{"id":520,"date":"2023-04-15T16:02:43","date_gmt":"2023-04-15T08:02:43","guid":{"rendered":"http:\/\/120.55.184.7\/?p=520"},"modified":"2025-04-16T00:44:23","modified_gmt":"2025-04-15T16:44:23","slug":"%e8%83%8c%e5%8c%85%e9%97%ae%e9%a2%98","status":"publish","type":"post","link":"https:\/\/beijian99.top\/?p=520","title":{"rendered":"\u80cc\u5305\u95ee\u9898"},"content":{"rendered":"\n

\u4e00\u3001\u80cc\u5305\u95ee\u9898\u6982\u8ff0<\/h2>\n\n\n\n

\u80cc\u5305\u95ee\u9898(Knapsack Problem)\u6e90\u4e8e\u4e00\u4e2a\u7b80\u5355\u573a\u666f\uff1a\u5047\u8bbe\u4f60\u6709\u4e00\u4e2a\u5bb9\u91cf\u6709\u9650\u7684\u80cc\u5305\uff0c\u9700\u8981\u4ece\u4e00\u7cfb\u5217\u7269\u54c1\u4e2d\u9009\u62e9\u4e00\u4e9b\u653e\u5165\u80cc\u5305\u3002\u6bcf\u4e2a\u7269\u54c1\u90fd\u6709\u81ea\u5df1\u7684\u91cd\u91cf\u548c\u4ef7\u503c\uff0c\u76ee\u6807\u662f\u5728\u4e0d\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\u7684\u524d\u63d0\u4e0b\uff0c\u6700\u5927\u5316\u80cc\u5305\u4e2d\u7269\u54c1\u7684\u603b\u4ef7\u503c\u3002<\/p>\n\n\n\n

\u6839\u636e\u7269\u54c1\u7684\u9009\u53d6\u89c4\u5219\uff0c\u80cc\u5305\u95ee\u9898\u4e3b\u8981\u5206\u4e3a\u4e09\u79cd\u7c7b\u578b\uff1a<\/p>\n\n\n\n

    \n
  1. 01\u80cc\u5305\u95ee\u9898<\/strong>\uff1a\u6bcf\u4e2a\u7269\u54c1\u53ea\u80fd\u9009\u62e9\u4e00\u6b21\uff0c\u8981\u4e48\u653e\u5165\u80cc\u5305\uff0c\u8981\u4e48\u4e0d\u653e\u5165<\/li>\n\n\n\n
  2. \u5b8c\u5168\u80cc\u5305\u95ee\u9898<\/strong>\uff1a\u6bcf\u4e2a\u7269\u54c1\u53ef\u4ee5\u65e0\u9650\u6b21\u653e\u5165\u80cc\u5305<\/li>\n\n\n\n
  3. \u591a\u91cd\u80cc\u5305\u95ee\u9898<\/strong>\uff1a\u6bcf\u4e2a\u7269\u54c1\u6709\u56fa\u5b9a\u7684\u6570\u91cf\u9650\u5236<\/li>\n<\/ol>\n\n\n\n
    \n\n\n\n

    \u4e8c\u300101\u80cc\u5305<\/h2>\n\n\n\n

    1. \u95ee\u9898\u63cf\u8ff0<\/h3>\n\n\n\n

    \u7ed9\u5b9a\u4e00\u4e2a\u5bb9\u91cf\u4e3aW\u7684\u80cc\u5305\u548cN\u4e2a\u7269\u54c1\uff0c\u6bcf\u4e2a\u7269\u54c1\u6709\u91cd\u91cfw_i\u548c\u4ef7\u503cv_i\u3002\u6c42\u5728\u4e0d\u8d85\u51fa\u80cc\u5305\u5bb9\u91cf\u7684\u60c5\u51b5\u4e0b\uff0c\u80fd\u88c5\u5165\u80cc\u5305\u7684\u6700\u5927\u4ef7\u503c\u3002<\/p>\n\n\n\n

    2. \u52a8\u6001\u89c4\u5212\u89e3\u6cd5<\/h3>\n\n\n\n

    01\u80cc\u5305\u95ee\u9898\u901a\u5e38\u91c7\u7528\u52a8\u6001\u89c4\u5212(Dynamic Programming)\u6765\u89e3\u51b3\uff0c\u5176\u6838\u5fc3\u601d\u60f3\u662f\u5c06\u95ee\u9898\u5206\u89e3\u4e3a\u5b50\u95ee\u9898\uff0c\u5e76\u901a\u8fc7\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\u6765\u907f\u514d\u91cd\u590d\u8ba1\u7b97\u3002<\/p>\n\n\n\n

    \u72b6\u6001\u5b9a\u4e49<\/h4>\n\n\n\n

    \u5b9a\u4e49\u4e8c\u7ef4\u6570\u7ec4dp[i][j]<\/code>\uff0c\u8868\u793a\u524di\u4e2a\u7269\u54c1\u5728\u5bb9\u91cf\u4e3aj\u7684\u80cc\u5305\u4e0b\u80fd\u83b7\u5f97\u7684\u6700\u5927\u4ef7\u503c\u3002<\/p>\n\n\n\n

    \u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b<\/h4>\n\n\n\n

    \u5bf9\u4e8e\u6bcf\u4e2a\u7269\u54c1i\uff0c\u6709\u4e24\u79cd\u9009\u62e9\uff1a<\/p>\n\n\n\n

      \n
    1. \u4e0d\u9009\u62e9\u7b2ci\u4e2a\u7269\u54c1\uff1adp[i][j] = dp[i-1][j]<\/code><\/li>\n\n\n\n
    2. \u9009\u62e9\u7b2ci\u4e2a\u7269\u54c1\uff08\u524d\u63d0\u662fj \u2265 w_i\uff09\uff1adp[i][j] = dp[i-1][j-w_i] + v_i<\/code><\/li>\n<\/ol>\n\n\n\n

      \u56e0\u6b64\uff0c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e3a\uff1a<\/p>\n\n\n

      \ndp[i][j] = \\max(dp[i-1][j], dp[i-1][j-w_i] + v_i)\n<\/pre><\/div>\n\n\n
      C++\u5b9e\u73b0<\/h4>\n\n\n
      \n#include <iostream>\n#include <vector>\n#include <algorithm>\nusing namespace std;\n\nint knapsack01(int W, const vector<int>& weights, const vector<int>& values) {\n    int n = weights.size();\n    vector<vector<int>> dp(n + 1, vector<int>(W + 1, 0));\n\n    for (int i = 1; i <= n; i++) {\n        for (int j = 1; j <= W; j++) {\n            if (j < weights[i-1]) {\n                dp[i][j] = dp[i-1][j];\n            } else {\n                dp[i][j] = max(dp[i-1][j], dp[i-1][j-weights[i-1]] + values[i-1]);\n            }\n        }\n    }\n    return dp[n][W];\n}\n\nint main() {\n    int W, n;\n    cout << "\u8bf7\u8f93\u5165\u80cc\u5305\u5bb9\u91cf\u548c\u7269\u54c1\u6570\u91cf: ";\n    cin >> W >> n;\n\n    vector<int> weights(n), values(n);\n    cout << "\u8bf7\u8f93\u5165\u6bcf\u4e2a\u7269\u54c1\u7684\u91cd\u91cf\u548c\u4ef7\u503c:" << endl;\n    for (int i = 0; i < n; i++) {\n        cin >> weights[i] >> values[i];\n    }\n\n    int maxValue = knapsack01(W, weights, values);\n    cout << "\u6700\u5927\u4ef7\u503c\u4e3a: " << maxValue << endl;\n    return 0;\n}\n<\/pre><\/div>\n\n\n
      3. \u7a7a\u95f4\u4f18\u5316<\/h3>\n\n\n\n
      \u89c2\u5bdf\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u53ef\u4ee5\u53d1\u73b0\uff0cdp[i][j]<\/code>\u53ea\u4f9d\u8d56\u4e8edp[i-1][...]<\/code>\uff0c\u56e0\u6b64\u53ef\u4ee5\u5c06\u4e8c\u7ef4\u6570\u7ec4\u4f18\u5316\u4e3a\u4e00\u7ef4\u6570\u7ec4\uff0c\u964d\u4f4e\u7a7a\u95f4\u590d\u6742\u5ea6\u3002<\/p>\n\n\n\n
      \u4f18\u5316\u540e\u7684C++\u5b9e\u73b0\uff1a<\/p>\n\n\n
      \nint knapsack01_optimized(int W, const vector<int>& weights, const vector<int>& values) {\n    int n = weights.size();\n    vector<int> dp(W + 1, 0);\n\n    for (int i = 0; i < n; i++) {\n        for (int j = W; j >= weights[i]; j--) {\n            dp[j] = max(dp[j], dp[j - weights[i]] + values[i]);\n        }\n    }\n    return dp[W];\n}\n<\/pre><\/div>\n\n\n
      \u6ce8\u610f<\/strong>\uff1a\u5185\u5c42\u5faa\u73af\u5fc5\u987b\u4ece\u5927\u5230\u5c0f\u904d\u5386\uff0c\u907f\u514d\u8986\u76d6\u672a\u8ba1\u7b97\u7684\u503c\u3002<\/p>\n\n\n\n
      \n\n\n\n
      \u4e09\u3001\u5b8c\u5168\u80cc\u5305<\/h2>\n\n\n\n
      1. \u95ee\u9898\u63cf\u8ff0<\/h3>\n\n\n\n
      \u4e0e01\u80cc\u5305\u4e0d\u540c\uff0c\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u4e2d\u6bcf\u79cd\u7269\u54c1\u6709\u65e0\u9650\u4e2a\u53ef\u7528\u3002\u5373\u5728\u4e0d\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\u7684\u524d\u63d0\u4e0b\uff0c\u6bcf\u79cd\u7269\u54c1\u53ef\u4ee5\u9009\u62e9\u591a\u6b21\u3002<\/p>\n\n\n\n
      2. \u52a8\u6001\u89c4\u5212\u89e3\u6cd5<\/h3>\n\n\n\n
      \u72b6\u6001\u5b9a\u4e49<\/h4>\n\n\n\n
      \u540c\u6837\u5b9a\u4e49dp[i][j]<\/code>\u4e3a\u524di\u79cd\u7269\u54c1\u5728\u5bb9\u91cf\u4e3aj\u7684\u80cc\u5305\u4e0b\u7684\u6700\u5927\u4ef7\u503c\u3002<\/p>\n\n\n\n
      \u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b<\/h4>\n\n\n\n
      \u5bf9\u4e8e\u5b8c\u5168\u80cc\u5305\u95ee\u9898\uff0c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e3a\uff1a<\/p>\n\n\n
      \ndp[i][j] = \\max(dp[i-1][j], dp[i][j-w_i] + v_i)\n<\/pre><\/div>\n\n\n
      \u4e0e01\u80cc\u5305\u7684\u533a\u522b\u5728\u4e8e\uff0c\u5f53\u9009\u62e9\u653e\u5165\u7269\u54c1\u65f6\uff0c\u662f\u4ecedp[i][j-w_i]<\/code>\u8f6c\u79fb\u800c\u6765\uff0c\u800c\u4e0d\u662fdp[i-1][j-w_i]<\/code>\u3002<\/p>\n\n\n\n
      C++\u5b9e\u73b0<\/h4>\n\n\n
      \nint knapsackComplete(int W, const vector<int>& weights, const vector<int>& values) {\n    int n = weights.size();\n    vector<vector<int>> dp(n + 1, vector<int>(W + 1, 0));\n\n    for (int i = 1; i <= n; i++) {\n        for (int j = 1; j <= W; j++) {\n            if (j < weights[i-1]) {\n                dp[i][j] = dp[i-1][j];\n            } else {\n                dp[i][j] = max(dp[i-1][j], dp[i][j-weights[i-1]] + values[i-1]);\n            }\n        }\n    }\n    return dp[n][W];\n}\n<\/pre><\/div>\n\n\n
      3. \u7a7a\u95f4\u4f18\u5316<\/h3>\n\n\n\n
      \u540c\u6837\u53ef\u4ee5\u4f18\u5316\u4e3a\u4e00\u7ef4\u6570\u7ec4\uff0c\u4f46\u4e0e01\u80cc\u5305\u4e0d\u540c\u7684\u662f\uff0c\u5185\u5c42\u5faa\u73af\u9700\u8981\u4ece\u5c0f\u5230\u5927\u904d\u5386\uff1a<\/p>\n\n\n
      \nint knapsackComplete_optimized(int W, const vector<int>& weights, const vector<int>& values) {\n    int n = weights.size();\n    vector<int> dp(W + 1, 0);\n\n    for (int i = 0; i < n; i++) {\n        for (int j = weights[i]; j <= W; j++) {\n            dp[j] = max(dp[j], dp[j - weights[i]] + values[i]);\n        }\n    }\n    return dp[W];\n}\n<\/pre><\/div>\n\n\n
      \n\n\n\n
      \u56db\u3001\u591a\u91cd\u80cc\u5305\u95ee\u9898<\/h2>\n\n\n\n
      \u591a\u91cd\u80cc\u5305\u95ee\u9898\u662f\u80cc\u5305\u95ee\u9898\u7684\u4e00\u79cd\u6269\u5c55\uff0c\u4e0e01\u80cc\u5305\u548c\u5b8c\u5168\u80cc\u5305\u4e0d\u540c\uff0c\u5728\u591a\u91cd\u80cc\u5305\u95ee\u9898\u4e2d\uff0c\u6bcf\u79cd\u7269\u54c1\u90fd\u6709\u6570\u91cf\u9650\u5236<\/strong>\u3002<\/p>\n\n\n\n
      1. \u95ee\u9898\u63cf\u8ff0<\/h3>\n\n\n\n
      \u7ed9\u5b9a\u4e00\u4e2a\u5bb9\u91cf\u4e3aC\u7684\u80cc\u5305\uff0c\u6709n\u79cd\u7269\u54c1\uff0c\u6bcf\u79cd\u7269\u54c1\u6709\u91cd\u91cfw[i]\u3001\u4ef7\u503cv[i]\u548c\u6570\u91cfs[i]\u3002\u76ee\u6807\u662f\u5728\u4e0d\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\u7684\u524d\u63d0\u4e0b\uff0c\u9009\u62e9\u7269\u54c1\u4f7f\u5f97\u603b\u4ef7\u503c\u6700\u5927\u3002<\/p>\n\n\n\n
      2. \u57fa\u672c\u89e3\u6cd5<\/h3>\n\n\n\n
      \u591a\u91cd\u80cc\u5305\u7684\u57fa\u672c\u89e3\u6cd5\u662f\u4e09\u91cd\u5faa\u73af\u7684\u52a8\u6001\u89c4\u5212\uff1a<\/p>\n\n\n
      \nint dp[V+1]; \/\/ dp[j]\u8868\u793a\u5bb9\u91cf\u4e3aj\u65f6\u7684\u6700\u5927\u4ef7\u503c\nmemset(dp, 0, sizeof(dp));\n\nfor(int i=1; i&lt;=n; i++) { \/\/ \u679a\u4e3e\u7269\u54c1\n    for(int j=V; j>=0; j--) { \/\/ \u679a\u4e3e\u5bb9\u91cf\n        for(int k=1; k&lt;=s[i] &amp;&amp; k*w[i]&lt;=j; k++) { \/\/ \u679a\u4e3e\u6570\u91cf\n            dp[j] = max(dp[j], dp[j-k*w[i]] + k*v[i]);\n        }\n    }\n}\n<\/pre><\/div>\n\n\n
      \u65f6\u95f4\u590d\u6742\u5ea6\u4e3aO(nV<\/em>s)\uff0c\u5f53s\u8f83\u5927\u65f6\u6548\u7387\u8f83\u4f4e\u3002<\/p>\n\n\n\n
      3. \u4e8c\u8fdb\u5236\u4f18\u5316<\/h3>\n\n\n\n
      \u901a\u8fc7\u4e8c\u8fdb\u5236\u62c6\u5206\u5c06\u591a\u91cd\u80cc\u5305\u8f6c\u5316\u4e3a01\u80cc\u5305\u95ee\u9898\uff1a<\/p>\n\n\n
      \nvector&lt;pair&lt;int, int>> items; \/\/ \u5b58\u50a8\u62c6\u5206\u540e\u7684\u7269\u54c1(\u91cd\u91cf,\u4ef7\u503c)\n\nfor(int i=1; i&lt;=n; i++) {\n    int cnt = s[i];\n    for(int k=1; k&lt;=cnt; k*=2) {\n        items.push_back({k*w[i], k*v[i]});\n        cnt -= k;\n    }\n    if(cnt > 0) {\n        items.push_back({cnt*w[i], cnt*v[i]});\n    }\n}\n\n\/\/ \u7136\u540e\u8fdb\u884c01\u80cc\u5305\u5904\u7406\nfor(auto item : items) {\n    for(int j=V; j>=item.first; j--) {\n        dp[j] = max(dp[j], dp[j-item.first] + item.second);\n    }\n}\n<\/pre><\/div>\n\n\n
      \u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u4e3aO(nV<\/em>log s)\u3002<\/p>\n\n\n\n
      \n\n\n\n
      \u4e94\u3001\u5206\u7ec4\u80cc\u5305\u95ee\u9898<\/h2>\n\n\n\n
      \u5206\u7ec4\u80cc\u5305\u95ee\u9898\u4e2d\uff0c\u7269\u54c1\u88ab\u5206\u6210\u82e5\u5e72\u7ec4\uff0c\u6bcf\u7ec4\u53ea\u80fd\u9009\u62e9\u4e00\u4e2a\u7269\u54c1<\/strong>\u3002<\/p>\n\n\n\n
      1. \u95ee\u9898\u63cf\u8ff0<\/h3>\n\n\n\n
      \u6709N\u7ec4\u7269\u54c1\u548c\u4e00\u4e2a\u5bb9\u91cf\u4e3aV\u7684\u80cc\u5305\u3002\u7b2ci\u7ec4\u7269\u54c1\u6709s[i]\u4e2a\u7269\u54c1\uff0c\u6bcf\u4e2a\u7269\u54c1\u6709\u91cd\u91cfw[i][j]\u548c\u4ef7\u503cv[i][j]\u3002\u6bcf\u7ec4\u53ea\u80fd\u9009\u4e00\u4e2a\u7269\u54c1\u6216\u4e0d\u9009\uff0c\u6c42\u6700\u5927\u4ef7\u503c\u3002<\/p>\n\n\n\n
      2. \u57fa\u672c\u89e3\u6cd5<\/h3>\n\n\n\n
      \u5206\u7ec4\u80cc\u5305\u7684\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e3a\uff1a<\/p>\n\n\n
      \nint dp[V+1] = {0};\n\nfor(int i=1; i<=N; i++) { \/\/ \u679a\u4e3e\u7ec4\n    for(int j=V; j>=0; j--) { \/\/ \u679a\u4e3e\u5bb9\u91cf\n        for(int k=0; k<s[i]; k++) { \/\/ \u679a\u4e3e\u7ec4\u5185\u7269\u54c1\n            if(j >= w[i][k]) {\n                dp[j] = max(dp[j], dp[j-w[i][k]] + v[i][k]);\n            }\n        }\n    }\n}\n<\/pre><\/div>\n\n\n
      3. \u4ee3\u7801\u5b9e\u73b0<\/h3>\n\n\n
      \n#include <vector>\n#include <algorithm>\nusing namespace std;\n\nint groupKnapsack(int V, vector<vector<pair<int, int>>>& groups) {\n    vector<int> dp(V+1, 0);\n\n    for(auto& group : groups) {\n        for(int j=V; j>=0; j--) {\n            for(auto& item : group) {\n                if(j >= item.first) {\n                    dp[j] = max(dp[j], dp[j-item.first] + item.second);\n                }\n            }\n        }\n    }\n    return dp[V];\n}\n<\/pre><\/div>\n\n\n
      <\/h2>\n\n\n\n
      <\/p>\n","protected":false},"excerpt":{"rendered":"
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