\/span> n<span class=\"token punctuation\">;<\/span> col<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n                    colSum<span class=\"token punctuation\">[<\/span>col<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#043;&#061;<\/span> matrix<span class=\"token punctuation\">[<\/span>bottom<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>col<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n                <span class=\"token punctuation\">}<\/span><\/p>\n<p>                <span class=\"token comment\">\/\/ \u5728\u538b\u7f29\u540e\u7684\u4e00\u7ef4\u6570\u7ec4\u4e0a\u4f7f\u7528\u524d\u7f00\u548c&#043;\u54c8\u5e0c\u8868<\/span><br \/>\n                count <span class=\"token operator\">&#043;&#061;<\/span> <span class=\"token function\">subarraySum<\/span><span class=\"token punctuation\">(<\/span>colSum<span class=\"token punctuation\">,<\/span> target<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><\/p>\n<p>        <span class=\"token keyword\">return<\/span> count<span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><\/p>\n<p>    <span class=\"token comment\">\/**<br \/>\n     * \u4e00\u7ef4\u6570\u7ec4\u7684\u5b50\u6570\u7ec4\u548c\u4e3atarget\u7684\u4e2a\u6570&#xff08;\u590d\u7528\u4e4b\u524d\u7684\u4ee3\u7801&#xff09;<br \/>\n     *\/<\/span><br \/>\n    <span class=\"token keyword\">private<\/span> <span class=\"token keyword\">int<\/span> <span class=\"token function\">subarraySum<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span><span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span> nums<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> k<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token keyword\">int<\/span> count <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">int<\/span> prefixSum <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token class-name\">Map<\/span><span class=\"token generics\"><span class=\"token punctuation\">&lt;<\/span><span class=\"token class-name\">Integer<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token class-name\">Integer<\/span><span class=\"token punctuation\">&gt;<\/span><\/span> map <span class=\"token operator\">&#061;<\/span> <span class=\"token keyword\">new<\/span> <span class=\"token class-name\">HashMap<\/span><span class=\"token generics\"><span class=\"token punctuation\">&lt;<\/span><span class=\"token punctuation\">&gt;<\/span><\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        map<span class=\"token punctuation\">.<\/span><span class=\"token function\">put<\/span><span class=\"token punctuation\">(<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> num <span class=\"token operator\">:<\/span> nums<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n            prefixSum <span class=\"token operator\">&#043;&#061;<\/span> num<span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token keyword\">int<\/span> target <span class=\"token operator\">&#061;<\/span> prefixSum <span class=\"token operator\">&#8211;<\/span> k<span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>map<span class=\"token punctuation\">.<\/span><span class=\"token function\">containsKey<\/span><span class=\"token punctuation\">(<\/span>target<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n                count <span class=\"token operator\">&#043;&#061;<\/span> map<span class=\"token punctuation\">.<\/span><span class=\"token function\">get<\/span><span class=\"token punctuation\">(<\/span>target<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><br \/>\n            map<span class=\"token punctuation\">.<\/span><span class=\"token function\">put<\/span><span class=\"token punctuation\">(<\/span>prefixSum<span class=\"token punctuation\">,<\/span> map<span class=\"token punctuation\">.<\/span><span class=\"token function\">getOrDefault<\/span><span class=\"token punctuation\">(<\/span>prefixSum<span class=\"token punctuation\">,<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token operator\">&#043;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><\/p>\n<p>        <span class=\"token keyword\">return<\/span> count<span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><\/p>\n<p>    <span class=\"token comment\">\/**<br \/>\n     * \u4f18\u5316\u7248\u672c&#xff1a;\u9884\u5904\u7406\u884c\u524d\u7f00\u548c<br \/>\n     *\/<\/span><br \/>\n    <span class=\"token keyword\">public<\/span> <span class=\"token keyword\">int<\/span> <span class=\"token function\">numSubmatrixSumTargetOptimized<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span><span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span> matrix<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> target<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token keyword\">int<\/span> m <span class=\"token operator\">&#061;<\/span> matrix<span class=\"token punctuation\">.<\/span>length<span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">int<\/span> n <span class=\"token operator\">&#061;<\/span> matrix<span class=\"token punctuation\">[<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">.<\/span>length<span class=\"token punctuation\">;<\/span><\/p>\n<p>        <span class=\"token comment\">\/\/ \u9884\u5904\u7406\u884c\u524d\u7f00\u548c<\/span><br \/>\n        <span class=\"token keyword\">int<\/span><span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span> rowPrefix <span class=\"token operator\">&#061;<\/span> <span class=\"token keyword\">new<\/span> <span class=\"token keyword\">int<\/span><span class=\"token punctuation\">[<\/span>m<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>n <span class=\"token operator\">&#043;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;<\/span> m<span class=\"token punctuation\">;<\/span> i<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n            <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> j <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> j <span class=\"token operator\">&lt;<\/span> n<span class=\"token punctuation\">;<\/span> j<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n                rowPrefix<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j <span class=\"token operator\">&#043;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> rowPrefix<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#043;<\/span> matrix<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><\/p>\n<p>        <span class=\"token keyword\">int<\/span> count <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>        <span class=\"token comment\">\/\/ \u679a\u4e3e\u5de6\u53f3\u8fb9\u754c<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> left <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> left <span class=\"token operator\">&lt;<\/span> n<span class=\"token punctuation\">;<\/span> left<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n            <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> right <span class=\"token operator\">&#061;<\/span> left<span class=\"token punctuation\">;<\/span> right <span class=\"token operator\">&lt;<\/span> n<span class=\"token punctuation\">;<\/span> right<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n                <span class=\"token comment\">\/\/ \u538b\u7f29\u5217&#xff1a;\u8ba1\u7b97\u6bcf\u4e00\u884c\u5728[left, right]\u8303\u56f4\u5185\u7684\u548c<\/span><br \/>\n                <span class=\"token keyword\">int<\/span><span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span> compressed <span class=\"token operator\">&#061;<\/span> <span class=\"token keyword\">new<\/span> <span class=\"token keyword\">int<\/span><span class=\"token punctuation\">[<\/span>m<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n                <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> row <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> row <span class=\"token operator\">&lt;<\/span> m<span class=\"token punctuation\">;<\/span> row<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n                    compressed<span class=\"token punctuation\">[<\/span>row<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> rowPrefix<span class=\"token punctuation\">[<\/span>row<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>right <span class=\"token operator\">&#043;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#8211;<\/span> rowPrefix<span class=\"token punctuation\">[<\/span>row<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>left<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n                <span class=\"token punctuation\">}<\/span><\/p>\n<p>                <span class=\"token comment\">\/\/ \u5728\u4e00\u7ef4\u6570\u7ec4\u4e0a\u4f7f\u7528\u524d\u7f00\u548c&#043;\u54c8\u5e0c\u8868<\/span><br \/>\n                count <span class=\"token operator\">&#043;&#061;<\/span> <span class=\"token function\">subarraySum<\/span><span class=\"token punctuation\">(<\/span>compressed<span class=\"token punctuation\">,<\/span> target<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><\/p>\n<p>        <span class=\"token keyword\">return<\/span> count<span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<h2>6. \u603b\u7ed3<\/h2>\n<h3>6.1 \u6838\u5fc3\u601d\u60f3\u603b\u7ed3<\/h3>\n<li>\u524d\u7f00\u548c\u8f6c\u6362&#xff1a;\u5c06\u5b50\u6570\u7ec4\u548c\u95ee\u9898\u8f6c\u5316\u4e3a\u524d\u7f00\u548c\u5dee\u503c\u95ee\u9898<\/li>\n<li>\u54c8\u5e0c\u8868\u4f18\u5316&#xff1a;\u901a\u8fc7\u8bb0\u5f55\u524d\u7f00\u548c\u51fa\u73b0\u6b21\u6570&#xff0c;\u5b9e\u73b0O(1)\u67e5\u627e<\/li>\n<li>\u7a7a\u95f4\u6362\u65f6\u95f4&#xff1a;\u4f7f\u7528\u989d\u5916\u7a7a\u95f4\u5b58\u50a8\u4e2d\u95f4\u7ed3\u679c&#xff0c;\u663e\u8457\u964d\u4f4e\u65f6\u95f4\u590d\u6742\u5ea6<\/li>\n<h3>6.2 \u7b97\u6cd5\u9009\u62e9\u6307\u5357<\/h3>\n<ul>\n<li>\u5c0f\u89c4\u6a21\u6570\u636e&#xff1a;\u53ef\u4ee5\u4f7f\u7528\u66b4\u529b\u89e3\u6cd5\u4f5c\u4e3a\u7406\u89e3\u57fa\u7840<\/li>\n<li>\u4e00\u822c\u573a\u666f&#xff1a;\u524d\u7f00\u548c&#043;\u54c8\u5e0c\u8868\u662f\u6700\u4f18\u9009\u62e9&#xff0c;\u65f6\u95f4\u590d\u6742\u5ea6O(n)<\/li>\n<li>\u7279\u6b8a\u7ea6\u675f&#xff1a;\u6839\u636e\u5177\u4f53\u95ee\u9898\u53d8\u4f53\u9009\u62e9\u5408\u9002\u7684\u4f18\u5316\u7b56\u7565<\/li>\n<li>\u9ad8\u7ef4\u6269\u5c55&#xff1a;\u901a\u8fc7\u538b\u7f29\u964d\u7ef4\u5c06\u9ad8\u7ef4\u95ee\u9898\u8f6c\u5316\u4e3a\u4e00\u7ef4\u95ee\u9898<\/li>\n<\/ul>\n<h3>6.3 \u5173\u952e\u5b9e\u73b0\u7ec6\u8282<\/h3>\n<li>\u521d\u59cb\u5316\u54c8\u5e0c\u8868&#xff1a;\u5fc5\u987b\u5305\u542b(0, 1)&#xff0c;\u8868\u793a\u7a7a\u6570\u7ec4\u7684\u524d\u7f00\u548c<\/li>\n<li>\u8d1f\u6570\u5904\u7406&#xff1a;\u7b97\u6cd5\u5929\u7136\u652f\u6301\u8d1f\u6570&#xff0c;\u65e0\u9700\u7279\u6b8a\u5904\u7406<\/li>\n<li>\u5927\u6570\u6ea2\u51fa&#xff1a;\u6ce8\u610f\u524d\u7f00\u548c\u53ef\u80fd\u8d85\u51faint\u8303\u56f4&#xff0c;\u4f7f\u7528long\u7c7b\u578b<\/li>\n<li>\u66f4\u65b0\u987a\u5e8f&#xff1a;\u5148\u68c0\u67e5\u76ee\u6807\u503c&#xff0c;\u518d\u66f4\u65b0\u5f53\u524d\u524d\u7f00\u548c\u8ba1\u6570<\/li>\n<h3>6.4 \u5e94\u7528\u573a\u666f<\/h3>\n<ul>\n<li>\u91d1\u878d\u5206\u6790&#xff1a;\u7edf\u8ba1\u8fbe\u5230\u7279\u5b9a\u6536\u76ca\u76ee\u6807\u7684\u8fde\u7eed\u4ea4\u6613\u65f6\u6bb5<\/li>\n<li>\u4fe1\u53f7\u5904\u7406&#xff1a;\u5bfb\u627e\u4fe1\u53f7\u5f3a\u5ea6\u8fbe\u5230\u7279\u5b9a\u9608\u503c\u7684\u8fde\u7eed\u65f6\u95f4\u6bb5<\/li>\n<li>\u6570\u636e\u5206\u6790&#xff1a;\u7edf\u8ba1\u6ee1\u8db3\u7279\u5b9a\u6761\u4ef6\u7684\u8fde\u7eed\u6570\u636e\u5b50\u5e8f\u5217<\/li>\n<li>\u7cfb\u7edf\u76d1\u63a7&#xff1a;\u68c0\u6d4b\u8fde\u7eed\u65f6\u95f4\u6bb5\u5185\u8d44\u6e90\u4f7f\u7528\u7387\u8fbe\u5230\u76ee\u6807\u503c<\/li>\n<\/ul>\n<h3>6.5 \u9762\u8bd5\u6280\u5de7<\/h3>\n<li>\u4ece\u66b4\u529b\u89e3\u6cd5\u5f00\u59cb&#xff0c;\u5c55\u793a\u601d\u8003\u8fc7\u7a0b<\/li>\n<li>\u5f15\u5165\u524d\u7f00\u548c\u6982\u5ff5\u8fdb\u884c\u4f18\u5316<\/li>\n<li>\u8fdb\u4e00\u6b65\u4f7f\u7528\u54c8\u5e0c\u8868\u5b9e\u73b0\u7ebf\u6027\u65f6\u95f4\u590d\u6742\u5ea6<\/li>\n<li>\u8ba8\u8bba\u8fb9\u754c\u6761\u4ef6\u548c\u7279\u6b8a\u60c5\u51b5<\/li>\n<li>\u5c55\u793a\u5bf9\u76f8\u5173\u53d8\u4f53\u95ee\u9898\u7684\u7406\u89e3<\/li>\n<p>\u901a\u8fc7\u672c\u9898\u7684\u5b66\u4e60&#xff0c;\u6211\u4eec\u4e0d\u4ec5\u638c\u63e1\u4e86\u89e3\u51b3&#034;\u548c\u4e3aK\u7684\u5b50\u6570\u7ec4&#034;\u95ee\u9898\u7684\u9ad8\u6548\u7b97\u6cd5&#xff0c;\u66f4\u91cd\u8981\u7684\u662f\u7406\u89e3\u4e86\u524d\u7f00\u548c&#043;\u54c8\u5e0c\u8868\u8fd9\u4e00\u91cd\u8981\u89e3\u9898\u6a21\u5f0f\u3002\u8fd9\u79cd\u601d\u60f3\u53ef\u4ee5\u5e7f\u6cdb\u5e94\u7528\u4e8e\u5404\u79cd\u8fde\u7eed\u5b50\u6570\u7ec4\u6c42\u548c\u95ee\u9898&#xff0c;\u662f\u7b97\u6cd5\u5b66\u4e60\u548c\u9762\u8bd5\u51c6\u5907\u4e2d\u7684\u91cd\u8981\u77e5\u8bc6\u70b9\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u76ee\u5f551. \u95ee\u9898\u63cf\u8ff02. \u95ee\u9898\u5206\u67902.1 \u9898\u76ee\u7406\u89e32.2 \u6838\u5fc3\u6d1e\u5bdf2.3 \u7834\u9898\u5173\u952e3. \u7b97\u6cd5\u8bbe\u8ba1\u4e0e\u5b9e\u73b03.1 \u66b4\u529b\u679a\u4e3e\u6cd53.2 \u524d\u7f00\u548c\u4f18\u53163.3 \u524d\u7f00\u548c\u54c8\u5e0c\u8868&#xff08;\u6700\u4f18&#xff09;3.4 \u7a7a\u95f4\u4f18\u5316\u7248\u672c4. \u6027\u80fd\u5bf9\u6bd45. \u6269\u5c55\u4e0e\u53d8\u4f535.1 \u548c\u4e3aK\u7684\u6700\u957f\u5b50\u6570\u7ec4\u957f\u5ea65.2 \u548c\u81f3\u5c11\u4e3aK\u7684\u6700\u77ed\u5b50\u6570\u7ec45.3 \u548c\u80fd\u88abK\u6574\u9664\u7684\u5b50\u6570\u7ec45.4 \u4e8c\u7ef4\u77e9\u9635\u4e2d\u548c\u4e3aK\u7684\u5b50\u77e9\u96356. \u603b\u7ed36.1 \u6838\u5fc3\u601d\u60f3\u603b\u7ed36.2 \u7b97\u6cd5\u9009\u62e9\u6307\u53576.3 \u5173\u952e\u5b9e\u73b0\u7ec6\u82826.4 \u5e94\u7528\u573a\u666f6.5 \u9762\u8bd5\u6280\u5de71. \u95ee\u9898\u63cf\u8ff0 <\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[6256,6254,6255,6253,2692,427],"topic":[],"class_list":["post-59359","post","type-post","status-publish","format-standard","hentry","category-server","tag-subarraysum","tag-6254","tag-6255","tag-6253","tag-leetcode","tag-427"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.3 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>LeetCode\u7b97\u6cd5\u9898\u8be6\u89e3 560\uff1a\u548c\u4e3aK\u7684\u5b50\u6570\u7ec4 - \u7f51\u7855\u4e92\u8054\u5e2e\u52a9\u4e2d\u5fc3<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.wsisp.com\/helps\/59359.html\" \/>\n<meta property=\"og:locale\" content=\"zh_CN\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"LeetCode\u7b97\u6cd5\u9898\u8be6\u89e3 560\uff1a\u548c\u4e3aK\u7684\u5b50\u6570\u7ec4 - \u7f51\u7855\u4e92\u8054\u5e2e\u52a9\u4e2d\u5fc3\" \/>\n<meta property=\"og:description\" content=\"\u76ee\u5f551. \u95ee\u9898\u63cf\u8ff02. \u95ee\u9898\u5206\u67902.1 \u9898\u76ee\u7406\u89e32.2 \u6838\u5fc3\u6d1e\u5bdf2.3 \u7834\u9898\u5173\u952e3. \u7b97\u6cd5\u8bbe\u8ba1\u4e0e\u5b9e\u73b03.1 \u66b4\u529b\u679a\u4e3e\u6cd53.2 \u524d\u7f00\u548c\u4f18\u53163.3 \u524d\u7f00\u548c\u54c8\u5e0c\u8868&#xff08;\u6700\u4f18&#xff09;3.4 \u7a7a\u95f4\u4f18\u5316\u7248\u672c4. \u6027\u80fd\u5bf9\u6bd45. \u6269\u5c55\u4e0e\u53d8\u4f535.1 \u548c\u4e3aK\u7684\u6700\u957f\u5b50\u6570\u7ec4\u957f\u5ea65.2 \u548c\u81f3\u5c11\u4e3aK\u7684\u6700\u77ed\u5b50\u6570\u7ec45.3 \u548c\u80fd\u88abK\u6574\u9664\u7684\u5b50\u6570\u7ec45.4 \u4e8c\u7ef4\u77e9\u9635\u4e2d\u548c\u4e3aK\u7684\u5b50\u77e9\u96356. \u603b\u7ed36.1 \u6838\u5fc3\u601d\u60f3\u603b\u7ed36.2 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