简单动态编程算法中的错误（经典背包）

Question

我看了http://rosettacode.org/wiki/Knapsack_problem/0-1做了基本的背包动态编程问题，我得到了一个有效的解决方案（knapsack1（）），但是当我尝试了一个不同的解决方案（knapsack2（））时，我觉得我不在在某个地方，因为我没有得到正确的价值。

有问题的代码是方法knapsack2，位于底部。我确信问题很小，但我觉得很荒谬，因为我找不到问题。正确的答案应该是1030而不是880。

（我知道一个大小为[n] [S]的数组应该足够了，如knapsack1（）所示，它可以工作，但knapsack2（）并不奇怪。）

提前感谢任何看过这个的人。

/**
 * http://rosettacode.org/wiki/Knapsack_problem/0-1
 */
public class Knapsack {

    public static void main(String[] args) {
        int n = 22;
        int S = 400;
        int s[] = new int[22];
        int v[] = new int[22];
        int i = 0;
        s[i] = 9;
        v[i] = 150;
        i++;
        s[i] = 13;
        v[i] = 35;
        i++;
        s[i] = 153;
        v[i] = 200;
        i++;
        s[i] = 50;
        v[i] = 160;
        i++;
        s[i] = 15;
        v[i] = 60;
        i++;
        s[i] = 68;
        v[i] = 45;
        i++;
        s[i] = 27;
        v[i] = 60;
        i++;
        s[i] = 39;
        v[i] = 40;
        i++;
        s[i] = 23;
        v[i] = 30;
        i++;
        s[i] = 52;
        v[i] = 10;
        i++;
        s[i] = 11;
        v[i] = 70;
        i++;
        s[i] = 32;
        v[i] = 30;
        i++;
        s[i] = 24;
        v[i] = 15;
        i++;
        s[i] = 48;
        v[i] = 10;
        i++;
        s[i] = 73;
        v[i] = 40;
        i++;
        s[i] = 42;
        v[i] = 70;
        i++;
        s[i] = 43;
        v[i] = 75;
        i++;
        s[i] = 22;
        v[i] = 80;
        i++;
        s[i] = 7;
        v[i] = 20;
        i++;
        s[i] = 18;
        v[i] = 12;
        i++;
        s[i] = 4;
        v[i] = 50;
        i++;
        s[i] = 30;
        v[i] = 10;

        System.out.println("--Given items with these values--");
        System.out.println("[item, weight (dag), value]");
        System.out.println("map  9   150");
        System.out.println("compass  13  35");
        System.out.println("water    153     200");
        System.out.println("sandwich     50  160");
        System.out.println("glucose  15  60");
        System.out.println("tin  68  45");
        System.out.println("banana   27  60");
        System.out.println("apple    39  40");
        System.out.println("cheese   23  30");
        System.out.println("beer     52  10");
        System.out.println("suntan cream     11  70");
        System.out.println("camera   32  30");
        System.out.println("T-shirt  24  15");
        System.out.println("trousers     48  10");
        System.out.println("umbrella     73  40");
        System.out.println("waterproof trousers  42  70");
        System.out.println("waterproof overclothes   43  75");
        System.out.println("note-case    22  80");
        System.out.println("sunglasses   7   20");
        System.out.println("towel    18  12");
        System.out.println("socks    4   50");
        System.out.println("book     30  10");
        System.out.println("--The max value you can achieve in your knapsack--");
        System.out.println(knapsack2(n, S, s, v));
        // proper value should be 1030, from website
    }

    /**
     * @param n items
     * @param S bag size/capacity
     * @param s array where s[i] is size/weight of item at index i
     * @param v array where v[i] is value of item at index i
     * @return best/max value possible
     */
    @SuppressWarnings("unused")
    private static int knapsack1(int n, int S, int s[], int v[]) {
        int dp[][] = new int[n][S];
        for (int i=n-1; i >= 0; i--) { // # items being left out + 1
            for (int j=0; j < S; j++) { // size/weight + 1
                if (i == n-1) {
                    dp[i][j] = 0;
                } else {
                    int choices[] = {0,0};
                    choices[0] = dp[i+1][j];
                    if (j >= s[i])
                        choices[1] = v[i] + dp[i+1][j-s[i]];
                    dp[i][j] = max(choices);
                }
            }
        }
        return dp[0][S-1];
    }

    private static int max(int choices[]) {
        if (choices[0] > choices[1])
            return choices[0];
        else
            return choices[1];
    }

    /**
     * @param n items
     * @param S bag size/capacity
     * @param s array where s[i] is size/weight of item at index i
     * @param v array where v[i] is value of item at index i
     * @return best/max value possible
     */
//  @SuppressWarnings("unused")
    private static int knapsack2(int n, int S, int s[], int v[]) {
        int dp[][] = new int[n][S]; // dp[i][j] holds max value using i items and not exceeding weight j+1 
        for (int i=0; i < n; i++) { // # items. don't need to reach n because we assume we can fit at most n-1 items.
            for (int j=0; j < S; j++) { // size/weight+1
                if (i == 0) {
                    dp[i][j] = 0;
                } else {
                    int choices[] = {0,0};
                    choices[0] = dp[i-1][j];
                    if (j >= s[i])
                        choices[1] = v[i] + dp[i-1][j-s[i]];
                    dp[i][j] = max(choices);
                }
            }
        }
        return dp[n-1][S-1]; // TODO: WHY IS THIS 880 AND NOT 1030?
    }
}

Answer 1

你的背包功能都存在问题。两者都不会通过简单测试 - S=1 n=1 s[0]=1 v[0]=10。答案应为10，但您的代码将返回0。

您可以看到问题出现在if (i==(n-1))（knapsack1）或if (i==0)（背包2）中。

此外，由于条件if (j >= s[i]，您会看到问题。您的j从0开始，因此，如果S为1，您将无法使用s[i] == 1项目，因为j将1永远不会到达{{1}}。

希望这会有所帮助。祝你好运。