01背包的改进实现

居然没想到背包问题还可以这么实现，在空间复杂度上达到O(W)次，有点不可思议啊

比较简单，不解释了，debug一下就明白了

/**
 * 背包问题<br/>
 * f[w]=max{ f[v],f[w-w[i]]+v[i]}<br/>
 * 也就是f[w][i] = max{f[w][i],f[w-w[i]+v[i]}<br/>
 * 时间复杂度O(VW) 空间复杂度O(W)<br/>
 * 也就是改进了空间复杂度了
 *
 * @author Joeson
 */
public class ZeroPackage {

    public static void main(String[] args) {
        int[] value = new int[]{12, 10, 20, 15, 20};
        int[] weight = new int[]{2, 1, 3, 2, 2};
        int W = 5;
        System.out.println(zeroPackageDiv(value, weight, 0, W));
        System.out.println(zeroPackage(value, weight, W));
    }

    /**
     * 非递归实现
     */
    public static int zeroPackage(int[] values, int[] weights, int totalWeight) {
        int[] dp = new int[totalWeight + 1];

        for(int i = 0; i < values.length; i++) {
            for (int t = totalWeight; t >= weights[i]; t--) {
                dp[t] = Math.max(dp[t - weights[i]] + values[i], dp[t]);
            }
        }

        return dp[totalWeight];
    }

    /**
     * 递归实现
     */
    public static int zeroPackageDiv(int[] values, int[] weights, int index, int leftWeight) {
        if (leftWeight <= 0 || index >= values.length) {
            return 0;
        }

        int val1 = zeroPackageDiv(values, weights, index + 1, leftWeight);
        if (leftWeight < weights[index]) {
            return val1;
        }

        int val2 = zeroPackageDiv(values, weights, index + 1, leftWeight - weights[index]) + values[index];

        return Math.max(val1,val2);
    }

}