演算法-動態規劃-完全揹包

0. 動態規劃五部曲：

1. 確定dp陣列（dp table）以及下標的含義
2. 確定遞推公式
3. dp陣列如何初始化
4. 確定遍歷順序
5. 舉例推導dp陣列

1. 完全揹包問題

完全揹包問題中，每個物品都有無數個，可以重複選擇。

• 二維dp陣列

      int[][] dp = new int[n][totalWeight+1];
      
      // 初始化第一行
      for(int j = weight[0]; j<=totalWeight; ++j) {
          dp[0][j] = value[0] * (j/weight[0]);
      }
      
      for(int i = 1; i<n; ++i) {
          for(int j = 0; j<=totalWeight; ++j) {
              if(j < weight[i])
                  dp[i][j] = dp[i-1][j];
              else 
                  dp[i][j] = Math.max(dp[i-1][j], dp[i][j-weight[i]]+value[i]);
          }
      }
  

• 一維dp陣列：順序遍歷，允許同種物品被重複選擇。

      int[] dp = new int[totalWeight+1];
      for(int i = 0; i<n; ++i) {
          for(int j = weight[i]; j<=totalWeight; ++j) {
              dp[j] = Math.max(dp[j], dp[j-weight[i]] + value[i]);
          }
      }
  

import java.util.Scanner;

public class Main{
    public static void main(String[] args) {
        Scanner scanner = new Scanner(System.in);
        int n = scanner.nextInt();
        int totalWeight = scanner.nextInt();
        int[] weight = new int[n];
        int[] value = new int[n];
        
        for(int i = 0; i<n; ++i) {
            weight[i] = scanner.nextInt();
            value[i] = scanner.nextInt();
        }
        
        int[] dp = new int[totalWeight+1];
        for(int i = 0; i<n; ++i) {
            // 順序遍歷，同一個物品可以重複選
            for(int j = weight[i]; j<=totalWeight; ++j) {
                dp[j] = Math.max(dp[j], dp[j-weight[i]]+value[i]);
            }
        }
        
        System.out.println(dp[totalWeight]);
    }
}